Research ArticleOn the Stability of Quadratic Functional Equations in 119865-Spaces

Xiuzhong Yang

College of Mathematics and Information Science Hebei Normal University and Hebei Key Laboratory ofComputational Mathematics and Applications Shijiazhuang 050024 China

Correspondence should be addressed to Xiuzhong Yang xiuzhongyang126com

Received 7 April 2016 Revised 22 May 2016 Accepted 27 June 2016

Academic Editor Krzysztof Cieplinski

Copyright copy 2016 Xiuzhong Yang This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The Hyers-Ulam-Rassias stability of quadratic functional equation 119891(2119909 + 119910) + 119891(2119909 minus 119910) = 119891(119909 + 119910) + 119891(119909 minus 119910) + 6119891(119909) andorthogonal stability of the Pexiderized quadratic functional equation 119891(119909 + 119910) + 119891(119909 minus 119910) = 2119892(119909) + 2ℎ(119910) in 119865-spaces are proved

1 Introduction

In 1940 Ulam [1] proposed the following stability problemgiven metric group 119866(sdot 120588) number 120576 gt 0 and mapping 119891 119866 rarr 119866 which satisfies inequality 120588(119891(119909 sdot 119910) 119891(119909) sdot 119891(119910)) lt 120576for all 119909 119910 in 119866 do automorphism 119886 of 119866 and constant 119896 gt 0depending only on119866 such that 120588(119886(119909) 119891(119909)) le 119896120576 for all 119909 in119866 exist If the answer is affirmative we call equation 119886(119909sdot119910) =119886(119909) sdot 119886(119910) of automorphism stable One year later Hyers [2]provided a positive partial answer to Ulamrsquos problem In 1978a generalized version of Hyersrsquo result was proved by Rassiasin [3] Since then the stability problems of several functionalequations have been extensively investigated by a numberof authors [4ndash17] In fact we also refer the readers to thepaper [18] for recent developments in Ulamrsquos type stability[19] for recent developments of the conditional stability ofthe homomorphism equation and books and [8 20] for thegeneral understanding of the stability theory

Another important stability problem is orthogonal sta-bility which is closely related to the notion of orthogonalityspaces we know that a number of definitions of orthogonalityin vector spaces in addition to the usual one for inner productspaces have appeared in the literature during the past halfcenturyMany of these arementioned in an article byDrljevic[21] Perhaps the best known of these is the ldquoBirkhoff-Jamesrdquoorthogonality (see James [22]) for real normed vector spaceswhere 119909 is orthogonal to 119910 meaning that 119909 + 120582119910 ge 119909 forall 120582 isin R In giving his axiomatic definition of orthogonalityRatz in a 1985 paper [23] modified the definition given on

pp 427-428 of Gudder and Strawther [24] and arrived at thefollowing

Definition 1 Suppose that 119883 is a real vector space withdim119883 ge 2 and perp is a binary relation on119883 with the followingproperties

(O1) Totality of perp for zero 119909 perp 0 0 perp 119909 for all 119909 isin 119883(O2) Independence if 119909 119910 isin 119883 0 119909 perp 119910 then 119909 119910 are

linearly independent(O3) Homogeneity if 119909 119910 isin 119883 119909 perp 119910 then 119886119909 perp 119887119910 for

all 119886 119887 isin R(O4) The Thalesian property Let 119875 be a 2-dimensional

subspace of 119883 If 119909 isin 119875 and 120582 isin R+ then there exists119910 isin 119875 such that 119909 perp 119910 and 119909 + 119910 perp 120582119909 minus 119910

The pair (119883 perp) is called an orthogonality space

Ratz points out that this definition ismore restrictive thanthat given by Gudder and Strawther [24] but he showed thathis definition includes the following basic examples

Example 2 The trivial orthogonality on vector space 119883 isdefined by (O1) and for nonzero elements 119909 119910 isin 119883 119909 perp 119910if and only if 119909 119910 are linearly independent

Example 3 The ordinary orthogonality on inner productspace (119883 ⟨sdot sdot⟩) is given by 119909 perp 119910 if and only if ⟨119909 119910⟩ = 0

Hindawi Publishing CorporationJournal of Function SpacesVolume 2016 Article ID 5636101 7 pageshttpdxdoiorg10115520165636101

2 Journal of Function Spaces

Example 4 The Birkhoff-James orthogonality on a normedspace (119883 sdot) is defined by 119909 perp 119910 if and only if 119909+120582119910 ge 119909for all 120582 isin R

It is well-known that two orthogonal vectors can not becommutated and it is necessary to introduce the followingdefinition

Definition 5 Relation perp is called symmetric if 119909 perp 119910 impliesthat 119910 perp 119909 for all 119909 119910 isin 119883

Clearly Examples 2 and 3 are symmetric but Example 4is not It is remarkable to note however that a real normedspace of dimension greater than or equal to 3 is an innerproduct space if and only if the Birkhoff-James orthogo-nality is symmetric Some notions of orthogonally additiveor orthogonally quadratic function equation are given asfollows

Definition 6 Let119883 be a vector space (an orthogonality space)and (119884 +) be an abelian group Mapping 119891 119883 rarr 119884is called (orthogonally) additive if it satisfies the so-called(orthogonal) additive functional equation

119891 (119909 + 119910) = 119891 (119909) + 119891 (119910) (1)

for all 119909 119910 isin 119883 with 119909 perp 119910

Definition 7 Mapping 119891 119883 rarr 119884 is said to be (orthogonally)quadratic if it satisfies the so-called (orthogonally) Jordan-von Neumann quadratic function equation

119891 (119909 + 119910) + 119891 (119909 minus 119910) = 2119891 (119909) + 2119891 (119910) (2)

for all 119909 119910 isin 119883 with 119909 perp 119910

The orthogonal quadratic equation (2) was first investi-gated by Vajzovic [25] when 119883 is a Hilbert space 119884 is thescalar field 119891 is continuous and perp means the Hilbert spaceorthogonality Later Drljevic [21] Fochi [26] and Szabo [27]generalized this result One of the significant conditionalequations is the so-called orthogonally quadratic functionalequation of Pexider type

119891 (119909 + 119910) + 119891 (119909 minus 119910) = 2119892 (119909) + 2ℎ (119910) 119909 perp 119910 (3)

Moslehian [28] considered the stability of this equationin the spirit of Hyers-Ulam under certain conditions Wewill examine the stability of this equation in more generalsetting such that the target spaces are 119865-spaces or complete120573-normed spaces We have obtained some results whichgeneralize work of Moslehian [28] and will be presented inSection 3 First we recall some notions of 119865-spaces and 120573-normed spaces for detailed understanding of the propertiesof the above spaces the readers are required to read the book[29]

Definition 8 Let 119883 be a linear space over K that denoteseither complex or real numbers A nonnegative valued

function sdot defined on119883 is called119865-norm (or briefly a norm)if it satisfies the following conditions

(n1) 119909 = 0 if and only if 119909 = 0(n2) 119886119909 = 119909 for all 119886 isin K |119886| = 1(n3) 119909 + 119910 le 119909 + 119910(n4) 119886

119899119909 rarr 0 provided 119886

119899rarr 0

(n5) 119886119909119899 rarr 0 provided 119909

119899rarr 0

(n6) 119886119899119909119899 rarr 0 provided 119886

119899rarr 0 119909

119899rarr 0

A linear space equipped with 119865-norm is called 119865lowast-spacewhich will be denoted by (119883 sdot ) or119883 A complete 119865lowast-spaceis called 119865-space

Definition 9 Let 119883 be a linear space over K that denoteseither complex or real numbers and 0 lt 120573 le 1 A nonnegativevalued function sdot defined on119883 is called120573-norm if it satisfiesthe following conditions

(n1) 119909 = 0 if and only if 119909 = 0

(n2) 119886119909 = |119886|120573119909 for all 119886 isin K(n3) 119909 + 119910 le 119909 + 119910

A linear space equipped with 120573-norm is called 120573-normedspace which will be denoted by (119883 sdot ) or briefly119883

Remark 10 It is clear that 120573-normed space is a special 119865lowast-space and when 120573 = 1 120573-normed space become normedspace Comparing with the normed space 119865lowast-space does notpossess good metric properties and the study of the stabilityof functional equations becomes more difficult

There are many forms of the quadratic functional equa-tion among them of great interest to us is the following

119891 (2119909 + 119910) + 119891 (2119909 minus 119910)

= 119891 (119909 + 119910) + 119891 (119909 minus 119910) + 6119891 (119909) (4)

The purpose of this paper is to study the stability ororthogonal stability of equations in 119865lowast-space or 120573-normedspace Section 2 is devoted to the study of stability of (4) inmore general setting such that the target spaces are 119865-spacesor complete 120573-normed spaces In Section 3 we focus on thestudy of orthogonal stability of (3) under the condition thatthe target spaces are 119865-spaces or complete 120573-normed spacesome open problems are also proposed

Throughout this paper let 119883 be a real 120573-normed spaceor orthogonality space and 119884 be complete 119865-spaces or 120573-normed space Also R K and N stand for the set of all realnumbers real numbers or complex numbers and naturalnumbers respectively

2 On the Hyers-Ulam-Rassias Stability of (4)

From now on let 119883 be a real vector space and let 119884 be 119865-space in which there exists 12 le 119888 lt 1 such that 1199102 le119888119910 for all 119909 isin 119884 unless we give any specific reference

Journal of Function Spaces 3

We will investigate the Hyers-Ulam-Rassias stability problemfor functional equation (4) Thus we find the condition thatthere exists a true quadratic function near an approximatelyquadratic function

Theorem 11 Let 119883 be a real vector space and 119884 be 119865-space inwhich there exists 12 le 119888 lt 1 such that 1199102 le 119888119910 for all119909 isin 119884 and let 120593 119883 times 119883 rarr R+ be a function such that

infin

sum119894=0

1198882119894120593 (2119894119909 0) (5)

converges and

lim119899rarrinfin

1198882119899120593 (2119899119909 2119899119910) = 0 (6)

for all 119909 119910 isin 119883 Suppose that 119891 satisfies

1003817100381710038171003817119891 (2119909 + 119910) + 119891 (2119909 minus 119910) minus 119891 (119909 + 119910) minus 119891 (119909 minus 119910)

minus 6119891 (119909)1003817100381710038171003817 le 120593 (119909 119910)

(7)

for all 119909 119910 isin 119883Then there exists unique quadratic function119892 119883 rarr 119884 which satisfies (4) and the inequality

1003817100381710038171003817119891 (119909) minus 119892 (119909)1003817100381710038171003817 le 1198883

infin

sum119894=0

1198882119894120593 (2119894119909 0) (8)

for all 119909 isin 119883 Function 119892 is given by

119892 (119909) = lim119899rarrinfin

2minus2119899119891 (2119899119909) (9)

for all 119909 isin 119883

Proof Putting 119910 = 0 in (7) we have

10038171003817100381710038172119891 (2119909) minus 8119891 (119909)1003817100381710038171003817 le 120593 (119909 0) (10)

It follows that

10038171003817100381710038171003817119891 (119909) minus 4minus1119891 (2119909)

10038171003817100381710038171003817 le 1198883120593 (119909 0) (11)

for all 119909 isin 119883 Replacing 119909 by 2119909 in (11) and by the assumptionon norm we get

100381710038171003817100381710038174minus1119891 (2119909) minus 4

minus2119891 (22119909)10038171003817100381710038171003817 le 1198883 sdot 1198882120593 (2119909 0) (12)

Hence

10038171003817100381710038171003817119891 (119909) minus 4minus2119891 (22119909)

10038171003817100381710038171003817 le 1198883 [120593 (119909 0) + 119888

2120593 (2119909 0)] (13)

for all 119909 isin 119883 Using the induction on positive integer 119899 weobtain that

1003817100381710038171003817119891 (119909) minus 4minus119899119891 (2119899119909)

1003817100381710038171003817 le 1198883

119899minus1

sum119894=0

1198882119894120593 (2119894119909 0) (14)

for all 119909 isin 119883 In order to prove convergence of sequence4minus119899119891(2119899119909) replace 119909 by 2119898119909 to find that for 119899119898 gt 0

100381710038171003817100381710038174minus(119899+119898)119891 (2119899+119898119909) minus 4minus119898119891 (2119898119909)

10038171003817100381710038171003817

=10038171003817100381710038174minus119898 [4minus119899119891 (2119899+119898119909) minus 119891 (2119898119909)]

1003817100381710038171003817

le 11988831198883119899

sum119894=0

1198882(119898+119894)120593 (2119898+119894119909 0)

= 11988831198883119899

sum119894=119898

1198882119894120593 (2119894119909 0)

(15)

Since the right hand side of the inequality tends to 0 as 119898tends to infinity sequence 4minus119899119891(2119899119909) is a Cauchy sequenceTherefore we may define 119892(119909) = lim

119899rarrinfin2minus2119899119891(2119899119909) for all

119909 isin 119883 By letting 119899 rarr infin in (14) we arrive at formula (8) Toshow that119879 satisfies (4) replace 119909 119910 by 2119899119909 2119899119910 respectivelythen it follows that10038171003817100381710038174minus119899 [119891 (2119899 (2119909 + 119910)) + 119891 (2119899 (2119909 minus 119910))

minus 119891 (2119899 (119909 + 119910)) minus 119891 (2119899 (119909 minus 119910)) minus 6119891 (2119899119909)]1003817100381710038171003817

le 1198882119899120593 (2119899119909 2119899119910)

(16)

Taking the limit as 119899 rarr infin we find that 119892 satisfies (4) forall 119909 119910 isin 119883 To prove the uniqueness of quadratic function119879 subject to (8) let us assume that there exists quadraticfunction 119878 119883 rarr 119884 which satisfies (4) and inequality (8)Obviously we have 119878(2119899119909) = 4119899119878(119909) and 119879(2119899119909) = 4119899119879(119909) forall 119909 isin 119883 and 119899 isin 119873 Hence it follows from (8) that

119878 (119909) minus 119879 (119909) =10038171003817100381710038174minus119899 [119878 (2119899119909) minus 119879 (2119899119909)]

1003817100381710038171003817

le 11988821198991003817100381710038171003817119878 (2119899119909) minus 119891 (2119899119909)

1003817100381710038171003817

+1003817100381710038171003817119891 (2119899119909) minus 119879 (2119899119909)

1003817100381710038171003817

le 1198883infin

sum119894=0

1198882(119899+119894)120593 (21198942119899119909 0)

(17)

for all 119909 isin 119883 By letting 119899 rarr infin in the preceding inequalitywe immediately find the uniqueness of 119892 This completes theproof of the theorem

Corollary 12 Let119883 be a real vector space and119884 be a complete120573-normed space (0 lt 120573 le 1) and let 120593 119883 times 119883 rarr 119877+ be afunction such that

infin

sum119894=0

4minus120573119894120593 (2119894119909 0) (18)

converges and

lim119899rarrinfin

4minus120573119899120593 (2119899119909 2119899119910) = 0 (19)

for all 119909 119910 isin 119883 Suppose that 119891 satisfies1003817100381710038171003817119891 (2119909 + 119910) + 119891 (2119909 minus 119910) minus 119891 (119909 + 119910) minus 119891 (119909 minus 119910)

minus 6119891 (119909)1003817100381710038171003817 le 120593 (119909 119910)

(20)

4 Journal of Function Spaces

for all 119909 119910 isin 119883Then there exists unique quadratic function119892 119883 rarr 119884 which satisfies (4) and inequality

1003817100381710038171003817119891 (119909) minus 119892 (119909)1003817100381710038171003817 le 8minus120573

infin

sum119894=0

4minus120573119894120593 (2119894119909 0) (21)

for all 119909 isin 119883 Function 119892 is given by

119892 (119909) = lim119899rarrinfin

2minus2119899119891 (2119899119909) (22)

for all 119909 isin 119883

3 On the Orthogonal Stability of (3)

Applying some ideas from [30] we deal with the conditionalstability problem of the following equation

119891 (119909 + 119910) + 119891 (119909 minus 119910) = 2119892 (119909) + 2ℎ (119910) for 119909 perp 119910 (23)

where 119891 is odd and perp is symmetric Throughout this section(119883 perp) denotes an orthogonality space in the sense of Ratz and(119884 sdot ) is a real 119865-space or 120573-Banach space (0 lt 120573 le 1) Firstwe give a technical lemma

Lemma 13 If 119860 119883 rarr 119884 fulfills 119860(119909 + 119910) + 119860(119909 minus 119910) =2119860(119909) for all 119909 119910 isin 119883 with 119909 perp 119910 and perp is symmetric then119860(119909) minus 119860(0) is orthogonally additive

Proof Assume that 119860(119909 + 119910) + 119860(119909 minus 119910) = 2119860(119909) for all119909 119910 isin 119883 with 119909 perp 119910 Putting 119909 = 0 we get minus119860(119910) =119860(minus119910) minus 2119860(0) 119910 isin 119883 Let 119909 perp 119910 Then 119910 perp 119909 and so119860(119910 minus 119909) = minus119860(119910 + 119909) + 2119860(119910) Hence

119860 (119909 + 119910) = minus119860 (119909 minus 119910) + 2119860 (119909)

= (119860 (119910 minus 119909) minus 2119860 (0)) + 2119860 (119909)

= (minus119860 (119910 + 119909) + 2119860 (119910)) minus 2119860 (0)

+ 2119860 (119909)

(24)

Thus

119860 (119909 + 119910) minus 119860 (0) = (119860 (119909) minus 119860 (0))

+ (119860 (119910) minus 119860 (0)) (25)

Therefore 119860(119909) minus 119860(0) is orthogonally additive

Remark 14 Ratz gave example to demonstrate that thereexists odd mapping 119860 from an orthogonality space into auniquely 2-divisible group (119884 +) (ie an abelian group inwhich map 120593 119884 rarr 119884 120593(119909) = 2119909 is bijective) satisfying119860(119909 + 119910) + 119860(119909 minus 119910) = 2119860(119909) 119909 perp 119910 such that 119860(0) = 0 Heconsidered 119884 = Z

2= 0 1 and 119860(119909) = 1 119909 isin 119883

Theorem 15 Let 119883 be an orthogonality space and 119884 be 119865-space in which there exists 12 le 119888 lt 1 such that 1199102 le 119888119910for all 119909 isin 119884 Suppose that perp is symmetric on 119883 and 119891 119892 ℎ 119883 rarr 119884 are mappings fulfilling

1003817100381710038171003817119891 (119909 + 119910) + 119891 (119909 minus 119910) minus 2119892 (119909) minus 2ℎ (119910)1003817100381710038171003817 le 120576 (26)

for some 120576 gt 0 and for all 119909 119910 isin 119883 with 119909 perp 119910 Assume that119891 is odd and 119892(0) = ℎ(0) = 0 Then there exist one additivemapping 119879 119883 rarr 119884 and one quadratic mapping 119876 119883 rarr 119884such that

1003817100381710038171003817119891 (119909) minus 119879 (119909) minus 119876 (119909)1003817100381710038171003817 le 119888120576 + (3120576 + 6119888120576)

119888

1 minus 119888

119909 isin 119883

1003817100381710038171003817119892 (119909) minus 119879 (119909) minus 119876 (119909)1003817100381710038171003817 le 119888120576 + (3120576 + 6119888120576)

119888

1 minus 119888

(27)

for all 119909 isin 119883

Proof Put119909 = 0 in (26)We can do this because of (O1)Then1003817100381710038171003817119891 (119910) + 119891 (minus119910) minus 2119892 (0) minus 2ℎ (119910)

1003817100381710038171003817 le 120576 (28)

Therefore10038171003817100381710038172ℎ (119910)

1003817100381710038171003817 le 120576 (29)

Similarly by putting 119910 = 0 in (26) we get1003817100381710038171003817119891 (119909) minus 119892 (119909)

1003817100381710038171003817 le 119888120576 (30)

for all 119909 isin 119883 Hence1003817100381710038171003817119891 (119909 + 119910) + 119891 (119909 minus 119910) minus 2119891 (119909)

1003817100381710038171003817

le1003817100381710038171003817119891 (119909 + 119910) + 119891 (119909 minus 119910) minus 2119892 (119909) minus 2ℎ (119910)

1003817100381710038171003817

+10038171003817100381710038172 [119891 (119909) minus 119892 (119909)]

1003817100381710038171003817 +10038171003817100381710038172ℎ (119910)

1003817100381710038171003817 le 3120576

(31)

for all 119909 119910 isin 119883 with 119909 perp 119910 Fix 119909 isin 119883 By (O4) there exists119910 isin 119883 such that119909 perp 119910 and119909+119910 perp 119909minus119910 Sinceperp is symmetric119909 minus 119910 perp 119909 + 119910 too Using inequality (31) and the oddness of119891 we get

1003817100381710038171003817119891 (119909 + 119910) + 119891 (119909 minus 119910) minus 2119891 (119909)1003817100381710038171003817 le 3120576

1003817100381710038171003817119891 (2119909) + 119891 (2119910) minus 2119891 (119909 + 119910)1003817100381710038171003817 le 3120576

1003817100381710038171003817119891 (2119909) minus 119891 (2119910) minus 2119891 (119909 minus 119910)1003817100381710038171003817 le 3120576

(32)

So that1003817100381710038171003817119891 (2119909) minus 2119891 (119909)

1003817100381710038171003817

le1003817100381710038171003817119891 (119909 + 119910) + 119891 (119909 minus 119910) minus 2119891 (119909)

1003817100381710038171003817

+1003817100381710038171003817100381710038171003817

1

2[119891 (2119909) + 119891 (2119910) minus 2119891 (119909 + 119910)]

1003817100381710038171003817100381710038171003817

+1003817100381710038171003817100381710038171003817

1

2[119891 (2119909) minus 119891 (2119910) minus 2119891 (119909 minus 119910)]

1003817100381710038171003817100381710038171003817

le 3120576 + 6119888120576

(33)

It is not hard to see that

10038171003817100381710038172minus119899119891 (2119899119909) minus 119891 (119909)

1003817100381710038171003817 le (3120576 + 6119888120576)119899

sum119896=1

119888119896 (34)

Journal of Function Spaces 5

for all 119899 In fact It follows from100381710038171003817100381710038172minus1119891 (2119909) minus 119891 (119909)

10038171003817100381710038171003817 le 119888 (3120576 + 6119888120576) (35)

that (34) holds for 119899 = 1 Assume that (34) holds for 119896 = 119899when 119896 = 119899 + 1 Replacing 119909 by 2119909 in (34) we get

100381710038171003817100381710038172minus119899119891 (2119899+1119909) minus 119891 (2119909)

10038171003817100381710038171003817 le (3120576 + 6119888120576)119899

sum119896=1

119888119896

100381710038171003817100381710038172minus119899minus1119891 (2119899+1119909) minus 2minus1119891 (119909)

10038171003817100381710038171003817 le (3120576 + 6119888120576)119899

sum119896=1

119888119896+1

(36)

Hence

100381710038171003817100381710038172minus119899minus1119891 (2119899+1119909) minus 119891 (119909)

10038171003817100381710038171003817 le (3120576 + 6119888120576)119899+1

sum119896=1

119888119896 (37)

So formula (34) is proved Replacing 119909 by 2119898119909 in inequality(34) we have

10038171003817100381710038172minus119899119891 (2119899+119898119909) minus 119891 (2119898119909)

1003817100381710038171003817 le (3120576 + 6119888120576)119899

sum119896=1

119888119896

10038171003817100381710038172minus119899minus119898119891 (2119899+119898119909) minus 2minus119898119891 (2119898119909)

1003817100381710038171003817 le (3120576 + 6119888120576)119888119898+1

1 minus 119888

(38)

which implies that 2minus119899119891(2119899119909) is a Cauchy sequence in 119865-space 119884 and lim

119899rarrinfin2minus119899119891(2119899119909) exists and map 119860(119909) fl

lim119899rarrinfin2minus119899119891(2119899119909) is well defined odd map from 119883 into 119884

satisfying

1003817100381710038171003817119891 (119909) minus 119860 (119909)1003817100381710038171003817 le (3120576 + 6119888120576)

119888

1 minus 119888 119909 isin 119883 (39)

For all 119909 119910 isin 119883 with 119909 perp 119910 by applying inequality (31) and(O3) we obtain

100381710038171003817100381710038172minus119899119891 (2119899 (119909 + 119910)) + 2minus119899119891 (2119899 (119909 minus 119910))

minus 2minus119899+1119891 (2119899119909)10038171003817100381710038171003817 le 119888119899 sdot 3120576

(40)

If 119899 rarr infin then we deduce that

119860 (119909 + 119910) + 119860 (119909 minus 119910) minus 2119860 (119909) = 0 (41)

for all 119909 119910 isin 119883 with 119909 perp 119910 Moreover 119860(0) =lim119899rarrinfin2minus119899119891(21198990) = 0 Using Lemma 13 we conclude that 119860

is an orthogonally additivemapping ByCorollary 7 of [23]119860is of form 119879 +119876 with 119879 additive and119876 quadratic If there areanother quadratic mapping1198761015840 and another additive mapping1198791015840 satisfying the required inequalities in our theorem and1198601015840 = 1198791015840 + 1198761015840 then

10038171003817100381710038171003817119860 (119909) minus 1198601015840

(119909)10038171003817100381710038171003817 le1003817100381710038171003817119891 (119909) minus 119860 (119909)

1003817100381710038171003817

+10038171003817100381710038171003817119891 (119909) minus 119860

1015840

(119909)10038171003817100381710038171003817

le (3120576 + 6119888120576)2119888

1 minus 119888

(42)

for all 119909 isin 119883 Using the fact that additive mappings are oddand quadratic mappings are even we obtain

10038171003817100381710038171003817119879 (119909) minus 1198791015840

(119909)10038171003817100381710038171003817 =1003817100381710038171003817100381710038171003817

1

2

sdot [(119879 (119909) + 119876 (119909) minus 1198791015840

(119909) minus 1198761015840

(119909))

+ (119879 (119909) minus 119876 (119909) minus 1198791015840

(119909) + 1198761015840

(119909))] le 11988810038171003817100381710038171003817119879 (119909)

+ 119876 (119909) minus 1198791015840

(119909) minus 1198761015840

(119909)10038171003817100381710038171003817 + 11988810038171003817100381710038171003817119879 (119909) minus 119876 (119909)

minus 1198791015840 (119909) + 1198761015840

(119909)10038171003817100381710038171003817 le 119888

10038171003817100381710038171003817119860 (119909) minus 1198601015840

(119909)10038171003817100381710038171003817

+ 11988810038171003817100381710038171003817119860 (minus119909) minus 119860

1015840

(minus119909)10038171003817100381710038171003817 le (3120576 + 6119888120576)

41198882

1 minus 119888

(43)

Hence10038171003817100381710038171003817119879 (119909) minus 119879

1015840

(119909)10038171003817100381710038171003817 le 119888119899 10038171003817100381710038171003817119879 (2

119899119909) minus 1198791015840 (2119899119909)10038171003817100381710038171003817

le (3120576 + 6119888120576)4119888119899+2

1 minus 119888

(44)

Letting 119899 tends toinfin we infer that 119879 = 1198791015840 Similarly10038171003817100381710038171003817119876 (119909) minus 119876

1015840

(119909)10038171003817100381710038171003817

=1003817100381710038171003817100381710038171003817

1

2[(119879 (119909) + 119876 (119909) minus 119879

1015840

(119909) minus 1198761015840

(119909))

minus (119879 (119909) minus 119876 (119909) minus 1198791015840

(119909) + 1198761015840

(119909))]1003817100381710038171003817100381710038171003817

le 2minus12057310038171003817100381710038171003817119879 (119909) + 119876 (119909) minus 119879

1015840

(119909) minus 1198761015840

(119909)10038171003817100381710038171003817 + 11988810038171003817100381710038171003817119879 (119909)

minus 119876 (119909) minus 1198791015840

(119909) + 1198761015840

(119909)10038171003817100381710038171003817 le 119888

10038171003817100381710038171003817119860 (119909) minus 1198601015840

(119909)10038171003817100381710038171003817

+ 11988810038171003817100381710038171003817119860 (minus119909) minus 119860

1015840

(minus119909)10038171003817100381710038171003817 le (3120576 + 6119888120576)

41198882

1 minus 119888

(45)

for all 119909 isin 119883 Hence10038171003817100381710038171003817119876 (119909) minus 119876

1015840

(119909)10038171003817100381710038171003817 le 1198882119899 10038171003817100381710038171003817119876 (2

119899119909) minus 1198761015840 (2119899119909)10038171003817100381710038171003817

le (3120576 + 6119888120576)41198882119899+2

1 minus 119888

(46)

for all 119909 isin 119883 Taking the limit as 119899 rarr infin we conclude that119876 = 1198761015840 Using (30) and (39) we infer that for all 119909 isin 1198831003817100381710038171003817119892 (119909) minus 119860 (119909)

1003817100381710038171003817 le1003817100381710038171003817119892 (119909) minus 119891 (119909)

1003817100381710038171003817 +1003817100381710038171003817119891 (119909) minus 119860 (119909)

1003817100381710038171003817

le 119888120576 + (3120576 + 6119888120576)119888

1 minus 119888

(47)

Corollary 16 Let 119883 be an orthogonality space and 119884 be 120573-Banach space Suppose that perp is symmetric on 119883 and 119891 119892 ℎ 119883 rarr 119884 are mappings fulfilling

1003817100381710038171003817119891 (119909 + 119910) + 119891 (119909 minus 119910) minus 2119892 (119909) minus 2ℎ (119910)1003817100381710038171003817 le 120576 (48)

6 Journal of Function Spaces

for some 120576 gt 0 and for all 119909 119910 isin 119883 with 119909 perp 119910 Assume that119891 is odd and 119892(0) = ℎ(0) = 0 Then there exist one additivemapping 119879 119883 rarr 119884 and one quadratic mapping 119876 119883 rarr 119884such that

1003817100381710038171003817119891 (119909) minus 119879 (119909) minus 119876 (119909)1003817100381710038171003817 le (3120576 +

6120576

2120573)1

2120573minus1 119909 isin 119883

1003817100381710038171003817119892 (119909) minus 119879 (119909) minus 119876 (119909)1003817100381710038171003817 le 2minus120573120576 +

1

2120573 minus 13 (120576 +

6120576

2120573)

(49)

for all 119909 isin 119883

Remark 17 (i) If 119892 = 120582119891 for some number 120582 = 1 theninequality (30) implies that (1 minus 120582)119891(119909) le 119888120576 119909 isin 119883Hence (1 minus 120582)2minus119899119891(2119899119909) le 119888119899+1120576 119909 isin 119883 So 119860(119909) =lim119899rarrinfin2minus119899119891(2119899119909) = 0 119909 isin 119883

(ii) Similarly if ℎ = 120582119891 for some number 120582 = 0 then itfollows from (29) that 119860(119909) = 0 for all 119909 isin 119883

As far as the author knows unlike orthogonally additivemaps (see Corollary 7 of [23]) there is no characterization fororthogonally quadratic maps Every orthogonally quadraticmapping 119902 into a uniquely 2-divisible abelian group (119884 +) iseven In fact 0 perp 0 so 119902(0)+119902(0) = 4119902(0)Therefore 119902(0) = 0For all 119910 isin 119883 we have 0 perp 119910 and hence 119902(119910) + 119902(minus119910) =2119902(0) + 2119902(119910)

Thus 119902(minus119910) = 119902(119910) There are some characterizations oforthogonally quadratic maps in various notions of orthogo-nality For example if 119860-orthogonality on Hilbert space119867 isdefined by perp

119860= (119909 119910) ⟨119860119909 119910⟩ = 0 where119860 is a bounded

self-adjoint operator on119867 then as shown by Fochi every119860-orthogonally quadratic functional is quadratic if dim119860(119867) ge3 (see [26 27])

To conclude this paper we propose the following prob-lem

Problem Let119883 be an orthogonality space and119884 be119865-space inwhich there exists 12 le 119888 lt 1 such that 1199102 le 119888119910 for all119909 isin 119884 Suppose thatperp is symmetric on119883 and119891 119892 ℎ 119883 rarr 119884are mappings fulfilling

1003817100381710038171003817119891 (119909 + 119910) + 119891 (119909 minus 119910) minus 2119892 (119909) minus 2ℎ (119910)1003817100381710038171003817 le 120576 (50)

for some 120576 and for all 119909 119910 isin 119883 with 119909 perp 119910 Assume that 119891 iseven and 119892(0) = ℎ(0) = 0 Does there exist an orthogonallyquadratic mapping 119876 119883 rarr 119884 under certain conditionssuch that

1003817100381710038171003817119891 (119909) minus 119876 (119909)1003817100381710038171003817 le 120572120576

1003817100381710038171003817119892 (119909) minus 119876 (119909)1003817100381710038171003817 le 120573120576

ℎ (119909) minus 119876 (119909) le 120574120576

(51)

for some scalars 120572 120573 120574 and for all 119909

Competing Interests

The author declares that there are no competing interestsregarding the publication of this paper

Acknowledgments

The paper is supported by the National Natural ScienceFoundation of China (Grant no 11371119) the Key Founda-tion of Education Department of Hebei Province (Grant noZD2016023) and by Natural Science Foundation of Educa-tion Department of Hebei Province (Grant no Z2014031)

References

[1] S M Ulam Problems in Modern Mathematics John Wiley ampSons New York NY USA 1960

[2] D H Hyers ldquoOn the stability of the linear functional equationrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 27 pp 222ndash224 1941

[3] T M Rassias ldquoOn the stability of the linear mapping in Banachspacesrdquo Proceedings of the American Mathematical Society vol72 no 2 pp 297ndash300 1978

[4] Z Gajda ldquoOn stability of additive mappingsrdquo InternationalJournal of Mathematics and Mathematical Sciences vol 14 no3 pp 431ndash434 1991

[5] P Gavruta ldquoA generalization of the Hyers-Ulam-Rassias stabil-ity of approximately additive mappingsrdquo Journal of Mathemati-cal Analysis and Applications vol 184 no 3 pp 431ndash436 1994

[6] R Ger and J Sikorska ldquoStability of the orthogonal additivityrdquoBulletin of the Polish Academy of Sciences Mathematics vol 43no 2 pp 143ndash151 1995

[7] S-M Jung and J M Rassias ldquoA fixed point approach to thestability of a functional equation of the spiral of TheodorusrdquoFixed Point Theory and Applications vol 2008 Article ID945010 7 pages 2008

[8] P L Kannappan Functional Equations and Inequalities withApplications Springer New York NY USA 2009

[9] C G Park ldquoOn the stability of the orthogonally quarticfunctionalrdquo Bulletin of the Iranian Mathematical Society vol 3no 1 pp 63ndash70 2005

[10] C Park and JMRassias ldquoStability of the Jensen-type functionalequation in Clowast-algebras a fixed point approachrdquo Abstract andApplied Analysis vol 2009 Article ID 360432 17 pages 2009

[11] J M Rassias andM J Rassias ldquoAsymptotic behavior of alterna-tive Jensen and Jensen type functional equationsrdquo Bulletin desSciences Mathematiques vol 129 no 7 pp 545ndash558 2005

[12] L G Wang B Liu and R Bai ldquoStability of a mixed typefunctional equation on multi-Banach spaces a fixed pointapproachrdquo Fixed Point Theory and Applications vol 2010Article ID 283827 9 pages 2010

[13] B Xu and J Brzdęk ldquoHyers-Ulam stability of a system of firstorder linear recurrences with constant coefficientsrdquo DiscreteDynamics in Nature and Society vol 2015 Article ID 2693565 pages 2015

[14] B Xu J Brzdęk and W Zhang ldquoFixed-point results and theHyers-Ulam stability of linear equations of higher ordersrdquoPacific Journal ofMathematics vol 273 no 2 pp 483ndash498 2015

[15] X Yang L Chang and G Liu ldquoOrthogonal stability of mixedadditive-quadratic Jensen type functional equation in multi-Banach spacesrdquo Advances in Pure Mathematics vol 5 no 6 pp325ndash332 2015

[16] X Yang L Chang G Liu and G Shen ldquoStability of functionalequations in (n120573)-normed spacesrdquo Journal of Inequalities andApplications vol 2015 article 112 18 pages 2015

Journal of Function Spaces 7

[17] X Zhao X Yang and C-T Pang ldquoSolution and stability ofa general mixed type cubic and quartic functional equationrdquoJournal of Function Spaces and Applications vol 2013 ArticleID 673810 8 pages 2013

[18] N Brillouet-Belluot J Brzdęk and K Cieplinski ldquoOn somerecent developments in Ulamrsquos type stabilityrdquo Abstract andApplied Analysis vol 2012 Article ID 716936 41 pages 2012

[19] J Brzdęk W Fechner M S Moslehian and J Sikorska ldquoRecentdevelopments of the conditional stability of the homomorphismequationrdquo Banach Journal of Mathematical Analysis vol 9 no3 pp 278ndash326 2015

[20] S-M Jung Hyers-Ulam-Rassias Stability of Functional Equa-tions in Nonlinear Analysis vol 48 of Springer Optimization andIts Applications Springer New York NY USA 2011

[21] FDrljevic ldquoOn a functionalwhich is quadratic onA-orthogonalvectorsrdquo Publications de lInstitutetut Mathematique vol 54 pp63ndash71 1986

[22] R C James ldquoOrthogonality and linear functionals in normedlinear spacesrdquo Transactions of the American MathematicalSociety vol 61 pp 265ndash292 1947

[23] J Ratz ldquoOn orthogonally additive mappingsrdquo AequationesMathematicae vol 28 no 1-2 pp 35ndash49 1985

[24] SGudder andD Strawther ldquoOrthogonally additive and orthog-onally increasing functions on vector spacesrdquo Pacific Journal ofMathematics vol 58 no 2 pp 427ndash436 1975

[25] F Vajzovic ldquoUber das funktional h mit der eigenschaft (119909 119910) =0 rArr 119867(119909+119910)+119867(119909minus119910) = 2119867(119909)+2119867(119910)rdquoGlasnikMatematickiSeries III vol 2 no 22 pp 73ndash81 1967

[26] M Fochi ldquoFunctional equations in A-orthogonal vectorsrdquoAequationes Mathematicae vol 38 no 1 pp 28ndash40 1989

[27] G Szabo ldquoSesquilinear-orthogonally quadratic mappingsrdquoAequationes Mathematicae vol 40 no 2-3 pp 190ndash200 1990

[28] M SMoslehian ldquoOn the orthogonal stability of the pexiderizedquadratic equationrdquo Journal of Difference Equations and Appli-cations vol 11 no 11 pp 999ndash1004 2005

[29] S Rolewicz Metric Linear Spaces Polish Scientific PublishersWarsaw Poland 1972

[30] S-M Jung and P K Sahoo ldquoHyers-Ulam stability of thequadratic equation of Pexider typerdquo Journal of the KoreanMathematical Society vol 38 no 3 pp 645ndash656 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Function Spaces

Example 4 The Birkhoff-James orthogonality on a normedspace (119883 sdot) is defined by 119909 perp 119910 if and only if 119909+120582119910 ge 119909for all 120582 isin R

It is well-known that two orthogonal vectors can not becommutated and it is necessary to introduce the followingdefinition

Definition 5 Relation perp is called symmetric if 119909 perp 119910 impliesthat 119910 perp 119909 for all 119909 119910 isin 119883

Clearly Examples 2 and 3 are symmetric but Example 4is not It is remarkable to note however that a real normedspace of dimension greater than or equal to 3 is an innerproduct space if and only if the Birkhoff-James orthogo-nality is symmetric Some notions of orthogonally additiveor orthogonally quadratic function equation are given asfollows

Definition 6 Let119883 be a vector space (an orthogonality space)and (119884 +) be an abelian group Mapping 119891 119883 rarr 119884is called (orthogonally) additive if it satisfies the so-called(orthogonal) additive functional equation

119891 (119909 + 119910) = 119891 (119909) + 119891 (119910) (1)

for all 119909 119910 isin 119883 with 119909 perp 119910

Definition 7 Mapping 119891 119883 rarr 119884 is said to be (orthogonally)quadratic if it satisfies the so-called (orthogonally) Jordan-von Neumann quadratic function equation

119891 (119909 + 119910) + 119891 (119909 minus 119910) = 2119891 (119909) + 2119891 (119910) (2)

for all 119909 119910 isin 119883 with 119909 perp 119910

The orthogonal quadratic equation (2) was first investi-gated by Vajzovic [25] when 119883 is a Hilbert space 119884 is thescalar field 119891 is continuous and perp means the Hilbert spaceorthogonality Later Drljevic [21] Fochi [26] and Szabo [27]generalized this result One of the significant conditionalequations is the so-called orthogonally quadratic functionalequation of Pexider type

119891 (119909 + 119910) + 119891 (119909 minus 119910) = 2119892 (119909) + 2ℎ (119910) 119909 perp 119910 (3)

Moslehian [28] considered the stability of this equationin the spirit of Hyers-Ulam under certain conditions Wewill examine the stability of this equation in more generalsetting such that the target spaces are 119865-spaces or complete120573-normed spaces We have obtained some results whichgeneralize work of Moslehian [28] and will be presented inSection 3 First we recall some notions of 119865-spaces and 120573-normed spaces for detailed understanding of the propertiesof the above spaces the readers are required to read the book[29]

Definition 8 Let 119883 be a linear space over K that denoteseither complex or real numbers A nonnegative valued

function sdot defined on119883 is called119865-norm (or briefly a norm)if it satisfies the following conditions

(n1) 119909 = 0 if and only if 119909 = 0(n2) 119886119909 = 119909 for all 119886 isin K |119886| = 1(n3) 119909 + 119910 le 119909 + 119910(n4) 119886

119899119909 rarr 0 provided 119886

119899rarr 0

(n5) 119886119909119899 rarr 0 provided 119909

119899rarr 0

(n6) 119886119899119909119899 rarr 0 provided 119886

119899rarr 0 119909

119899rarr 0

A linear space equipped with 119865-norm is called 119865lowast-spacewhich will be denoted by (119883 sdot ) or119883 A complete 119865lowast-spaceis called 119865-space

Definition 9 Let 119883 be a linear space over K that denoteseither complex or real numbers and 0 lt 120573 le 1 A nonnegativevalued function sdot defined on119883 is called120573-norm if it satisfiesthe following conditions

(n1) 119909 = 0 if and only if 119909 = 0

(n2) 119886119909 = |119886|120573119909 for all 119886 isin K(n3) 119909 + 119910 le 119909 + 119910

A linear space equipped with 120573-norm is called 120573-normedspace which will be denoted by (119883 sdot ) or briefly119883

Remark 10 It is clear that 120573-normed space is a special 119865lowast-space and when 120573 = 1 120573-normed space become normedspace Comparing with the normed space 119865lowast-space does notpossess good metric properties and the study of the stabilityof functional equations becomes more difficult

There are many forms of the quadratic functional equa-tion among them of great interest to us is the following

119891 (2119909 + 119910) + 119891 (2119909 minus 119910)

= 119891 (119909 + 119910) + 119891 (119909 minus 119910) + 6119891 (119909) (4)

The purpose of this paper is to study the stability ororthogonal stability of equations in 119865lowast-space or 120573-normedspace Section 2 is devoted to the study of stability of (4) inmore general setting such that the target spaces are 119865-spacesor complete 120573-normed spaces In Section 3 we focus on thestudy of orthogonal stability of (3) under the condition thatthe target spaces are 119865-spaces or complete 120573-normed spacesome open problems are also proposed

Throughout this paper let 119883 be a real 120573-normed spaceor orthogonality space and 119884 be complete 119865-spaces or 120573-normed space Also R K and N stand for the set of all realnumbers real numbers or complex numbers and naturalnumbers respectively

2 On the Hyers-Ulam-Rassias Stability of (4)

From now on let 119883 be a real vector space and let 119884 be 119865-space in which there exists 12 le 119888 lt 1 such that 1199102 le119888119910 for all 119909 isin 119884 unless we give any specific reference

Journal of Function Spaces 3

We will investigate the Hyers-Ulam-Rassias stability problemfor functional equation (4) Thus we find the condition thatthere exists a true quadratic function near an approximatelyquadratic function

Theorem 11 Let 119883 be a real vector space and 119884 be 119865-space inwhich there exists 12 le 119888 lt 1 such that 1199102 le 119888119910 for all119909 isin 119884 and let 120593 119883 times 119883 rarr R+ be a function such that

infin

sum119894=0

1198882119894120593 (2119894119909 0) (5)

converges and

lim119899rarrinfin

1198882119899120593 (2119899119909 2119899119910) = 0 (6)

for all 119909 119910 isin 119883 Suppose that 119891 satisfies

1003817100381710038171003817119891 (2119909 + 119910) + 119891 (2119909 minus 119910) minus 119891 (119909 + 119910) minus 119891 (119909 minus 119910)

minus 6119891 (119909)1003817100381710038171003817 le 120593 (119909 119910)

(7)

for all 119909 119910 isin 119883Then there exists unique quadratic function119892 119883 rarr 119884 which satisfies (4) and the inequality

1003817100381710038171003817119891 (119909) minus 119892 (119909)1003817100381710038171003817 le 1198883

infin

sum119894=0

1198882119894120593 (2119894119909 0) (8)

for all 119909 isin 119883 Function 119892 is given by

119892 (119909) = lim119899rarrinfin

2minus2119899119891 (2119899119909) (9)

for all 119909 isin 119883

Proof Putting 119910 = 0 in (7) we have

10038171003817100381710038172119891 (2119909) minus 8119891 (119909)1003817100381710038171003817 le 120593 (119909 0) (10)

It follows that

10038171003817100381710038171003817119891 (119909) minus 4minus1119891 (2119909)

10038171003817100381710038171003817 le 1198883120593 (119909 0) (11)

for all 119909 isin 119883 Replacing 119909 by 2119909 in (11) and by the assumptionon norm we get

100381710038171003817100381710038174minus1119891 (2119909) minus 4

minus2119891 (22119909)10038171003817100381710038171003817 le 1198883 sdot 1198882120593 (2119909 0) (12)

Hence

10038171003817100381710038171003817119891 (119909) minus 4minus2119891 (22119909)

10038171003817100381710038171003817 le 1198883 [120593 (119909 0) + 119888

2120593 (2119909 0)] (13)

for all 119909 isin 119883 Using the induction on positive integer 119899 weobtain that

1003817100381710038171003817119891 (119909) minus 4minus119899119891 (2119899119909)

1003817100381710038171003817 le 1198883

119899minus1

sum119894=0

1198882119894120593 (2119894119909 0) (14)

for all 119909 isin 119883 In order to prove convergence of sequence4minus119899119891(2119899119909) replace 119909 by 2119898119909 to find that for 119899119898 gt 0

100381710038171003817100381710038174minus(119899+119898)119891 (2119899+119898119909) minus 4minus119898119891 (2119898119909)

10038171003817100381710038171003817

=10038171003817100381710038174minus119898 [4minus119899119891 (2119899+119898119909) minus 119891 (2119898119909)]

1003817100381710038171003817

le 11988831198883119899

sum119894=0

1198882(119898+119894)120593 (2119898+119894119909 0)

= 11988831198883119899

sum119894=119898

1198882119894120593 (2119894119909 0)

(15)

Since the right hand side of the inequality tends to 0 as 119898tends to infinity sequence 4minus119899119891(2119899119909) is a Cauchy sequenceTherefore we may define 119892(119909) = lim

119899rarrinfin2minus2119899119891(2119899119909) for all

119909 isin 119883 By letting 119899 rarr infin in (14) we arrive at formula (8) Toshow that119879 satisfies (4) replace 119909 119910 by 2119899119909 2119899119910 respectivelythen it follows that10038171003817100381710038174minus119899 [119891 (2119899 (2119909 + 119910)) + 119891 (2119899 (2119909 minus 119910))

minus 119891 (2119899 (119909 + 119910)) minus 119891 (2119899 (119909 minus 119910)) minus 6119891 (2119899119909)]1003817100381710038171003817

le 1198882119899120593 (2119899119909 2119899119910)

(16)

Taking the limit as 119899 rarr infin we find that 119892 satisfies (4) forall 119909 119910 isin 119883 To prove the uniqueness of quadratic function119879 subject to (8) let us assume that there exists quadraticfunction 119878 119883 rarr 119884 which satisfies (4) and inequality (8)Obviously we have 119878(2119899119909) = 4119899119878(119909) and 119879(2119899119909) = 4119899119879(119909) forall 119909 isin 119883 and 119899 isin 119873 Hence it follows from (8) that

119878 (119909) minus 119879 (119909) =10038171003817100381710038174minus119899 [119878 (2119899119909) minus 119879 (2119899119909)]

1003817100381710038171003817

le 11988821198991003817100381710038171003817119878 (2119899119909) minus 119891 (2119899119909)

1003817100381710038171003817

+1003817100381710038171003817119891 (2119899119909) minus 119879 (2119899119909)

1003817100381710038171003817

le 1198883infin

sum119894=0

1198882(119899+119894)120593 (21198942119899119909 0)

(17)

for all 119909 isin 119883 By letting 119899 rarr infin in the preceding inequalitywe immediately find the uniqueness of 119892 This completes theproof of the theorem

Corollary 12 Let119883 be a real vector space and119884 be a complete120573-normed space (0 lt 120573 le 1) and let 120593 119883 times 119883 rarr 119877+ be afunction such that

infin

sum119894=0

4minus120573119894120593 (2119894119909 0) (18)

converges and

lim119899rarrinfin

4minus120573119899120593 (2119899119909 2119899119910) = 0 (19)

for all 119909 119910 isin 119883 Suppose that 119891 satisfies1003817100381710038171003817119891 (2119909 + 119910) + 119891 (2119909 minus 119910) minus 119891 (119909 + 119910) minus 119891 (119909 minus 119910)

minus 6119891 (119909)1003817100381710038171003817 le 120593 (119909 119910)

(20)

4 Journal of Function Spaces

for all 119909 119910 isin 119883Then there exists unique quadratic function119892 119883 rarr 119884 which satisfies (4) and inequality

1003817100381710038171003817119891 (119909) minus 119892 (119909)1003817100381710038171003817 le 8minus120573

infin

sum119894=0

4minus120573119894120593 (2119894119909 0) (21)

for all 119909 isin 119883 Function 119892 is given by

119892 (119909) = lim119899rarrinfin

2minus2119899119891 (2119899119909) (22)

for all 119909 isin 119883

3 On the Orthogonal Stability of (3)

Applying some ideas from [30] we deal with the conditionalstability problem of the following equation

119891 (119909 + 119910) + 119891 (119909 minus 119910) = 2119892 (119909) + 2ℎ (119910) for 119909 perp 119910 (23)

where 119891 is odd and perp is symmetric Throughout this section(119883 perp) denotes an orthogonality space in the sense of Ratz and(119884 sdot ) is a real 119865-space or 120573-Banach space (0 lt 120573 le 1) Firstwe give a technical lemma

Lemma 13 If 119860 119883 rarr 119884 fulfills 119860(119909 + 119910) + 119860(119909 minus 119910) =2119860(119909) for all 119909 119910 isin 119883 with 119909 perp 119910 and perp is symmetric then119860(119909) minus 119860(0) is orthogonally additive

Proof Assume that 119860(119909 + 119910) + 119860(119909 minus 119910) = 2119860(119909) for all119909 119910 isin 119883 with 119909 perp 119910 Putting 119909 = 0 we get minus119860(119910) =119860(minus119910) minus 2119860(0) 119910 isin 119883 Let 119909 perp 119910 Then 119910 perp 119909 and so119860(119910 minus 119909) = minus119860(119910 + 119909) + 2119860(119910) Hence

119860 (119909 + 119910) = minus119860 (119909 minus 119910) + 2119860 (119909)

= (119860 (119910 minus 119909) minus 2119860 (0)) + 2119860 (119909)

= (minus119860 (119910 + 119909) + 2119860 (119910)) minus 2119860 (0)

+ 2119860 (119909)

(24)

Thus

119860 (119909 + 119910) minus 119860 (0) = (119860 (119909) minus 119860 (0))

+ (119860 (119910) minus 119860 (0)) (25)

Therefore 119860(119909) minus 119860(0) is orthogonally additive

Remark 14 Ratz gave example to demonstrate that thereexists odd mapping 119860 from an orthogonality space into auniquely 2-divisible group (119884 +) (ie an abelian group inwhich map 120593 119884 rarr 119884 120593(119909) = 2119909 is bijective) satisfying119860(119909 + 119910) + 119860(119909 minus 119910) = 2119860(119909) 119909 perp 119910 such that 119860(0) = 0 Heconsidered 119884 = Z

2= 0 1 and 119860(119909) = 1 119909 isin 119883

Theorem 15 Let 119883 be an orthogonality space and 119884 be 119865-space in which there exists 12 le 119888 lt 1 such that 1199102 le 119888119910for all 119909 isin 119884 Suppose that perp is symmetric on 119883 and 119891 119892 ℎ 119883 rarr 119884 are mappings fulfilling

1003817100381710038171003817119891 (119909 + 119910) + 119891 (119909 minus 119910) minus 2119892 (119909) minus 2ℎ (119910)1003817100381710038171003817 le 120576 (26)

for some 120576 gt 0 and for all 119909 119910 isin 119883 with 119909 perp 119910 Assume that119891 is odd and 119892(0) = ℎ(0) = 0 Then there exist one additivemapping 119879 119883 rarr 119884 and one quadratic mapping 119876 119883 rarr 119884such that

1003817100381710038171003817119891 (119909) minus 119879 (119909) minus 119876 (119909)1003817100381710038171003817 le 119888120576 + (3120576 + 6119888120576)

119888

1 minus 119888

119909 isin 119883

1003817100381710038171003817119892 (119909) minus 119879 (119909) minus 119876 (119909)1003817100381710038171003817 le 119888120576 + (3120576 + 6119888120576)

119888

1 minus 119888

(27)

for all 119909 isin 119883

Proof Put119909 = 0 in (26)We can do this because of (O1)Then1003817100381710038171003817119891 (119910) + 119891 (minus119910) minus 2119892 (0) minus 2ℎ (119910)

1003817100381710038171003817 le 120576 (28)

Therefore10038171003817100381710038172ℎ (119910)

1003817100381710038171003817 le 120576 (29)

Similarly by putting 119910 = 0 in (26) we get1003817100381710038171003817119891 (119909) minus 119892 (119909)

1003817100381710038171003817 le 119888120576 (30)

for all 119909 isin 119883 Hence1003817100381710038171003817119891 (119909 + 119910) + 119891 (119909 minus 119910) minus 2119891 (119909)

1003817100381710038171003817

le1003817100381710038171003817119891 (119909 + 119910) + 119891 (119909 minus 119910) minus 2119892 (119909) minus 2ℎ (119910)

1003817100381710038171003817

+10038171003817100381710038172 [119891 (119909) minus 119892 (119909)]

1003817100381710038171003817 +10038171003817100381710038172ℎ (119910)

1003817100381710038171003817 le 3120576

(31)

for all 119909 119910 isin 119883 with 119909 perp 119910 Fix 119909 isin 119883 By (O4) there exists119910 isin 119883 such that119909 perp 119910 and119909+119910 perp 119909minus119910 Sinceperp is symmetric119909 minus 119910 perp 119909 + 119910 too Using inequality (31) and the oddness of119891 we get

1003817100381710038171003817119891 (119909 + 119910) + 119891 (119909 minus 119910) minus 2119891 (119909)1003817100381710038171003817 le 3120576

1003817100381710038171003817119891 (2119909) + 119891 (2119910) minus 2119891 (119909 + 119910)1003817100381710038171003817 le 3120576

1003817100381710038171003817119891 (2119909) minus 119891 (2119910) minus 2119891 (119909 minus 119910)1003817100381710038171003817 le 3120576

(32)

So that1003817100381710038171003817119891 (2119909) minus 2119891 (119909)

1003817100381710038171003817

le1003817100381710038171003817119891 (119909 + 119910) + 119891 (119909 minus 119910) minus 2119891 (119909)

1003817100381710038171003817

+1003817100381710038171003817100381710038171003817

1

2[119891 (2119909) + 119891 (2119910) minus 2119891 (119909 + 119910)]

1003817100381710038171003817100381710038171003817

+1003817100381710038171003817100381710038171003817

1

2[119891 (2119909) minus 119891 (2119910) minus 2119891 (119909 minus 119910)]

1003817100381710038171003817100381710038171003817

le 3120576 + 6119888120576

(33)

It is not hard to see that

10038171003817100381710038172minus119899119891 (2119899119909) minus 119891 (119909)

1003817100381710038171003817 le (3120576 + 6119888120576)119899

sum119896=1

119888119896 (34)

Journal of Function Spaces 5

for all 119899 In fact It follows from100381710038171003817100381710038172minus1119891 (2119909) minus 119891 (119909)

10038171003817100381710038171003817 le 119888 (3120576 + 6119888120576) (35)

that (34) holds for 119899 = 1 Assume that (34) holds for 119896 = 119899when 119896 = 119899 + 1 Replacing 119909 by 2119909 in (34) we get

100381710038171003817100381710038172minus119899119891 (2119899+1119909) minus 119891 (2119909)

10038171003817100381710038171003817 le (3120576 + 6119888120576)119899

sum119896=1

119888119896

100381710038171003817100381710038172minus119899minus1119891 (2119899+1119909) minus 2minus1119891 (119909)

10038171003817100381710038171003817 le (3120576 + 6119888120576)119899

sum119896=1

119888119896+1

(36)

Hence

100381710038171003817100381710038172minus119899minus1119891 (2119899+1119909) minus 119891 (119909)

10038171003817100381710038171003817 le (3120576 + 6119888120576)119899+1

sum119896=1

119888119896 (37)

So formula (34) is proved Replacing 119909 by 2119898119909 in inequality(34) we have

10038171003817100381710038172minus119899119891 (2119899+119898119909) minus 119891 (2119898119909)

1003817100381710038171003817 le (3120576 + 6119888120576)119899

sum119896=1

119888119896

10038171003817100381710038172minus119899minus119898119891 (2119899+119898119909) minus 2minus119898119891 (2119898119909)

1003817100381710038171003817 le (3120576 + 6119888120576)119888119898+1

1 minus 119888

(38)

which implies that 2minus119899119891(2119899119909) is a Cauchy sequence in 119865-space 119884 and lim

119899rarrinfin2minus119899119891(2119899119909) exists and map 119860(119909) fl

lim119899rarrinfin2minus119899119891(2119899119909) is well defined odd map from 119883 into 119884

satisfying

1003817100381710038171003817119891 (119909) minus 119860 (119909)1003817100381710038171003817 le (3120576 + 6119888120576)

119888

1 minus 119888 119909 isin 119883 (39)

For all 119909 119910 isin 119883 with 119909 perp 119910 by applying inequality (31) and(O3) we obtain

100381710038171003817100381710038172minus119899119891 (2119899 (119909 + 119910)) + 2minus119899119891 (2119899 (119909 minus 119910))

minus 2minus119899+1119891 (2119899119909)10038171003817100381710038171003817 le 119888119899 sdot 3120576

(40)

If 119899 rarr infin then we deduce that

119860 (119909 + 119910) + 119860 (119909 minus 119910) minus 2119860 (119909) = 0 (41)

for all 119909 119910 isin 119883 with 119909 perp 119910 Moreover 119860(0) =lim119899rarrinfin2minus119899119891(21198990) = 0 Using Lemma 13 we conclude that 119860

is an orthogonally additivemapping ByCorollary 7 of [23]119860is of form 119879 +119876 with 119879 additive and119876 quadratic If there areanother quadratic mapping1198761015840 and another additive mapping1198791015840 satisfying the required inequalities in our theorem and1198601015840 = 1198791015840 + 1198761015840 then

10038171003817100381710038171003817119860 (119909) minus 1198601015840

(119909)10038171003817100381710038171003817 le1003817100381710038171003817119891 (119909) minus 119860 (119909)

1003817100381710038171003817

+10038171003817100381710038171003817119891 (119909) minus 119860

1015840

(119909)10038171003817100381710038171003817

le (3120576 + 6119888120576)2119888

1 minus 119888

(42)

for all 119909 isin 119883 Using the fact that additive mappings are oddand quadratic mappings are even we obtain

10038171003817100381710038171003817119879 (119909) minus 1198791015840

(119909)10038171003817100381710038171003817 =1003817100381710038171003817100381710038171003817

1

2

sdot [(119879 (119909) + 119876 (119909) minus 1198791015840

(119909) minus 1198761015840

(119909))

+ (119879 (119909) minus 119876 (119909) minus 1198791015840

(119909) + 1198761015840

(119909))] le 11988810038171003817100381710038171003817119879 (119909)

+ 119876 (119909) minus 1198791015840

(119909) minus 1198761015840

(119909)10038171003817100381710038171003817 + 11988810038171003817100381710038171003817119879 (119909) minus 119876 (119909)

minus 1198791015840 (119909) + 1198761015840

(119909)10038171003817100381710038171003817 le 119888

10038171003817100381710038171003817119860 (119909) minus 1198601015840

(119909)10038171003817100381710038171003817

+ 11988810038171003817100381710038171003817119860 (minus119909) minus 119860

1015840

(minus119909)10038171003817100381710038171003817 le (3120576 + 6119888120576)

41198882

1 minus 119888

(43)

Hence10038171003817100381710038171003817119879 (119909) minus 119879

1015840

(119909)10038171003817100381710038171003817 le 119888119899 10038171003817100381710038171003817119879 (2

119899119909) minus 1198791015840 (2119899119909)10038171003817100381710038171003817

le (3120576 + 6119888120576)4119888119899+2

1 minus 119888

(44)

Letting 119899 tends toinfin we infer that 119879 = 1198791015840 Similarly10038171003817100381710038171003817119876 (119909) minus 119876

1015840

(119909)10038171003817100381710038171003817

=1003817100381710038171003817100381710038171003817

1

2[(119879 (119909) + 119876 (119909) minus 119879

1015840

(119909) minus 1198761015840

(119909))

minus (119879 (119909) minus 119876 (119909) minus 1198791015840

(119909) + 1198761015840

(119909))]1003817100381710038171003817100381710038171003817

le 2minus12057310038171003817100381710038171003817119879 (119909) + 119876 (119909) minus 119879

1015840

(119909) minus 1198761015840

(119909)10038171003817100381710038171003817 + 11988810038171003817100381710038171003817119879 (119909)

minus 119876 (119909) minus 1198791015840

(119909) + 1198761015840

(119909)10038171003817100381710038171003817 le 119888

10038171003817100381710038171003817119860 (119909) minus 1198601015840

(119909)10038171003817100381710038171003817

+ 11988810038171003817100381710038171003817119860 (minus119909) minus 119860

1015840

(minus119909)10038171003817100381710038171003817 le (3120576 + 6119888120576)

41198882

1 minus 119888

(45)

for all 119909 isin 119883 Hence10038171003817100381710038171003817119876 (119909) minus 119876

1015840

(119909)10038171003817100381710038171003817 le 1198882119899 10038171003817100381710038171003817119876 (2

119899119909) minus 1198761015840 (2119899119909)10038171003817100381710038171003817

le (3120576 + 6119888120576)41198882119899+2

1 minus 119888

(46)

for all 119909 isin 119883 Taking the limit as 119899 rarr infin we conclude that119876 = 1198761015840 Using (30) and (39) we infer that for all 119909 isin 1198831003817100381710038171003817119892 (119909) minus 119860 (119909)

1003817100381710038171003817 le1003817100381710038171003817119892 (119909) minus 119891 (119909)

1003817100381710038171003817 +1003817100381710038171003817119891 (119909) minus 119860 (119909)

1003817100381710038171003817

le 119888120576 + (3120576 + 6119888120576)119888

1 minus 119888

(47)

Corollary 16 Let 119883 be an orthogonality space and 119884 be 120573-Banach space Suppose that perp is symmetric on 119883 and 119891 119892 ℎ 119883 rarr 119884 are mappings fulfilling

1003817100381710038171003817119891 (119909 + 119910) + 119891 (119909 minus 119910) minus 2119892 (119909) minus 2ℎ (119910)1003817100381710038171003817 le 120576 (48)

6 Journal of Function Spaces

for some 120576 gt 0 and for all 119909 119910 isin 119883 with 119909 perp 119910 Assume that119891 is odd and 119892(0) = ℎ(0) = 0 Then there exist one additivemapping 119879 119883 rarr 119884 and one quadratic mapping 119876 119883 rarr 119884such that

1003817100381710038171003817119891 (119909) minus 119879 (119909) minus 119876 (119909)1003817100381710038171003817 le (3120576 +

6120576

2120573)1

2120573minus1 119909 isin 119883

1003817100381710038171003817119892 (119909) minus 119879 (119909) minus 119876 (119909)1003817100381710038171003817 le 2minus120573120576 +

1

2120573 minus 13 (120576 +

6120576

2120573)

(49)

for all 119909 isin 119883

Remark 17 (i) If 119892 = 120582119891 for some number 120582 = 1 theninequality (30) implies that (1 minus 120582)119891(119909) le 119888120576 119909 isin 119883Hence (1 minus 120582)2minus119899119891(2119899119909) le 119888119899+1120576 119909 isin 119883 So 119860(119909) =lim119899rarrinfin2minus119899119891(2119899119909) = 0 119909 isin 119883

(ii) Similarly if ℎ = 120582119891 for some number 120582 = 0 then itfollows from (29) that 119860(119909) = 0 for all 119909 isin 119883

As far as the author knows unlike orthogonally additivemaps (see Corollary 7 of [23]) there is no characterization fororthogonally quadratic maps Every orthogonally quadraticmapping 119902 into a uniquely 2-divisible abelian group (119884 +) iseven In fact 0 perp 0 so 119902(0)+119902(0) = 4119902(0)Therefore 119902(0) = 0For all 119910 isin 119883 we have 0 perp 119910 and hence 119902(119910) + 119902(minus119910) =2119902(0) + 2119902(119910)

Thus 119902(minus119910) = 119902(119910) There are some characterizations oforthogonally quadratic maps in various notions of orthogo-nality For example if 119860-orthogonality on Hilbert space119867 isdefined by perp

119860= (119909 119910) ⟨119860119909 119910⟩ = 0 where119860 is a bounded

self-adjoint operator on119867 then as shown by Fochi every119860-orthogonally quadratic functional is quadratic if dim119860(119867) ge3 (see [26 27])

To conclude this paper we propose the following prob-lem

Problem Let119883 be an orthogonality space and119884 be119865-space inwhich there exists 12 le 119888 lt 1 such that 1199102 le 119888119910 for all119909 isin 119884 Suppose thatperp is symmetric on119883 and119891 119892 ℎ 119883 rarr 119884are mappings fulfilling

1003817100381710038171003817119891 (119909 + 119910) + 119891 (119909 minus 119910) minus 2119892 (119909) minus 2ℎ (119910)1003817100381710038171003817 le 120576 (50)

for some 120576 and for all 119909 119910 isin 119883 with 119909 perp 119910 Assume that 119891 iseven and 119892(0) = ℎ(0) = 0 Does there exist an orthogonallyquadratic mapping 119876 119883 rarr 119884 under certain conditionssuch that

1003817100381710038171003817119891 (119909) minus 119876 (119909)1003817100381710038171003817 le 120572120576

1003817100381710038171003817119892 (119909) minus 119876 (119909)1003817100381710038171003817 le 120573120576

ℎ (119909) minus 119876 (119909) le 120574120576

(51)

for some scalars 120572 120573 120574 and for all 119909

Competing Interests

The author declares that there are no competing interestsregarding the publication of this paper

Acknowledgments

The paper is supported by the National Natural ScienceFoundation of China (Grant no 11371119) the Key Founda-tion of Education Department of Hebei Province (Grant noZD2016023) and by Natural Science Foundation of Educa-tion Department of Hebei Province (Grant no Z2014031)

References

[1] S M Ulam Problems in Modern Mathematics John Wiley ampSons New York NY USA 1960

[2] D H Hyers ldquoOn the stability of the linear functional equationrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 27 pp 222ndash224 1941

[3] T M Rassias ldquoOn the stability of the linear mapping in Banachspacesrdquo Proceedings of the American Mathematical Society vol72 no 2 pp 297ndash300 1978

[4] Z Gajda ldquoOn stability of additive mappingsrdquo InternationalJournal of Mathematics and Mathematical Sciences vol 14 no3 pp 431ndash434 1991

[5] P Gavruta ldquoA generalization of the Hyers-Ulam-Rassias stabil-ity of approximately additive mappingsrdquo Journal of Mathemati-cal Analysis and Applications vol 184 no 3 pp 431ndash436 1994

[6] R Ger and J Sikorska ldquoStability of the orthogonal additivityrdquoBulletin of the Polish Academy of Sciences Mathematics vol 43no 2 pp 143ndash151 1995

[7] S-M Jung and J M Rassias ldquoA fixed point approach to thestability of a functional equation of the spiral of TheodorusrdquoFixed Point Theory and Applications vol 2008 Article ID945010 7 pages 2008

[8] P L Kannappan Functional Equations and Inequalities withApplications Springer New York NY USA 2009

[9] C G Park ldquoOn the stability of the orthogonally quarticfunctionalrdquo Bulletin of the Iranian Mathematical Society vol 3no 1 pp 63ndash70 2005

[10] C Park and JMRassias ldquoStability of the Jensen-type functionalequation in Clowast-algebras a fixed point approachrdquo Abstract andApplied Analysis vol 2009 Article ID 360432 17 pages 2009

[11] J M Rassias andM J Rassias ldquoAsymptotic behavior of alterna-tive Jensen and Jensen type functional equationsrdquo Bulletin desSciences Mathematiques vol 129 no 7 pp 545ndash558 2005

[12] L G Wang B Liu and R Bai ldquoStability of a mixed typefunctional equation on multi-Banach spaces a fixed pointapproachrdquo Fixed Point Theory and Applications vol 2010Article ID 283827 9 pages 2010

[13] B Xu and J Brzdęk ldquoHyers-Ulam stability of a system of firstorder linear recurrences with constant coefficientsrdquo DiscreteDynamics in Nature and Society vol 2015 Article ID 2693565 pages 2015

[14] B Xu J Brzdęk and W Zhang ldquoFixed-point results and theHyers-Ulam stability of linear equations of higher ordersrdquoPacific Journal ofMathematics vol 273 no 2 pp 483ndash498 2015

[15] X Yang L Chang and G Liu ldquoOrthogonal stability of mixedadditive-quadratic Jensen type functional equation in multi-Banach spacesrdquo Advances in Pure Mathematics vol 5 no 6 pp325ndash332 2015

[16] X Yang L Chang G Liu and G Shen ldquoStability of functionalequations in (n120573)-normed spacesrdquo Journal of Inequalities andApplications vol 2015 article 112 18 pages 2015

Journal of Function Spaces 7

[17] X Zhao X Yang and C-T Pang ldquoSolution and stability ofa general mixed type cubic and quartic functional equationrdquoJournal of Function Spaces and Applications vol 2013 ArticleID 673810 8 pages 2013

[18] N Brillouet-Belluot J Brzdęk and K Cieplinski ldquoOn somerecent developments in Ulamrsquos type stabilityrdquo Abstract andApplied Analysis vol 2012 Article ID 716936 41 pages 2012

[19] J Brzdęk W Fechner M S Moslehian and J Sikorska ldquoRecentdevelopments of the conditional stability of the homomorphismequationrdquo Banach Journal of Mathematical Analysis vol 9 no3 pp 278ndash326 2015

[20] S-M Jung Hyers-Ulam-Rassias Stability of Functional Equa-tions in Nonlinear Analysis vol 48 of Springer Optimization andIts Applications Springer New York NY USA 2011

[21] FDrljevic ldquoOn a functionalwhich is quadratic onA-orthogonalvectorsrdquo Publications de lInstitutetut Mathematique vol 54 pp63ndash71 1986

[22] R C James ldquoOrthogonality and linear functionals in normedlinear spacesrdquo Transactions of the American MathematicalSociety vol 61 pp 265ndash292 1947

[23] J Ratz ldquoOn orthogonally additive mappingsrdquo AequationesMathematicae vol 28 no 1-2 pp 35ndash49 1985

[24] SGudder andD Strawther ldquoOrthogonally additive and orthog-onally increasing functions on vector spacesrdquo Pacific Journal ofMathematics vol 58 no 2 pp 427ndash436 1975

[25] F Vajzovic ldquoUber das funktional h mit der eigenschaft (119909 119910) =0 rArr 119867(119909+119910)+119867(119909minus119910) = 2119867(119909)+2119867(119910)rdquoGlasnikMatematickiSeries III vol 2 no 22 pp 73ndash81 1967

[26] M Fochi ldquoFunctional equations in A-orthogonal vectorsrdquoAequationes Mathematicae vol 38 no 1 pp 28ndash40 1989

[27] G Szabo ldquoSesquilinear-orthogonally quadratic mappingsrdquoAequationes Mathematicae vol 40 no 2-3 pp 190ndash200 1990

[28] M SMoslehian ldquoOn the orthogonal stability of the pexiderizedquadratic equationrdquo Journal of Difference Equations and Appli-cations vol 11 no 11 pp 999ndash1004 2005

[29] S Rolewicz Metric Linear Spaces Polish Scientific PublishersWarsaw Poland 1972

[30] S-M Jung and P K Sahoo ldquoHyers-Ulam stability of thequadratic equation of Pexider typerdquo Journal of the KoreanMathematical Society vol 38 no 3 pp 645ndash656 2001

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Stochastic AnalysisInternational Journal of

Journal of Function Spaces 3

We will investigate the Hyers-Ulam-Rassias stability problemfor functional equation (4) Thus we find the condition thatthere exists a true quadratic function near an approximatelyquadratic function

Theorem 11 Let 119883 be a real vector space and 119884 be 119865-space inwhich there exists 12 le 119888 lt 1 such that 1199102 le 119888119910 for all119909 isin 119884 and let 120593 119883 times 119883 rarr R+ be a function such that

infin

sum119894=0

1198882119894120593 (2119894119909 0) (5)

converges and

lim119899rarrinfin

1198882119899120593 (2119899119909 2119899119910) = 0 (6)

for all 119909 119910 isin 119883 Suppose that 119891 satisfies

1003817100381710038171003817119891 (2119909 + 119910) + 119891 (2119909 minus 119910) minus 119891 (119909 + 119910) minus 119891 (119909 minus 119910)

minus 6119891 (119909)1003817100381710038171003817 le 120593 (119909 119910)

(7)

for all 119909 119910 isin 119883Then there exists unique quadratic function119892 119883 rarr 119884 which satisfies (4) and the inequality

1003817100381710038171003817119891 (119909) minus 119892 (119909)1003817100381710038171003817 le 1198883

infin

sum119894=0

1198882119894120593 (2119894119909 0) (8)

for all 119909 isin 119883 Function 119892 is given by

119892 (119909) = lim119899rarrinfin

2minus2119899119891 (2119899119909) (9)

for all 119909 isin 119883

Proof Putting 119910 = 0 in (7) we have

10038171003817100381710038172119891 (2119909) minus 8119891 (119909)1003817100381710038171003817 le 120593 (119909 0) (10)

It follows that

10038171003817100381710038171003817119891 (119909) minus 4minus1119891 (2119909)

10038171003817100381710038171003817 le 1198883120593 (119909 0) (11)

for all 119909 isin 119883 Replacing 119909 by 2119909 in (11) and by the assumptionon norm we get

100381710038171003817100381710038174minus1119891 (2119909) minus 4

minus2119891 (22119909)10038171003817100381710038171003817 le 1198883 sdot 1198882120593 (2119909 0) (12)

Hence

10038171003817100381710038171003817119891 (119909) minus 4minus2119891 (22119909)

10038171003817100381710038171003817 le 1198883 [120593 (119909 0) + 119888

2120593 (2119909 0)] (13)

for all 119909 isin 119883 Using the induction on positive integer 119899 weobtain that

1003817100381710038171003817119891 (119909) minus 4minus119899119891 (2119899119909)

1003817100381710038171003817 le 1198883

119899minus1

sum119894=0

1198882119894120593 (2119894119909 0) (14)

for all 119909 isin 119883 In order to prove convergence of sequence4minus119899119891(2119899119909) replace 119909 by 2119898119909 to find that for 119899119898 gt 0

100381710038171003817100381710038174minus(119899+119898)119891 (2119899+119898119909) minus 4minus119898119891 (2119898119909)

10038171003817100381710038171003817

=10038171003817100381710038174minus119898 [4minus119899119891 (2119899+119898119909) minus 119891 (2119898119909)]

1003817100381710038171003817

le 11988831198883119899

sum119894=0

1198882(119898+119894)120593 (2119898+119894119909 0)

= 11988831198883119899

sum119894=119898

1198882119894120593 (2119894119909 0)

(15)

Since the right hand side of the inequality tends to 0 as 119898tends to infinity sequence 4minus119899119891(2119899119909) is a Cauchy sequenceTherefore we may define 119892(119909) = lim

119899rarrinfin2minus2119899119891(2119899119909) for all

119909 isin 119883 By letting 119899 rarr infin in (14) we arrive at formula (8) Toshow that119879 satisfies (4) replace 119909 119910 by 2119899119909 2119899119910 respectivelythen it follows that10038171003817100381710038174minus119899 [119891 (2119899 (2119909 + 119910)) + 119891 (2119899 (2119909 minus 119910))

minus 119891 (2119899 (119909 + 119910)) minus 119891 (2119899 (119909 minus 119910)) minus 6119891 (2119899119909)]1003817100381710038171003817

le 1198882119899120593 (2119899119909 2119899119910)

(16)

Taking the limit as 119899 rarr infin we find that 119892 satisfies (4) forall 119909 119910 isin 119883 To prove the uniqueness of quadratic function119879 subject to (8) let us assume that there exists quadraticfunction 119878 119883 rarr 119884 which satisfies (4) and inequality (8)Obviously we have 119878(2119899119909) = 4119899119878(119909) and 119879(2119899119909) = 4119899119879(119909) forall 119909 isin 119883 and 119899 isin 119873 Hence it follows from (8) that

119878 (119909) minus 119879 (119909) =10038171003817100381710038174minus119899 [119878 (2119899119909) minus 119879 (2119899119909)]

1003817100381710038171003817

le 11988821198991003817100381710038171003817119878 (2119899119909) minus 119891 (2119899119909)

1003817100381710038171003817

+1003817100381710038171003817119891 (2119899119909) minus 119879 (2119899119909)

1003817100381710038171003817

le 1198883infin

sum119894=0

1198882(119899+119894)120593 (21198942119899119909 0)

(17)

for all 119909 isin 119883 By letting 119899 rarr infin in the preceding inequalitywe immediately find the uniqueness of 119892 This completes theproof of the theorem

Corollary 12 Let119883 be a real vector space and119884 be a complete120573-normed space (0 lt 120573 le 1) and let 120593 119883 times 119883 rarr 119877+ be afunction such that

infin

sum119894=0

4minus120573119894120593 (2119894119909 0) (18)

converges and

lim119899rarrinfin

4minus120573119899120593 (2119899119909 2119899119910) = 0 (19)

for all 119909 119910 isin 119883 Suppose that 119891 satisfies1003817100381710038171003817119891 (2119909 + 119910) + 119891 (2119909 minus 119910) minus 119891 (119909 + 119910) minus 119891 (119909 minus 119910)

minus 6119891 (119909)1003817100381710038171003817 le 120593 (119909 119910)

(20)

4 Journal of Function Spaces

for all 119909 119910 isin 119883Then there exists unique quadratic function119892 119883 rarr 119884 which satisfies (4) and inequality

1003817100381710038171003817119891 (119909) minus 119892 (119909)1003817100381710038171003817 le 8minus120573

infin

sum119894=0

4minus120573119894120593 (2119894119909 0) (21)

for all 119909 isin 119883 Function 119892 is given by

119892 (119909) = lim119899rarrinfin

2minus2119899119891 (2119899119909) (22)

for all 119909 isin 119883

3 On the Orthogonal Stability of (3)

Applying some ideas from [30] we deal with the conditionalstability problem of the following equation

119891 (119909 + 119910) + 119891 (119909 minus 119910) = 2119892 (119909) + 2ℎ (119910) for 119909 perp 119910 (23)

where 119891 is odd and perp is symmetric Throughout this section(119883 perp) denotes an orthogonality space in the sense of Ratz and(119884 sdot ) is a real 119865-space or 120573-Banach space (0 lt 120573 le 1) Firstwe give a technical lemma

Lemma 13 If 119860 119883 rarr 119884 fulfills 119860(119909 + 119910) + 119860(119909 minus 119910) =2119860(119909) for all 119909 119910 isin 119883 with 119909 perp 119910 and perp is symmetric then119860(119909) minus 119860(0) is orthogonally additive

Proof Assume that 119860(119909 + 119910) + 119860(119909 minus 119910) = 2119860(119909) for all119909 119910 isin 119883 with 119909 perp 119910 Putting 119909 = 0 we get minus119860(119910) =119860(minus119910) minus 2119860(0) 119910 isin 119883 Let 119909 perp 119910 Then 119910 perp 119909 and so119860(119910 minus 119909) = minus119860(119910 + 119909) + 2119860(119910) Hence

119860 (119909 + 119910) = minus119860 (119909 minus 119910) + 2119860 (119909)

= (119860 (119910 minus 119909) minus 2119860 (0)) + 2119860 (119909)

= (minus119860 (119910 + 119909) + 2119860 (119910)) minus 2119860 (0)

+ 2119860 (119909)

(24)

Thus

119860 (119909 + 119910) minus 119860 (0) = (119860 (119909) minus 119860 (0))

+ (119860 (119910) minus 119860 (0)) (25)

Therefore 119860(119909) minus 119860(0) is orthogonally additive

Remark 14 Ratz gave example to demonstrate that thereexists odd mapping 119860 from an orthogonality space into auniquely 2-divisible group (119884 +) (ie an abelian group inwhich map 120593 119884 rarr 119884 120593(119909) = 2119909 is bijective) satisfying119860(119909 + 119910) + 119860(119909 minus 119910) = 2119860(119909) 119909 perp 119910 such that 119860(0) = 0 Heconsidered 119884 = Z

2= 0 1 and 119860(119909) = 1 119909 isin 119883

Theorem 15 Let 119883 be an orthogonality space and 119884 be 119865-space in which there exists 12 le 119888 lt 1 such that 1199102 le 119888119910for all 119909 isin 119884 Suppose that perp is symmetric on 119883 and 119891 119892 ℎ 119883 rarr 119884 are mappings fulfilling

1003817100381710038171003817119891 (119909 + 119910) + 119891 (119909 minus 119910) minus 2119892 (119909) minus 2ℎ (119910)1003817100381710038171003817 le 120576 (26)

for some 120576 gt 0 and for all 119909 119910 isin 119883 with 119909 perp 119910 Assume that119891 is odd and 119892(0) = ℎ(0) = 0 Then there exist one additivemapping 119879 119883 rarr 119884 and one quadratic mapping 119876 119883 rarr 119884such that

1003817100381710038171003817119891 (119909) minus 119879 (119909) minus 119876 (119909)1003817100381710038171003817 le 119888120576 + (3120576 + 6119888120576)

119888

1 minus 119888

119909 isin 119883

1003817100381710038171003817119892 (119909) minus 119879 (119909) minus 119876 (119909)1003817100381710038171003817 le 119888120576 + (3120576 + 6119888120576)

119888

1 minus 119888

(27)

for all 119909 isin 119883

Proof Put119909 = 0 in (26)We can do this because of (O1)Then1003817100381710038171003817119891 (119910) + 119891 (minus119910) minus 2119892 (0) minus 2ℎ (119910)

1003817100381710038171003817 le 120576 (28)

Therefore10038171003817100381710038172ℎ (119910)

1003817100381710038171003817 le 120576 (29)

Similarly by putting 119910 = 0 in (26) we get1003817100381710038171003817119891 (119909) minus 119892 (119909)

1003817100381710038171003817 le 119888120576 (30)

for all 119909 isin 119883 Hence1003817100381710038171003817119891 (119909 + 119910) + 119891 (119909 minus 119910) minus 2119891 (119909)

1003817100381710038171003817

le1003817100381710038171003817119891 (119909 + 119910) + 119891 (119909 minus 119910) minus 2119892 (119909) minus 2ℎ (119910)

1003817100381710038171003817

+10038171003817100381710038172 [119891 (119909) minus 119892 (119909)]

1003817100381710038171003817 +10038171003817100381710038172ℎ (119910)

1003817100381710038171003817 le 3120576

(31)

for all 119909 119910 isin 119883 with 119909 perp 119910 Fix 119909 isin 119883 By (O4) there exists119910 isin 119883 such that119909 perp 119910 and119909+119910 perp 119909minus119910 Sinceperp is symmetric119909 minus 119910 perp 119909 + 119910 too Using inequality (31) and the oddness of119891 we get

1003817100381710038171003817119891 (119909 + 119910) + 119891 (119909 minus 119910) minus 2119891 (119909)1003817100381710038171003817 le 3120576

1003817100381710038171003817119891 (2119909) + 119891 (2119910) minus 2119891 (119909 + 119910)1003817100381710038171003817 le 3120576

1003817100381710038171003817119891 (2119909) minus 119891 (2119910) minus 2119891 (119909 minus 119910)1003817100381710038171003817 le 3120576

(32)

So that1003817100381710038171003817119891 (2119909) minus 2119891 (119909)

1003817100381710038171003817

le1003817100381710038171003817119891 (119909 + 119910) + 119891 (119909 minus 119910) minus 2119891 (119909)

1003817100381710038171003817

+1003817100381710038171003817100381710038171003817

1

2[119891 (2119909) + 119891 (2119910) minus 2119891 (119909 + 119910)]

1003817100381710038171003817100381710038171003817

+1003817100381710038171003817100381710038171003817

1

2[119891 (2119909) minus 119891 (2119910) minus 2119891 (119909 minus 119910)]

1003817100381710038171003817100381710038171003817

le 3120576 + 6119888120576

(33)

It is not hard to see that

10038171003817100381710038172minus119899119891 (2119899119909) minus 119891 (119909)

1003817100381710038171003817 le (3120576 + 6119888120576)119899

sum119896=1

119888119896 (34)

Journal of Function Spaces 5

for all 119899 In fact It follows from100381710038171003817100381710038172minus1119891 (2119909) minus 119891 (119909)

10038171003817100381710038171003817 le 119888 (3120576 + 6119888120576) (35)

that (34) holds for 119899 = 1 Assume that (34) holds for 119896 = 119899when 119896 = 119899 + 1 Replacing 119909 by 2119909 in (34) we get

100381710038171003817100381710038172minus119899119891 (2119899+1119909) minus 119891 (2119909)

10038171003817100381710038171003817 le (3120576 + 6119888120576)119899

sum119896=1

119888119896

100381710038171003817100381710038172minus119899minus1119891 (2119899+1119909) minus 2minus1119891 (119909)

10038171003817100381710038171003817 le (3120576 + 6119888120576)119899

sum119896=1

119888119896+1

(36)

Hence

100381710038171003817100381710038172minus119899minus1119891 (2119899+1119909) minus 119891 (119909)

10038171003817100381710038171003817 le (3120576 + 6119888120576)119899+1

sum119896=1

119888119896 (37)

So formula (34) is proved Replacing 119909 by 2119898119909 in inequality(34) we have

10038171003817100381710038172minus119899119891 (2119899+119898119909) minus 119891 (2119898119909)

1003817100381710038171003817 le (3120576 + 6119888120576)119899

sum119896=1

119888119896

10038171003817100381710038172minus119899minus119898119891 (2119899+119898119909) minus 2minus119898119891 (2119898119909)

1003817100381710038171003817 le (3120576 + 6119888120576)119888119898+1

1 minus 119888

(38)

which implies that 2minus119899119891(2119899119909) is a Cauchy sequence in 119865-space 119884 and lim

119899rarrinfin2minus119899119891(2119899119909) exists and map 119860(119909) fl

lim119899rarrinfin2minus119899119891(2119899119909) is well defined odd map from 119883 into 119884

satisfying

1003817100381710038171003817119891 (119909) minus 119860 (119909)1003817100381710038171003817 le (3120576 + 6119888120576)

119888

1 minus 119888 119909 isin 119883 (39)

For all 119909 119910 isin 119883 with 119909 perp 119910 by applying inequality (31) and(O3) we obtain

100381710038171003817100381710038172minus119899119891 (2119899 (119909 + 119910)) + 2minus119899119891 (2119899 (119909 minus 119910))

minus 2minus119899+1119891 (2119899119909)10038171003817100381710038171003817 le 119888119899 sdot 3120576

(40)

If 119899 rarr infin then we deduce that

119860 (119909 + 119910) + 119860 (119909 minus 119910) minus 2119860 (119909) = 0 (41)

for all 119909 119910 isin 119883 with 119909 perp 119910 Moreover 119860(0) =lim119899rarrinfin2minus119899119891(21198990) = 0 Using Lemma 13 we conclude that 119860

is an orthogonally additivemapping ByCorollary 7 of [23]119860is of form 119879 +119876 with 119879 additive and119876 quadratic If there areanother quadratic mapping1198761015840 and another additive mapping1198791015840 satisfying the required inequalities in our theorem and1198601015840 = 1198791015840 + 1198761015840 then

10038171003817100381710038171003817119860 (119909) minus 1198601015840

(119909)10038171003817100381710038171003817 le1003817100381710038171003817119891 (119909) minus 119860 (119909)

1003817100381710038171003817

+10038171003817100381710038171003817119891 (119909) minus 119860

1015840

(119909)10038171003817100381710038171003817

le (3120576 + 6119888120576)2119888

1 minus 119888

(42)

for all 119909 isin 119883 Using the fact that additive mappings are oddand quadratic mappings are even we obtain

10038171003817100381710038171003817119879 (119909) minus 1198791015840

(119909)10038171003817100381710038171003817 =1003817100381710038171003817100381710038171003817

1

2

sdot [(119879 (119909) + 119876 (119909) minus 1198791015840

(119909) minus 1198761015840

(119909))

+ (119879 (119909) minus 119876 (119909) minus 1198791015840

(119909) + 1198761015840

(119909))] le 11988810038171003817100381710038171003817119879 (119909)

+ 119876 (119909) minus 1198791015840

(119909) minus 1198761015840

(119909)10038171003817100381710038171003817 + 11988810038171003817100381710038171003817119879 (119909) minus 119876 (119909)

minus 1198791015840 (119909) + 1198761015840

(119909)10038171003817100381710038171003817 le 119888

10038171003817100381710038171003817119860 (119909) minus 1198601015840

(119909)10038171003817100381710038171003817

+ 11988810038171003817100381710038171003817119860 (minus119909) minus 119860

1015840

(minus119909)10038171003817100381710038171003817 le (3120576 + 6119888120576)

41198882

1 minus 119888

(43)

Hence10038171003817100381710038171003817119879 (119909) minus 119879

1015840

(119909)10038171003817100381710038171003817 le 119888119899 10038171003817100381710038171003817119879 (2

119899119909) minus 1198791015840 (2119899119909)10038171003817100381710038171003817

le (3120576 + 6119888120576)4119888119899+2

1 minus 119888

(44)

Letting 119899 tends toinfin we infer that 119879 = 1198791015840 Similarly10038171003817100381710038171003817119876 (119909) minus 119876

1015840

(119909)10038171003817100381710038171003817

=1003817100381710038171003817100381710038171003817

1

2[(119879 (119909) + 119876 (119909) minus 119879

1015840

(119909) minus 1198761015840

(119909))

minus (119879 (119909) minus 119876 (119909) minus 1198791015840

(119909) + 1198761015840

(119909))]1003817100381710038171003817100381710038171003817

le 2minus12057310038171003817100381710038171003817119879 (119909) + 119876 (119909) minus 119879

1015840

(119909) minus 1198761015840

(119909)10038171003817100381710038171003817 + 11988810038171003817100381710038171003817119879 (119909)

minus 119876 (119909) minus 1198791015840

(119909) + 1198761015840

(119909)10038171003817100381710038171003817 le 119888

10038171003817100381710038171003817119860 (119909) minus 1198601015840

(119909)10038171003817100381710038171003817

+ 11988810038171003817100381710038171003817119860 (minus119909) minus 119860

1015840

(minus119909)10038171003817100381710038171003817 le (3120576 + 6119888120576)

41198882

1 minus 119888

(45)

for all 119909 isin 119883 Hence10038171003817100381710038171003817119876 (119909) minus 119876

1015840

(119909)10038171003817100381710038171003817 le 1198882119899 10038171003817100381710038171003817119876 (2

119899119909) minus 1198761015840 (2119899119909)10038171003817100381710038171003817

le (3120576 + 6119888120576)41198882119899+2

1 minus 119888

(46)

for all 119909 isin 119883 Taking the limit as 119899 rarr infin we conclude that119876 = 1198761015840 Using (30) and (39) we infer that for all 119909 isin 1198831003817100381710038171003817119892 (119909) minus 119860 (119909)

1003817100381710038171003817 le1003817100381710038171003817119892 (119909) minus 119891 (119909)

1003817100381710038171003817 +1003817100381710038171003817119891 (119909) minus 119860 (119909)

1003817100381710038171003817

le 119888120576 + (3120576 + 6119888120576)119888

1 minus 119888

(47)

Corollary 16 Let 119883 be an orthogonality space and 119884 be 120573-Banach space Suppose that perp is symmetric on 119883 and 119891 119892 ℎ 119883 rarr 119884 are mappings fulfilling

1003817100381710038171003817119891 (119909 + 119910) + 119891 (119909 minus 119910) minus 2119892 (119909) minus 2ℎ (119910)1003817100381710038171003817 le 120576 (48)

6 Journal of Function Spaces

for some 120576 gt 0 and for all 119909 119910 isin 119883 with 119909 perp 119910 Assume that119891 is odd and 119892(0) = ℎ(0) = 0 Then there exist one additivemapping 119879 119883 rarr 119884 and one quadratic mapping 119876 119883 rarr 119884such that

1003817100381710038171003817119891 (119909) minus 119879 (119909) minus 119876 (119909)1003817100381710038171003817 le (3120576 +

6120576

2120573)1

2120573minus1 119909 isin 119883

1003817100381710038171003817119892 (119909) minus 119879 (119909) minus 119876 (119909)1003817100381710038171003817 le 2minus120573120576 +

1

2120573 minus 13 (120576 +

6120576

2120573)

(49)

for all 119909 isin 119883

Remark 17 (i) If 119892 = 120582119891 for some number 120582 = 1 theninequality (30) implies that (1 minus 120582)119891(119909) le 119888120576 119909 isin 119883Hence (1 minus 120582)2minus119899119891(2119899119909) le 119888119899+1120576 119909 isin 119883 So 119860(119909) =lim119899rarrinfin2minus119899119891(2119899119909) = 0 119909 isin 119883

(ii) Similarly if ℎ = 120582119891 for some number 120582 = 0 then itfollows from (29) that 119860(119909) = 0 for all 119909 isin 119883

As far as the author knows unlike orthogonally additivemaps (see Corollary 7 of [23]) there is no characterization fororthogonally quadratic maps Every orthogonally quadraticmapping 119902 into a uniquely 2-divisible abelian group (119884 +) iseven In fact 0 perp 0 so 119902(0)+119902(0) = 4119902(0)Therefore 119902(0) = 0For all 119910 isin 119883 we have 0 perp 119910 and hence 119902(119910) + 119902(minus119910) =2119902(0) + 2119902(119910)

Thus 119902(minus119910) = 119902(119910) There are some characterizations oforthogonally quadratic maps in various notions of orthogo-nality For example if 119860-orthogonality on Hilbert space119867 isdefined by perp

119860= (119909 119910) ⟨119860119909 119910⟩ = 0 where119860 is a bounded

self-adjoint operator on119867 then as shown by Fochi every119860-orthogonally quadratic functional is quadratic if dim119860(119867) ge3 (see [26 27])

To conclude this paper we propose the following prob-lem

Problem Let119883 be an orthogonality space and119884 be119865-space inwhich there exists 12 le 119888 lt 1 such that 1199102 le 119888119910 for all119909 isin 119884 Suppose thatperp is symmetric on119883 and119891 119892 ℎ 119883 rarr 119884are mappings fulfilling

1003817100381710038171003817119891 (119909 + 119910) + 119891 (119909 minus 119910) minus 2119892 (119909) minus 2ℎ (119910)1003817100381710038171003817 le 120576 (50)

for some 120576 and for all 119909 119910 isin 119883 with 119909 perp 119910 Assume that 119891 iseven and 119892(0) = ℎ(0) = 0 Does there exist an orthogonallyquadratic mapping 119876 119883 rarr 119884 under certain conditionssuch that

1003817100381710038171003817119891 (119909) minus 119876 (119909)1003817100381710038171003817 le 120572120576

1003817100381710038171003817119892 (119909) minus 119876 (119909)1003817100381710038171003817 le 120573120576

ℎ (119909) minus 119876 (119909) le 120574120576

(51)

for some scalars 120572 120573 120574 and for all 119909

Competing Interests

The author declares that there are no competing interestsregarding the publication of this paper

Acknowledgments

The paper is supported by the National Natural ScienceFoundation of China (Grant no 11371119) the Key Founda-tion of Education Department of Hebei Province (Grant noZD2016023) and by Natural Science Foundation of Educa-tion Department of Hebei Province (Grant no Z2014031)

References

[1] S M Ulam Problems in Modern Mathematics John Wiley ampSons New York NY USA 1960

[2] D H Hyers ldquoOn the stability of the linear functional equationrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 27 pp 222ndash224 1941

[3] T M Rassias ldquoOn the stability of the linear mapping in Banachspacesrdquo Proceedings of the American Mathematical Society vol72 no 2 pp 297ndash300 1978

[4] Z Gajda ldquoOn stability of additive mappingsrdquo InternationalJournal of Mathematics and Mathematical Sciences vol 14 no3 pp 431ndash434 1991

[5] P Gavruta ldquoA generalization of the Hyers-Ulam-Rassias stabil-ity of approximately additive mappingsrdquo Journal of Mathemati-cal Analysis and Applications vol 184 no 3 pp 431ndash436 1994

[6] R Ger and J Sikorska ldquoStability of the orthogonal additivityrdquoBulletin of the Polish Academy of Sciences Mathematics vol 43no 2 pp 143ndash151 1995

[7] S-M Jung and J M Rassias ldquoA fixed point approach to thestability of a functional equation of the spiral of TheodorusrdquoFixed Point Theory and Applications vol 2008 Article ID945010 7 pages 2008

[8] P L Kannappan Functional Equations and Inequalities withApplications Springer New York NY USA 2009

[9] C G Park ldquoOn the stability of the orthogonally quarticfunctionalrdquo Bulletin of the Iranian Mathematical Society vol 3no 1 pp 63ndash70 2005

[10] C Park and JMRassias ldquoStability of the Jensen-type functionalequation in Clowast-algebras a fixed point approachrdquo Abstract andApplied Analysis vol 2009 Article ID 360432 17 pages 2009

[11] J M Rassias andM J Rassias ldquoAsymptotic behavior of alterna-tive Jensen and Jensen type functional equationsrdquo Bulletin desSciences Mathematiques vol 129 no 7 pp 545ndash558 2005

[12] L G Wang B Liu and R Bai ldquoStability of a mixed typefunctional equation on multi-Banach spaces a fixed pointapproachrdquo Fixed Point Theory and Applications vol 2010Article ID 283827 9 pages 2010

[13] B Xu and J Brzdęk ldquoHyers-Ulam stability of a system of firstorder linear recurrences with constant coefficientsrdquo DiscreteDynamics in Nature and Society vol 2015 Article ID 2693565 pages 2015

[14] B Xu J Brzdęk and W Zhang ldquoFixed-point results and theHyers-Ulam stability of linear equations of higher ordersrdquoPacific Journal ofMathematics vol 273 no 2 pp 483ndash498 2015

[15] X Yang L Chang and G Liu ldquoOrthogonal stability of mixedadditive-quadratic Jensen type functional equation in multi-Banach spacesrdquo Advances in Pure Mathematics vol 5 no 6 pp325ndash332 2015

[16] X Yang L Chang G Liu and G Shen ldquoStability of functionalequations in (n120573)-normed spacesrdquo Journal of Inequalities andApplications vol 2015 article 112 18 pages 2015

Journal of Function Spaces 7

[17] X Zhao X Yang and C-T Pang ldquoSolution and stability ofa general mixed type cubic and quartic functional equationrdquoJournal of Function Spaces and Applications vol 2013 ArticleID 673810 8 pages 2013

[18] N Brillouet-Belluot J Brzdęk and K Cieplinski ldquoOn somerecent developments in Ulamrsquos type stabilityrdquo Abstract andApplied Analysis vol 2012 Article ID 716936 41 pages 2012

[19] J Brzdęk W Fechner M S Moslehian and J Sikorska ldquoRecentdevelopments of the conditional stability of the homomorphismequationrdquo Banach Journal of Mathematical Analysis vol 9 no3 pp 278ndash326 2015

[20] S-M Jung Hyers-Ulam-Rassias Stability of Functional Equa-tions in Nonlinear Analysis vol 48 of Springer Optimization andIts Applications Springer New York NY USA 2011

[21] FDrljevic ldquoOn a functionalwhich is quadratic onA-orthogonalvectorsrdquo Publications de lInstitutetut Mathematique vol 54 pp63ndash71 1986

[22] R C James ldquoOrthogonality and linear functionals in normedlinear spacesrdquo Transactions of the American MathematicalSociety vol 61 pp 265ndash292 1947

[23] J Ratz ldquoOn orthogonally additive mappingsrdquo AequationesMathematicae vol 28 no 1-2 pp 35ndash49 1985

[24] SGudder andD Strawther ldquoOrthogonally additive and orthog-onally increasing functions on vector spacesrdquo Pacific Journal ofMathematics vol 58 no 2 pp 427ndash436 1975

[25] F Vajzovic ldquoUber das funktional h mit der eigenschaft (119909 119910) =0 rArr 119867(119909+119910)+119867(119909minus119910) = 2119867(119909)+2119867(119910)rdquoGlasnikMatematickiSeries III vol 2 no 22 pp 73ndash81 1967

[26] M Fochi ldquoFunctional equations in A-orthogonal vectorsrdquoAequationes Mathematicae vol 38 no 1 pp 28ndash40 1989

[27] G Szabo ldquoSesquilinear-orthogonally quadratic mappingsrdquoAequationes Mathematicae vol 40 no 2-3 pp 190ndash200 1990

[28] M SMoslehian ldquoOn the orthogonal stability of the pexiderizedquadratic equationrdquo Journal of Difference Equations and Appli-cations vol 11 no 11 pp 999ndash1004 2005

[29] S Rolewicz Metric Linear Spaces Polish Scientific PublishersWarsaw Poland 1972

[30] S-M Jung and P K Sahoo ldquoHyers-Ulam stability of thequadratic equation of Pexider typerdquo Journal of the KoreanMathematical Society vol 38 no 3 pp 645ndash656 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Function Spaces

for all 119909 119910 isin 119883Then there exists unique quadratic function119892 119883 rarr 119884 which satisfies (4) and inequality

1003817100381710038171003817119891 (119909) minus 119892 (119909)1003817100381710038171003817 le 8minus120573

infin

sum119894=0

4minus120573119894120593 (2119894119909 0) (21)

for all 119909 isin 119883 Function 119892 is given by

119892 (119909) = lim119899rarrinfin

2minus2119899119891 (2119899119909) (22)

for all 119909 isin 119883

3 On the Orthogonal Stability of (3)

Applying some ideas from [30] we deal with the conditionalstability problem of the following equation

119891 (119909 + 119910) + 119891 (119909 minus 119910) = 2119892 (119909) + 2ℎ (119910) for 119909 perp 119910 (23)

where 119891 is odd and perp is symmetric Throughout this section(119883 perp) denotes an orthogonality space in the sense of Ratz and(119884 sdot ) is a real 119865-space or 120573-Banach space (0 lt 120573 le 1) Firstwe give a technical lemma

Lemma 13 If 119860 119883 rarr 119884 fulfills 119860(119909 + 119910) + 119860(119909 minus 119910) =2119860(119909) for all 119909 119910 isin 119883 with 119909 perp 119910 and perp is symmetric then119860(119909) minus 119860(0) is orthogonally additive

Proof Assume that 119860(119909 + 119910) + 119860(119909 minus 119910) = 2119860(119909) for all119909 119910 isin 119883 with 119909 perp 119910 Putting 119909 = 0 we get minus119860(119910) =119860(minus119910) minus 2119860(0) 119910 isin 119883 Let 119909 perp 119910 Then 119910 perp 119909 and so119860(119910 minus 119909) = minus119860(119910 + 119909) + 2119860(119910) Hence

119860 (119909 + 119910) = minus119860 (119909 minus 119910) + 2119860 (119909)

= (119860 (119910 minus 119909) minus 2119860 (0)) + 2119860 (119909)

= (minus119860 (119910 + 119909) + 2119860 (119910)) minus 2119860 (0)

+ 2119860 (119909)

(24)

Thus

119860 (119909 + 119910) minus 119860 (0) = (119860 (119909) minus 119860 (0))

+ (119860 (119910) minus 119860 (0)) (25)

Therefore 119860(119909) minus 119860(0) is orthogonally additive

Remark 14 Ratz gave example to demonstrate that thereexists odd mapping 119860 from an orthogonality space into auniquely 2-divisible group (119884 +) (ie an abelian group inwhich map 120593 119884 rarr 119884 120593(119909) = 2119909 is bijective) satisfying119860(119909 + 119910) + 119860(119909 minus 119910) = 2119860(119909) 119909 perp 119910 such that 119860(0) = 0 Heconsidered 119884 = Z

2= 0 1 and 119860(119909) = 1 119909 isin 119883

Theorem 15 Let 119883 be an orthogonality space and 119884 be 119865-space in which there exists 12 le 119888 lt 1 such that 1199102 le 119888119910for all 119909 isin 119884 Suppose that perp is symmetric on 119883 and 119891 119892 ℎ 119883 rarr 119884 are mappings fulfilling

1003817100381710038171003817119891 (119909 + 119910) + 119891 (119909 minus 119910) minus 2119892 (119909) minus 2ℎ (119910)1003817100381710038171003817 le 120576 (26)

for some 120576 gt 0 and for all 119909 119910 isin 119883 with 119909 perp 119910 Assume that119891 is odd and 119892(0) = ℎ(0) = 0 Then there exist one additivemapping 119879 119883 rarr 119884 and one quadratic mapping 119876 119883 rarr 119884such that

1003817100381710038171003817119891 (119909) minus 119879 (119909) minus 119876 (119909)1003817100381710038171003817 le 119888120576 + (3120576 + 6119888120576)

119888

1 minus 119888

119909 isin 119883

1003817100381710038171003817119892 (119909) minus 119879 (119909) minus 119876 (119909)1003817100381710038171003817 le 119888120576 + (3120576 + 6119888120576)

119888

1 minus 119888

(27)

for all 119909 isin 119883

Proof Put119909 = 0 in (26)We can do this because of (O1)Then1003817100381710038171003817119891 (119910) + 119891 (minus119910) minus 2119892 (0) minus 2ℎ (119910)

1003817100381710038171003817 le 120576 (28)

Therefore10038171003817100381710038172ℎ (119910)

1003817100381710038171003817 le 120576 (29)

Similarly by putting 119910 = 0 in (26) we get1003817100381710038171003817119891 (119909) minus 119892 (119909)

1003817100381710038171003817 le 119888120576 (30)

for all 119909 isin 119883 Hence1003817100381710038171003817119891 (119909 + 119910) + 119891 (119909 minus 119910) minus 2119891 (119909)

1003817100381710038171003817

le1003817100381710038171003817119891 (119909 + 119910) + 119891 (119909 minus 119910) minus 2119892 (119909) minus 2ℎ (119910)

1003817100381710038171003817

+10038171003817100381710038172 [119891 (119909) minus 119892 (119909)]

1003817100381710038171003817 +10038171003817100381710038172ℎ (119910)

1003817100381710038171003817 le 3120576

(31)

for all 119909 119910 isin 119883 with 119909 perp 119910 Fix 119909 isin 119883 By (O4) there exists119910 isin 119883 such that119909 perp 119910 and119909+119910 perp 119909minus119910 Sinceperp is symmetric119909 minus 119910 perp 119909 + 119910 too Using inequality (31) and the oddness of119891 we get

1003817100381710038171003817119891 (119909 + 119910) + 119891 (119909 minus 119910) minus 2119891 (119909)1003817100381710038171003817 le 3120576

1003817100381710038171003817119891 (2119909) + 119891 (2119910) minus 2119891 (119909 + 119910)1003817100381710038171003817 le 3120576

1003817100381710038171003817119891 (2119909) minus 119891 (2119910) minus 2119891 (119909 minus 119910)1003817100381710038171003817 le 3120576

(32)

So that1003817100381710038171003817119891 (2119909) minus 2119891 (119909)

1003817100381710038171003817

le1003817100381710038171003817119891 (119909 + 119910) + 119891 (119909 minus 119910) minus 2119891 (119909)

1003817100381710038171003817

+1003817100381710038171003817100381710038171003817

1

2[119891 (2119909) + 119891 (2119910) minus 2119891 (119909 + 119910)]

1003817100381710038171003817100381710038171003817

+1003817100381710038171003817100381710038171003817

1

2[119891 (2119909) minus 119891 (2119910) minus 2119891 (119909 minus 119910)]

1003817100381710038171003817100381710038171003817

le 3120576 + 6119888120576

(33)

It is not hard to see that

10038171003817100381710038172minus119899119891 (2119899119909) minus 119891 (119909)

1003817100381710038171003817 le (3120576 + 6119888120576)119899

sum119896=1

119888119896 (34)

Journal of Function Spaces 5

for all 119899 In fact It follows from100381710038171003817100381710038172minus1119891 (2119909) minus 119891 (119909)

10038171003817100381710038171003817 le 119888 (3120576 + 6119888120576) (35)

that (34) holds for 119899 = 1 Assume that (34) holds for 119896 = 119899when 119896 = 119899 + 1 Replacing 119909 by 2119909 in (34) we get

100381710038171003817100381710038172minus119899119891 (2119899+1119909) minus 119891 (2119909)

10038171003817100381710038171003817 le (3120576 + 6119888120576)119899

sum119896=1

119888119896

100381710038171003817100381710038172minus119899minus1119891 (2119899+1119909) minus 2minus1119891 (119909)

10038171003817100381710038171003817 le (3120576 + 6119888120576)119899

sum119896=1

119888119896+1

(36)

Hence

100381710038171003817100381710038172minus119899minus1119891 (2119899+1119909) minus 119891 (119909)

10038171003817100381710038171003817 le (3120576 + 6119888120576)119899+1

sum119896=1

119888119896 (37)

So formula (34) is proved Replacing 119909 by 2119898119909 in inequality(34) we have

10038171003817100381710038172minus119899119891 (2119899+119898119909) minus 119891 (2119898119909)

1003817100381710038171003817 le (3120576 + 6119888120576)119899

sum119896=1

119888119896

10038171003817100381710038172minus119899minus119898119891 (2119899+119898119909) minus 2minus119898119891 (2119898119909)

1003817100381710038171003817 le (3120576 + 6119888120576)119888119898+1

1 minus 119888

(38)

which implies that 2minus119899119891(2119899119909) is a Cauchy sequence in 119865-space 119884 and lim

119899rarrinfin2minus119899119891(2119899119909) exists and map 119860(119909) fl

lim119899rarrinfin2minus119899119891(2119899119909) is well defined odd map from 119883 into 119884

satisfying

1003817100381710038171003817119891 (119909) minus 119860 (119909)1003817100381710038171003817 le (3120576 + 6119888120576)

119888

1 minus 119888 119909 isin 119883 (39)

For all 119909 119910 isin 119883 with 119909 perp 119910 by applying inequality (31) and(O3) we obtain

100381710038171003817100381710038172minus119899119891 (2119899 (119909 + 119910)) + 2minus119899119891 (2119899 (119909 minus 119910))

minus 2minus119899+1119891 (2119899119909)10038171003817100381710038171003817 le 119888119899 sdot 3120576

(40)

If 119899 rarr infin then we deduce that

119860 (119909 + 119910) + 119860 (119909 minus 119910) minus 2119860 (119909) = 0 (41)

for all 119909 119910 isin 119883 with 119909 perp 119910 Moreover 119860(0) =lim119899rarrinfin2minus119899119891(21198990) = 0 Using Lemma 13 we conclude that 119860

is an orthogonally additivemapping ByCorollary 7 of [23]119860is of form 119879 +119876 with 119879 additive and119876 quadratic If there areanother quadratic mapping1198761015840 and another additive mapping1198791015840 satisfying the required inequalities in our theorem and1198601015840 = 1198791015840 + 1198761015840 then

10038171003817100381710038171003817119860 (119909) minus 1198601015840

(119909)10038171003817100381710038171003817 le1003817100381710038171003817119891 (119909) minus 119860 (119909)

1003817100381710038171003817

+10038171003817100381710038171003817119891 (119909) minus 119860

1015840

(119909)10038171003817100381710038171003817

le (3120576 + 6119888120576)2119888

1 minus 119888

(42)

for all 119909 isin 119883 Using the fact that additive mappings are oddand quadratic mappings are even we obtain

10038171003817100381710038171003817119879 (119909) minus 1198791015840

(119909)10038171003817100381710038171003817 =1003817100381710038171003817100381710038171003817

1

2

sdot [(119879 (119909) + 119876 (119909) minus 1198791015840

(119909) minus 1198761015840

(119909))

+ (119879 (119909) minus 119876 (119909) minus 1198791015840

(119909) + 1198761015840

(119909))] le 11988810038171003817100381710038171003817119879 (119909)

+ 119876 (119909) minus 1198791015840

(119909) minus 1198761015840

(119909)10038171003817100381710038171003817 + 11988810038171003817100381710038171003817119879 (119909) minus 119876 (119909)

minus 1198791015840 (119909) + 1198761015840

(119909)10038171003817100381710038171003817 le 119888

10038171003817100381710038171003817119860 (119909) minus 1198601015840

(119909)10038171003817100381710038171003817

+ 11988810038171003817100381710038171003817119860 (minus119909) minus 119860

1015840

(minus119909)10038171003817100381710038171003817 le (3120576 + 6119888120576)

41198882

1 minus 119888

(43)

Hence10038171003817100381710038171003817119879 (119909) minus 119879

1015840

(119909)10038171003817100381710038171003817 le 119888119899 10038171003817100381710038171003817119879 (2

119899119909) minus 1198791015840 (2119899119909)10038171003817100381710038171003817

le (3120576 + 6119888120576)4119888119899+2

1 minus 119888

(44)

Letting 119899 tends toinfin we infer that 119879 = 1198791015840 Similarly10038171003817100381710038171003817119876 (119909) minus 119876

1015840

(119909)10038171003817100381710038171003817

=1003817100381710038171003817100381710038171003817

1

2[(119879 (119909) + 119876 (119909) minus 119879

1015840

(119909) minus 1198761015840

(119909))

minus (119879 (119909) minus 119876 (119909) minus 1198791015840

(119909) + 1198761015840

(119909))]1003817100381710038171003817100381710038171003817

le 2minus12057310038171003817100381710038171003817119879 (119909) + 119876 (119909) minus 119879

1015840

(119909) minus 1198761015840

(119909)10038171003817100381710038171003817 + 11988810038171003817100381710038171003817119879 (119909)

minus 119876 (119909) minus 1198791015840

(119909) + 1198761015840

(119909)10038171003817100381710038171003817 le 119888

10038171003817100381710038171003817119860 (119909) minus 1198601015840

(119909)10038171003817100381710038171003817

+ 11988810038171003817100381710038171003817119860 (minus119909) minus 119860

1015840

(minus119909)10038171003817100381710038171003817 le (3120576 + 6119888120576)

41198882

1 minus 119888

(45)

for all 119909 isin 119883 Hence10038171003817100381710038171003817119876 (119909) minus 119876

1015840

(119909)10038171003817100381710038171003817 le 1198882119899 10038171003817100381710038171003817119876 (2

119899119909) minus 1198761015840 (2119899119909)10038171003817100381710038171003817

le (3120576 + 6119888120576)41198882119899+2

1 minus 119888

(46)

for all 119909 isin 119883 Taking the limit as 119899 rarr infin we conclude that119876 = 1198761015840 Using (30) and (39) we infer that for all 119909 isin 1198831003817100381710038171003817119892 (119909) minus 119860 (119909)

1003817100381710038171003817 le1003817100381710038171003817119892 (119909) minus 119891 (119909)

1003817100381710038171003817 +1003817100381710038171003817119891 (119909) minus 119860 (119909)

1003817100381710038171003817

le 119888120576 + (3120576 + 6119888120576)119888

1 minus 119888

(47)

Corollary 16 Let 119883 be an orthogonality space and 119884 be 120573-Banach space Suppose that perp is symmetric on 119883 and 119891 119892 ℎ 119883 rarr 119884 are mappings fulfilling

1003817100381710038171003817119891 (119909 + 119910) + 119891 (119909 minus 119910) minus 2119892 (119909) minus 2ℎ (119910)1003817100381710038171003817 le 120576 (48)

6 Journal of Function Spaces

for some 120576 gt 0 and for all 119909 119910 isin 119883 with 119909 perp 119910 Assume that119891 is odd and 119892(0) = ℎ(0) = 0 Then there exist one additivemapping 119879 119883 rarr 119884 and one quadratic mapping 119876 119883 rarr 119884such that

1003817100381710038171003817119891 (119909) minus 119879 (119909) minus 119876 (119909)1003817100381710038171003817 le (3120576 +

6120576

2120573)1

2120573minus1 119909 isin 119883

1003817100381710038171003817119892 (119909) minus 119879 (119909) minus 119876 (119909)1003817100381710038171003817 le 2minus120573120576 +

1

2120573 minus 13 (120576 +

6120576

2120573)

(49)

for all 119909 isin 119883

Remark 17 (i) If 119892 = 120582119891 for some number 120582 = 1 theninequality (30) implies that (1 minus 120582)119891(119909) le 119888120576 119909 isin 119883Hence (1 minus 120582)2minus119899119891(2119899119909) le 119888119899+1120576 119909 isin 119883 So 119860(119909) =lim119899rarrinfin2minus119899119891(2119899119909) = 0 119909 isin 119883

(ii) Similarly if ℎ = 120582119891 for some number 120582 = 0 then itfollows from (29) that 119860(119909) = 0 for all 119909 isin 119883

As far as the author knows unlike orthogonally additivemaps (see Corollary 7 of [23]) there is no characterization fororthogonally quadratic maps Every orthogonally quadraticmapping 119902 into a uniquely 2-divisible abelian group (119884 +) iseven In fact 0 perp 0 so 119902(0)+119902(0) = 4119902(0)Therefore 119902(0) = 0For all 119910 isin 119883 we have 0 perp 119910 and hence 119902(119910) + 119902(minus119910) =2119902(0) + 2119902(119910)

Thus 119902(minus119910) = 119902(119910) There are some characterizations oforthogonally quadratic maps in various notions of orthogo-nality For example if 119860-orthogonality on Hilbert space119867 isdefined by perp

119860= (119909 119910) ⟨119860119909 119910⟩ = 0 where119860 is a bounded

self-adjoint operator on119867 then as shown by Fochi every119860-orthogonally quadratic functional is quadratic if dim119860(119867) ge3 (see [26 27])

To conclude this paper we propose the following prob-lem

Problem Let119883 be an orthogonality space and119884 be119865-space inwhich there exists 12 le 119888 lt 1 such that 1199102 le 119888119910 for all119909 isin 119884 Suppose thatperp is symmetric on119883 and119891 119892 ℎ 119883 rarr 119884are mappings fulfilling

1003817100381710038171003817119891 (119909 + 119910) + 119891 (119909 minus 119910) minus 2119892 (119909) minus 2ℎ (119910)1003817100381710038171003817 le 120576 (50)

for some 120576 and for all 119909 119910 isin 119883 with 119909 perp 119910 Assume that 119891 iseven and 119892(0) = ℎ(0) = 0 Does there exist an orthogonallyquadratic mapping 119876 119883 rarr 119884 under certain conditionssuch that

1003817100381710038171003817119891 (119909) minus 119876 (119909)1003817100381710038171003817 le 120572120576

1003817100381710038171003817119892 (119909) minus 119876 (119909)1003817100381710038171003817 le 120573120576

ℎ (119909) minus 119876 (119909) le 120574120576

(51)

for some scalars 120572 120573 120574 and for all 119909

Competing Interests

The author declares that there are no competing interestsregarding the publication of this paper

Acknowledgments

The paper is supported by the National Natural ScienceFoundation of China (Grant no 11371119) the Key Founda-tion of Education Department of Hebei Province (Grant noZD2016023) and by Natural Science Foundation of Educa-tion Department of Hebei Province (Grant no Z2014031)

References

[1] S M Ulam Problems in Modern Mathematics John Wiley ampSons New York NY USA 1960

[2] D H Hyers ldquoOn the stability of the linear functional equationrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 27 pp 222ndash224 1941

[3] T M Rassias ldquoOn the stability of the linear mapping in Banachspacesrdquo Proceedings of the American Mathematical Society vol72 no 2 pp 297ndash300 1978

[4] Z Gajda ldquoOn stability of additive mappingsrdquo InternationalJournal of Mathematics and Mathematical Sciences vol 14 no3 pp 431ndash434 1991

[5] P Gavruta ldquoA generalization of the Hyers-Ulam-Rassias stabil-ity of approximately additive mappingsrdquo Journal of Mathemati-cal Analysis and Applications vol 184 no 3 pp 431ndash436 1994

[6] R Ger and J Sikorska ldquoStability of the orthogonal additivityrdquoBulletin of the Polish Academy of Sciences Mathematics vol 43no 2 pp 143ndash151 1995

[7] S-M Jung and J M Rassias ldquoA fixed point approach to thestability of a functional equation of the spiral of TheodorusrdquoFixed Point Theory and Applications vol 2008 Article ID945010 7 pages 2008

[8] P L Kannappan Functional Equations and Inequalities withApplications Springer New York NY USA 2009

[9] C G Park ldquoOn the stability of the orthogonally quarticfunctionalrdquo Bulletin of the Iranian Mathematical Society vol 3no 1 pp 63ndash70 2005

[10] C Park and JMRassias ldquoStability of the Jensen-type functionalequation in Clowast-algebras a fixed point approachrdquo Abstract andApplied Analysis vol 2009 Article ID 360432 17 pages 2009

[11] J M Rassias andM J Rassias ldquoAsymptotic behavior of alterna-tive Jensen and Jensen type functional equationsrdquo Bulletin desSciences Mathematiques vol 129 no 7 pp 545ndash558 2005

[12] L G Wang B Liu and R Bai ldquoStability of a mixed typefunctional equation on multi-Banach spaces a fixed pointapproachrdquo Fixed Point Theory and Applications vol 2010Article ID 283827 9 pages 2010

[13] B Xu and J Brzdęk ldquoHyers-Ulam stability of a system of firstorder linear recurrences with constant coefficientsrdquo DiscreteDynamics in Nature and Society vol 2015 Article ID 2693565 pages 2015

[14] B Xu J Brzdęk and W Zhang ldquoFixed-point results and theHyers-Ulam stability of linear equations of higher ordersrdquoPacific Journal ofMathematics vol 273 no 2 pp 483ndash498 2015

[15] X Yang L Chang and G Liu ldquoOrthogonal stability of mixedadditive-quadratic Jensen type functional equation in multi-Banach spacesrdquo Advances in Pure Mathematics vol 5 no 6 pp325ndash332 2015

[16] X Yang L Chang G Liu and G Shen ldquoStability of functionalequations in (n120573)-normed spacesrdquo Journal of Inequalities andApplications vol 2015 article 112 18 pages 2015

Journal of Function Spaces 7

[17] X Zhao X Yang and C-T Pang ldquoSolution and stability ofa general mixed type cubic and quartic functional equationrdquoJournal of Function Spaces and Applications vol 2013 ArticleID 673810 8 pages 2013

[18] N Brillouet-Belluot J Brzdęk and K Cieplinski ldquoOn somerecent developments in Ulamrsquos type stabilityrdquo Abstract andApplied Analysis vol 2012 Article ID 716936 41 pages 2012

[19] J Brzdęk W Fechner M S Moslehian and J Sikorska ldquoRecentdevelopments of the conditional stability of the homomorphismequationrdquo Banach Journal of Mathematical Analysis vol 9 no3 pp 278ndash326 2015

[20] S-M Jung Hyers-Ulam-Rassias Stability of Functional Equa-tions in Nonlinear Analysis vol 48 of Springer Optimization andIts Applications Springer New York NY USA 2011

[21] FDrljevic ldquoOn a functionalwhich is quadratic onA-orthogonalvectorsrdquo Publications de lInstitutetut Mathematique vol 54 pp63ndash71 1986

[22] R C James ldquoOrthogonality and linear functionals in normedlinear spacesrdquo Transactions of the American MathematicalSociety vol 61 pp 265ndash292 1947

[23] J Ratz ldquoOn orthogonally additive mappingsrdquo AequationesMathematicae vol 28 no 1-2 pp 35ndash49 1985

[24] SGudder andD Strawther ldquoOrthogonally additive and orthog-onally increasing functions on vector spacesrdquo Pacific Journal ofMathematics vol 58 no 2 pp 427ndash436 1975

[25] F Vajzovic ldquoUber das funktional h mit der eigenschaft (119909 119910) =0 rArr 119867(119909+119910)+119867(119909minus119910) = 2119867(119909)+2119867(119910)rdquoGlasnikMatematickiSeries III vol 2 no 22 pp 73ndash81 1967

[26] M Fochi ldquoFunctional equations in A-orthogonal vectorsrdquoAequationes Mathematicae vol 38 no 1 pp 28ndash40 1989

[27] G Szabo ldquoSesquilinear-orthogonally quadratic mappingsrdquoAequationes Mathematicae vol 40 no 2-3 pp 190ndash200 1990

[28] M SMoslehian ldquoOn the orthogonal stability of the pexiderizedquadratic equationrdquo Journal of Difference Equations and Appli-cations vol 11 no 11 pp 999ndash1004 2005

[29] S Rolewicz Metric Linear Spaces Polish Scientific PublishersWarsaw Poland 1972

[30] S-M Jung and P K Sahoo ldquoHyers-Ulam stability of thequadratic equation of Pexider typerdquo Journal of the KoreanMathematical Society vol 38 no 3 pp 645ndash656 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces 5

for all 119899 In fact It follows from100381710038171003817100381710038172minus1119891 (2119909) minus 119891 (119909)

10038171003817100381710038171003817 le 119888 (3120576 + 6119888120576) (35)

that (34) holds for 119899 = 1 Assume that (34) holds for 119896 = 119899when 119896 = 119899 + 1 Replacing 119909 by 2119909 in (34) we get

100381710038171003817100381710038172minus119899119891 (2119899+1119909) minus 119891 (2119909)

10038171003817100381710038171003817 le (3120576 + 6119888120576)119899

sum119896=1

119888119896

100381710038171003817100381710038172minus119899minus1119891 (2119899+1119909) minus 2minus1119891 (119909)

10038171003817100381710038171003817 le (3120576 + 6119888120576)119899

sum119896=1

119888119896+1

(36)

Hence

100381710038171003817100381710038172minus119899minus1119891 (2119899+1119909) minus 119891 (119909)

10038171003817100381710038171003817 le (3120576 + 6119888120576)119899+1

sum119896=1

119888119896 (37)

So formula (34) is proved Replacing 119909 by 2119898119909 in inequality(34) we have

10038171003817100381710038172minus119899119891 (2119899+119898119909) minus 119891 (2119898119909)

1003817100381710038171003817 le (3120576 + 6119888120576)119899

sum119896=1

119888119896

10038171003817100381710038172minus119899minus119898119891 (2119899+119898119909) minus 2minus119898119891 (2119898119909)

1003817100381710038171003817 le (3120576 + 6119888120576)119888119898+1

1 minus 119888

(38)

which implies that 2minus119899119891(2119899119909) is a Cauchy sequence in 119865-space 119884 and lim

119899rarrinfin2minus119899119891(2119899119909) exists and map 119860(119909) fl

lim119899rarrinfin2minus119899119891(2119899119909) is well defined odd map from 119883 into 119884

satisfying

1003817100381710038171003817119891 (119909) minus 119860 (119909)1003817100381710038171003817 le (3120576 + 6119888120576)

119888

1 minus 119888 119909 isin 119883 (39)

For all 119909 119910 isin 119883 with 119909 perp 119910 by applying inequality (31) and(O3) we obtain

100381710038171003817100381710038172minus119899119891 (2119899 (119909 + 119910)) + 2minus119899119891 (2119899 (119909 minus 119910))

minus 2minus119899+1119891 (2119899119909)10038171003817100381710038171003817 le 119888119899 sdot 3120576

(40)

If 119899 rarr infin then we deduce that

119860 (119909 + 119910) + 119860 (119909 minus 119910) minus 2119860 (119909) = 0 (41)

for all 119909 119910 isin 119883 with 119909 perp 119910 Moreover 119860(0) =lim119899rarrinfin2minus119899119891(21198990) = 0 Using Lemma 13 we conclude that 119860

is an orthogonally additivemapping ByCorollary 7 of [23]119860is of form 119879 +119876 with 119879 additive and119876 quadratic If there areanother quadratic mapping1198761015840 and another additive mapping1198791015840 satisfying the required inequalities in our theorem and1198601015840 = 1198791015840 + 1198761015840 then

10038171003817100381710038171003817119860 (119909) minus 1198601015840

(119909)10038171003817100381710038171003817 le1003817100381710038171003817119891 (119909) minus 119860 (119909)

1003817100381710038171003817

+10038171003817100381710038171003817119891 (119909) minus 119860

1015840

(119909)10038171003817100381710038171003817

le (3120576 + 6119888120576)2119888

1 minus 119888

(42)

for all 119909 isin 119883 Using the fact that additive mappings are oddand quadratic mappings are even we obtain

10038171003817100381710038171003817119879 (119909) minus 1198791015840

(119909)10038171003817100381710038171003817 =1003817100381710038171003817100381710038171003817

1

2

sdot [(119879 (119909) + 119876 (119909) minus 1198791015840

(119909) minus 1198761015840

(119909))

+ (119879 (119909) minus 119876 (119909) minus 1198791015840

(119909) + 1198761015840

(119909))] le 11988810038171003817100381710038171003817119879 (119909)

+ 119876 (119909) minus 1198791015840

(119909) minus 1198761015840

(119909)10038171003817100381710038171003817 + 11988810038171003817100381710038171003817119879 (119909) minus 119876 (119909)

minus 1198791015840 (119909) + 1198761015840

(119909)10038171003817100381710038171003817 le 119888

10038171003817100381710038171003817119860 (119909) minus 1198601015840

(119909)10038171003817100381710038171003817

+ 11988810038171003817100381710038171003817119860 (minus119909) minus 119860

1015840

(minus119909)10038171003817100381710038171003817 le (3120576 + 6119888120576)

41198882

1 minus 119888

(43)

Hence10038171003817100381710038171003817119879 (119909) minus 119879

1015840

(119909)10038171003817100381710038171003817 le 119888119899 10038171003817100381710038171003817119879 (2

119899119909) minus 1198791015840 (2119899119909)10038171003817100381710038171003817

le (3120576 + 6119888120576)4119888119899+2

1 minus 119888

(44)

Letting 119899 tends toinfin we infer that 119879 = 1198791015840 Similarly10038171003817100381710038171003817119876 (119909) minus 119876

1015840

(119909)10038171003817100381710038171003817

=1003817100381710038171003817100381710038171003817

1

2[(119879 (119909) + 119876 (119909) minus 119879

1015840

(119909) minus 1198761015840

(119909))

minus (119879 (119909) minus 119876 (119909) minus 1198791015840

(119909) + 1198761015840

(119909))]1003817100381710038171003817100381710038171003817

le 2minus12057310038171003817100381710038171003817119879 (119909) + 119876 (119909) minus 119879

1015840

(119909) minus 1198761015840

(119909)10038171003817100381710038171003817 + 11988810038171003817100381710038171003817119879 (119909)

minus 119876 (119909) minus 1198791015840

(119909) + 1198761015840

(119909)10038171003817100381710038171003817 le 119888

10038171003817100381710038171003817119860 (119909) minus 1198601015840

(119909)10038171003817100381710038171003817

+ 11988810038171003817100381710038171003817119860 (minus119909) minus 119860

1015840

(minus119909)10038171003817100381710038171003817 le (3120576 + 6119888120576)

41198882

1 minus 119888

(45)

for all 119909 isin 119883 Hence10038171003817100381710038171003817119876 (119909) minus 119876

1015840

(119909)10038171003817100381710038171003817 le 1198882119899 10038171003817100381710038171003817119876 (2

119899119909) minus 1198761015840 (2119899119909)10038171003817100381710038171003817

le (3120576 + 6119888120576)41198882119899+2

1 minus 119888

(46)

for all 119909 isin 119883 Taking the limit as 119899 rarr infin we conclude that119876 = 1198761015840 Using (30) and (39) we infer that for all 119909 isin 1198831003817100381710038171003817119892 (119909) minus 119860 (119909)

1003817100381710038171003817 le1003817100381710038171003817119892 (119909) minus 119891 (119909)

1003817100381710038171003817 +1003817100381710038171003817119891 (119909) minus 119860 (119909)

1003817100381710038171003817

le 119888120576 + (3120576 + 6119888120576)119888

1 minus 119888

(47)

Corollary 16 Let 119883 be an orthogonality space and 119884 be 120573-Banach space Suppose that perp is symmetric on 119883 and 119891 119892 ℎ 119883 rarr 119884 are mappings fulfilling

1003817100381710038171003817119891 (119909 + 119910) + 119891 (119909 minus 119910) minus 2119892 (119909) minus 2ℎ (119910)1003817100381710038171003817 le 120576 (48)

6 Journal of Function Spaces

for some 120576 gt 0 and for all 119909 119910 isin 119883 with 119909 perp 119910 Assume that119891 is odd and 119892(0) = ℎ(0) = 0 Then there exist one additivemapping 119879 119883 rarr 119884 and one quadratic mapping 119876 119883 rarr 119884such that

1003817100381710038171003817119891 (119909) minus 119879 (119909) minus 119876 (119909)1003817100381710038171003817 le (3120576 +

6120576

2120573)1

2120573minus1 119909 isin 119883

1003817100381710038171003817119892 (119909) minus 119879 (119909) minus 119876 (119909)1003817100381710038171003817 le 2minus120573120576 +

1

2120573 minus 13 (120576 +

6120576

2120573)

(49)

for all 119909 isin 119883

Remark 17 (i) If 119892 = 120582119891 for some number 120582 = 1 theninequality (30) implies that (1 minus 120582)119891(119909) le 119888120576 119909 isin 119883Hence (1 minus 120582)2minus119899119891(2119899119909) le 119888119899+1120576 119909 isin 119883 So 119860(119909) =lim119899rarrinfin2minus119899119891(2119899119909) = 0 119909 isin 119883

(ii) Similarly if ℎ = 120582119891 for some number 120582 = 0 then itfollows from (29) that 119860(119909) = 0 for all 119909 isin 119883

As far as the author knows unlike orthogonally additivemaps (see Corollary 7 of [23]) there is no characterization fororthogonally quadratic maps Every orthogonally quadraticmapping 119902 into a uniquely 2-divisible abelian group (119884 +) iseven In fact 0 perp 0 so 119902(0)+119902(0) = 4119902(0)Therefore 119902(0) = 0For all 119910 isin 119883 we have 0 perp 119910 and hence 119902(119910) + 119902(minus119910) =2119902(0) + 2119902(119910)

Thus 119902(minus119910) = 119902(119910) There are some characterizations oforthogonally quadratic maps in various notions of orthogo-nality For example if 119860-orthogonality on Hilbert space119867 isdefined by perp

119860= (119909 119910) ⟨119860119909 119910⟩ = 0 where119860 is a bounded

self-adjoint operator on119867 then as shown by Fochi every119860-orthogonally quadratic functional is quadratic if dim119860(119867) ge3 (see [26 27])

To conclude this paper we propose the following prob-lem

Problem Let119883 be an orthogonality space and119884 be119865-space inwhich there exists 12 le 119888 lt 1 such that 1199102 le 119888119910 for all119909 isin 119884 Suppose thatperp is symmetric on119883 and119891 119892 ℎ 119883 rarr 119884are mappings fulfilling

1003817100381710038171003817119891 (119909 + 119910) + 119891 (119909 minus 119910) minus 2119892 (119909) minus 2ℎ (119910)1003817100381710038171003817 le 120576 (50)

for some 120576 and for all 119909 119910 isin 119883 with 119909 perp 119910 Assume that 119891 iseven and 119892(0) = ℎ(0) = 0 Does there exist an orthogonallyquadratic mapping 119876 119883 rarr 119884 under certain conditionssuch that

1003817100381710038171003817119891 (119909) minus 119876 (119909)1003817100381710038171003817 le 120572120576

1003817100381710038171003817119892 (119909) minus 119876 (119909)1003817100381710038171003817 le 120573120576

ℎ (119909) minus 119876 (119909) le 120574120576

(51)

for some scalars 120572 120573 120574 and for all 119909

Competing Interests

The author declares that there are no competing interestsregarding the publication of this paper

Acknowledgments

The paper is supported by the National Natural ScienceFoundation of China (Grant no 11371119) the Key Founda-tion of Education Department of Hebei Province (Grant noZD2016023) and by Natural Science Foundation of Educa-tion Department of Hebei Province (Grant no Z2014031)

References

[1] S M Ulam Problems in Modern Mathematics John Wiley ampSons New York NY USA 1960

[2] D H Hyers ldquoOn the stability of the linear functional equationrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 27 pp 222ndash224 1941

[3] T M Rassias ldquoOn the stability of the linear mapping in Banachspacesrdquo Proceedings of the American Mathematical Society vol72 no 2 pp 297ndash300 1978

[4] Z Gajda ldquoOn stability of additive mappingsrdquo InternationalJournal of Mathematics and Mathematical Sciences vol 14 no3 pp 431ndash434 1991

[5] P Gavruta ldquoA generalization of the Hyers-Ulam-Rassias stabil-ity of approximately additive mappingsrdquo Journal of Mathemati-cal Analysis and Applications vol 184 no 3 pp 431ndash436 1994

[6] R Ger and J Sikorska ldquoStability of the orthogonal additivityrdquoBulletin of the Polish Academy of Sciences Mathematics vol 43no 2 pp 143ndash151 1995

[7] S-M Jung and J M Rassias ldquoA fixed point approach to thestability of a functional equation of the spiral of TheodorusrdquoFixed Point Theory and Applications vol 2008 Article ID945010 7 pages 2008

[8] P L Kannappan Functional Equations and Inequalities withApplications Springer New York NY USA 2009

[9] C G Park ldquoOn the stability of the orthogonally quarticfunctionalrdquo Bulletin of the Iranian Mathematical Society vol 3no 1 pp 63ndash70 2005

[10] C Park and JMRassias ldquoStability of the Jensen-type functionalequation in Clowast-algebras a fixed point approachrdquo Abstract andApplied Analysis vol 2009 Article ID 360432 17 pages 2009

[11] J M Rassias andM J Rassias ldquoAsymptotic behavior of alterna-tive Jensen and Jensen type functional equationsrdquo Bulletin desSciences Mathematiques vol 129 no 7 pp 545ndash558 2005

[12] L G Wang B Liu and R Bai ldquoStability of a mixed typefunctional equation on multi-Banach spaces a fixed pointapproachrdquo Fixed Point Theory and Applications vol 2010Article ID 283827 9 pages 2010

[13] B Xu and J Brzdęk ldquoHyers-Ulam stability of a system of firstorder linear recurrences with constant coefficientsrdquo DiscreteDynamics in Nature and Society vol 2015 Article ID 2693565 pages 2015

[14] B Xu J Brzdęk and W Zhang ldquoFixed-point results and theHyers-Ulam stability of linear equations of higher ordersrdquoPacific Journal ofMathematics vol 273 no 2 pp 483ndash498 2015

[15] X Yang L Chang and G Liu ldquoOrthogonal stability of mixedadditive-quadratic Jensen type functional equation in multi-Banach spacesrdquo Advances in Pure Mathematics vol 5 no 6 pp325ndash332 2015

[16] X Yang L Chang G Liu and G Shen ldquoStability of functionalequations in (n120573)-normed spacesrdquo Journal of Inequalities andApplications vol 2015 article 112 18 pages 2015

Journal of Function Spaces 7

[17] X Zhao X Yang and C-T Pang ldquoSolution and stability ofa general mixed type cubic and quartic functional equationrdquoJournal of Function Spaces and Applications vol 2013 ArticleID 673810 8 pages 2013

[18] N Brillouet-Belluot J Brzdęk and K Cieplinski ldquoOn somerecent developments in Ulamrsquos type stabilityrdquo Abstract andApplied Analysis vol 2012 Article ID 716936 41 pages 2012

[19] J Brzdęk W Fechner M S Moslehian and J Sikorska ldquoRecentdevelopments of the conditional stability of the homomorphismequationrdquo Banach Journal of Mathematical Analysis vol 9 no3 pp 278ndash326 2015

[20] S-M Jung Hyers-Ulam-Rassias Stability of Functional Equa-tions in Nonlinear Analysis vol 48 of Springer Optimization andIts Applications Springer New York NY USA 2011

[21] FDrljevic ldquoOn a functionalwhich is quadratic onA-orthogonalvectorsrdquo Publications de lInstitutetut Mathematique vol 54 pp63ndash71 1986

[22] R C James ldquoOrthogonality and linear functionals in normedlinear spacesrdquo Transactions of the American MathematicalSociety vol 61 pp 265ndash292 1947

[23] J Ratz ldquoOn orthogonally additive mappingsrdquo AequationesMathematicae vol 28 no 1-2 pp 35ndash49 1985

[24] SGudder andD Strawther ldquoOrthogonally additive and orthog-onally increasing functions on vector spacesrdquo Pacific Journal ofMathematics vol 58 no 2 pp 427ndash436 1975

[25] F Vajzovic ldquoUber das funktional h mit der eigenschaft (119909 119910) =0 rArr 119867(119909+119910)+119867(119909minus119910) = 2119867(119909)+2119867(119910)rdquoGlasnikMatematickiSeries III vol 2 no 22 pp 73ndash81 1967

[26] M Fochi ldquoFunctional equations in A-orthogonal vectorsrdquoAequationes Mathematicae vol 38 no 1 pp 28ndash40 1989

[27] G Szabo ldquoSesquilinear-orthogonally quadratic mappingsrdquoAequationes Mathematicae vol 40 no 2-3 pp 190ndash200 1990

[28] M SMoslehian ldquoOn the orthogonal stability of the pexiderizedquadratic equationrdquo Journal of Difference Equations and Appli-cations vol 11 no 11 pp 999ndash1004 2005

[29] S Rolewicz Metric Linear Spaces Polish Scientific PublishersWarsaw Poland 1972

[30] S-M Jung and P K Sahoo ldquoHyers-Ulam stability of thequadratic equation of Pexider typerdquo Journal of the KoreanMathematical Society vol 38 no 3 pp 645ndash656 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Function Spaces

for some 120576 gt 0 and for all 119909 119910 isin 119883 with 119909 perp 119910 Assume that119891 is odd and 119892(0) = ℎ(0) = 0 Then there exist one additivemapping 119879 119883 rarr 119884 and one quadratic mapping 119876 119883 rarr 119884such that

1003817100381710038171003817119891 (119909) minus 119879 (119909) minus 119876 (119909)1003817100381710038171003817 le (3120576 +

6120576

2120573)1

2120573minus1 119909 isin 119883

1003817100381710038171003817119892 (119909) minus 119879 (119909) minus 119876 (119909)1003817100381710038171003817 le 2minus120573120576 +

1

2120573 minus 13 (120576 +

6120576

2120573)

(49)

for all 119909 isin 119883

Remark 17 (i) If 119892 = 120582119891 for some number 120582 = 1 theninequality (30) implies that (1 minus 120582)119891(119909) le 119888120576 119909 isin 119883Hence (1 minus 120582)2minus119899119891(2119899119909) le 119888119899+1120576 119909 isin 119883 So 119860(119909) =lim119899rarrinfin2minus119899119891(2119899119909) = 0 119909 isin 119883

(ii) Similarly if ℎ = 120582119891 for some number 120582 = 0 then itfollows from (29) that 119860(119909) = 0 for all 119909 isin 119883

As far as the author knows unlike orthogonally additivemaps (see Corollary 7 of [23]) there is no characterization fororthogonally quadratic maps Every orthogonally quadraticmapping 119902 into a uniquely 2-divisible abelian group (119884 +) iseven In fact 0 perp 0 so 119902(0)+119902(0) = 4119902(0)Therefore 119902(0) = 0For all 119910 isin 119883 we have 0 perp 119910 and hence 119902(119910) + 119902(minus119910) =2119902(0) + 2119902(119910)

Thus 119902(minus119910) = 119902(119910) There are some characterizations oforthogonally quadratic maps in various notions of orthogo-nality For example if 119860-orthogonality on Hilbert space119867 isdefined by perp

119860= (119909 119910) ⟨119860119909 119910⟩ = 0 where119860 is a bounded

self-adjoint operator on119867 then as shown by Fochi every119860-orthogonally quadratic functional is quadratic if dim119860(119867) ge3 (see [26 27])

To conclude this paper we propose the following prob-lem

Problem Let119883 be an orthogonality space and119884 be119865-space inwhich there exists 12 le 119888 lt 1 such that 1199102 le 119888119910 for all119909 isin 119884 Suppose thatperp is symmetric on119883 and119891 119892 ℎ 119883 rarr 119884are mappings fulfilling

1003817100381710038171003817119891 (119909 + 119910) + 119891 (119909 minus 119910) minus 2119892 (119909) minus 2ℎ (119910)1003817100381710038171003817 le 120576 (50)

for some 120576 and for all 119909 119910 isin 119883 with 119909 perp 119910 Assume that 119891 iseven and 119892(0) = ℎ(0) = 0 Does there exist an orthogonallyquadratic mapping 119876 119883 rarr 119884 under certain conditionssuch that

1003817100381710038171003817119891 (119909) minus 119876 (119909)1003817100381710038171003817 le 120572120576

1003817100381710038171003817119892 (119909) minus 119876 (119909)1003817100381710038171003817 le 120573120576

ℎ (119909) minus 119876 (119909) le 120574120576

(51)

for some scalars 120572 120573 120574 and for all 119909

Competing Interests

The author declares that there are no competing interestsregarding the publication of this paper

Acknowledgments

The paper is supported by the National Natural ScienceFoundation of China (Grant no 11371119) the Key Founda-tion of Education Department of Hebei Province (Grant noZD2016023) and by Natural Science Foundation of Educa-tion Department of Hebei Province (Grant no Z2014031)

References

[1] S M Ulam Problems in Modern Mathematics John Wiley ampSons New York NY USA 1960

[2] D H Hyers ldquoOn the stability of the linear functional equationrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 27 pp 222ndash224 1941

[3] T M Rassias ldquoOn the stability of the linear mapping in Banachspacesrdquo Proceedings of the American Mathematical Society vol72 no 2 pp 297ndash300 1978

[4] Z Gajda ldquoOn stability of additive mappingsrdquo InternationalJournal of Mathematics and Mathematical Sciences vol 14 no3 pp 431ndash434 1991

[5] P Gavruta ldquoA generalization of the Hyers-Ulam-Rassias stabil-ity of approximately additive mappingsrdquo Journal of Mathemati-cal Analysis and Applications vol 184 no 3 pp 431ndash436 1994

[6] R Ger and J Sikorska ldquoStability of the orthogonal additivityrdquoBulletin of the Polish Academy of Sciences Mathematics vol 43no 2 pp 143ndash151 1995

[7] S-M Jung and J M Rassias ldquoA fixed point approach to thestability of a functional equation of the spiral of TheodorusrdquoFixed Point Theory and Applications vol 2008 Article ID945010 7 pages 2008

[8] P L Kannappan Functional Equations and Inequalities withApplications Springer New York NY USA 2009

[9] C G Park ldquoOn the stability of the orthogonally quarticfunctionalrdquo Bulletin of the Iranian Mathematical Society vol 3no 1 pp 63ndash70 2005

[10] C Park and JMRassias ldquoStability of the Jensen-type functionalequation in Clowast-algebras a fixed point approachrdquo Abstract andApplied Analysis vol 2009 Article ID 360432 17 pages 2009

[11] J M Rassias andM J Rassias ldquoAsymptotic behavior of alterna-tive Jensen and Jensen type functional equationsrdquo Bulletin desSciences Mathematiques vol 129 no 7 pp 545ndash558 2005

[12] L G Wang B Liu and R Bai ldquoStability of a mixed typefunctional equation on multi-Banach spaces a fixed pointapproachrdquo Fixed Point Theory and Applications vol 2010Article ID 283827 9 pages 2010

[13] B Xu and J Brzdęk ldquoHyers-Ulam stability of a system of firstorder linear recurrences with constant coefficientsrdquo DiscreteDynamics in Nature and Society vol 2015 Article ID 2693565 pages 2015

[14] B Xu J Brzdęk and W Zhang ldquoFixed-point results and theHyers-Ulam stability of linear equations of higher ordersrdquoPacific Journal ofMathematics vol 273 no 2 pp 483ndash498 2015

[15] X Yang L Chang and G Liu ldquoOrthogonal stability of mixedadditive-quadratic Jensen type functional equation in multi-Banach spacesrdquo Advances in Pure Mathematics vol 5 no 6 pp325ndash332 2015

[16] X Yang L Chang G Liu and G Shen ldquoStability of functionalequations in (n120573)-normed spacesrdquo Journal of Inequalities andApplications vol 2015 article 112 18 pages 2015

Journal of Function Spaces 7

[17] X Zhao X Yang and C-T Pang ldquoSolution and stability ofa general mixed type cubic and quartic functional equationrdquoJournal of Function Spaces and Applications vol 2013 ArticleID 673810 8 pages 2013

[18] N Brillouet-Belluot J Brzdęk and K Cieplinski ldquoOn somerecent developments in Ulamrsquos type stabilityrdquo Abstract andApplied Analysis vol 2012 Article ID 716936 41 pages 2012

[19] J Brzdęk W Fechner M S Moslehian and J Sikorska ldquoRecentdevelopments of the conditional stability of the homomorphismequationrdquo Banach Journal of Mathematical Analysis vol 9 no3 pp 278ndash326 2015

[20] S-M Jung Hyers-Ulam-Rassias Stability of Functional Equa-tions in Nonlinear Analysis vol 48 of Springer Optimization andIts Applications Springer New York NY USA 2011

[21] FDrljevic ldquoOn a functionalwhich is quadratic onA-orthogonalvectorsrdquo Publications de lInstitutetut Mathematique vol 54 pp63ndash71 1986

[22] R C James ldquoOrthogonality and linear functionals in normedlinear spacesrdquo Transactions of the American MathematicalSociety vol 61 pp 265ndash292 1947

[23] J Ratz ldquoOn orthogonally additive mappingsrdquo AequationesMathematicae vol 28 no 1-2 pp 35ndash49 1985

[24] SGudder andD Strawther ldquoOrthogonally additive and orthog-onally increasing functions on vector spacesrdquo Pacific Journal ofMathematics vol 58 no 2 pp 427ndash436 1975

[25] F Vajzovic ldquoUber das funktional h mit der eigenschaft (119909 119910) =0 rArr 119867(119909+119910)+119867(119909minus119910) = 2119867(119909)+2119867(119910)rdquoGlasnikMatematickiSeries III vol 2 no 22 pp 73ndash81 1967

[26] M Fochi ldquoFunctional equations in A-orthogonal vectorsrdquoAequationes Mathematicae vol 38 no 1 pp 28ndash40 1989

[27] G Szabo ldquoSesquilinear-orthogonally quadratic mappingsrdquoAequationes Mathematicae vol 40 no 2-3 pp 190ndash200 1990

[28] M SMoslehian ldquoOn the orthogonal stability of the pexiderizedquadratic equationrdquo Journal of Difference Equations and Appli-cations vol 11 no 11 pp 999ndash1004 2005

[29] S Rolewicz Metric Linear Spaces Polish Scientific PublishersWarsaw Poland 1972

[30] S-M Jung and P K Sahoo ldquoHyers-Ulam stability of thequadratic equation of Pexider typerdquo Journal of the KoreanMathematical Society vol 38 no 3 pp 645ndash656 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces 7

[17] X Zhao X Yang and C-T Pang ldquoSolution and stability ofa general mixed type cubic and quartic functional equationrdquoJournal of Function Spaces and Applications vol 2013 ArticleID 673810 8 pages 2013

[18] N Brillouet-Belluot J Brzdęk and K Cieplinski ldquoOn somerecent developments in Ulamrsquos type stabilityrdquo Abstract andApplied Analysis vol 2012 Article ID 716936 41 pages 2012

[19] J Brzdęk W Fechner M S Moslehian and J Sikorska ldquoRecentdevelopments of the conditional stability of the homomorphismequationrdquo Banach Journal of Mathematical Analysis vol 9 no3 pp 278ndash326 2015

[20] S-M Jung Hyers-Ulam-Rassias Stability of Functional Equa-tions in Nonlinear Analysis vol 48 of Springer Optimization andIts Applications Springer New York NY USA 2011

[21] FDrljevic ldquoOn a functionalwhich is quadratic onA-orthogonalvectorsrdquo Publications de lInstitutetut Mathematique vol 54 pp63ndash71 1986

[22] R C James ldquoOrthogonality and linear functionals in normedlinear spacesrdquo Transactions of the American MathematicalSociety vol 61 pp 265ndash292 1947

[23] J Ratz ldquoOn orthogonally additive mappingsrdquo AequationesMathematicae vol 28 no 1-2 pp 35ndash49 1985

[24] SGudder andD Strawther ldquoOrthogonally additive and orthog-onally increasing functions on vector spacesrdquo Pacific Journal ofMathematics vol 58 no 2 pp 427ndash436 1975

[25] F Vajzovic ldquoUber das funktional h mit der eigenschaft (119909 119910) =0 rArr 119867(119909+119910)+119867(119909minus119910) = 2119867(119909)+2119867(119910)rdquoGlasnikMatematickiSeries III vol 2 no 22 pp 73ndash81 1967

[26] M Fochi ldquoFunctional equations in A-orthogonal vectorsrdquoAequationes Mathematicae vol 38 no 1 pp 28ndash40 1989

[27] G Szabo ldquoSesquilinear-orthogonally quadratic mappingsrdquoAequationes Mathematicae vol 40 no 2-3 pp 190ndash200 1990

[28] M SMoslehian ldquoOn the orthogonal stability of the pexiderizedquadratic equationrdquo Journal of Difference Equations and Appli-cations vol 11 no 11 pp 999ndash1004 2005

[29] S Rolewicz Metric Linear Spaces Polish Scientific PublishersWarsaw Poland 1972

[30] S-M Jung and P K Sahoo ldquoHyers-Ulam stability of thequadratic equation of Pexider typerdquo Journal of the KoreanMathematical Society vol 38 no 3 pp 645ndash656 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of