Hindawi Publishing CorporationISRN Computational MathematicsVolume 2013 Article ID 184026 6 pageshttpdxdoiorg1011552013184026

Research ArticleFixed Point Theorems and Error Bounds inPartial Metric Spaces

S A M Mohsenalhosseini

Faculty of Mathematics Vali-e-Asr University Rafsanjan Iran

Correspondence should be addressed to S A M Mohsenalhosseini mohsenialhosseinigmailcom

Received 4 November 2013 Accepted 11 December 2013

Academic Editors R El-Khazali and Q-W Wang

Copyright copy 2013 S A M Mohsenalhosseini This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper is introduced as a survey of result on some generalization of Banachrsquos fixed point and their approximations to the fixedpoint and error bounds and it contains some new fixed point theorems and applications on dualistic partial metric spaces

1 Introduction

The partial metric spaces were introduced in [1] as a partof the study of denotational semantics of dataflow networksHe established the precise relationship between partialmetricspaces and the weightable quasi-metric spaces and proved apartial metric generalization of Banach contraction mappingtheorem

A partial metric [1] on a set119883 is a function 119901 119883 times 119883 rarr

[0infin) such that for all 119909 119910 119911 isin 119883

(1) 119909 = 119910 hArr 119901(119909 119909) = 119901(119909 119910) = 119901(119910 119910)(2) 119901(119909 119909) le 119901(119909 119910)(3) 119901(119909 119910) = 119901(119910 119909)(4) 119901(119909 119911) le 119901(119909 119910) + 119901(119910 119911) minus 119901(119910 119910)

A partial metric space is a pair (119883 119901) where 119901 is a partialmetric on 119883 If 119901 is a partial metric on 119883 then the function119901119904

119883times119883 rarr [0infin) given by 119901119904(119909 119910) = 2119901(119909 119910)minus119901(119909 119909)minus119901(119910 119910) is a (usual) metric on119883

Each partial metric 119901 on119883 induces a1198790topology 120591

119901on119883

which has as a basis the family of open 119901-balls 119861119901(119909 120598) 119909 isin

119883 120598 gt 0 where 119861119901(119909 120598) = 119910 isin 119883 119901(119909 119910) lt 119901(119909 119909) + 120598

for all 119909 isin 119883 and 120598 gt 0 Similarly closed 119901-ball is defined as119861119901(119909 120598) = 119910 isin 119883 119901(119909 119910) le 119901(119909 119909) + 120598A sequence 119909

119899119899isin119873

in a partial metric space (119883 119901)

is called a Cauchy sequence if there exists (and is finite)lim119899119898119901(119909119899 119909119898) [1]

Note that 119909119899119899isin119873

is a Cauchy sequence in (119883 119901) if andonly if it is a Cauchy sequence in the metric space (119883 119901119904) [1]

A partialmetric space (119883 119901) is said to be complete if everyCauchy sequence 119909

119899119899isin119873

in 119883 converges with respect to 120591119901

to a point 119909 isin 119883 such that 119901(119909 119909) = lim119899119898119901(119909119899 119909119898) [1]

A mapping 119879 119883 rarr 119883 is said to be continuous at 1199090isin

119883 if for 120598 gt 0 there exists 120575 gt 0 such that 119879(119861119901(1199090 120575)) sub

119861119901(119879(1199090) 120598) [2]

Definition 1 (see [1]) An open ball for a partial metric 119901

119883 times 119883 rarr [0infin) is a set of the form 119861119901

120598(119909) = 119910 isin 119883

119901(119909 119910) lt 120598 for each 120598 gt 0 and 119909 isin 119883

In [3] OrsquoNeill proposed one significant change to Matt-hews definition of the partial metrics and that was to extendtheir range from 119877

+ to 119877 In the following partial metrics inthe OrsquoNeill sense will be called dualistic partial metrics and apair (119883 119901) such that 119883 is a nonempty set and 119901 is a dualisticpartial metric on 119883 will be called a dualistic partial metricspace

A dualistic partial metric on a set 119883 is a partial metric119901 119883 times 119883 rarr 119877 A dualistic partial metric space is a pair(119883 119901) where 119901 is a dualistic partial metric on119883

A quasi-metric on a set 119883 is a nonnegative real-valuedfunction 119889 on119883 times 119883 such that for all 119909 119910 119911 isin 119883

(i) 119889(119909 119910) = 119889(119910 119909) = 0 hArr 119909 = 119910(ii) 119889(119909 119910) le 119889(119909 119911) + 119889(119911 119910)

2 ISRN Computational Mathematics

Lemma 2 (see [1]) If (119883 119901) is a dualistic partial metric spacethen the function 119889

119901 119883 times 119883 rarr 119877

+ defined by 119889119901(119909 119910) =

119901(119909 119910)minus119901(119909 119909) is a quasi-metric on119883 such that 120591(119901) = 120591(119889119901)

Lemma 3 (see [1]) A dualistic partial metric space (119883 119901) iscomplete if and only if the metric space (119883 (119889

119901)119904

) is completeFurthermore lim

119899rarrinfin(119889119901)119904

(119886 119909119899) = 0 if and only if 119901(119886 119886) =

lim119899rarrinfin

119901(119886 119909119899) = lim

119899119898rarrinfin119901(119909119899 119909119898)

Before stating our main results we establish some (essen-tially known) correspondences between dualistic partial met-rics and quasi-metric spaces Also refer to definition of 120598-Fixed point and the existence of 120598-Fixed point for 120598 gt 0 Ourbasic references for quasi-metric spaces are [4 5] and for 120598-Fixed point is [6]

If 119889 is a quasi-metric on 119883 then the function 119889119904 definedon 119883 times 119883 by 119889119904(119909 119910) = max119889(119909 119910) 119889(119910 119909) is a metric on119883

Definition 4 (see [6]) Let (119883 119901) be a dualistic partial metricspace and 119879 119883 rarr 119883 be a mapThen 119909

0isin 119883 is 120598-fixed point

for 119879 if

119889119901(1198791199090 1199090) le 120598 (1)

We say 119879 has the 120598-fixed point property if for every 120598 gt 0119860119865(119879) = 0 where

119860119865 (119879) = 1199090isin 119883 119889 (119879119909

0 1199090) le 120598 (2)

Theorem 5 (see [6]) Let (119883 119901) be a dualistic partial metricspace and 119879 119883 rarr 119883 be a map 119909

0isin 119883 and 120598 gt 0 If

119889119901(119879119899

(1199090) 119879119899+119896(119909

0)) rarr 0 as 119899 rarr infin for some 119896 gt 0 then

119879119896 has an 120598-fixed point

Definition 6 (see [7]) Let 119879 119860 cup 119861 rarr 119860 cup 119861 be continuesmap such that 119879(119860) sube 119861 119879(119861) sube 119860 and 120598 gt 0 We definediameter 119875119886

119879(119860 119861) by

diam (119875119886

119879(119860 119861)) = sup 119889 (119909 119910) 119909 119910 isin 119875119886

119879(119860 119861) (3)

Theorem 7 (see [1]) The partial metric contraction mappingtheorem Let (119883 119901) be a complete partial metric space and 119879

119883 rarr 119883 be a map such that for all 119909 119910 isin 119883

119901 (119879119909 119879119910) le 119871119901 (119909 119910) 0 le 119871 lt 1 (4)

then 119879 has a unique fixed point 119906 and 119879119899(119909) rarr 119906 as 119899 rarr infin

for each 119909 isin 119883

2 Some Result Fixed Point on Partial Metric

In this section we give some result on fixed point and 120598-fixedpoint in dualistic partial metric space and its diameter

Definition 8 An open ball for a dualistic partial metric 119901

119883 times 119883 rarr 119877 is a set of the form 119861119901

120598(119909) = 119910 isin 119883 119901(119909 119910) lt

120598 for each 120598 gt 0 and 119909 isin 119883

Definition 9 Let 119879 be a mapping of a complete dualistic par-tial metric 119883 into itself [119879 119883 rarr 119883] then 119879 is called

a partial metric contractionmapping if there exists a constant119871 0 le 119871 lt 1 such that 119901(119879119909 119879119910) le 119871119901(119909 119910) for all 119909 119910 isin 119883

Theorem 10 Every contraction mapping 119879 defined on acomplete dualistic partial metric119883 into itself has a unique fixedpoint 119906 isin 119883 Moreover if 119909

0is any point in119883 and the sequence

119909119899is defined by

1199091= 119879 (119909

0) 119909

2= 119879 (119909

1) 119909

119899= 119879 (119909

119899minus1) (5)

then lim119899rarrinfin

119909119899= 119906 and

119901 (119909119899 119906) le

119871119899

1 minus 119871119901 (1199091 1199090) 0 le 119871 lt 1 (6)

Proof Existence of a fixed point Let 1199090be an arbitrary point

in 119883 and we defined by 1199091= 119879(119909

0) 1199092= 119879(119909

1) 119909

119899=

119879(119909119899minus1

) Then

1199092= 119879 (119909

1) = 119879 (119879 (119909

0)) = 119879

2

(1199090)

1199093= 119879 (119909

2) = 119879 (119879 (119909

1)) = 119879 (119879 (119879 (119909

0))) = 119879

3

(1199090)

119909119899= 119879119899

(1199090)

(7)

If119898 gt 119899 say119898 = 119899 + 120572 120572 = 1 2 3 Then

119901 (119909119899+ 120572 119909

119899) = 119901 (119879

119899+120572

(1199090) 119879119899

(1199090))

= 119901 (119879 (119879119899+120572minus1

(1199090)) 119879 (119879

119899minus1

(1199090)))

le 119871119901 (119879119899+120572minus1

(1199090) 119879119899minus1

(1199090))

(8)

Continuing this process 119899 minus 1 times we have

119901 (119909119899+120572

119909119899) le 119871119899

119901 (119879120572

(1199090) 1199090) (9)

for 119899 = 0 1 2 and all 120572 ge 1However

119901 (119879120572

(1199090) 1199090) le 119901 (119879

120572

(1199090) 119879120572minus1

(1199090))

+ 119901 (119879120572minus1

(1199090) 119879120572minus2

(1199090)) + sdot sdot sdot

+ 119901 (119879 (1199090) 1199090)

(10)

Therefore we see that

119901 (119909119899+120572

119909119899)

le 119871119899

[119871120572minus1

119901 (1199091 1199090) + 119871120572minus2

119901 (1199091 1199090) + sdot sdot sdot + 119901 (119909

1 1199090)]

le 119871119899

119901 (1199091 1199090) [1 + 119871 + 119871

2

+ sdot sdot sdot + 119871120572minus1

+ 119871120572

]

(11)

Hence

119901 (119909119899+120572

119909119899) le 119871119899

119901 (1199091 1199090)

1

1 minus 119871 (12)

ISRN Computational Mathematics 3

As 119899 119898 = 119899 + 120572 rarr infin from (12) we see that119901(119909119899+120572

119909119899) rarr 0 that is 119909

119899 is a Cauchy sequence in

the metric space 119883 Hence 119909119899 must be convergent say

lim119899rarrinfin

119909119899= 119906

Since 119879 is continuous we have

119879119906 = 119879( lim119899rarrinfin

119909119899) = lim119899rarrinfin

119879 (119909119899) = lim119899rarrinfin

119909119899+1

(13)

or 119879119906 = 119906 Thus 119906 is a fixed point of 119879Uniqueness of the fixed point 119879Let V be another fixed point of 119879 Then 119879V = V We also

have 119901(119879(119906) 119879(V)) le 119871119901(119906 V) But 119901(119879(119906) 119879(V)) = 119901(119906 V)which implies that 119901(119906 V) le 119871119901(119906 V) where 0 le 119871 lt 1 Thisis possible only when 119901(119906 V) = 0 that is 119906 = V This provesthat the fixed point of 119879 is unique

Corollary 11 Let (119883 119901) be a complete dualistic partial metricspace and 119861119901

119903(1199090) = 119910 isin 119883 119901(119909

0 119910) lt 119903 Let 119879 119861

119901

119903rarr 119883

be a partial metric contraction mapping If 119901(1198791199100 1199100) lt (1 minus

119871)119903 then 119879 has a fixed point

Proof Choose 119903 lt 120598 so that 119901(1198791199100 1199100) le (1 minus 119871)119903 lt (1 minus 119871)120598

We show that 119879 maps the closed ball 119870 = 119910 119901(119910 1199100) le 120598

into itself for if 119910 isin 119870 then

119901 (119879 (119910) 1199100) le 119901 (119879 (119910) 119879 (119910

0)) + 119901 (119879 (119910

0) 1199100)

le 119871119901 (119910 1199100) + (1 minus 119871) 120598

le 119871120598 + 120598 minus 119871120598 = 120598

(14)

Since 119870 is complete and 119879 119870 rarr 119870 satisfy in (5) thus byTheorem 10 119879 has a fixed point

Theorem 12 If 119883 is a complete dualistic partial metric spaceand 119879 119883 rarr 119883 is such that 119879119903 is contraction for some integer119903 gt 0 then 119879119903 has an unique fixed point

Proof Since 119879119903 where 119903 is a positive integer is a contractionmapping by Theorem 10 there exists an unique fixed point 119906of 119879119903 that is 119879119903(119906) = 119906 We want to show that 119906 ia a fixedpoint of 119879 that is 119879(119906) = 119906 Let 119878 = 119879

119903 therefore 119878(119906) = 119906This implies that

1198782

(119906) = 119878 (119878 (119906)) = 119878 (119906) = 119906

1198783

(119906) = 1198782

(119878 (119906)) = 1198782

(119906) ℎ = 119906

119878119899

(119906) = 119878119899minus1

(119878 (119906)) = 119878119899minus1

(119906) = sdot sdot sdot = 119906

(15)

and so 119879(119906) = 119879(119878119899(119906)) = 119904119899(119879(119906)) = 119904119899119910 say

119901 (119878119899

(119910) 119906) = 119901 (119878119899

(119910) 119878119899

(119906))

le 119871119901 (119878119899minus1

(119910) 119878119899minus1

(119906))

le 1198712

119901 (119878119899minus2

(119910) 119878119899minus2

(119906))

le 119871119899

119901 (119910 119906)

lim119899rarrinfin

119901 (119878119899

(119910) 119906) = 0

(16)

as 119871119899 rarr 0 Thus lim119899rarrinfin

119878119899

(119910) = 119906 and we have 119879(119906) =119906

Theorem 13 Let (119883 119901) is a complete dualistic partial metricspace and let 119879 119883 rarr 119883 and 119878 119883 rarr 119883 be two mapscontraction If for every 119909 isin 119883

119901 (119879119909 119878119910) le 120582 (17)

120582 gt 0 chosen suitably Then for every 119909 isin 119883

119901 (119879119898

(119909) 119878119898

(119909)) le 1205821 minus 119871119898

1 minus 119871 0 le 119871 lt 1

119898 = 1 2

(18)

Proof The relation is true for 119898 = 1 We use the principle ofinduction in order to prove this relation Let it be true for all119898 ge 1 Then

119901 (119879119899+1

(119909) minus 119878119899+1

(119909)) = 119901 (119879119879119898

(119909) 119878119878119898

(119909))

le 119901 (119879119879119898

(119909) 119879119878119898

(119909))

+ 119901 (119879119878119898

(119909) 119878119878119898

(119909))

le 119871119901 (119879119898

(119909) 119878119898

(119909)) + 120582

le 1198711205821 minus 119871119898

1 minus 119871+ 120582

=119871120582 minus 119871

119898+1

120582 + 120582 minus 119871120582

1 minus 119871

= 1205821 minus 119871119898+1

1 minus 119871

(19)

Thus the relation is true for119898 + 1

Corollary 14 Let (119883 119901) is a complete dualistic partial metricspace and let 119879 119883 rarr 119883 be a partial contractive map on 119883Moreover the iteration sequence 119909

1= 119879(119909

0) 1199092= 119879(119909

1) =

1198792

1199090 119909

119899= 119879119899

1199090 with arbitrary 119909

0isin 119883 converges to

the unique fixed point 119906 of 119879 Error estimate is the followingestimate (prior estimate)

119901 (119909119898 119906) le

119871119898

1 minus 119871119901 (1199090 1199091) (20)

and the posterior estimate

119901 (119909119898 119906) le

119871

1 minus 119871119901 (119909119898minus1

119909119898) (21)

4 ISRN Computational Mathematics

Proof The First statement is obvious by

119901 (119909119898 119909119899) le 119901 (119909

119898 119909119898+1

) + 119901 (119909119898+1

119909119898+2

) + sdot sdot sdot

+ 119901 (119909119899minus1

119909119899)

le (119871119898

+ 119871119898+1

+ sdot sdot sdot + 119871119899minus1

) 119901 (1199090 1199091)

(22)

Thus

119901 (119909119898 119909119899) le

119871119898

1 minus 119871119901 (1199090 1199091) (119899 gt 119898) (23)

Now inequality (20) follows from (23) by letting 119899 rarr infinWe have

119901 (119909119898 119906) le

119871119898

1 minus 119871119901 (1199090 1199091) (24)

Now for inequality (21) taking 119898 = 1 and writing 1199100for

1199090and 119910

1for 1199091 we have from (23)

119901 (1199101 119906) le

119871

1 minus 119871119901 (1199100 1199101) (25)

Setting 1199100= 119909119898minus1

we have 1199101= 1198791199100= 119909119898and obtain (21)

Corollary 15 Let (119883 119901) be a complete dualistic partial metricspace and let 119879 119883 rarr 119883 be a contraction on a closed ball119861 = 119861(119909

0 119903) = 119909 119901(119909 119909

0) le 119903 Moreover assume that

119901(1199090 119879x0) lt (1minus119871)119903Then prior error estimate is the following

estimate

119901 (119909119898 119906) le 119871

119898

119903 (26)

and the posterior estimate

119901 (119909119898 119906) le 119871119903 (27)

Proof By Corollary 11 the iteration sequence as (5) is con-verges to the unique fixed point 119906 of119879 hence byCorollary 14we have

119901 (119909119898 119906) le

119871119898

1 minus 119871119901 (1199090 1199091) (28)

since 1199091= 1198791199090and 119901(119909

0 1198791199090) lt (1 minus 119871)119903 we have

119901 (119909119898 119906) le

119871119898

1 minus 119871119901 (1199090 1199091)

=119871119898

1 minus 119871119901 (1199090 1198791199090)

lt119871119898

1 minus 119871sdot (1 minus 119871) 119903

= 119871119898

sdot 119903

(29)

Therefore 119901(119909119898 119906) lt 119871

119898

119903 Also by Corollary 14 we have

119901 (119909119898 119906) le

119871

1 minus 119871119901 (119909119898minus1

119909119898) (30)

since 119909119898minus1

= 119879119898minus1

1199090and 119901(119909

0 1198791199090) lt (1 minus 119871)119903 we have

119901 (119909119898 119906) le

119871

1 minus 119871119901 (119909119898minus1

119909119898)

=119871

1 minus 119871119901 (119879119898minus1

1199090 119879119898

1199090)

=119871

1 minus 119871119901 (119879119898minus1

1199090 119879 (119879

119898minus1

1199090))

lt119871

1 minus 119871(1 minus 119871) 119903

lt 119871119903

(31)

Therefore 119901(119909119898 119906) lt 119871119903

3 Applications of Banach ContractionPrinciple on Complete Dualistic PartialMetric Space

In this section we apply Theorem 10 to prove existence ofthe solutions a system of 119899 linear algebraic equations with 119899unknowns and we show that applied of Corollaries 14 and 15in numerical analysis

31 Application 31 Suppose we want to find the solution of asystem of 119899 linear algebraic equations with 119899 unknowns then

119886111199091+ 119886121199092+ sdot sdot sdot + 119886

1119899119909119899= 1198871

119886211199091+ 119886221199092+ sdot sdot sdot + 119886

2119899119909119899= 1198872

11988611989911199091+ 11988611989921199092+ sdot sdot sdot + 119886

119899119899119909119899= 119887119899

(32)

This system can be written as

1199091= (1 minus 119886

11) 1199091minus 119886121199092minus 119886131199093minus sdot sdot sdot minus 119886

1119899119909119899+ 1198871

1199092= minus119886211199091+ (1 minus 119886

22) 1199092minus 119886231199093minus sdot sdot sdot minus 119886

2119899119909119899+ 1198872

1199093= minus119886311199091minus 119886321199093+ (1 minus 119886

33) 1199093minus sdot sdot sdot minus 119886

3119899119909119899+ 1198873

119909119899= minus11988611989911199091minus 11988611989921199092minus 11988611989931199093minus sdot sdot sdot + (1 minus 119886

119899119899) 119909119899+ 119887119899

(33)

By assuming 120572119894119895= minus119886119894119895+ 120575119894119895 where

120575119894119895=

0 119894 = 119895

1 119894 = 119895(34)

Equation (33) can bewritten in the following equivalent form

119909119894=

119899

sum

119895=1

120572119894119895119909119895+ 119887119894 119894 = 1 2 3 119899 (35)

ISRN Computational Mathematics 5

If 119909 = (1199091 1199092 119909

119899) isin 119877119899 then (35) can be written in the

form 119879119909 = 119909 where 119879 is defined by

119879119909 = 119910 where 119910 = (1199101 1199102 119910

119899) (36)

119910119894=

119899

sum

119895=1

120572119894119895119909119895+ 119887119894 119894 = 1 2 3 119899

119879 119877119899

997888rarr 119877119899

(120572119894119895) is a 119899 times 119899 matrix

(37)

Finding solutions of the system described by (32) or (35)is thus equivalent to finding the fixed point of the operatorequation (36) In order to find a unique solution of 119879 thatis a unique solution of (32) we apply Theorem 10 In factwe prove the following result Equation (32) has a uniquesolution if

119899

sum

119895=1

10038161003816100381610038161003816120572119894119895

10038161003816100381610038161003816=

119899

sum

119895=1

10038161003816100381610038161003816minus119886119894119895+ 120575119894119895

10038161003816100381610038161003816le 119871 lt 1 119894 = 1 119899 (38)

For 119909 = (1199091 119909

119899) and 1199091015840 = (1199091015840

1 1199091015840

2 119909

1015840

119899) we have

119901 (119879119909 1198791199091015840

) = 119901 (119910 1199101015840

) (39)

where

119910 = (1199101 1199102 119910

119899) isin 119877119899

1199101015840

= (1199101015840

1 1199101015840

2 119910

1015840

119899) isin 119877119899

119910119894=

119899

sum

119895=1

120572119894119895119909119895+ 119887119894

1199101015840

119894=

119899

sum

119895=1

1205721198941198951199091015840

119895+ 119887119894 119894 = 1 2 119899

(40)

If 119910 = (1199101 1199102 119910

119899) isin 119877

119899 then 119901(119910 119910) = sup1le119894le119899

|119910119894|

Therefore

119901 (119879119909 1198791199091015840

) = sup1le119894le119899

10038161003816100381610038161003816119910119894minus 1199101015840

119894

10038161003816100381610038161003816

= sup1le119894le119899

10038161003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119895=1

120572119894119895119909119895+ 119887119894minus

119899

sum

119895=1

1205721198941198951199091015840

119895minus 119887119894

10038161003816100381610038161003816100381610038161003816100381610038161003816

= sup1le119894le119899

10038161003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119895=1

120572119894119895(119909119895minus 1199091015840

119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le sup1le119894le119899

119899

sum

119895=1

10038161003816100381610038161003816120572119894119895

10038161003816100381610038161003816

10038161003816100381610038161003816119909119895minus 1199091015840

119895

10038161003816100381610038161003816

le sup1le119895le119899

10038161003816100381610038161003816119909119895minus 1199091015840

119895

10038161003816100381610038161003816sup1le119894le119899

119899

sum

119895=1

10038161003816100381610038161003816120572119894119895

10038161003816100381610038161003816

le 119871 sup1le119895le119899

10038161003816100381610038161003816119909119895minus 1199091015840

119895

10038161003816100381610038161003816

(41)

Since 119901(119909 1199091015840) = sup1le119895le119899

|119909119895minus 1199091015840

119895| we have 119901(119879119909 1198791199091015840) le

119871119901(119909 1199091015840

) 0 le 119871 lt 1 that is 119879 is a contraction mapping of

the complete dualistic partial metric space 119877119899 into itself

Hence by Theorem 10 there exists a unique fixed point 119906 of119879 in 119877119899 that is 119906 is a unique solution of (32)

Theorem 16 If 119891(119905) is a nonlinear integral equation as thefollowing

119891 (119905) = int

119905

0

119890minusV119905 cos (120572119891 (V)) 119889V 0 le 119905 le 1 0 lt 120572 lt 1

(42)

then it has a unique solution

Proof We apply Theorem 10 and we can prove that thisequation has a unique continuous real-valued solution 119891(119905)Let 119883 = 119862[0 1] and the mapping 119879 119883 rarr 119883 defined by119879(119891) = 119891 for 119891 isin 119883 where 119883 is a complete dualistic partialmetric space with sup 119901(119909 119910) is a contraction mapping

cos (120572119886) minus cos (120572119887) = 120572 (119887 minus 119886) sin120573 (43)

where 120573 lies between 120572119886 and 120572119887 Therefore | cos(120572119886) minuscos(120572119887)| le 120572|119887 minus 119886| For functions 119886(119905) and 119887(119905) we get

|cos120572119886 (119905) minus cos120572119887 (119905)| le sup0le119905le1

|119886 (119905) minus 119887 (119905)| = 119901 (119886 119887)

(44)

For 119891 = 119879119891 and 119892 = 119879119892 we have

1003816100381610038161003816119879119891 minus 1198791198921003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816

int

119905

0

119890minusV119905 1003816100381610038161003816cos (120572119891 (V)) minus cos (120572119892 (V))1003816100381610038161003816 119889V

10038161003816100381610038161003816100381610038161003816

le int

119905

0

119890minusV119905 1003816100381610038161003816cos (120572119891 (V)) minus cos (120572119892 (V))1003816100381610038161003816 119889V

le 120572119901 (119891 119892) int

119905

0

119890minusV119905119889V le 120572119901 (119891 119892)

(45)

Taking sup over 0 le 119905 le 1 we get

sup119905

1003816100381610038161003816119879119891 (119905) minus 119879119892 (119905)1003816100381610038161003816 le 120572119901 (119891 119892) (46)

or

119901 (119879 (119891) 119879 (119892)) le 120572119901 (119891 119892) (47)

Theorem 17 Let 1199090be an initial value and the iterative

sequence 119909119899 as the following

119909119899= 119892 (119909

119899minus1) 119899 = 1 2 (48)

If 119892 is continuously differentiable on some interval 119870 = [1199090minus

119903 1199090+ 119903] and satisfies |1198921015840(119909)| le 119871 lt 1 on 119870 as well as

1003816100381610038161003816119892 (1199090) minus 11990901003816100381610038161003816 lt (1 minus 119871) 119903 (49)

then 119909 = 119892(119909) has a unique solution 119906 on 119870 the iterativesequence 119909

119898 converges to that solution and one has the error

estimates1003816100381610038161003816119909 minus 119909119898

1003816100381610038161003816 lt 119871119898

1199031003816100381610038161003816119909 minus 119909119898

1003816100381610038161003816 lt 119871119903 (50)

6 ISRN Computational Mathematics

Proof Suppose that 119901(119909 119892(119909)) = |119909 minus 119892(119909)| for 119909 isin 119870By the mean-value theorem and the given condition 119892(119909)is a contraction mapping of the complete dualistic partialmetric space119870 into itself Hence by Corollary 11 there existsa unique fixed point 119906 of 119892 in119870 that is 119906 is a unique solutionof 119909 = 119892(119909) Also the iteration sequence 119909

119898 converges to 119906

Moreover by 119901(119909 119892(119909)) = |119909 minus 119892(119909)| and Corollary 15 it hasthe prior error estimate

1003816100381610038161003816119909 minus 1199091198981003816100381610038161003816 le 119871119898

119903 (51)

and the posterior estimate1003816100381610038161003816119909 minus 119909119898

1003816100381610038161003816 le 119871119903 (52)

References

[1] S G Matthews ldquoPartial metric topologyrdquo in Proceedings of the11th Summer Conference on General Topology and Applicationsvol 728 pp 183ndash197 The New York Academy of SciencesAugusts 1995

[2] I Altun and A Erduran ldquoFixed point theorems for monotonemappings on partial metric spacesrdquo Fixed Point Theory andApplications vol 2011 Article ID 508730 10 pages 2011

[3] S J OrsquoNeill ldquoPartial metrics valuations and domain theoryrdquoAnnals of the New York Academy of Sciences vol 806 pp 304ndash315 1996

[4] P Fletcher and W F Lindgren Quasi-Uniform Spaces MarcelDekker New York NY USA 1982

[5] H P A Kunzi ldquoHandbook of the history of general topologyrdquoin Non-Symmetric Distances and Their Associated TopologiesAbout the Origins of Basic Ideas in the Area of AsymmetricTopology vol 3 pp 853ndash968 Kluwer Academic 2001

[6] S A M Mohsenalhosseini H Mazaheri M A Dehghan andA Zareh ldquoFixed point for partial metric spacesrdquo ISRN AppliedMathematics vol 2011 Article ID 657868 6 pages 2011

[7] S A M Mohsenalhosseini H Mazaheri and M A DehghanldquoApproximate best proximity pairs in metric spacerdquo Abstractand Applied Analysis vol 2011 Article ID 596971 9 pages 2011

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Stochastic AnalysisInternational Journal of

2 ISRN Computational Mathematics

Lemma 2 (see [1]) If (119883 119901) is a dualistic partial metric spacethen the function 119889

119901 119883 times 119883 rarr 119877

+ defined by 119889119901(119909 119910) =

119901(119909 119910)minus119901(119909 119909) is a quasi-metric on119883 such that 120591(119901) = 120591(119889119901)

Lemma 3 (see [1]) A dualistic partial metric space (119883 119901) iscomplete if and only if the metric space (119883 (119889

119901)119904

) is completeFurthermore lim

119899rarrinfin(119889119901)119904

(119886 119909119899) = 0 if and only if 119901(119886 119886) =

lim119899rarrinfin

119901(119886 119909119899) = lim

119899119898rarrinfin119901(119909119899 119909119898)

Before stating our main results we establish some (essen-tially known) correspondences between dualistic partial met-rics and quasi-metric spaces Also refer to definition of 120598-Fixed point and the existence of 120598-Fixed point for 120598 gt 0 Ourbasic references for quasi-metric spaces are [4 5] and for 120598-Fixed point is [6]

If 119889 is a quasi-metric on 119883 then the function 119889119904 definedon 119883 times 119883 by 119889119904(119909 119910) = max119889(119909 119910) 119889(119910 119909) is a metric on119883

Definition 4 (see [6]) Let (119883 119901) be a dualistic partial metricspace and 119879 119883 rarr 119883 be a mapThen 119909

0isin 119883 is 120598-fixed point

for 119879 if

119889119901(1198791199090 1199090) le 120598 (1)

We say 119879 has the 120598-fixed point property if for every 120598 gt 0119860119865(119879) = 0 where

119860119865 (119879) = 1199090isin 119883 119889 (119879119909

0 1199090) le 120598 (2)

Theorem 5 (see [6]) Let (119883 119901) be a dualistic partial metricspace and 119879 119883 rarr 119883 be a map 119909

0isin 119883 and 120598 gt 0 If

119889119901(119879119899

(1199090) 119879119899+119896(119909

0)) rarr 0 as 119899 rarr infin for some 119896 gt 0 then

119879119896 has an 120598-fixed point

Definition 6 (see [7]) Let 119879 119860 cup 119861 rarr 119860 cup 119861 be continuesmap such that 119879(119860) sube 119861 119879(119861) sube 119860 and 120598 gt 0 We definediameter 119875119886

119879(119860 119861) by

diam (119875119886

119879(119860 119861)) = sup 119889 (119909 119910) 119909 119910 isin 119875119886

119879(119860 119861) (3)

Theorem 7 (see [1]) The partial metric contraction mappingtheorem Let (119883 119901) be a complete partial metric space and 119879

119883 rarr 119883 be a map such that for all 119909 119910 isin 119883

119901 (119879119909 119879119910) le 119871119901 (119909 119910) 0 le 119871 lt 1 (4)

then 119879 has a unique fixed point 119906 and 119879119899(119909) rarr 119906 as 119899 rarr infin

for each 119909 isin 119883

2 Some Result Fixed Point on Partial Metric

In this section we give some result on fixed point and 120598-fixedpoint in dualistic partial metric space and its diameter

Definition 8 An open ball for a dualistic partial metric 119901

119883 times 119883 rarr 119877 is a set of the form 119861119901

120598(119909) = 119910 isin 119883 119901(119909 119910) lt

120598 for each 120598 gt 0 and 119909 isin 119883

Definition 9 Let 119879 be a mapping of a complete dualistic par-tial metric 119883 into itself [119879 119883 rarr 119883] then 119879 is called

a partial metric contractionmapping if there exists a constant119871 0 le 119871 lt 1 such that 119901(119879119909 119879119910) le 119871119901(119909 119910) for all 119909 119910 isin 119883

Theorem 10 Every contraction mapping 119879 defined on acomplete dualistic partial metric119883 into itself has a unique fixedpoint 119906 isin 119883 Moreover if 119909

0is any point in119883 and the sequence

119909119899is defined by

1199091= 119879 (119909

0) 119909

2= 119879 (119909

1) 119909

119899= 119879 (119909

119899minus1) (5)

then lim119899rarrinfin

119909119899= 119906 and

119901 (119909119899 119906) le

119871119899

1 minus 119871119901 (1199091 1199090) 0 le 119871 lt 1 (6)

Proof Existence of a fixed point Let 1199090be an arbitrary point

in 119883 and we defined by 1199091= 119879(119909

0) 1199092= 119879(119909

1) 119909

119899=

119879(119909119899minus1

) Then

1199092= 119879 (119909

1) = 119879 (119879 (119909

0)) = 119879

2

(1199090)

1199093= 119879 (119909

2) = 119879 (119879 (119909

1)) = 119879 (119879 (119879 (119909

0))) = 119879

3

(1199090)

119909119899= 119879119899

(1199090)

(7)

If119898 gt 119899 say119898 = 119899 + 120572 120572 = 1 2 3 Then

119901 (119909119899+ 120572 119909

119899) = 119901 (119879

119899+120572

(1199090) 119879119899

(1199090))

= 119901 (119879 (119879119899+120572minus1

(1199090)) 119879 (119879

119899minus1

(1199090)))

le 119871119901 (119879119899+120572minus1

(1199090) 119879119899minus1

(1199090))

(8)

Continuing this process 119899 minus 1 times we have

119901 (119909119899+120572

119909119899) le 119871119899

119901 (119879120572

(1199090) 1199090) (9)

for 119899 = 0 1 2 and all 120572 ge 1However

119901 (119879120572

(1199090) 1199090) le 119901 (119879

120572

(1199090) 119879120572minus1

(1199090))

+ 119901 (119879120572minus1

(1199090) 119879120572minus2

(1199090)) + sdot sdot sdot

+ 119901 (119879 (1199090) 1199090)

(10)

Therefore we see that

119901 (119909119899+120572

119909119899)

le 119871119899

[119871120572minus1

119901 (1199091 1199090) + 119871120572minus2

119901 (1199091 1199090) + sdot sdot sdot + 119901 (119909

1 1199090)]

le 119871119899

119901 (1199091 1199090) [1 + 119871 + 119871

2

+ sdot sdot sdot + 119871120572minus1

+ 119871120572

]

(11)

Hence

119901 (119909119899+120572

119909119899) le 119871119899

119901 (1199091 1199090)

1

1 minus 119871 (12)

ISRN Computational Mathematics 3

As 119899 119898 = 119899 + 120572 rarr infin from (12) we see that119901(119909119899+120572

119909119899) rarr 0 that is 119909

119899 is a Cauchy sequence in

the metric space 119883 Hence 119909119899 must be convergent say

lim119899rarrinfin

119909119899= 119906

Since 119879 is continuous we have

119879119906 = 119879( lim119899rarrinfin

119909119899) = lim119899rarrinfin

119879 (119909119899) = lim119899rarrinfin

119909119899+1

(13)

or 119879119906 = 119906 Thus 119906 is a fixed point of 119879Uniqueness of the fixed point 119879Let V be another fixed point of 119879 Then 119879V = V We also

have 119901(119879(119906) 119879(V)) le 119871119901(119906 V) But 119901(119879(119906) 119879(V)) = 119901(119906 V)which implies that 119901(119906 V) le 119871119901(119906 V) where 0 le 119871 lt 1 Thisis possible only when 119901(119906 V) = 0 that is 119906 = V This provesthat the fixed point of 119879 is unique

Corollary 11 Let (119883 119901) be a complete dualistic partial metricspace and 119861119901

119903(1199090) = 119910 isin 119883 119901(119909

0 119910) lt 119903 Let 119879 119861

119901

119903rarr 119883

be a partial metric contraction mapping If 119901(1198791199100 1199100) lt (1 minus

119871)119903 then 119879 has a fixed point

Proof Choose 119903 lt 120598 so that 119901(1198791199100 1199100) le (1 minus 119871)119903 lt (1 minus 119871)120598

We show that 119879 maps the closed ball 119870 = 119910 119901(119910 1199100) le 120598

into itself for if 119910 isin 119870 then

119901 (119879 (119910) 1199100) le 119901 (119879 (119910) 119879 (119910

0)) + 119901 (119879 (119910

0) 1199100)

le 119871119901 (119910 1199100) + (1 minus 119871) 120598

le 119871120598 + 120598 minus 119871120598 = 120598

(14)

Since 119870 is complete and 119879 119870 rarr 119870 satisfy in (5) thus byTheorem 10 119879 has a fixed point

Theorem 12 If 119883 is a complete dualistic partial metric spaceand 119879 119883 rarr 119883 is such that 119879119903 is contraction for some integer119903 gt 0 then 119879119903 has an unique fixed point

Proof Since 119879119903 where 119903 is a positive integer is a contractionmapping by Theorem 10 there exists an unique fixed point 119906of 119879119903 that is 119879119903(119906) = 119906 We want to show that 119906 ia a fixedpoint of 119879 that is 119879(119906) = 119906 Let 119878 = 119879

119903 therefore 119878(119906) = 119906This implies that

1198782

(119906) = 119878 (119878 (119906)) = 119878 (119906) = 119906

1198783

(119906) = 1198782

(119878 (119906)) = 1198782

(119906) ℎ = 119906

119878119899

(119906) = 119878119899minus1

(119878 (119906)) = 119878119899minus1

(119906) = sdot sdot sdot = 119906

(15)

and so 119879(119906) = 119879(119878119899(119906)) = 119904119899(119879(119906)) = 119904119899119910 say

119901 (119878119899

(119910) 119906) = 119901 (119878119899

(119910) 119878119899

(119906))

le 119871119901 (119878119899minus1

(119910) 119878119899minus1

(119906))

le 1198712

119901 (119878119899minus2

(119910) 119878119899minus2

(119906))

le 119871119899

119901 (119910 119906)

lim119899rarrinfin

119901 (119878119899

(119910) 119906) = 0

(16)

as 119871119899 rarr 0 Thus lim119899rarrinfin

119878119899

(119910) = 119906 and we have 119879(119906) =119906

Theorem 13 Let (119883 119901) is a complete dualistic partial metricspace and let 119879 119883 rarr 119883 and 119878 119883 rarr 119883 be two mapscontraction If for every 119909 isin 119883

119901 (119879119909 119878119910) le 120582 (17)

120582 gt 0 chosen suitably Then for every 119909 isin 119883

119901 (119879119898

(119909) 119878119898

(119909)) le 1205821 minus 119871119898

1 minus 119871 0 le 119871 lt 1

119898 = 1 2

(18)

Proof The relation is true for 119898 = 1 We use the principle ofinduction in order to prove this relation Let it be true for all119898 ge 1 Then

119901 (119879119899+1

(119909) minus 119878119899+1

(119909)) = 119901 (119879119879119898

(119909) 119878119878119898

(119909))

le 119901 (119879119879119898

(119909) 119879119878119898

(119909))

+ 119901 (119879119878119898

(119909) 119878119878119898

(119909))

le 119871119901 (119879119898

(119909) 119878119898

(119909)) + 120582

le 1198711205821 minus 119871119898

1 minus 119871+ 120582

=119871120582 minus 119871

119898+1

120582 + 120582 minus 119871120582

1 minus 119871

= 1205821 minus 119871119898+1

1 minus 119871

(19)

Thus the relation is true for119898 + 1

Corollary 14 Let (119883 119901) is a complete dualistic partial metricspace and let 119879 119883 rarr 119883 be a partial contractive map on 119883Moreover the iteration sequence 119909

1= 119879(119909

0) 1199092= 119879(119909

1) =

1198792

1199090 119909

119899= 119879119899

1199090 with arbitrary 119909

0isin 119883 converges to

the unique fixed point 119906 of 119879 Error estimate is the followingestimate (prior estimate)

119901 (119909119898 119906) le

119871119898

1 minus 119871119901 (1199090 1199091) (20)

and the posterior estimate

119901 (119909119898 119906) le

119871

1 minus 119871119901 (119909119898minus1

119909119898) (21)

4 ISRN Computational Mathematics

Proof The First statement is obvious by

119901 (119909119898 119909119899) le 119901 (119909

119898 119909119898+1

) + 119901 (119909119898+1

119909119898+2

) + sdot sdot sdot

+ 119901 (119909119899minus1

119909119899)

le (119871119898

+ 119871119898+1

+ sdot sdot sdot + 119871119899minus1

) 119901 (1199090 1199091)

(22)

Thus

119901 (119909119898 119909119899) le

119871119898

1 minus 119871119901 (1199090 1199091) (119899 gt 119898) (23)

Now inequality (20) follows from (23) by letting 119899 rarr infinWe have

119901 (119909119898 119906) le

119871119898

1 minus 119871119901 (1199090 1199091) (24)

Now for inequality (21) taking 119898 = 1 and writing 1199100for

1199090and 119910

1for 1199091 we have from (23)

119901 (1199101 119906) le

119871

1 minus 119871119901 (1199100 1199101) (25)

Setting 1199100= 119909119898minus1

we have 1199101= 1198791199100= 119909119898and obtain (21)

Corollary 15 Let (119883 119901) be a complete dualistic partial metricspace and let 119879 119883 rarr 119883 be a contraction on a closed ball119861 = 119861(119909

0 119903) = 119909 119901(119909 119909

0) le 119903 Moreover assume that

119901(1199090 119879x0) lt (1minus119871)119903Then prior error estimate is the following

estimate

119901 (119909119898 119906) le 119871

119898

119903 (26)

and the posterior estimate

119901 (119909119898 119906) le 119871119903 (27)

Proof By Corollary 11 the iteration sequence as (5) is con-verges to the unique fixed point 119906 of119879 hence byCorollary 14we have

119901 (119909119898 119906) le

119871119898

1 minus 119871119901 (1199090 1199091) (28)

since 1199091= 1198791199090and 119901(119909

0 1198791199090) lt (1 minus 119871)119903 we have

119901 (119909119898 119906) le

119871119898

1 minus 119871119901 (1199090 1199091)

=119871119898

1 minus 119871119901 (1199090 1198791199090)

lt119871119898

1 minus 119871sdot (1 minus 119871) 119903

= 119871119898

sdot 119903

(29)

Therefore 119901(119909119898 119906) lt 119871

119898

119903 Also by Corollary 14 we have

119901 (119909119898 119906) le

119871

1 minus 119871119901 (119909119898minus1

119909119898) (30)

since 119909119898minus1

= 119879119898minus1

1199090and 119901(119909

0 1198791199090) lt (1 minus 119871)119903 we have

119901 (119909119898 119906) le

119871

1 minus 119871119901 (119909119898minus1

119909119898)

=119871

1 minus 119871119901 (119879119898minus1

1199090 119879119898

1199090)

=119871

1 minus 119871119901 (119879119898minus1

1199090 119879 (119879

119898minus1

1199090))

lt119871

1 minus 119871(1 minus 119871) 119903

lt 119871119903

(31)

Therefore 119901(119909119898 119906) lt 119871119903

3 Applications of Banach ContractionPrinciple on Complete Dualistic PartialMetric Space

In this section we apply Theorem 10 to prove existence ofthe solutions a system of 119899 linear algebraic equations with 119899unknowns and we show that applied of Corollaries 14 and 15in numerical analysis

31 Application 31 Suppose we want to find the solution of asystem of 119899 linear algebraic equations with 119899 unknowns then

119886111199091+ 119886121199092+ sdot sdot sdot + 119886

1119899119909119899= 1198871

119886211199091+ 119886221199092+ sdot sdot sdot + 119886

2119899119909119899= 1198872

11988611989911199091+ 11988611989921199092+ sdot sdot sdot + 119886

119899119899119909119899= 119887119899

(32)

This system can be written as

1199091= (1 minus 119886

11) 1199091minus 119886121199092minus 119886131199093minus sdot sdot sdot minus 119886

1119899119909119899+ 1198871

1199092= minus119886211199091+ (1 minus 119886

22) 1199092minus 119886231199093minus sdot sdot sdot minus 119886

2119899119909119899+ 1198872

1199093= minus119886311199091minus 119886321199093+ (1 minus 119886

33) 1199093minus sdot sdot sdot minus 119886

3119899119909119899+ 1198873

119909119899= minus11988611989911199091minus 11988611989921199092minus 11988611989931199093minus sdot sdot sdot + (1 minus 119886

119899119899) 119909119899+ 119887119899

(33)

By assuming 120572119894119895= minus119886119894119895+ 120575119894119895 where

120575119894119895=

0 119894 = 119895

1 119894 = 119895(34)

Equation (33) can bewritten in the following equivalent form

119909119894=

119899

sum

119895=1

120572119894119895119909119895+ 119887119894 119894 = 1 2 3 119899 (35)

ISRN Computational Mathematics 5

If 119909 = (1199091 1199092 119909

119899) isin 119877119899 then (35) can be written in the

form 119879119909 = 119909 where 119879 is defined by

119879119909 = 119910 where 119910 = (1199101 1199102 119910

119899) (36)

119910119894=

119899

sum

119895=1

120572119894119895119909119895+ 119887119894 119894 = 1 2 3 119899

119879 119877119899

997888rarr 119877119899

(120572119894119895) is a 119899 times 119899 matrix

(37)

Finding solutions of the system described by (32) or (35)is thus equivalent to finding the fixed point of the operatorequation (36) In order to find a unique solution of 119879 thatis a unique solution of (32) we apply Theorem 10 In factwe prove the following result Equation (32) has a uniquesolution if

119899

sum

119895=1

10038161003816100381610038161003816120572119894119895

10038161003816100381610038161003816=

119899

sum

119895=1

10038161003816100381610038161003816minus119886119894119895+ 120575119894119895

10038161003816100381610038161003816le 119871 lt 1 119894 = 1 119899 (38)

For 119909 = (1199091 119909

119899) and 1199091015840 = (1199091015840

1 1199091015840

2 119909

1015840

119899) we have

119901 (119879119909 1198791199091015840

) = 119901 (119910 1199101015840

) (39)

where

119910 = (1199101 1199102 119910

119899) isin 119877119899

1199101015840

= (1199101015840

1 1199101015840

2 119910

1015840

119899) isin 119877119899

119910119894=

119899

sum

119895=1

120572119894119895119909119895+ 119887119894

1199101015840

119894=

119899

sum

119895=1

1205721198941198951199091015840

119895+ 119887119894 119894 = 1 2 119899

(40)

If 119910 = (1199101 1199102 119910

119899) isin 119877

119899 then 119901(119910 119910) = sup1le119894le119899

|119910119894|

Therefore

119901 (119879119909 1198791199091015840

) = sup1le119894le119899

10038161003816100381610038161003816119910119894minus 1199101015840

119894

10038161003816100381610038161003816

= sup1le119894le119899

10038161003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119895=1

120572119894119895119909119895+ 119887119894minus

119899

sum

119895=1

1205721198941198951199091015840

119895minus 119887119894

10038161003816100381610038161003816100381610038161003816100381610038161003816

= sup1le119894le119899

10038161003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119895=1

120572119894119895(119909119895minus 1199091015840

119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le sup1le119894le119899

119899

sum

119895=1

10038161003816100381610038161003816120572119894119895

10038161003816100381610038161003816

10038161003816100381610038161003816119909119895minus 1199091015840

119895

10038161003816100381610038161003816

le sup1le119895le119899

10038161003816100381610038161003816119909119895minus 1199091015840

119895

10038161003816100381610038161003816sup1le119894le119899

119899

sum

119895=1

10038161003816100381610038161003816120572119894119895

10038161003816100381610038161003816

le 119871 sup1le119895le119899

10038161003816100381610038161003816119909119895minus 1199091015840

119895

10038161003816100381610038161003816

(41)

Since 119901(119909 1199091015840) = sup1le119895le119899

|119909119895minus 1199091015840

119895| we have 119901(119879119909 1198791199091015840) le

119871119901(119909 1199091015840

) 0 le 119871 lt 1 that is 119879 is a contraction mapping of

the complete dualistic partial metric space 119877119899 into itself

Hence by Theorem 10 there exists a unique fixed point 119906 of119879 in 119877119899 that is 119906 is a unique solution of (32)

Theorem 16 If 119891(119905) is a nonlinear integral equation as thefollowing

119891 (119905) = int

119905

0

119890minusV119905 cos (120572119891 (V)) 119889V 0 le 119905 le 1 0 lt 120572 lt 1

(42)

then it has a unique solution

Proof We apply Theorem 10 and we can prove that thisequation has a unique continuous real-valued solution 119891(119905)Let 119883 = 119862[0 1] and the mapping 119879 119883 rarr 119883 defined by119879(119891) = 119891 for 119891 isin 119883 where 119883 is a complete dualistic partialmetric space with sup 119901(119909 119910) is a contraction mapping

cos (120572119886) minus cos (120572119887) = 120572 (119887 minus 119886) sin120573 (43)

where 120573 lies between 120572119886 and 120572119887 Therefore | cos(120572119886) minuscos(120572119887)| le 120572|119887 minus 119886| For functions 119886(119905) and 119887(119905) we get

|cos120572119886 (119905) minus cos120572119887 (119905)| le sup0le119905le1

|119886 (119905) minus 119887 (119905)| = 119901 (119886 119887)

(44)

For 119891 = 119879119891 and 119892 = 119879119892 we have

1003816100381610038161003816119879119891 minus 1198791198921003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816

int

119905

0

119890minusV119905 1003816100381610038161003816cos (120572119891 (V)) minus cos (120572119892 (V))1003816100381610038161003816 119889V

10038161003816100381610038161003816100381610038161003816

le int

119905

0

119890minusV119905 1003816100381610038161003816cos (120572119891 (V)) minus cos (120572119892 (V))1003816100381610038161003816 119889V

le 120572119901 (119891 119892) int

119905

0

119890minusV119905119889V le 120572119901 (119891 119892)

(45)

Taking sup over 0 le 119905 le 1 we get

sup119905

1003816100381610038161003816119879119891 (119905) minus 119879119892 (119905)1003816100381610038161003816 le 120572119901 (119891 119892) (46)

or

119901 (119879 (119891) 119879 (119892)) le 120572119901 (119891 119892) (47)

Theorem 17 Let 1199090be an initial value and the iterative

sequence 119909119899 as the following

119909119899= 119892 (119909

119899minus1) 119899 = 1 2 (48)

If 119892 is continuously differentiable on some interval 119870 = [1199090minus

119903 1199090+ 119903] and satisfies |1198921015840(119909)| le 119871 lt 1 on 119870 as well as

1003816100381610038161003816119892 (1199090) minus 11990901003816100381610038161003816 lt (1 minus 119871) 119903 (49)

then 119909 = 119892(119909) has a unique solution 119906 on 119870 the iterativesequence 119909

119898 converges to that solution and one has the error

estimates1003816100381610038161003816119909 minus 119909119898

1003816100381610038161003816 lt 119871119898

1199031003816100381610038161003816119909 minus 119909119898

1003816100381610038161003816 lt 119871119903 (50)

6 ISRN Computational Mathematics

Proof Suppose that 119901(119909 119892(119909)) = |119909 minus 119892(119909)| for 119909 isin 119870By the mean-value theorem and the given condition 119892(119909)is a contraction mapping of the complete dualistic partialmetric space119870 into itself Hence by Corollary 11 there existsa unique fixed point 119906 of 119892 in119870 that is 119906 is a unique solutionof 119909 = 119892(119909) Also the iteration sequence 119909

119898 converges to 119906

Moreover by 119901(119909 119892(119909)) = |119909 minus 119892(119909)| and Corollary 15 it hasthe prior error estimate

1003816100381610038161003816119909 minus 1199091198981003816100381610038161003816 le 119871119898

119903 (51)

and the posterior estimate1003816100381610038161003816119909 minus 119909119898

1003816100381610038161003816 le 119871119903 (52)

References

[1] S G Matthews ldquoPartial metric topologyrdquo in Proceedings of the11th Summer Conference on General Topology and Applicationsvol 728 pp 183ndash197 The New York Academy of SciencesAugusts 1995

[2] I Altun and A Erduran ldquoFixed point theorems for monotonemappings on partial metric spacesrdquo Fixed Point Theory andApplications vol 2011 Article ID 508730 10 pages 2011

[3] S J OrsquoNeill ldquoPartial metrics valuations and domain theoryrdquoAnnals of the New York Academy of Sciences vol 806 pp 304ndash315 1996

[4] P Fletcher and W F Lindgren Quasi-Uniform Spaces MarcelDekker New York NY USA 1982

[5] H P A Kunzi ldquoHandbook of the history of general topologyrdquoin Non-Symmetric Distances and Their Associated TopologiesAbout the Origins of Basic Ideas in the Area of AsymmetricTopology vol 3 pp 853ndash968 Kluwer Academic 2001

[6] S A M Mohsenalhosseini H Mazaheri M A Dehghan andA Zareh ldquoFixed point for partial metric spacesrdquo ISRN AppliedMathematics vol 2011 Article ID 657868 6 pages 2011

[7] S A M Mohsenalhosseini H Mazaheri and M A DehghanldquoApproximate best proximity pairs in metric spacerdquo Abstractand Applied Analysis vol 2011 Article ID 596971 9 pages 2011

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Stochastic AnalysisInternational Journal of

ISRN Computational Mathematics 3

As 119899 119898 = 119899 + 120572 rarr infin from (12) we see that119901(119909119899+120572

119909119899) rarr 0 that is 119909

119899 is a Cauchy sequence in

the metric space 119883 Hence 119909119899 must be convergent say

lim119899rarrinfin

119909119899= 119906

Since 119879 is continuous we have

119879119906 = 119879( lim119899rarrinfin

119909119899) = lim119899rarrinfin

119879 (119909119899) = lim119899rarrinfin

119909119899+1

(13)

or 119879119906 = 119906 Thus 119906 is a fixed point of 119879Uniqueness of the fixed point 119879Let V be another fixed point of 119879 Then 119879V = V We also

have 119901(119879(119906) 119879(V)) le 119871119901(119906 V) But 119901(119879(119906) 119879(V)) = 119901(119906 V)which implies that 119901(119906 V) le 119871119901(119906 V) where 0 le 119871 lt 1 Thisis possible only when 119901(119906 V) = 0 that is 119906 = V This provesthat the fixed point of 119879 is unique

Corollary 11 Let (119883 119901) be a complete dualistic partial metricspace and 119861119901

119903(1199090) = 119910 isin 119883 119901(119909

0 119910) lt 119903 Let 119879 119861

119901

119903rarr 119883

be a partial metric contraction mapping If 119901(1198791199100 1199100) lt (1 minus

119871)119903 then 119879 has a fixed point

Proof Choose 119903 lt 120598 so that 119901(1198791199100 1199100) le (1 minus 119871)119903 lt (1 minus 119871)120598

We show that 119879 maps the closed ball 119870 = 119910 119901(119910 1199100) le 120598

into itself for if 119910 isin 119870 then

119901 (119879 (119910) 1199100) le 119901 (119879 (119910) 119879 (119910

0)) + 119901 (119879 (119910

0) 1199100)

le 119871119901 (119910 1199100) + (1 minus 119871) 120598

le 119871120598 + 120598 minus 119871120598 = 120598

(14)

Since 119870 is complete and 119879 119870 rarr 119870 satisfy in (5) thus byTheorem 10 119879 has a fixed point

Theorem 12 If 119883 is a complete dualistic partial metric spaceand 119879 119883 rarr 119883 is such that 119879119903 is contraction for some integer119903 gt 0 then 119879119903 has an unique fixed point

Proof Since 119879119903 where 119903 is a positive integer is a contractionmapping by Theorem 10 there exists an unique fixed point 119906of 119879119903 that is 119879119903(119906) = 119906 We want to show that 119906 ia a fixedpoint of 119879 that is 119879(119906) = 119906 Let 119878 = 119879

119903 therefore 119878(119906) = 119906This implies that

1198782

(119906) = 119878 (119878 (119906)) = 119878 (119906) = 119906

1198783

(119906) = 1198782

(119878 (119906)) = 1198782

(119906) ℎ = 119906

119878119899

(119906) = 119878119899minus1

(119878 (119906)) = 119878119899minus1

(119906) = sdot sdot sdot = 119906

(15)

and so 119879(119906) = 119879(119878119899(119906)) = 119904119899(119879(119906)) = 119904119899119910 say

119901 (119878119899

(119910) 119906) = 119901 (119878119899

(119910) 119878119899

(119906))

le 119871119901 (119878119899minus1

(119910) 119878119899minus1

(119906))

le 1198712

119901 (119878119899minus2

(119910) 119878119899minus2

(119906))

le 119871119899

119901 (119910 119906)

lim119899rarrinfin

119901 (119878119899

(119910) 119906) = 0

(16)

as 119871119899 rarr 0 Thus lim119899rarrinfin

119878119899

(119910) = 119906 and we have 119879(119906) =119906

Theorem 13 Let (119883 119901) is a complete dualistic partial metricspace and let 119879 119883 rarr 119883 and 119878 119883 rarr 119883 be two mapscontraction If for every 119909 isin 119883

119901 (119879119909 119878119910) le 120582 (17)

120582 gt 0 chosen suitably Then for every 119909 isin 119883

119901 (119879119898

(119909) 119878119898

(119909)) le 1205821 minus 119871119898

1 minus 119871 0 le 119871 lt 1

119898 = 1 2

(18)

Proof The relation is true for 119898 = 1 We use the principle ofinduction in order to prove this relation Let it be true for all119898 ge 1 Then

119901 (119879119899+1

(119909) minus 119878119899+1

(119909)) = 119901 (119879119879119898

(119909) 119878119878119898

(119909))

le 119901 (119879119879119898

(119909) 119879119878119898

(119909))

+ 119901 (119879119878119898

(119909) 119878119878119898

(119909))

le 119871119901 (119879119898

(119909) 119878119898

(119909)) + 120582

le 1198711205821 minus 119871119898

1 minus 119871+ 120582

=119871120582 minus 119871

119898+1

120582 + 120582 minus 119871120582

1 minus 119871

= 1205821 minus 119871119898+1

1 minus 119871

(19)

Thus the relation is true for119898 + 1

Corollary 14 Let (119883 119901) is a complete dualistic partial metricspace and let 119879 119883 rarr 119883 be a partial contractive map on 119883Moreover the iteration sequence 119909

1= 119879(119909

0) 1199092= 119879(119909

1) =

1198792

1199090 119909

119899= 119879119899

1199090 with arbitrary 119909

0isin 119883 converges to

the unique fixed point 119906 of 119879 Error estimate is the followingestimate (prior estimate)

119901 (119909119898 119906) le

119871119898

1 minus 119871119901 (1199090 1199091) (20)

and the posterior estimate

119901 (119909119898 119906) le

119871

1 minus 119871119901 (119909119898minus1

119909119898) (21)

4 ISRN Computational Mathematics

Proof The First statement is obvious by

119901 (119909119898 119909119899) le 119901 (119909

119898 119909119898+1

) + 119901 (119909119898+1

119909119898+2

) + sdot sdot sdot

+ 119901 (119909119899minus1

119909119899)

le (119871119898

+ 119871119898+1

+ sdot sdot sdot + 119871119899minus1

) 119901 (1199090 1199091)

(22)

Thus

119901 (119909119898 119909119899) le

119871119898

1 minus 119871119901 (1199090 1199091) (119899 gt 119898) (23)

Now inequality (20) follows from (23) by letting 119899 rarr infinWe have

119901 (119909119898 119906) le

119871119898

1 minus 119871119901 (1199090 1199091) (24)

Now for inequality (21) taking 119898 = 1 and writing 1199100for

1199090and 119910

1for 1199091 we have from (23)

119901 (1199101 119906) le

119871

1 minus 119871119901 (1199100 1199101) (25)

Setting 1199100= 119909119898minus1

we have 1199101= 1198791199100= 119909119898and obtain (21)

Corollary 15 Let (119883 119901) be a complete dualistic partial metricspace and let 119879 119883 rarr 119883 be a contraction on a closed ball119861 = 119861(119909

0 119903) = 119909 119901(119909 119909

0) le 119903 Moreover assume that

119901(1199090 119879x0) lt (1minus119871)119903Then prior error estimate is the following

estimate

119901 (119909119898 119906) le 119871

119898

119903 (26)

and the posterior estimate

119901 (119909119898 119906) le 119871119903 (27)

Proof By Corollary 11 the iteration sequence as (5) is con-verges to the unique fixed point 119906 of119879 hence byCorollary 14we have

119901 (119909119898 119906) le

119871119898

1 minus 119871119901 (1199090 1199091) (28)

since 1199091= 1198791199090and 119901(119909

0 1198791199090) lt (1 minus 119871)119903 we have

119901 (119909119898 119906) le

119871119898

1 minus 119871119901 (1199090 1199091)

=119871119898

1 minus 119871119901 (1199090 1198791199090)

lt119871119898

1 minus 119871sdot (1 minus 119871) 119903

= 119871119898

sdot 119903

(29)

Therefore 119901(119909119898 119906) lt 119871

119898

119903 Also by Corollary 14 we have

119901 (119909119898 119906) le

119871

1 minus 119871119901 (119909119898minus1

119909119898) (30)

since 119909119898minus1

= 119879119898minus1

1199090and 119901(119909

0 1198791199090) lt (1 minus 119871)119903 we have

119901 (119909119898 119906) le

119871

1 minus 119871119901 (119909119898minus1

119909119898)

=119871

1 minus 119871119901 (119879119898minus1

1199090 119879119898

1199090)

=119871

1 minus 119871119901 (119879119898minus1

1199090 119879 (119879

119898minus1

1199090))

lt119871

1 minus 119871(1 minus 119871) 119903

lt 119871119903

(31)

Therefore 119901(119909119898 119906) lt 119871119903

3 Applications of Banach ContractionPrinciple on Complete Dualistic PartialMetric Space

In this section we apply Theorem 10 to prove existence ofthe solutions a system of 119899 linear algebraic equations with 119899unknowns and we show that applied of Corollaries 14 and 15in numerical analysis

31 Application 31 Suppose we want to find the solution of asystem of 119899 linear algebraic equations with 119899 unknowns then

119886111199091+ 119886121199092+ sdot sdot sdot + 119886

1119899119909119899= 1198871

119886211199091+ 119886221199092+ sdot sdot sdot + 119886

2119899119909119899= 1198872

11988611989911199091+ 11988611989921199092+ sdot sdot sdot + 119886

119899119899119909119899= 119887119899

(32)

This system can be written as

1199091= (1 minus 119886

11) 1199091minus 119886121199092minus 119886131199093minus sdot sdot sdot minus 119886

1119899119909119899+ 1198871

1199092= minus119886211199091+ (1 minus 119886

22) 1199092minus 119886231199093minus sdot sdot sdot minus 119886

2119899119909119899+ 1198872

1199093= minus119886311199091minus 119886321199093+ (1 minus 119886

33) 1199093minus sdot sdot sdot minus 119886

3119899119909119899+ 1198873

119909119899= minus11988611989911199091minus 11988611989921199092minus 11988611989931199093minus sdot sdot sdot + (1 minus 119886

119899119899) 119909119899+ 119887119899

(33)

By assuming 120572119894119895= minus119886119894119895+ 120575119894119895 where

120575119894119895=

0 119894 = 119895

1 119894 = 119895(34)

Equation (33) can bewritten in the following equivalent form

119909119894=

119899

sum

119895=1

120572119894119895119909119895+ 119887119894 119894 = 1 2 3 119899 (35)

ISRN Computational Mathematics 5

If 119909 = (1199091 1199092 119909

119899) isin 119877119899 then (35) can be written in the

form 119879119909 = 119909 where 119879 is defined by

119879119909 = 119910 where 119910 = (1199101 1199102 119910

119899) (36)

119910119894=

119899

sum

119895=1

120572119894119895119909119895+ 119887119894 119894 = 1 2 3 119899

119879 119877119899

997888rarr 119877119899

(120572119894119895) is a 119899 times 119899 matrix

(37)

Finding solutions of the system described by (32) or (35)is thus equivalent to finding the fixed point of the operatorequation (36) In order to find a unique solution of 119879 thatis a unique solution of (32) we apply Theorem 10 In factwe prove the following result Equation (32) has a uniquesolution if

119899

sum

119895=1

10038161003816100381610038161003816120572119894119895

10038161003816100381610038161003816=

119899

sum

119895=1

10038161003816100381610038161003816minus119886119894119895+ 120575119894119895

10038161003816100381610038161003816le 119871 lt 1 119894 = 1 119899 (38)

For 119909 = (1199091 119909

119899) and 1199091015840 = (1199091015840

1 1199091015840

2 119909

1015840

119899) we have

119901 (119879119909 1198791199091015840

) = 119901 (119910 1199101015840

) (39)

where

119910 = (1199101 1199102 119910

119899) isin 119877119899

1199101015840

= (1199101015840

1 1199101015840

2 119910

1015840

119899) isin 119877119899

119910119894=

119899

sum

119895=1

120572119894119895119909119895+ 119887119894

1199101015840

119894=

119899

sum

119895=1

1205721198941198951199091015840

119895+ 119887119894 119894 = 1 2 119899

(40)

If 119910 = (1199101 1199102 119910

119899) isin 119877

119899 then 119901(119910 119910) = sup1le119894le119899

|119910119894|

Therefore

119901 (119879119909 1198791199091015840

) = sup1le119894le119899

10038161003816100381610038161003816119910119894minus 1199101015840

119894

10038161003816100381610038161003816

= sup1le119894le119899

10038161003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119895=1

120572119894119895119909119895+ 119887119894minus

119899

sum

119895=1

1205721198941198951199091015840

119895minus 119887119894

10038161003816100381610038161003816100381610038161003816100381610038161003816

= sup1le119894le119899

10038161003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119895=1

120572119894119895(119909119895minus 1199091015840

119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le sup1le119894le119899

119899

sum

119895=1

10038161003816100381610038161003816120572119894119895

10038161003816100381610038161003816

10038161003816100381610038161003816119909119895minus 1199091015840

119895

10038161003816100381610038161003816

le sup1le119895le119899

10038161003816100381610038161003816119909119895minus 1199091015840

119895

10038161003816100381610038161003816sup1le119894le119899

119899

sum

119895=1

10038161003816100381610038161003816120572119894119895

10038161003816100381610038161003816

le 119871 sup1le119895le119899

10038161003816100381610038161003816119909119895minus 1199091015840

119895

10038161003816100381610038161003816

(41)

Since 119901(119909 1199091015840) = sup1le119895le119899

|119909119895minus 1199091015840

119895| we have 119901(119879119909 1198791199091015840) le

119871119901(119909 1199091015840

) 0 le 119871 lt 1 that is 119879 is a contraction mapping of

the complete dualistic partial metric space 119877119899 into itself

Hence by Theorem 10 there exists a unique fixed point 119906 of119879 in 119877119899 that is 119906 is a unique solution of (32)

Theorem 16 If 119891(119905) is a nonlinear integral equation as thefollowing

119891 (119905) = int

119905

0

119890minusV119905 cos (120572119891 (V)) 119889V 0 le 119905 le 1 0 lt 120572 lt 1

(42)

then it has a unique solution

Proof We apply Theorem 10 and we can prove that thisequation has a unique continuous real-valued solution 119891(119905)Let 119883 = 119862[0 1] and the mapping 119879 119883 rarr 119883 defined by119879(119891) = 119891 for 119891 isin 119883 where 119883 is a complete dualistic partialmetric space with sup 119901(119909 119910) is a contraction mapping

cos (120572119886) minus cos (120572119887) = 120572 (119887 minus 119886) sin120573 (43)

where 120573 lies between 120572119886 and 120572119887 Therefore | cos(120572119886) minuscos(120572119887)| le 120572|119887 minus 119886| For functions 119886(119905) and 119887(119905) we get

|cos120572119886 (119905) minus cos120572119887 (119905)| le sup0le119905le1

|119886 (119905) minus 119887 (119905)| = 119901 (119886 119887)

(44)

For 119891 = 119879119891 and 119892 = 119879119892 we have

1003816100381610038161003816119879119891 minus 1198791198921003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816

int

119905

0

119890minusV119905 1003816100381610038161003816cos (120572119891 (V)) minus cos (120572119892 (V))1003816100381610038161003816 119889V

10038161003816100381610038161003816100381610038161003816

le int

119905

0

119890minusV119905 1003816100381610038161003816cos (120572119891 (V)) minus cos (120572119892 (V))1003816100381610038161003816 119889V

le 120572119901 (119891 119892) int

119905

0

119890minusV119905119889V le 120572119901 (119891 119892)

(45)

Taking sup over 0 le 119905 le 1 we get

sup119905

1003816100381610038161003816119879119891 (119905) minus 119879119892 (119905)1003816100381610038161003816 le 120572119901 (119891 119892) (46)

or

119901 (119879 (119891) 119879 (119892)) le 120572119901 (119891 119892) (47)

Theorem 17 Let 1199090be an initial value and the iterative

sequence 119909119899 as the following

119909119899= 119892 (119909

119899minus1) 119899 = 1 2 (48)

If 119892 is continuously differentiable on some interval 119870 = [1199090minus

119903 1199090+ 119903] and satisfies |1198921015840(119909)| le 119871 lt 1 on 119870 as well as

1003816100381610038161003816119892 (1199090) minus 11990901003816100381610038161003816 lt (1 minus 119871) 119903 (49)

then 119909 = 119892(119909) has a unique solution 119906 on 119870 the iterativesequence 119909

119898 converges to that solution and one has the error

estimates1003816100381610038161003816119909 minus 119909119898

1003816100381610038161003816 lt 119871119898

1199031003816100381610038161003816119909 minus 119909119898

1003816100381610038161003816 lt 119871119903 (50)

6 ISRN Computational Mathematics

Proof Suppose that 119901(119909 119892(119909)) = |119909 minus 119892(119909)| for 119909 isin 119870By the mean-value theorem and the given condition 119892(119909)is a contraction mapping of the complete dualistic partialmetric space119870 into itself Hence by Corollary 11 there existsa unique fixed point 119906 of 119892 in119870 that is 119906 is a unique solutionof 119909 = 119892(119909) Also the iteration sequence 119909

119898 converges to 119906

Moreover by 119901(119909 119892(119909)) = |119909 minus 119892(119909)| and Corollary 15 it hasthe prior error estimate

1003816100381610038161003816119909 minus 1199091198981003816100381610038161003816 le 119871119898

119903 (51)

and the posterior estimate1003816100381610038161003816119909 minus 119909119898

1003816100381610038161003816 le 119871119903 (52)

References

[1] S G Matthews ldquoPartial metric topologyrdquo in Proceedings of the11th Summer Conference on General Topology and Applicationsvol 728 pp 183ndash197 The New York Academy of SciencesAugusts 1995

[2] I Altun and A Erduran ldquoFixed point theorems for monotonemappings on partial metric spacesrdquo Fixed Point Theory andApplications vol 2011 Article ID 508730 10 pages 2011

[3] S J OrsquoNeill ldquoPartial metrics valuations and domain theoryrdquoAnnals of the New York Academy of Sciences vol 806 pp 304ndash315 1996

[4] P Fletcher and W F Lindgren Quasi-Uniform Spaces MarcelDekker New York NY USA 1982

[5] H P A Kunzi ldquoHandbook of the history of general topologyrdquoin Non-Symmetric Distances and Their Associated TopologiesAbout the Origins of Basic Ideas in the Area of AsymmetricTopology vol 3 pp 853ndash968 Kluwer Academic 2001

[6] S A M Mohsenalhosseini H Mazaheri M A Dehghan andA Zareh ldquoFixed point for partial metric spacesrdquo ISRN AppliedMathematics vol 2011 Article ID 657868 6 pages 2011

[7] S A M Mohsenalhosseini H Mazaheri and M A DehghanldquoApproximate best proximity pairs in metric spacerdquo Abstractand Applied Analysis vol 2011 Article ID 596971 9 pages 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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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 ISRN Computational Mathematics

Proof The First statement is obvious by

119901 (119909119898 119909119899) le 119901 (119909

119898 119909119898+1

) + 119901 (119909119898+1

119909119898+2

) + sdot sdot sdot

+ 119901 (119909119899minus1

119909119899)

le (119871119898

+ 119871119898+1

+ sdot sdot sdot + 119871119899minus1

) 119901 (1199090 1199091)

(22)

Thus

119901 (119909119898 119909119899) le

119871119898

1 minus 119871119901 (1199090 1199091) (119899 gt 119898) (23)

Now inequality (20) follows from (23) by letting 119899 rarr infinWe have

119901 (119909119898 119906) le

119871119898

1 minus 119871119901 (1199090 1199091) (24)

Now for inequality (21) taking 119898 = 1 and writing 1199100for

1199090and 119910

1for 1199091 we have from (23)

119901 (1199101 119906) le

119871

1 minus 119871119901 (1199100 1199101) (25)

Setting 1199100= 119909119898minus1

we have 1199101= 1198791199100= 119909119898and obtain (21)

Corollary 15 Let (119883 119901) be a complete dualistic partial metricspace and let 119879 119883 rarr 119883 be a contraction on a closed ball119861 = 119861(119909

0 119903) = 119909 119901(119909 119909

0) le 119903 Moreover assume that

119901(1199090 119879x0) lt (1minus119871)119903Then prior error estimate is the following

estimate

119901 (119909119898 119906) le 119871

119898

119903 (26)

and the posterior estimate

119901 (119909119898 119906) le 119871119903 (27)

Proof By Corollary 11 the iteration sequence as (5) is con-verges to the unique fixed point 119906 of119879 hence byCorollary 14we have

119901 (119909119898 119906) le

119871119898

1 minus 119871119901 (1199090 1199091) (28)

since 1199091= 1198791199090and 119901(119909

0 1198791199090) lt (1 minus 119871)119903 we have

119901 (119909119898 119906) le

119871119898

1 minus 119871119901 (1199090 1199091)

=119871119898

1 minus 119871119901 (1199090 1198791199090)

lt119871119898

1 minus 119871sdot (1 minus 119871) 119903

= 119871119898

sdot 119903

(29)

Therefore 119901(119909119898 119906) lt 119871

119898

119903 Also by Corollary 14 we have

119901 (119909119898 119906) le

119871

1 minus 119871119901 (119909119898minus1

119909119898) (30)

since 119909119898minus1

= 119879119898minus1

1199090and 119901(119909

0 1198791199090) lt (1 minus 119871)119903 we have

119901 (119909119898 119906) le

119871

1 minus 119871119901 (119909119898minus1

119909119898)

=119871

1 minus 119871119901 (119879119898minus1

1199090 119879119898

1199090)

=119871

1 minus 119871119901 (119879119898minus1

1199090 119879 (119879

119898minus1

1199090))

lt119871

1 minus 119871(1 minus 119871) 119903

lt 119871119903

(31)

Therefore 119901(119909119898 119906) lt 119871119903

3 Applications of Banach ContractionPrinciple on Complete Dualistic PartialMetric Space

In this section we apply Theorem 10 to prove existence ofthe solutions a system of 119899 linear algebraic equations with 119899unknowns and we show that applied of Corollaries 14 and 15in numerical analysis

31 Application 31 Suppose we want to find the solution of asystem of 119899 linear algebraic equations with 119899 unknowns then

119886111199091+ 119886121199092+ sdot sdot sdot + 119886

1119899119909119899= 1198871

119886211199091+ 119886221199092+ sdot sdot sdot + 119886

2119899119909119899= 1198872

11988611989911199091+ 11988611989921199092+ sdot sdot sdot + 119886

119899119899119909119899= 119887119899

(32)

This system can be written as

1199091= (1 minus 119886

11) 1199091minus 119886121199092minus 119886131199093minus sdot sdot sdot minus 119886

1119899119909119899+ 1198871

1199092= minus119886211199091+ (1 minus 119886

22) 1199092minus 119886231199093minus sdot sdot sdot minus 119886

2119899119909119899+ 1198872

1199093= minus119886311199091minus 119886321199093+ (1 minus 119886

33) 1199093minus sdot sdot sdot minus 119886

3119899119909119899+ 1198873

119909119899= minus11988611989911199091minus 11988611989921199092minus 11988611989931199093minus sdot sdot sdot + (1 minus 119886

119899119899) 119909119899+ 119887119899

(33)

By assuming 120572119894119895= minus119886119894119895+ 120575119894119895 where

120575119894119895=

0 119894 = 119895

1 119894 = 119895(34)

Equation (33) can bewritten in the following equivalent form

119909119894=

119899

sum

119895=1

120572119894119895119909119895+ 119887119894 119894 = 1 2 3 119899 (35)

ISRN Computational Mathematics 5

If 119909 = (1199091 1199092 119909

119899) isin 119877119899 then (35) can be written in the

form 119879119909 = 119909 where 119879 is defined by

119879119909 = 119910 where 119910 = (1199101 1199102 119910

119899) (36)

119910119894=

119899

sum

119895=1

120572119894119895119909119895+ 119887119894 119894 = 1 2 3 119899

119879 119877119899

997888rarr 119877119899

(120572119894119895) is a 119899 times 119899 matrix

(37)

Finding solutions of the system described by (32) or (35)is thus equivalent to finding the fixed point of the operatorequation (36) In order to find a unique solution of 119879 thatis a unique solution of (32) we apply Theorem 10 In factwe prove the following result Equation (32) has a uniquesolution if

119899

sum

119895=1

10038161003816100381610038161003816120572119894119895

10038161003816100381610038161003816=

119899

sum

119895=1

10038161003816100381610038161003816minus119886119894119895+ 120575119894119895

10038161003816100381610038161003816le 119871 lt 1 119894 = 1 119899 (38)

For 119909 = (1199091 119909

119899) and 1199091015840 = (1199091015840

1 1199091015840

2 119909

1015840

119899) we have

119901 (119879119909 1198791199091015840

) = 119901 (119910 1199101015840

) (39)

where

119910 = (1199101 1199102 119910

119899) isin 119877119899

1199101015840

= (1199101015840

1 1199101015840

2 119910

1015840

119899) isin 119877119899

119910119894=

119899

sum

119895=1

120572119894119895119909119895+ 119887119894

1199101015840

119894=

119899

sum

119895=1

1205721198941198951199091015840

119895+ 119887119894 119894 = 1 2 119899

(40)

If 119910 = (1199101 1199102 119910

119899) isin 119877

119899 then 119901(119910 119910) = sup1le119894le119899

|119910119894|

Therefore

119901 (119879119909 1198791199091015840

) = sup1le119894le119899

10038161003816100381610038161003816119910119894minus 1199101015840

119894

10038161003816100381610038161003816

= sup1le119894le119899

10038161003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119895=1

120572119894119895119909119895+ 119887119894minus

119899

sum

119895=1

1205721198941198951199091015840

119895minus 119887119894

10038161003816100381610038161003816100381610038161003816100381610038161003816

= sup1le119894le119899

10038161003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119895=1

120572119894119895(119909119895minus 1199091015840

119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le sup1le119894le119899

119899

sum

119895=1

10038161003816100381610038161003816120572119894119895

10038161003816100381610038161003816

10038161003816100381610038161003816119909119895minus 1199091015840

119895

10038161003816100381610038161003816

le sup1le119895le119899

10038161003816100381610038161003816119909119895minus 1199091015840

119895

10038161003816100381610038161003816sup1le119894le119899

119899

sum

119895=1

10038161003816100381610038161003816120572119894119895

10038161003816100381610038161003816

le 119871 sup1le119895le119899

10038161003816100381610038161003816119909119895minus 1199091015840

119895

10038161003816100381610038161003816

(41)

Since 119901(119909 1199091015840) = sup1le119895le119899

|119909119895minus 1199091015840

119895| we have 119901(119879119909 1198791199091015840) le

119871119901(119909 1199091015840

) 0 le 119871 lt 1 that is 119879 is a contraction mapping of

the complete dualistic partial metric space 119877119899 into itself

Hence by Theorem 10 there exists a unique fixed point 119906 of119879 in 119877119899 that is 119906 is a unique solution of (32)

Theorem 16 If 119891(119905) is a nonlinear integral equation as thefollowing

119891 (119905) = int

119905

0

119890minusV119905 cos (120572119891 (V)) 119889V 0 le 119905 le 1 0 lt 120572 lt 1

(42)

then it has a unique solution

Proof We apply Theorem 10 and we can prove that thisequation has a unique continuous real-valued solution 119891(119905)Let 119883 = 119862[0 1] and the mapping 119879 119883 rarr 119883 defined by119879(119891) = 119891 for 119891 isin 119883 where 119883 is a complete dualistic partialmetric space with sup 119901(119909 119910) is a contraction mapping

cos (120572119886) minus cos (120572119887) = 120572 (119887 minus 119886) sin120573 (43)

where 120573 lies between 120572119886 and 120572119887 Therefore | cos(120572119886) minuscos(120572119887)| le 120572|119887 minus 119886| For functions 119886(119905) and 119887(119905) we get

|cos120572119886 (119905) minus cos120572119887 (119905)| le sup0le119905le1

|119886 (119905) minus 119887 (119905)| = 119901 (119886 119887)

(44)

For 119891 = 119879119891 and 119892 = 119879119892 we have

1003816100381610038161003816119879119891 minus 1198791198921003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816

int

119905

0

119890minusV119905 1003816100381610038161003816cos (120572119891 (V)) minus cos (120572119892 (V))1003816100381610038161003816 119889V

10038161003816100381610038161003816100381610038161003816

le int

119905

0

119890minusV119905 1003816100381610038161003816cos (120572119891 (V)) minus cos (120572119892 (V))1003816100381610038161003816 119889V

le 120572119901 (119891 119892) int

119905

0

119890minusV119905119889V le 120572119901 (119891 119892)

(45)

Taking sup over 0 le 119905 le 1 we get

sup119905

1003816100381610038161003816119879119891 (119905) minus 119879119892 (119905)1003816100381610038161003816 le 120572119901 (119891 119892) (46)

or

119901 (119879 (119891) 119879 (119892)) le 120572119901 (119891 119892) (47)

Theorem 17 Let 1199090be an initial value and the iterative

sequence 119909119899 as the following

119909119899= 119892 (119909

119899minus1) 119899 = 1 2 (48)

If 119892 is continuously differentiable on some interval 119870 = [1199090minus

119903 1199090+ 119903] and satisfies |1198921015840(119909)| le 119871 lt 1 on 119870 as well as

1003816100381610038161003816119892 (1199090) minus 11990901003816100381610038161003816 lt (1 minus 119871) 119903 (49)

then 119909 = 119892(119909) has a unique solution 119906 on 119870 the iterativesequence 119909

119898 converges to that solution and one has the error

estimates1003816100381610038161003816119909 minus 119909119898

1003816100381610038161003816 lt 119871119898

1199031003816100381610038161003816119909 minus 119909119898

1003816100381610038161003816 lt 119871119903 (50)

6 ISRN Computational Mathematics

Proof Suppose that 119901(119909 119892(119909)) = |119909 minus 119892(119909)| for 119909 isin 119870By the mean-value theorem and the given condition 119892(119909)is a contraction mapping of the complete dualistic partialmetric space119870 into itself Hence by Corollary 11 there existsa unique fixed point 119906 of 119892 in119870 that is 119906 is a unique solutionof 119909 = 119892(119909) Also the iteration sequence 119909

119898 converges to 119906

Moreover by 119901(119909 119892(119909)) = |119909 minus 119892(119909)| and Corollary 15 it hasthe prior error estimate

1003816100381610038161003816119909 minus 1199091198981003816100381610038161003816 le 119871119898

119903 (51)

and the posterior estimate1003816100381610038161003816119909 minus 119909119898

1003816100381610038161003816 le 119871119903 (52)

References

[1] S G Matthews ldquoPartial metric topologyrdquo in Proceedings of the11th Summer Conference on General Topology and Applicationsvol 728 pp 183ndash197 The New York Academy of SciencesAugusts 1995

[2] I Altun and A Erduran ldquoFixed point theorems for monotonemappings on partial metric spacesrdquo Fixed Point Theory andApplications vol 2011 Article ID 508730 10 pages 2011

[3] S J OrsquoNeill ldquoPartial metrics valuations and domain theoryrdquoAnnals of the New York Academy of Sciences vol 806 pp 304ndash315 1996

[4] P Fletcher and W F Lindgren Quasi-Uniform Spaces MarcelDekker New York NY USA 1982

[5] H P A Kunzi ldquoHandbook of the history of general topologyrdquoin Non-Symmetric Distances and Their Associated TopologiesAbout the Origins of Basic Ideas in the Area of AsymmetricTopology vol 3 pp 853ndash968 Kluwer Academic 2001

[6] S A M Mohsenalhosseini H Mazaheri M A Dehghan andA Zareh ldquoFixed point for partial metric spacesrdquo ISRN AppliedMathematics vol 2011 Article ID 657868 6 pages 2011

[7] S A M Mohsenalhosseini H Mazaheri and M A DehghanldquoApproximate best proximity pairs in metric spacerdquo Abstractand Applied Analysis vol 2011 Article ID 596971 9 pages 2011

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

ISRN Computational Mathematics 5

If 119909 = (1199091 1199092 119909

119899) isin 119877119899 then (35) can be written in the

form 119879119909 = 119909 where 119879 is defined by

119879119909 = 119910 where 119910 = (1199101 1199102 119910

119899) (36)

119910119894=

119899

sum

119895=1

120572119894119895119909119895+ 119887119894 119894 = 1 2 3 119899

119879 119877119899

997888rarr 119877119899

(120572119894119895) is a 119899 times 119899 matrix

(37)

Finding solutions of the system described by (32) or (35)is thus equivalent to finding the fixed point of the operatorequation (36) In order to find a unique solution of 119879 thatis a unique solution of (32) we apply Theorem 10 In factwe prove the following result Equation (32) has a uniquesolution if

119899

sum

119895=1

10038161003816100381610038161003816120572119894119895

10038161003816100381610038161003816=

119899

sum

119895=1

10038161003816100381610038161003816minus119886119894119895+ 120575119894119895

10038161003816100381610038161003816le 119871 lt 1 119894 = 1 119899 (38)

For 119909 = (1199091 119909

119899) and 1199091015840 = (1199091015840

1 1199091015840

2 119909

1015840

119899) we have

119901 (119879119909 1198791199091015840

) = 119901 (119910 1199101015840

) (39)

where

119910 = (1199101 1199102 119910

119899) isin 119877119899

1199101015840

= (1199101015840

1 1199101015840

2 119910

1015840

119899) isin 119877119899

119910119894=

119899

sum

119895=1

120572119894119895119909119895+ 119887119894

1199101015840

119894=

119899

sum

119895=1

1205721198941198951199091015840

119895+ 119887119894 119894 = 1 2 119899

(40)

If 119910 = (1199101 1199102 119910

119899) isin 119877

119899 then 119901(119910 119910) = sup1le119894le119899

|119910119894|

Therefore

119901 (119879119909 1198791199091015840

) = sup1le119894le119899

10038161003816100381610038161003816119910119894minus 1199101015840

119894

10038161003816100381610038161003816

= sup1le119894le119899

10038161003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119895=1

120572119894119895119909119895+ 119887119894minus

119899

sum

119895=1

1205721198941198951199091015840

119895minus 119887119894

10038161003816100381610038161003816100381610038161003816100381610038161003816

= sup1le119894le119899

10038161003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119895=1

120572119894119895(119909119895minus 1199091015840

119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le sup1le119894le119899

119899

sum

119895=1

10038161003816100381610038161003816120572119894119895

10038161003816100381610038161003816

10038161003816100381610038161003816119909119895minus 1199091015840

119895

10038161003816100381610038161003816

le sup1le119895le119899

10038161003816100381610038161003816119909119895minus 1199091015840

119895

10038161003816100381610038161003816sup1le119894le119899

119899

sum

119895=1

10038161003816100381610038161003816120572119894119895

10038161003816100381610038161003816

le 119871 sup1le119895le119899

10038161003816100381610038161003816119909119895minus 1199091015840

119895

10038161003816100381610038161003816

(41)

Since 119901(119909 1199091015840) = sup1le119895le119899

|119909119895minus 1199091015840

119895| we have 119901(119879119909 1198791199091015840) le

119871119901(119909 1199091015840

) 0 le 119871 lt 1 that is 119879 is a contraction mapping of

the complete dualistic partial metric space 119877119899 into itself

Hence by Theorem 10 there exists a unique fixed point 119906 of119879 in 119877119899 that is 119906 is a unique solution of (32)

Theorem 16 If 119891(119905) is a nonlinear integral equation as thefollowing

119891 (119905) = int

119905

0

119890minusV119905 cos (120572119891 (V)) 119889V 0 le 119905 le 1 0 lt 120572 lt 1

(42)

then it has a unique solution

Proof We apply Theorem 10 and we can prove that thisequation has a unique continuous real-valued solution 119891(119905)Let 119883 = 119862[0 1] and the mapping 119879 119883 rarr 119883 defined by119879(119891) = 119891 for 119891 isin 119883 where 119883 is a complete dualistic partialmetric space with sup 119901(119909 119910) is a contraction mapping

cos (120572119886) minus cos (120572119887) = 120572 (119887 minus 119886) sin120573 (43)

where 120573 lies between 120572119886 and 120572119887 Therefore | cos(120572119886) minuscos(120572119887)| le 120572|119887 minus 119886| For functions 119886(119905) and 119887(119905) we get

|cos120572119886 (119905) minus cos120572119887 (119905)| le sup0le119905le1

|119886 (119905) minus 119887 (119905)| = 119901 (119886 119887)

(44)

For 119891 = 119879119891 and 119892 = 119879119892 we have

1003816100381610038161003816119879119891 minus 1198791198921003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816

int

119905

0

119890minusV119905 1003816100381610038161003816cos (120572119891 (V)) minus cos (120572119892 (V))1003816100381610038161003816 119889V

10038161003816100381610038161003816100381610038161003816

le int

119905

0

119890minusV119905 1003816100381610038161003816cos (120572119891 (V)) minus cos (120572119892 (V))1003816100381610038161003816 119889V

le 120572119901 (119891 119892) int

119905

0

119890minusV119905119889V le 120572119901 (119891 119892)

(45)

Taking sup over 0 le 119905 le 1 we get

sup119905

1003816100381610038161003816119879119891 (119905) minus 119879119892 (119905)1003816100381610038161003816 le 120572119901 (119891 119892) (46)

or

119901 (119879 (119891) 119879 (119892)) le 120572119901 (119891 119892) (47)

Theorem 17 Let 1199090be an initial value and the iterative

sequence 119909119899 as the following

119909119899= 119892 (119909

119899minus1) 119899 = 1 2 (48)

If 119892 is continuously differentiable on some interval 119870 = [1199090minus

119903 1199090+ 119903] and satisfies |1198921015840(119909)| le 119871 lt 1 on 119870 as well as

1003816100381610038161003816119892 (1199090) minus 11990901003816100381610038161003816 lt (1 minus 119871) 119903 (49)

then 119909 = 119892(119909) has a unique solution 119906 on 119870 the iterativesequence 119909

119898 converges to that solution and one has the error

estimates1003816100381610038161003816119909 minus 119909119898

1003816100381610038161003816 lt 119871119898

1199031003816100381610038161003816119909 minus 119909119898

1003816100381610038161003816 lt 119871119903 (50)

6 ISRN Computational Mathematics

Proof Suppose that 119901(119909 119892(119909)) = |119909 minus 119892(119909)| for 119909 isin 119870By the mean-value theorem and the given condition 119892(119909)is a contraction mapping of the complete dualistic partialmetric space119870 into itself Hence by Corollary 11 there existsa unique fixed point 119906 of 119892 in119870 that is 119906 is a unique solutionof 119909 = 119892(119909) Also the iteration sequence 119909

119898 converges to 119906

Moreover by 119901(119909 119892(119909)) = |119909 minus 119892(119909)| and Corollary 15 it hasthe prior error estimate

1003816100381610038161003816119909 minus 1199091198981003816100381610038161003816 le 119871119898

119903 (51)

and the posterior estimate1003816100381610038161003816119909 minus 119909119898

1003816100381610038161003816 le 119871119903 (52)

References

[1] S G Matthews ldquoPartial metric topologyrdquo in Proceedings of the11th Summer Conference on General Topology and Applicationsvol 728 pp 183ndash197 The New York Academy of SciencesAugusts 1995

[2] I Altun and A Erduran ldquoFixed point theorems for monotonemappings on partial metric spacesrdquo Fixed Point Theory andApplications vol 2011 Article ID 508730 10 pages 2011

[3] S J OrsquoNeill ldquoPartial metrics valuations and domain theoryrdquoAnnals of the New York Academy of Sciences vol 806 pp 304ndash315 1996

[4] P Fletcher and W F Lindgren Quasi-Uniform Spaces MarcelDekker New York NY USA 1982

[5] H P A Kunzi ldquoHandbook of the history of general topologyrdquoin Non-Symmetric Distances and Their Associated TopologiesAbout the Origins of Basic Ideas in the Area of AsymmetricTopology vol 3 pp 853ndash968 Kluwer Academic 2001

[6] S A M Mohsenalhosseini H Mazaheri M A Dehghan andA Zareh ldquoFixed point for partial metric spacesrdquo ISRN AppliedMathematics vol 2011 Article ID 657868 6 pages 2011

[7] S A M Mohsenalhosseini H Mazaheri and M A DehghanldquoApproximate best proximity pairs in metric spacerdquo Abstractand Applied Analysis vol 2011 Article ID 596971 9 pages 2011

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 ISRN Computational Mathematics

Proof Suppose that 119901(119909 119892(119909)) = |119909 minus 119892(119909)| for 119909 isin 119870By the mean-value theorem and the given condition 119892(119909)is a contraction mapping of the complete dualistic partialmetric space119870 into itself Hence by Corollary 11 there existsa unique fixed point 119906 of 119892 in119870 that is 119906 is a unique solutionof 119909 = 119892(119909) Also the iteration sequence 119909

119898 converges to 119906

Moreover by 119901(119909 119892(119909)) = |119909 minus 119892(119909)| and Corollary 15 it hasthe prior error estimate

1003816100381610038161003816119909 minus 1199091198981003816100381610038161003816 le 119871119898

119903 (51)

and the posterior estimate1003816100381610038161003816119909 minus 119909119898

1003816100381610038161003816 le 119871119903 (52)

References

[1] S G Matthews ldquoPartial metric topologyrdquo in Proceedings of the11th Summer Conference on General Topology and Applicationsvol 728 pp 183ndash197 The New York Academy of SciencesAugusts 1995

[2] I Altun and A Erduran ldquoFixed point theorems for monotonemappings on partial metric spacesrdquo Fixed Point Theory andApplications vol 2011 Article ID 508730 10 pages 2011

[3] S J OrsquoNeill ldquoPartial metrics valuations and domain theoryrdquoAnnals of the New York Academy of Sciences vol 806 pp 304ndash315 1996

[4] P Fletcher and W F Lindgren Quasi-Uniform Spaces MarcelDekker New York NY USA 1982

[5] H P A Kunzi ldquoHandbook of the history of general topologyrdquoin Non-Symmetric Distances and Their Associated TopologiesAbout the Origins of Basic Ideas in the Area of AsymmetricTopology vol 3 pp 853ndash968 Kluwer Academic 2001

[6] S A M Mohsenalhosseini H Mazaheri M A Dehghan andA Zareh ldquoFixed point for partial metric spacesrdquo ISRN AppliedMathematics vol 2011 Article ID 657868 6 pages 2011

[7] S A M Mohsenalhosseini H Mazaheri and M A DehghanldquoApproximate best proximity pairs in metric spacerdquo Abstractand Applied Analysis vol 2011 Article ID 596971 9 pages 2011

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