Applied Mathematics and Computation 216 (2010) 80–86

Contents lists available at ScienceDirect

Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

Common fixed points for four maps in cone metric spaces

Mujahid Abbas a, B.E. Rhoades b,*, Talat Nazir a

a Department of Mathematics, Lahore University of Management Sciences, 54792 Lahore, Pakistanb Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, United States

a r t i c l e i n f o

Keywords:Common fixed pointCoincidence pointCone metric spacesNon-normal cone

0096-3003/$ - see front matter � 2010 Elsevier Incdoi:10.1016/j.amc.2010.01.003

* Corresponding author.E-mail addresses: mujahid@lum.edu.pk (M. Abba

a b s t r a c t

The existence of coincidence points and common fixed points for four mappings satisfyinggeneralized contractive conditions without exploiting the notion of continuity of any mapinvolved therein, in a cone metric space is proved. These results extend, unify and gener-alize several well known comparable results in the existing literature.

� 2010 Elsevier Inc. All rights reserved.

1. Introduction and preliminaries

Jungck [9] defined a pair of self-mappings to be weakly compatible if they commute at their coincidence points. In recentyears, several authors have obtained coincidence point results for various classes of mappings on a metric space, utilizingthese concepts. For a survey of coincidence point theory, its applications, comparison of different contractive conditionsand related results, we refer to [4,6,11] and the references contained therein. Huang and Zhang [5] generalized the conceptof a metric space, replacing the set of real numbers by an ordered Banach space and obtained some fixed point theorems formappings satisfying different contractive conditions. Subsequently, Abbas and Jungck [2] and Abbas and Rhoades [1] studiedcommon fixed point theorems in cone metric spaces ( see also, [3,5,8,13] and the references mentioned therein). In this pa-per, common fixed point theorems for two pairs of weakly compatible maps, which are more general than R-weakly com-muting and compatible mappings, are obtained in the setting of cone metric spaces, without exploiting the notion ofcontinuity. It is worth mentioning that our results do not require the assumption that the cone is normal. Our results extendand unify various comparable results in the literature [2–4,7,8]. Consistent with Huang and Zhang [5], the following defini-tions and results will be needed in the sequel.

Let E be a real Banach space. A subset P of E is called a cone if and only if:

(a) P is closed, non-empty and P–f0g;(b) a; b 2 R; a; b P 0; x; y 2 P imply that axþ by 2 P;(c) P \ ð�PÞ ¼ f0g.

Given a cone P � E; we define a partial ordering 6 with respect to P by x 6 y if and only if y� x 2 P; where x� y meansthat y� x 2 int P (interior of P). A cone P is said to be normal if there is a number K > 0 such that for all x; y 2 E,

0 6 x 6 y implies kxk 6 Kkyk:

The least positive number satisfying the above inequality is called the normal constant of P.Rezapour and Hamlbarani [12] proved that there is no normal cone with normal constant K < 1 and for each k > 1 there

are cones with normal constants K > k.

. All rights reserved.

s), rhoades@indiana.edu (B.E. Rhoades), talat@lums.edu.pk (T. Nazir).

M. Abbas et al. / Applied Mathematics and Computation 216 (2010) 80–86 81

Definition 1.1. Let X be a non-empty set. Suppose that the mapping d : X � X ! E satisfies:

(d1) 0 6 dðx; yÞ for all x; y 2 X and dðx; yÞ ¼ 0 if and only if x ¼ y;(d2) dðx; yÞ ¼ dðy; xÞ for all x; y 2 X;(d3) dðx; yÞ 6 dðx; zÞ þ dðz; yÞ for all x; y; z 2 X.

Then d is called a cone metric on X and ðX; dÞ is called a cone metric space. The concept of a cone metric space is moregeneral than that of a metric space.

Definition 1.2. Let ðX; dÞ be a cone metric space, fxng a sequence in X and x 2 X: For every c 2 E with 0� c; we say that fxngis

(i) a Cauchy sequence if there is an N such that, for all n;m > N, dðxn; xmÞ � c:(ii) a convergent sequence if there is an N such that, for all n > N; dðxn; xÞ � c for some x in X.

A cone metric space X is said to be complete if every Cauchy sequence in X is convergent in X. It is known that fxngconverges to x 2 X if and only if dðxn; xÞ ! 0 as n!1. A subset A of X is closed if every Cauchy sequence in A has its limitpoint in A.

Definition 1.3. Let f and g be self-maps on a set X . If w ¼ fx ¼ gx, for some x in X, then x is called coincidence point of f and g,where w is called a point of coincidence of f and g.

Definition 1.4. Let f and g be two self-maps defined on a set X. Then f and g are said to be weakly compatible if they com-mute at every coincidence point.

Remark 1.5. If E is a real Banach space with a cone Pand if a 6 ha where a 2 P and h 2 ð0;1Þ, then a ¼ 0.

Remark 1.6. If 0 6 u� c for each 0� c then u ¼ 0.

2. Common fixed point results

The following Lemma not only improves but also extends Lemma 1 of [10] cone metric spaces.

Lemma 2.1. Let f, g, S and T be self-maps on a cone metric space X with cone P having non-empty interior, satisfying f ðXÞ � TðXÞand gðXÞ � SðXÞ. Define fxng and fyng by y2nþ1 ¼ fx2n ¼ Tx2nþ1; y2nþ2 ¼ gx2nþ1 ¼ Sx2nþ2; n P 0. Suppose that there exist ak 2 ½0;1Þ such that

dðyn; ynþ1Þ 6 kdðyn�1; ynÞ for each n P 1: ð2:1Þ

Then either

(a) ff ; Sg and fg; Tg have coincidence points, and fyng converges, or(b) fyng is Cauchy.

Moreover, if X is complete, then fyng converges to a point z 2 X and

dðyn; zÞ 6kn

1� kdðy0; y1Þ for each n > 0: ð2:2Þ

Proof. To prove part (a), suppose that there exists an n such that y2n ¼ y2nþ1. Then, from the definition offyng; gx2n�1 ¼ Sx2n ¼ fx2n ¼ Tx2nþ1, and f and S have a coincidence point. Moreover, from (2.1),

dðy2nþ1; y2nþ2Þ 6 kdðy2n; y2nþ1Þ ¼ 0;

so that y2nþ1 ¼ y2nþ2; i.e., fx2n ¼ Tx2nþ1 ¼ gx2nþ1 ¼ Sx2nþ2, and g and T have a coincidence point. In addition, repeated use of(2.1) yields y2n ¼ ym for each m > 2n, and hence fyng converges.

The same conclusion holds if y2nþ1 ¼ y2nþ2 for some n.

For part (b), assume that y2n–y2nþ1 for all n. Then (2.1) implies that

dðyn; ynþ1Þ 6 kndðy0; y1Þ:

For any m;n 2 N with m > n it follows that

82 M. Abbas et al. / Applied Mathematics and Computation 216 (2010) 80–86

dðyn; ymÞ 6Xm�1

i¼n

dðyi; yiþ1Þ 6Xm�1

i¼n

kidðy0; y1Þ ¼ kndðy0; y1ÞXm�n�1

j¼0kj6

kn

1� kdðy0; y1Þ; ð2:3Þ

Let 0� c be given. Choose d > 0 such that c þ Ndð0Þ# P, where Ndð0Þ ¼ fy 2 E : y < dg. Also, choose N1 2 Nsuch thatkn

1�k dðy0; y1Þ 2 Ndð0Þ, for all n P N1. Then kn

1�k dðy0; y1Þ � c, for all n > N1. Thus, for all m;n P N, dðyn; ymÞ � c, and fyng isCauchy.If X is complete, there exists a point z 2 X such that fymg converges to z as m!1. Choose N2 2 N such that dðym; zÞ 6 c for allm > N2. Thus

dðyn; zÞ 6 dðyn; ymÞ þ dðym; zÞ6

kn

1�k dðy0; y1Þ þ dðym; zÞ6

kn

1�k dðy0; y1Þ þ c;

yields (2.2) by using Remark 1.5(c).The following theorem extends and improves Theorem 2.1. of [8] h

Theorem 2.2. Let f, g, S and T be self-maps of a cone metric space X with cone P having non-empty interior, satisfyingf ðXÞ � TðXÞ; gðXÞ � SðXÞ and

dðfx; gyÞ 6 hux;yðf ; g; S; TÞ; ð2:4Þ

where h 2 ð0;1Þ and

ux;yðf ; g; S; TÞ 2 dðSx; TyÞ;dðfx; SxÞ;dðgy; TyÞ;dðfx; TyÞ þ dðgy; SxÞ2

� �

for all x; y 2 X. If one of f ðXÞ; gðXÞ; SðXÞ, or TðXÞ is a complete subspace of X, then ff ; Sg and fg; Tg have a unique point ofcoincidence in X. Moreover if ff ; Sg and fg; Tg are weakly compatible, then f, g, S and T have a unique common fixed point.

Proof. For any arbitrary point x0 in X, construct sequences fxng and fyng in X such that

fx2n�2 ¼ Tx2n�1 ¼ y2n�1; and gx2n�1 ¼ Sx2n ¼ y2n:

We have from (2.4),

dðy2nþ1; y2nþ2Þ ¼ dðfx2n; gx2nþ1Þ 6 hux2n ;x2nþ1 ðf ; g; S; TÞ

for n ¼ 1;2;3; . . ., where

ux2n ;x2nþ1 ðf ; g; S; TÞ 2 dðSx2n; Tx2nþ1Þ;dðfx2n; Sx2nÞ;dðgx2nþ1; Tx2nþ1Þ;½dðfx2n; Tx2nþ1Þ þ dðgx2nþ1; Sx2nÞ�

2

� �

¼ dðy2n; y2nþ1Þ; dðy2nþ1; y2nÞ; dðy2nþ2; y2nþ1Þ;½dðy2nþ1; y2nþ1Þ þ dðy2nþ2; y2nÞ�

2

� �

¼ dðy2n; y2nþ1Þ; dðy2nþ1; y2nþ2Þ;½dðy2n; y2nþ1Þ þ dðy2nþ1; y2nþ2Þ�

2

� �:

Now if ux2n ;x2nþ1 ðf ; g; S; TÞ ¼ dðy2n; y2nþ1Þ, then dðy2nþ1; y2nþ2Þ 6 hdðy2n; y2nþ1Þ. And if ux2n ;x2nþ1 ðf ; g; S; TÞ ¼ dðy2nþ1; y2nþ2Þ, thendðy2nþ1; y2nþ2Þ 6 hdðy2nþ1; y2nþ2Þ which implies that dðy2nþ1; y2nþ2Þ ¼ 0, and y2nþ1 ¼ y2nþ2. For

ux2n ;x2nþ1 ðf ; g; S; TÞ ¼½dðy2n ;y2nþ1Þþdðy2nþ1 ;y2nþ2Þ�

2 we obtain

dðy2nþ1; y2nþ2Þ 6h2½dðy2n; y2nþ1Þ þ dðy2nþ1; y2nþ2Þ� 6

h2

dðy2n; y2nþ1Þ þ12

dðy2nþ1; y2nþ2Þ

which implies that,

dðy2nþ1; y2nþ2Þ 6 hdðy2n; y2nþ1Þ:

Hence condition (2.1) of Lemma 2.1 is satisfied. Now we show that ff ; Sg and fg; Tg have coincidence points in X. Withoutloss of generality we may assume that yn–ynþ1 for any n, for if we have equality for some n, then (a) of Lemma 2.1 applies.Now from Lemma 2.1 fyng is a Cauchy sequence. Suppose that SðXÞ is complete. Then there exists a u in SðXÞ, such thatSx2n ¼ y2n ! u as n!1. Consequently, we can find a v in X such that Sv ¼ u. We claim that fv ¼ u. For this, consider

dðf v ;uÞ 6 dðfv ; gx2n�1Þ þ dðgx2n�1; uÞ 6 huv;x2n�1 ðf ; g; S; TÞ þ dðgx2n�1; uÞ;

where

uv;x2n�1 ðf ; g; S; TÞ 2 dðSv ; Tx2n�1Þ;dðf v; SvÞ;dðgx2n�1; Tx2n�1Þ;dðf v; Tx2n�1Þ þ dðgx2n�1; SvÞ

2

� �

M. Abbas et al. / Applied Mathematics and Computation 216 (2010) 80–86 83

for all n 2 N. Then at least one of these four elements from the set uv ;x2n�1 ðf ; g; S; TÞ occurs infinitely often. There are fourpossibilities:

uv ;x2nk�1 ðf ; g; S; TÞ ¼ dðSv ; Tx2nk�1Þ; k 2 N so that dðf v;uÞ 6 hdðSv ; Tx2nk�1Þ þ dðgx2nk�1; uÞ � c as k!1,uv ;x2nk�1 ðf ; g; S; TÞ ¼ dðfv ; SvÞ which implies that dðf v ;uÞ 6 hdðfv ; SvÞ þ dðgx2nk�1;uÞ � c as k!1 ,uv ;x2nk�1 ðf ; g; S; TÞ ¼ dðgx2nk�1; Tx2nk�1Þ and

dðfv ;uÞ 6 hdðgx2nk�1; Tx2nk�1Þ þ dðgx2nk�1;uÞ � c

as k!1,or uv ;x2nk�1 ðf ; g; S; TÞ ¼

dðfv ;Tx2nk�1Þþdðgx2nk�1 ;svÞ2 , which implies that

dðfv ;uÞ 6 hdðf v; Tx2nk�1Þ þ dðgx2nk�1; svÞ

2þ dðgx2nk�1;uÞ 6

h2

dðfv ; Tx2nk�1Þ þh2

dðgx2nk�1; svÞ þ dðgx2nk�1;uÞ

� c as k!1:

So in all cases we have dðfv ;uÞ � c. Thus dðu; f vÞ � cm for all m P 1, and c

m� dðu; fvÞ 2 P, for all m P 1. Since cm! 0 as

m!1 and P is closed, �dðu; f vÞ 2 P. But also dðu; f vÞ 2 P, so that dðu; f vÞ ¼ 0; f v ¼ Sv ¼ u.Since u 2 f ðXÞ � TðXÞ, there exists a w 2 X such that Tw ¼ u. Now we shall show that gw ¼ u. Consider

dðgw; uÞ 6 dðgw; fx2nÞ þ dðfx2n;uÞ ¼ dðfx2n; gwÞ þ dðfx2n;uÞ 6 hux2n ;wðf ; g; S; TÞ þ dðfx2n;uÞ;

where

ux2n ;wðf ; g; S; TÞ 2 dðSx2n; TwÞ;dðfx2n; Sx2nÞ;dðgw; TwÞ; dðfx2n; TwÞ þ dðgw; Sx2nÞ2

� �

for all n 2 N.At least one of these four elements from the set ux2n ;wðf ; g; S; TÞ occurs infinitely often. Thus there exists a subsequence

fnkg of fng such that one of the following is true:ux2nk

;wðf ; g; S; TÞ ¼ dðSx2nk; TwÞ; k 2 N, so that

dðgw; uÞ 6 hdðSx2nk; TwÞ þ dðfx2nk

; uÞ � c as k!1;

ux2nk;wðf ; g; S; TÞ ¼ dðfx2nk

; Sx2nkÞ, which implies that

dðgw; uÞ 6 hdðfx2nk; Sx2nk

Þ þ dðfx2nk;uÞ � c as k!1;

ux2nk;wðf ; g; S; TÞ ¼ dðgw; TwÞ, and

dðgw; uÞ 6 hdðgw; TwÞ þ dðfx2nk;uÞ ¼ hdðgw;uÞ þ dðfx2nk

;uÞ � c as k!1;

or ux2nk;wðf ; g; S; TÞ ¼

dðfx2nk;TwÞþdðgw;Sx2nk

Þ2 , which implies that

dðgw; uÞ 6 hdðfx2nk

; TwÞ þ dðgw; Sx2nkÞ

2þ dðfx2nk

;uÞ 6 h2

dðfx2nk; uÞ þ 1

2dðgw; Sx2nk

Þ þ dðfx2nk;uÞ � c as k!1:

Thus dðgw;uÞ � c. Following similar arguments to those given above we obtain gw ¼ Tw ¼ u. Thus ff ; Sg and fg; Tg have acommon point of coincidence in X. Now, if ff ; Sg and fg; Tg are weakly compatible, fu ¼ fSv ¼ Sfv ¼ Su ¼ w1 (say) andgu ¼ gTw ¼ Tgw ¼ Tu ¼ w2 (say). Now

dðw1;w2Þ ¼ dðfu; guÞ 6 huu;uðf ; g; S; TÞ;

where

uu;uðf ; g; S; TÞ 2 dðSu; TuÞ;dðfu; SuÞ; dðgu; TuÞ; dðfu; TuÞ þ dðgu; SuÞ2

� �¼ fdðw1;w2Þg:

Therefore dðw1;w2Þ 6 hdðw1;w2Þ which implies that w1 ¼ w2; and hence

fu ¼ gu ¼ Su ¼ Tu:

Now we shall show that u ¼ gu.

dðu; guÞ ¼ dðf v; guÞ 6 huv;uðf ; g; S; TÞ;

where

uv;uðf ; g; S; TÞ 2 dðSv ; TuÞ; dðfv ; SvÞ;dðgu; TuÞ;dðfv ; TuÞ þ dðgu; SvÞ2

� �¼ fdðu; guÞg:

Thus dðu; guÞ 6 hdðu; guÞ, which implies that gu ¼ u, and u is a common fixed point of f, g, S and T.

84 M. Abbas et al. / Applied Mathematics and Computation 216 (2010) 80–86

For uniqueness, suppose that u� is also a fixed point of f, g, S, and T. From (2.4),

dðu;u�Þ ¼ dðfu; gu�Þ 6 huu;u� ðf ; g; S; TÞ;

where

uu;u� ðf ; g; S; TÞ 2 dðSu; Tu�Þ; dðfu; SuÞ; dðgu�; Tu�Þ; dðfu; Tu�Þ þ dðgu�; SuÞ2

� �¼ fdðu;u�Þg;

which is possible only if u ¼ u�. The proofs for the cases in which gðXÞ, SðXÞ or TðXÞ is complete are similar, and are thereforeomitted. h

Corollary 2.3. Let f, g, S and T be self-maps on a cone metric space X with cone P having non-empty interior, satisfyingf ðXÞ � TðXÞ; gðXÞ � SðXÞ and for some m; n 2 N,

dðf mx; gnyÞ 6 hux;yðf m; gn; Sm; TnÞ; ð2:5Þ

where h 2 ð0;1Þ and

ux;yðf m; gn; Sm; TnÞ 2 dðSmx; TnyÞ;dðf mx; SmxÞ; dðgny; TnyÞ;dðfmx; TnyÞ þ dðgny; SmxÞ

2

� �

for all x; y 2 X. If one of f ðXÞ; gðXÞ; SðXÞ, or TðXÞ is complete subspace of X, then ff ; Sg and fg; Tg have a unique point of coin-cidence in X. Moreover if ff ; Sg and fg; Tg are weakly compatible, then f, g, S and T have a unique common fixed point.

Proof. It follows from Theorem 2.2, that ff m; Smg and fgm; Tng have a unique common fixed point p. Nowf ðpÞ ¼ f ðf mðpÞÞ ¼ f mþ1ðpÞ ¼ f mðf ðpÞÞ, and SðpÞ ¼ SðSmðpÞÞ ¼ Smþ1ðpÞ ¼ SmðSðpÞÞ implies that f ðpÞ and SðpÞ are also fixed pointsfor f m and Sm. Hence f ðpÞ ¼ SðpÞ ¼ p. By using the same argument, we obtain gðpÞ ¼ TðpÞ ¼ p. h

The following corollary extends well known Fisher’s result [6] to cone metric spaces.

Corollary 2.4. Let f, g, S and T be self-maps on a cone metric space X with cone P having non-empty interior, satisfyingf ðXÞ � TðXÞ; gðXÞ � SðXÞ

dðfx; gyÞ 6 hdðSx; TyÞ

for all x; y 2 X; where h 2 ð0;1Þ. If one of f ðXÞ; gðXÞ; SðXÞ, or TðXÞ is a complete subspace of X, then ff ; Sg and fg; Tg have aunique point of coincidence in X. Moreover if ff ; Sg and fg; Tg are weakly compatible, then f, g, S and T have a unique commonfixed point.

Corollary 2.5. Let f, g and T be self-maps on a cone metric space X with cone P having non-empty interior, satisfyingf ðXÞ [ gðXÞ � TðXÞ and

dðfx; gyÞ 6 hux;yðf ; g; TÞ;

where h 2 ð0;1Þ and

ux;yðf ; g; TÞ 2 dðSx; TyÞ; dðfx; TxÞ; dðgy; TyÞ;dðfx; TyÞ þ dðgy; TxÞ2

� �

for all x; y 2 X. If one of f ðXÞ; gðXÞ or TðXÞ is a complete subspace of X, then ff ; Tg and fg; Tg have a unique point of coinci-dence in X. Moreover if ff ; Tg and fg; Tg are weakly compatible, then f, g and T have a unique common fixed point.

Corollary 2.6. Let f and T be self-maps on a cone metric space X with cone P having non-empty interior, satisfying f ðXÞ � TðXÞ and

dðfx; fyÞ 6 hux;yðf ; TÞ;

where h 2 ð0;1Þ and

ux;yðf ; TÞ 2 dðTx; TyÞ; dðfx; TxÞ; dðfy; TyÞ; dðfx; TyÞ þ dðfy; TxÞ2

� �

for all x; y 2 X: If one of f ðXÞ or TðXÞ is a complete subspace of X , then ff ; Tg have a unique point of coincidence in X. Moreoverif f and T are weakly compatible, then f and T has a unique common fixed point.

Corollary 2.7. Let f be a self-map on a cone metric space X with cone P having non-empty interior, satisfying

dðfx; fyÞ 6 hux;yðf Þ;

where h 2 ð0;1Þ and

M. Abbas et al. / Applied Mathematics and Computation 216 (2010) 80–86 85

ux;yðf Þ 2 dðx; yÞ;dðfx; xÞ; dðfy; yÞ;dðfx; yÞ þ dðfy; xÞ2

� �

for all x; y 2 X. If f ðXÞ is a complete subspace of X, then f has a unique fixed point.

The following theorem extends and improves Theorem 2 of [3].

Theorem 2.8. Let f, g, S and T be self-maps on a cone metric space X with cone P having non-empty interior, satisfyingf ðXÞ � TðXÞ; gðXÞ � SðXÞ and

dðfx; gyÞ 6 pdðSx; TyÞ þ qdðfx; SxÞ þ rdðgy; TyÞ þ t½dðfx; TyÞ þ dðgy; SxÞ� ð2:6Þ

for all x; y 2 X, where p; q; r; t 2 ½0;1Þ satisfying

pþ qþ r þ 2t < 1:

If one of f ðXÞ; gðXÞ; SðXÞ, or TðXÞ is a complete subspace of X, then ff ; Sg and fg; Tg have a unique point of coincidence in X.Moreover if ff ; Sg and fg; Tg are weakly compatible, then f, g, S and T have a unique common fixed point.

Proof. For any arbitrary point x0 in X, construct sequences fxng and fyng in X as in the proof of Theorem 2.2. Then

dðy2nþ1; y2nþ2Þ ¼ dðfx2n; gx2nþ1Þ 6 pdðSx2n; Tx2nþ1Þ þ qdðfx2n; Sx2nÞ þ rdðgx2nþ1; Tx2nþ1Þ þ t½dðfx2n; Tx2nþ1Þ þ dðgx2nþ1; Sx2nÞ�¼ pdðy2n; y2nþ1Þ þ qdðy2nþ1; y2nÞ þ rdðy2nþ2; y2nþ1Þ þ t½dðy2nþ1; y2nþ1Þ þ dðy2nþ2; y2nÞ�¼ ðpþ qþ r þ tÞdðy2nþ1; y2nþ2Þ þ tdðy2n; y2nþ1Þ;

or dðy2nþ1; y2nþ2Þ 6 ddðy2n; y2nþ1Þ, where d ¼ t=ð1� p� q� r � tÞ < 1. From Lemma 2.1, it follows that fyng is a Cauchy se-quence. Suppose that SðXÞ is complete. Then there exists a point u in SðXÞ such that Sx2n ¼ y2n ! u as n!1. Consequently,we can find a v in X such that Sv ¼ u.

Now we shall show that f v ¼ u. Using (2.6),

dðf v;uÞ6 dðfv ;gx2n�1Þþdðgx2n�1;uÞ6pdðSv ;Tx2n�1Þþqdðf v;SvÞþ rdðgx2n�1;Tx2n�1Þþ t½dðf v ;Tx2n�1Þþdðgx2n�1;SvÞ�þdðgx2n�1;uÞ;6pðdðSv;uÞþdðu;Tx2nþ1ÞÞþqdðfu;SvÞþ rdðgx2n�1;uÞþ t½dðf v ;uÞþdðu;Tx2n�1Þþdðgx2n�1;uÞþdðu;SvÞ�þdðgx2n�1;uÞ;

which, on taking n!1, yields

dðfv ;uÞ 6 pdðSv; uÞ þ qdðfv ; SvÞ þ tdðf v ;uÞ þ rdðu; SuÞ þ c

¼ ðqþ tÞdðfv ;uÞ þ c;

and Remark 1.5(c) implies that dðfv ;uÞ 6 ðqþ tÞdðfv ;uÞ, and fu ¼ v .Since u 2 TðXÞ, we can find a point w in X such that Tw ¼ u. Now we shall show that gw ¼ u. Using (2.6),

dðgw; uÞ 6 dðfx2n; gwÞ þ dðfx2n;uÞ 6 pdðSx2n; TwÞ þ qdðfx2n; Sx2nÞ þ rdðgw; TwÞ þ t½dðfx2n; TwÞ þ dðgw; Sx2nÞ� þ dðfx2n;uÞ:

Following arguments similar to those given above, we obtain dðgw;uÞ 6 ðr þ tÞdðgw;uÞ and gw ¼ u. Hence gu ¼ Tw ¼ fv ¼ u.If the pairs ff ; sg and fg; Tg are weakly compatible, then

fu ¼ fSv ¼ Sf v ¼ Su ¼ w1 ðsayÞ; andgu ¼ gTw ¼ Tgw ¼ Tu ¼ w2 ðsayÞ:

Again using (2.6),

dðw1;w2Þ ¼ dðfu; guÞ 6 pdðSu; TuÞ þ qdðfu; SuÞ þ rdðgu; TuÞ þ t½dðfu; TuÞ þ dðgu; SuÞ� ¼ ðpþ 2tÞdðw1;w2Þ

which implies that w1 ¼ w2. Therefore fu ¼ gu ¼ Su ¼ Tu. Now we show that u ¼ gu. Using (2.6),

dðu; guÞ ¼ dðf v; guÞ 6 pdðSv ; TuÞ þ qdðfv ; SvÞ þ rdðgu; TuÞ þ t½dðf v; TuÞ þ dðgu; SvÞ� ¼ ðpþ 2tÞðgu; uÞ

and gu ¼ u. Thus u is a common fixed point of f, g, S and T. h

Clearly (2.6) implies the uniqueness of the common fixed point.

Corollary 2.9. Let f, g, S and T be self-maps on a cone metric space X with cone P having non-empty interior, satisfyingf ðXÞ � TðXÞ; gðXÞ � SðXÞ and for some m;n 2 N,

dðf mx; gnyÞ 6 pdðSmx; TnyÞ þ qdðf mx; SmxÞ þ rdðgny; TnyÞ þ t½dðf mx; TnyÞ þ dðgny; SmxÞ� ð2:7Þ

for all x; y 2 X, where p; q; r; t 2 ½0;1½ satisfying

pþ qþ r þ 2t < 1:

86 M. Abbas et al. / Applied Mathematics and Computation 216 (2010) 80–86

If one of f ðXÞ; gðXÞ; SðXÞ, or TðXÞ is complete subspace of X, then ff ; Sg and fg; Tg have a unique point of coincidence in X.Moreover if ff ; Sg and fg; Tg are weakly compatible, then f, g, S and T have a unique common fixed point.

The following corollary extends and improves results in [2].

Corollary 2.10. Let f, g and T be self-maps on a cone metric space X with cone P having non-empty interior, satisfyingf ðXÞ [ gðXÞ � TðXÞ and

dðfx; gyÞ 6 pdðTx; TyÞ þ qdðfx; TxÞ þ rdðgy; TyÞ þ t½dðfx; TyÞ þ dðgy; TxÞ� ð2:8Þ

for all x; y 2 X, where p; q; r; t 2 ½0;1½ satisfying

pþ qþ r þ 2t < 1:

If one of f ðXÞ; gðXÞ or TðXÞ is a complete subspace of X, then ff ; Tg and fg; Tg have a unique point of coincidence in X. More-over if ff ; Tg and fg; Tg are weakly compatible, then f, g and T have a unique common fixed point.

Corollary 2.11. Let f and T be self-maps on a cone metric space X with cone P having non-empty interior, satisfying f ðXÞ � TðXÞand

dðfx; fyÞ 6 pdðTx; TyÞ þ qdðfx; TxÞ þ rdðfy; TyÞ þ t½dðfx; TyÞ þ dðfy; TxÞ�

for all x; y 2 X, where p; q; r; t 2 ½0;1½ satisfying

pþ qþ r þ 2t < 1:

If one of f ðXÞ or TðXÞ is a complete subspace of X, then f and T have a unique point of coincidence in X: Moreover if f and T areweakly compatible, then f and T have a unique common fixed point.

Corollary 2.12. Let f be a self-maps on a cone metric space X with cone P having non-empty interior, satisfying

dðfx; fyÞ 6 pdðx; yÞ þ qdðfx; xÞ þ rdðfy; yÞ þ t½dðfx; yÞ þ dðfy; xÞ�

for all x; y 2 X; where p; q; r; t 2 ½0;1½ satisfying

pþ qþ r þ 2t < 1:

If f ðXÞ is a complete subspace of X, then f has a unique fixed point in X.

Acknowledgement

We thank the referee for the careful reading of the manuscript.

References

[1] M. Abbas, B.E. Rhoades, Fixed and periodic point results in cone metric spaces, Appl. Math. Lett. 22 (2009) 511–515.[2] M. Abbas, G. Jungck, Common fixed point results of noncommuting mappings without continuity in cone metric spaces, J. Math. Anal. Appl. 341 (2008)

418–420.[3] M. Arshad, A. Azam, P. Vetro, Some common fixed point results in cone metric spaces, Fixed Point Theory Appl., 2009 (Article ID 493965, 11 pages).[4] V. Berinde, A common fixed point theorem for compatible quasi contractive self mappings in metric spaces, Appl. Math. Comput. 213 (2009) 348–354.[5] L.G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1467–1475.[6] B. Fisher, Four mappings with a common fixed point, J. Univ. Kuwait Sci. 8 (1981) 131–139.[7] L. Gajic, V. Rakocevic, Pair of non-self-mappings and common fixed points, Appl. Math. Comput. 187 (2007) 999–1006.[8] G. Jungck, Common fixed points for commuting and compatible maps on compacta, Proc. Am. Math. Soc. 103 (1988) 977–983.[9] G. Jungck, Common fixed points for noncontinuous nonself maps on nonmetric spaces, Far East J. Math. Sci. 4 (1996) 199–215.

[10] S. Radenovich, B.E. Rhoades, Fixed point theorem for two non-self mappings in cone metric spaces, Comput. Math. Appl. 57 (2009) 1701–1707.[11] S. Rezapour, A review on topological properties of cone metric spaces, Analysis, Topology and Applications 2008, Vrnjacka Banja, Serbia, from May 30

to June 4, 2008.[12] S. Rezapour, R. Hamlbarani, Some note on the paper cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 345

(2008) 719–724.[13] C. Di Bari, P. Vetro, u-Pairs and common fixed points in cone metric spaces, Rend. Circ. Mat. Palermo 57 (2008) 279–285.