Research Article Common Fixed Point Theorem for Four Non-Self Mappings in Cone Metric Spaces

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1 Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 010, Article ID 98380, 14 pages doi: /010/98380 Research Article Common Fixed Point Theorem for Four Non-Self Mappings in Cone Metric Spaces Xianjiu Huang, 1 Chuanxi Zhu, 1 and Xi Wen 1 Department of Mathematics, Nanchang University, Nanchang, Jiangxi , China Department of Computer Science, Nanchang University, Nanchang, Jiangxi , China Correspondence should be addressed to Xianjiu Huang, xjhuangxwen@163.com Received 13 June 009; Revised 1 March 010; Accepted 18 April 010 Academic Editor: Lai Jiu Lin Copyright 010 Xianjiu Huang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We extend a common fixed point theorem of Radenovic and Rhoades for four non-self-mappings in cone metric spaces. 1. Introduction and Preliminaries Recently, Huang and Zhang 1 generalized the concept of a metric space, replacing the set of real numbers by ordered Banach space and obtained some fixed point theorems for mappings satisfying different contractive conditions. Subseuently, the study of fixed point theorems in such spaces is followed by some other mathematicians; see 8. The aim of this paper is to prove a common fixed point theorem for four non-self-mappings on cone metric spaces in which the cone need not be normal. This result generalizes the result of Radenović and Rhoades 5. Consistent with Huang and Zhang 1, the following definitions and results will be needed in the seuel. Let E be a real Banach space. A subset P of E is called a cone if and only if a P is closed, nonempty and P / {θ}; b a, b R, a, b 0, x, y P implies ax by P; c P P {θ}.

2 Fixed Point Theory and Applications Given a cone P E, we define a partial ordering with respect to P by x y if and only if y x P. A cone P is called normal if there is a number K>0 such that for all x, y E, θ x y implies x K y. 1.1 The least positive number satisfying the above ineuality is called the normal constant of P, while x y stands for y x int P interior of P. Definition 1.1 see 1. LetX be a nonempty set. Suppose that the mapping d : X X E satisfies d1 θ d x, y for all x, y X and d x, y θ if and only if x y; d d x, y d y, x for all x, y X; d3 d x, y d x, z d z, y for all x, y, z X. Then d is called a cone metric on X,and X, d is called a cone metric space. The concept of a cone metric space is more general than that of a metric space. Definition 1. see 1. Let X, d be a cone metric space. One says that {x n } is e a Cauchy seuence if for every c E with θ c, there is an N such that for all n, m > N, d x n,x m c; f a Convergent seuence if for every c E with θ c, there is an N such that for all n>n, d x n,x c for some fixed x X. A cone metric space X is said to be complete if every Cauchy seuence in X is convergent in X. It is known that {x n } converges to x X if and only if d x n,x θ as n. It is a Cauchy seuence if and only if d x n,x m θ n, m. Remark 1.3 see 9. LetE be an ordered Banach normed space. Then c is an interior point of P if and only if c, c is a neighborhood of θ. Corollary 1.4 see If a b and b c,thena c. Indeed, c a c b b a c b implies c a,c a c b,c b. If a b and b c,thena c. Indeed, c a c b b a c b implies c a,c a c b,c b. 3 If θ u c for each c int P,thenu θ. Remark 1.5 see 5, 11. Ifc int P, θ a n,anda n θ, then there exists an n 0 such that for all n>n 0 we have a n c. Remark 1.6 see 6, 10. IfE is a real Banach space with cone P and if a ka where a P and 0 <k<1, then a θ. We find it convenient to introduce the following definition. Definition 1.7 see 5. Let X, d be a complete cone metric space and C a nonempty closed subset of X,andf, g : C X satisfying d fx,fy ) λu, 1.

3 Fixed Point Theory and Applications 3 where { ) d gx,gy u,d fx,gx ),d fy,gy ), d fx,gy fy,gx ) }, 1.3 for all x, y C,0<λ<1/, λ, then f is called a generalized g-contractive mapping of C into X. Definition 1.8 see. Letf and g be self-maps on a set X i.e., f, g : X X.Ifw fx gx for some x in X, then x is called a coincidence point of f and g, andw is called a point of coincidence of f and g. Self-maps f and g are said to be weakly compatible if they commute at their coincidence point; that is, if fx gx for some x X, then fgx gfx.. Main Result The following theorem is Radenović and Rhoades 5 generalization of Imdad and Kumar s 1 result in cone metric spaces. Theorem.1. Let X, d be a complete cone metric space and C a nonempty closed subset of X such that for each x C and y / C there exists a point z C the boundary of C) such that d x, z d z, y x, y.1 Suppose that f, g : C X are such that f is a generalized g-contractive mapping of C into X, and i C gc,fc C gc, ii gx C fx C, iii gc is closed in X. Then the pair f, g has a coincidence point. Moreover, if pair f, g is weakly compatible, then f and g have a uniue common fixed point. The purpose of this paper is to extend the above theorem for four non-self-mappings in cone metric spaces. We begin with the following definition. Definition.. Let X, d be a complete cone metric space and C a nonempty closed subset of X, andf, G, S, T : C X satisfying d Fx,Gy ) λu,. where { ) d Tx,Sy u,d Tx,Fx,d Sy, Gy ), d Tx,Gy Sy, Fx ) },.3 for all x, y C, 0<λ<1/, λ, then F, G is called a generalized T, S -contractive mappings pair of C into X.

4 4 Fixed Point Theory and Applications Notice that by setting G F f and T S g in., one deduces the slightly generalized form of 1.3. We state and prove our main result as follows. Theorem.3. Let X, d be a complete cone metric space and C a nonempty closed subset of X such that for each x C and y / C there exists a point z C the boundary of C) such that d x, z d z, y x, y.4 Suppose that F, G, S, T : C X are such that F, G is a generalized T, S -contractive mappings pair of C into X, and I C SC TC,FC C SC, GC C TC, II Tx C Fx C, Sx C Gx C, III SC and TC or FC and GC) are closed in X. Then IV F, T has a point of coincidence, V G, S has a point of coincidence. Moreover, if F, T and G, S are weakly compatible pairs, then F, G, S, and T have a uniue common fixed point. Proof. Firstly, we proceed to construct two seuences {x n } and {y n } in the following way. Let x C be arbitrary. Then due to C TC there exists a point x 0 C such that x Tx 0. Since Tx C Fx C, one concludes that Fx 0 FC C SC. Thus, there exists x 1 C such that y 1 Sx 1 Fx 0 C. Since y 1 Fx 0 there exists a point y Gx 1 such that d y 1,y Fx0,Gx 1..5 Suppose that y C. Then y GC C TC which implies that there exists a point x C such that y Tx. Otherwise, if y / C, then there exists a point p C such that d Sx 1,p p, y Sx1,y.6 Since p C TC there exists a point x C with p Tx,sothat d Sx 1,Tx d Tx,y Sx1,y.7 Let y 3 Fx be such that d y,y 3 d Gx 1,Fx. Thus, repeating the foregoing arguments, one obtains two seuences {x n } and {y n } such that a y n Gx n 1, y n 1 Fx n, b y n C y n Tx n or y n / C Tx n C, d Sx n 1,Tx n d Tx n,y n Sxn 1,y n.8

5 Fixed Point Theory and Applications 5 c y n 1 C y n 1 Sx n 1 or y n 1 / C Sx n 1 C, d Tx n,sx n 1 d Sx n 1,y n 1 Txn,y n 1.9 We denote that P 0 { Tx i {Tx n } : Tx i y i }, P 1 { Tx i {Tx n } : Tx i / y i }, Q 0 { Sx i 1 {Sx n 1 } : Sx i 1 y i 1 },.10 Q 1 { Sx i 1 {Sx n 1 } : Sx i 1 / y i 1 }. Note that Tx n,sx n 1 / P 1 Q 1,asifTx n P 1, then y n / Tx n, and one infers that Tx n C which implies that y n 1 Fx n C. Hence y n 1 Sx n 1 Q 0. Similarly, one can argue that Sx n 1,Tx n / Q 1 P 1. Now, we distinguish the following three cases. Case 1. If Tx n,sx n 1 P 0 Q 0, then from. d Tx n,sx n 1 d Fx n,gx n 1 λu n 1,.11 where { d Sxn 1,Tx n u n 1,d Sx n 1,Gx n 1,d Tx n,fx n, d Tx } n,gx n 1 d Sx n 1,Fx n { ) d yn 1,y n,d ) ) d )} y n 1,y n 1 y n 1,y n,d yn,y n 1,..1 Clearly, there are infinite many n such that at least one of the following four cases holds: 1 d Tx n,sx n 1 λ d y n 1,y n ) λd Sx n 1,Tx n,.13 d Tx n,sx n 1 λd y n 1,y n ) λd Sxn 1,Tx n,.14

6 6 Fixed Point Theory and Applications 3 d Tx n,sx n 1 λd y n,y n 1 ) d Txn,Sx n 1 θ λd Sx n 1,Tx n,.15 4 d Tx n,sx n 1 λ d ) y n 1,y n 1 λ d ) ) y n 1,y n d yn,y n 1 λ d Sx n 1,Tx n d Tx n,sx n 1,.16 which implies 1 λ/ d Tx n,sx n 1 λ/ d Sx n 1,Tx n,thatis, d Tx n,sx n 1 λ λ d Sx n 1,Tx n λd Sx n 1,Tx n..17 From 1,, 3, and 4 it follows that d Tx n,sx n 1 λd Sx n 1,Tx n..18 Similarly, if Sx n 1,Tx n Q 0 P 0, we have d Sx n 1,Tx n d Fx n,gx n 1 λd Tx n,sx n If Sx n 1,Tx n Q 0 P 0, we have d Sx n 1,Tx n d Fx n,gx n 1 λd Tx n,sx n 1..0 Case. If Tx n,sx n 1 P 0 Q 1, then Sx n 1 Q 1 and d Tx n,sx n 1 d Sx n 1,y n 1 Txn,y n 1 ).1 which in turn yields and hence d Tx n,sx n 1 d Tx n,y n 1 yn,y n 1 ) d Tx n,sx n 1 d y n,y n 1 Fxn,Gx n Now, proceeding as in Case 1, we have that.18 holds.

7 Fixed Point Theory and Applications 7 If Sx n 1,Tx n Q 1 P 0, then Tx n P 0. We show that d Sx n 1,Tx n λd Tx n,sx n 1..4 Using.1,weget d Sx n 1,Tx n d Sx n 1,y n 1 yn 1,Tx n ) d Tx n,y n 1 ) d Txn,Sx n 1 d y n 1,Tx n.5 By noting that Tx n,tx n P 0, one can conclude that d y n 1,Tx n yn 1,y n Fxn,Gx n 1 λd Tx n,sx n 1, d Tx n,y n 1 yn,y n 1 Fxn,Gx n 1 λd Sx n 1,Tx n,.6 in view of Case 1. Thus, d Sx n 1,Tx n λd Sx n 1,Tx n 1 λ d Tx n,sx n 1 λd Sx n 1,Tx n,.7 and we proved.4. Case 3. If Tx n,sx n 1 P 1 Q 0, then Sx n 1 Q 0. We show that d Tx n,sx n 1 λd Sx n 1,Tx n..8 Since Tx n P 1, then d Sx n 1,Tx n d Tx n,y n Sxn 1,y n.9 From this, we get d Tx n,sx n 1 d Tx n,y n yn,sx n 1 ) d Sx n 1,y n ) d Sxn 1,Tx n d y n,sx n 1.30 By noting that Sx n 1,Sx n 1 Q 0, one can conclude that d y n,sx n 1 yn,y n 1 Fxn,Gx n 1 λd Sx n 1,Tx n, d Sx n 1,y n yn 1,y n Fxn,Gx n 1 λd Sx n 1,Tx n,.31 in view of Case 1.

8 8 Fixed Point Theory and Applications Thus, d Tx n,sx n 1 λd Sx n 1,Tx n 1 λ d Sx n 1,Tx n λd Sx n 1,Tx n,.3 and we proved.8. Similarly, if Sx n 1,Tx n Q 0 P 1, then Tx n P 1,and d Sx n 1,Tx n d Tx n,y n Sxn 1,y n.33 From this, we have d Sx n 1,Tx n d Sx n 1,y n yn,tx n ) d Sx n 1,y n Sxn 1,y n ) d Sxn 1,Tx n d Sx n 1,y n ) d Sxn 1,Tx n d Sx n 1,Tx n.34 d Sx n 1,y n By noting that Sx n 1 Q 0, one can conclude that d Sx n 1,Tx n d Sx n 1,y n Fxn,Gx n 1 λd Tx n,sx n 1,.35 in view of Case 1. Thus, in all Cases 1 3, there exists w n {d Sx n 1,Tx n,d Tx n,sx n 1 } such that d Tx n,sx n 1 λw n,.36 and there exists w n 1 {d Sx n 1,Tx n,d Tx n,sx n 1 } such that d Sx n 1,Tx n λw n Following the procedure of Assad and Kirk 13, it can easily be shown by induction that, for n 1, there exists w {d Tx 0,Sx 1,d Sx 1,Tx } such that d Tx n,sx n 1 λ n 1/ w, d Sx n 1,Tx n λ n w..38 From.38 and by the triangle ineuality, for n>m, we have d Tx n,sx m 1 d Tx n,sx n 1 d Sx n 1,Tx n d Tx m,sx m 1 λ m λ m 1/ λ n 1) w λ m 1 λ w θ, as m..39 From Remark 1.5 and Corollary 1.4 1, d Tx n,sx m 1 c.

9 Fixed Point Theory and Applications 9 Thus, the seuence {Tx 0,Sx 1,Tx,Sx 3,...,Sx n 1,Tx n,sx n 1,...} is a Cauchy seuence. Then, as noted in 14, there exists at least one subseuence {Tx nk } or {Sx nk 1} which is contained in P 0 or Q 0, respectively, and finds its limit z C. Furthermore, subseuences {Tx nk } and {Sx nk 1} both converge to z C as C is a closed subset of complete cone metric space X, d. We assume that there exists a subseuence {Tx nk } P 0 for each k N, then Tx nk y nk Gx nk 1 C GC TC. Since TCas well as SC are closed in X,and {Tx nk } is Cauchy in TC, it converges to a point z TC.Letw T 1 z, then Tw z. Similarly, {Sx nk 1} a subseuence of Cauchy seuence {Tx 0,Sx 1,Tx,Sx 3,...,Sx n 1,Tx n,sx n 1,...} also converges to z as SC is closed. Using., one can write d Fw,z d Fw,Gx nk 1 d Gx nk 1,z λu nk 1 d Gx nk 1,z,.40 where { d Tw,Sxnk 1 u nk 1,d Tw,Fw,d Sx nk 1,Gx nk 1, d Tw,Gx } n k 1 d Fw,Sx nk 1 { d z, Sxnk 1,d z, Fw,d Sx nk 1,Gx nk 1, d z, Gx } n k 1 d Fw,Sx nk Let θ c. Clearly at least one of the following four cases holds for infinitely many n: 1 d Fw,z λ d z, Sx n k 1 d Gx nk 1,z λ c λ c c;.4 d Fw,z λd z, Fw d Gx nk 1,z d Fw,z 1 1 λ d Gx n k 1,z λ 1 λ c c; 3 d Fw,z λd Sx nk 1,Gx nk 1 d Gx nk 1,z λ d Sx nk 1,z d z, Gx nk 1 d Gx nk 1,z λ 1 d Gx nk 1,z λd Sx nk 1,z c λ 1 λ 1 λ c c; λ.44

10 10 Fixed Point Theory and Applications 4 d Fw,z λ d z, Gx n k 1 d Fw,Sx nk 1 d Gx nk 1,z λ d z, Gx n k 1 d Fw,z d z, Sx nk 1 λ λ d Gx n k 1,z λ λ d z, Sx n k 1 λ c λ λ ) / λ c )) λ λ λ/ )) c. λ d Gx nk 1,z d Fw,z.45 In all cases we obtain d Fw,z c for each c int P.UsingCorollary it follows that d Fw,z θ or Fw z.thus,fw z Tw,thatis,z is a coincidence point of F, T. Further, since Cauchy seuence {Tx 0,Sx 1,Tx,Sx 3,...,Sx n 1,Tx n,sx n 1,...} converges to z C and z Fw, z FC C SC, there exists v C such that Sv z. Again using., weget d Sv, Gv d z, Gv d Fw,Gv λu,.46 where { d Tw,Sv u,d Tw,Fw,d Sv, Gv, { } d z, Gv θ θ, θ, d Sv, Gv, { } θ, d Sv, Gv,. d Sv, Gv } d Tw,Gv d Fw,Sv.47 Hence, we get the following cases: d Sv, Gv λθ θ, d Sv, Gv λd Sv, Gv, d Sv, Gv λ d Sv, Gv..48 Since λ/ λ/ λ λ/ 1 1 λ <λ,usingremark 1.6 and Corollary 1.4 3, it follows that Sv Gv; therefore, Sv z Gv, thatis,z is a coincidence point of G, S. In case FC and GC are closed in X, z FC C SC or z GC C TC.The analogous arguments establish IV and V. If we assume that there exists a subseuence {Sx nk 1} Q 0 with TC as well SC being closed in X, then noting that {Sx nk 1} is a Cauchy seuence in SC, foregoing arguments establish IV and V. Suppose now that F, T and G, S are weakly compatible pairs, then z Fw Tw Fz FTw TFw Tz, z Gv Sv Gz GSv SGv Sz..49

11 Fixed Point Theory and Applications 11 Then, from., d Fz,z d Fz,Gv λu,.50 where { } d Sv, Tz d Tz,Gv d Sv, Fz u,d Tz,Fz,d Sv, Gv, { } d z, Fz d Fz,z d z, Fz,d Fz,Fz,d z, z, { } d z, Fz d z, Fz,θ,..51 Hence, we get the following cases: d Fz,z λ d z, Fz, d Fz,z λθ θ and d Fz,z λd z, Fz,.5 Since λ/ λ/ λ λ/ 1 1 λ < λ < 1, using Remark 1.6 and Corollary 1.4 3, it follows that Fz z. Thus,Fz z Tz. Similarly, we can prove that Gz z Sz. Therefore z Fz Gz Sz Tz,thatis,z is a common fixed point of F, G, S,andT. Uniueness of the common fixed point follows easily from.. The following example shows that in general F, G, S, andt satisfying the hypotheses of Theorem.3 need not have a common coincidence justifying two separate conclusions IV and V. Example.4. Let E C 1 0, 1,R, P {ϕ E : ϕ t 0, t 0, 1 }, X 0,, C 0,, and d : X X E defined by d x, y x y ϕ, where ϕ P is a fixed function, for example, ϕ t e t. Then X, d is a complete cone metric space with a nonnormal cone having the nonempty interior. Define F, G, S, andt : C X as Fx x 4 5, Gx x 4 5, Tx 5x, Sx 5x, x C..53 Since C {0, }. Clearly, for each x C and y / C there exists a point z C such that d x, z d z, y d x, y. Further, SC TC 0, 0 0, 10 0, 10 {0, } C, FC C 4/5, 14/5 0, 4/5, SC, GC C 4/5, 4/5 0, 4/5, TC, and SC, TC, FC,andGC are closed in X.

12 1 Fixed Point Theory and Applications Also, T T0 0 C F0 4 5 C, S0 0 C G0 4 5 C, ) ) C F C, S C G C..54 Moreover, for each x, y C, d Fx,Gy ) x y 1 ϕ 5 d Tx,Sy ) ),.55 that is,. is satisfied with λ /5. Evidently, 1 T 1/5 F 1/5 / 1/5 and1 S 1/ 5 G 1/ 5 / 1/ 5. Notice that two separate coincidence points are not common fixed points as FT 1/5 / TF 1/5 and SG 1/ 5 / GS 1/ 5, which shows necessity of weakly compatible property in Theorem.3. Next, we furnish an illustrate example in support of our result. In doing so, we are essentially inspired by Imdad and Kumar 1. Example.5. Let E C 1 0, 1,R,P {ϕ E : ϕ t 0,t 0, 1 }, X 1,, C 1, 3, and d : X X E defined by d x, y x y ϕ, where ϕ P is a fixed function, for example, ϕ t e t. Then X, d is a complete cone metric space with a nonnormal cone having the nonempty interior. Define F, G, S, andt : C X as x if 1 x, Fx if <x 3, x 3 if 1 x, Gx if <x 3, 4x 4 3 if 1 x, Tx 13 if <x 3, 4x 6 3 if 1 x, Sx 13 if <x Since C {1, 3}. Clearly, for each x C and y / C there exists a point z 3 C such that d x, z d z, y d x, y. Further, SC TC 1, 53 1, 61 1, 61 {1, 3} C, FC C 1, 4 1, 3 1, 3 SC,andGC C 1, 8 1, 3 1, 3 TC. Also, T1 1 C F1 1 C, S1 1 C G1 1 C, T C F C, S C G C..57

13 Fixed Point Theory and Applications 13 Moreover, if x 1, and y, 3, then d Fx,Gy ) x x 4 4 ϕ x ϕ 4 x 4 4 / x ϕ 1 x d Tx,Sy )..58 Next, if x, y, 3, then Finally, if x, y 1,, then d Fx,Gy ) 0 λ d Tx,Sy )..59 d Fx,Gy ) x y 3 x 4 y 6 ϕ x y 3 ϕ 4 x 4 y 6 / x y 3 ϕ 1 d ) Tx,Sy x y 3.60 Therefore, condition. is satisfied if we choose λ max{1/ x, 1/ x y 3 } 0, 1/. Moreover 1 is a point of coincidence as T1 F1 as well as S1 G1 whereas both the pairs F, T and G, S are weakly compatible as TF1 1 FT1 andsg1 1 GS1. Also, SC, TC, FC, and GC are closed in X. Thus, all the conditions of Theorem.3 are satisfied and 1 is the uniue common fixed point of F, G, S, andt. One may note that 1 is also a point of coincidence for both the pairs F, T and G, S. Remark.6. 1 Setting G F f and T S g in Theorem.3, one deduces Theorem.1 due to 5. Setting G F f and T S I X in Theorem.3, we obtain the following result. Corollary.7. Let X, d be a complete cone metric space and C a nonempty closed subset of X such that for each x C and y / C there exists a point z C the boundary of C) such that d x, z d z, y x, y.61 Suppose that f : C X satisfies the condition d fx,fy ) λu x, y ),.6 where u x, y ) { ) d x, y,d x, fx ),d y, fy ), d x, fy y, fx ) }.63 for all x, y C, 0 <λ<1/, λ, and f has the additional property that for each x C, fx C, f has a uniue fixed point.

14 14 Fixed Point Theory and Applications Acknowledgments The authors would like to express their sincere appreciation to the referees for their very helpful suggestions and many kind comments. This project was supported by the National Natural Science Foundation of China and and supported partly by the Provincial Natural Science Foundation of Jiangxi, China 008GZS0076 and 009GZS0019. References 1 L.-G. Huang and X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, Journal of Mathematical Analysis and Applications, vol. 33, no., pp , 007. M. Abbas and G. Jungck, Common fixed point results for noncommuting mappings without continuity in cone metric spaces, Journal of Mathematical Analysis and Applications, vol. 341, no. 1, pp , M. Abbas and B. E. Rhoades, Fixed and periodic point results in cone metric spaces, Applied Mathematics Letters, vol., no. 4, pp , D. Ilić and V. Rakočević, Common fixed points for maps on cone metric space, Journal of Mathematical Analysis and Applications, vol. 341, no., pp , S. Radenović and B. E. Rhoades, Fixed point theorem for two non-self mappings in cone metric spaces, Computers & Mathematics with Applications, vol. 57, no. 10, pp , Sh. Rezapour and R. Hamlbarani, Some notes on the paper: Cone metric spaces and fixed point theorems of contractive mappings, Journal of Mathematical Analysis and Applications, vol. 345, no., pp , P. Vetro, Common fixed points in cone metric spaces, Rendiconti del Circolo Matematico di Palermo, vol. 56, no. 3, pp , D. Wardowski, Endpoints and fixed points of set-valued contractions in cone metric spaces, Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 1-, pp , Y. C. Wong and K. F. Ng, Partially Ordered Topological Vector Spaces, Oxford Mathematical Monographs, Clarendon Press, Oxford, UK, Sh. Rezapour, A review on topological properties of cone metric spaces, in Analysis, Topology and Applications ATA 08), Vrnjacka Banja, Serbia, May-June G. Jungck, S. Radenović, S. Radojević, and V. Rakočević, Common fixed point theorems for weakly compatible pairs on cone metric spaces, Fixed Point Theory and Applications, vol. 009, Article ID , 13 pages, M. Imdad and S. Kumar, Rhoades-type fixed-point theorems for a pair of nonself mappings, Computers & Mathematics with Applications, vol. 46, no. 5-6, pp , N. A. Assad and W. A. Kirk, Fixed point theorems for set-valued mappings of contractive type, Pacific Journal of Mathematics, vol. 43, pp , O. Hadžić andl.gajić, Coincidence points for set-valued mappings in convex metric spaces, Univerzitet u Novom Sadu. Zbornik Radova Prirodno-Matematičkog Fakulteta. Serija za Matemati, vol. 16, no. 1, pp. 13 5, 1986.