SOME RESULTS ON FIXED POINT THEOREMS FOR MULTIVALUED MAPPINGS IN COMPLETE METRIC SPACES

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1 IJMMS 30: PII. S Hindawi Publishing Corp. SOME RESULTS ON FIXED POINT THEOREMS FOR MULTIVALUED MAPPINGS IN COMPLETE METRIC SPACES JEONG SHEOK UME, BYUNG SOO LEE, and SUNG JIN CHO Received 17 October 2001 Using the concept of w-distance, we improve some well-known fixed point theorems Mathematics Subject Classification: 47H Introduction. Recently, Ume [3] improved the fixed point theorems in a complete metric space using the concept of w-distance, introduced by Kada, Suzuki, and Takahashi [2], and more general contractive mappings than quasi-contractive mappings. In this paper, using the concept of w-distance, we first prove common fixed point theorems for multivalued mappings in complete metric spaces, then these theorems are used to improve Ćirić s fixed point theorem [1], Kada-Suzuki-Takahashi s fixed point theorem [2], and Ume s fixed point theorem [3]. 2. Preliminaries. Throughout, we denote by N the set of all positive integers and by R the set of all real numbers. Definition 2.1 see [2]. Let X,d be a metric space, then a function p : X X [0, is called a w-distance on X if the following are satisfied: 1 px,z px,y+py,z for all x,y,z X; 2 for any x X, px, : X [0, is lower semicontinuous; 3 for any ɛ>0, there exists δ>0 such that pz,x δ and pz,y δ imply dx,y ɛ. Definition 2.2. Let X,d be a metric space with a w-distance p, then 1 for any x X and A X, dx,a := inf{dx,y : y A} and da,x := inf{dy,x : y A}; 2 for any x X and A X, px,a := inf{px,y : y A} and pa,x := inf{py,x : y A}; 3 for any A,B X, pa,b := inf{px,y : x A, y B}; 4 CB p X ={A A is nonempty closed subset of X and sup x,y A px,y < }. The following lemmas are fundamental. Lemma 2.3 see [2]. Let X be a metric space with a metric d, let p be a w-distance on X. Let{x n } and {y n } be sequences in X, let {α n } and {β n } be sequences in [0, converging to 0, and let x,y,z X. Then the following hold:

2 320 JEONG SHEOK UME ET AL. 1 if px n,y α n and px n,z β n for any n N, then y = z. In particular, if px,y = 0 and px,z = 0, then y = z; 2 if px n,y n α n and px n,z β n for any n N, then {y n } converges to z; 3 if px n,x m α n for any n,m N with m>n, then {x n } is a Cauchy sequence; 4 if py,x n α n for any n N, then {x n } is a Cauchy sequence. Lemma 2.4 see [3]. Let X be a metric space with a metric d, let p be a w-distance on X, and let T be a mapping of X into itself satisfying ptx,ty q max { px,y,px,tx,py,ty,px,ty,py,tx } 2.1 for all x,y X and some q [0,1. Then 1 for each x X, n N, and i,j N with i,j n, p T i x,t j x q δ Ox,n ; for each x X and n N, there exist k,l N with k,l n such that δ Ox,n = max { px,x,p x,t k x,p T l x,x } ; for each x X, δ Ox, 1 { } px,x+px,tx+ptx,x ; q 4 for each x X, {T n x} n=1 is a Cauchy sequence. 3. Main results Theorem 3.1. Let X be a complete metric space with a metric d and let p be a w-distance on X. Suppose that S and T are two mappings of X into CB p X and ϕ : X X [0, is a mapping such that max { p u 1,u 2,p v1,v 2 } q ϕx,y 3.1 for all nonempty subsets A, B of X, u 1 SA, u 2 S 2 A, v 1 TB, v 2 T 2 B, x A, y B, and some q [0,1, { ϕx,y sup sup min [ px,sa,py,tb ] : x A, y B } : A,B X < 1 q, 3.2 inf { py,u+px,sx+py,ty : x,y X } > 0, 3.3 for every u X with u Su or u Tu, where SA means a A Sa.ThenS and T have a common fixed point in X. Let { ϕx,y β = sup sup min [ px,sa,py,tb ] : x A, y B } : A, B X, 3.4

3 FIXED POINT THEOREMS 321 and k = βq. Define x n+1 Sx n and y n+1 Ty n for all n N. Thenx n Sx n 1, x n+1 S 2 x n 1, y n Ty n 1, and y n+1 T 2 y n 1. From 3.1 and 3.2, we have p x n,x n+1 kp xn 1,x n k n 1 p x 1,x 2, 3.5 p y n,y n+1 kp yn 1,y n k n 1 p y 1,y 2, 3.6 for all n N and some k [0,1. Letn and m be any positive integers such that n<m. Then, from 3.6, we obtain p y n,y m p yn,y n+1 + +p ym 1,y m m n 1 = i=0 m n 1 i=0 p y n+i,y n+i+1 k n+i 1 p y 1,y kn 1 1 k p y 1,y 2. By Lemma 2.3, {y n } is a Cauchy sequence. Since X is complete, {y n } converges to u X. Then, since py n, is lower semicontinuous, from 3.7 we have p y n,u lim inf p k n 1 y n,y m m 1 k p y 1,y Suppose that u Su or u Tu. Then, by 3.3, 3.5, 3.6, and 3.8, we have 0 < inf { py,u+px,sx+py,ty : x,y X } inf { p y n,u +p } x n,x n+1 +p yn,y n+1 : n N { k n 1 inf 1 k p y 1,y 2 +k n 1 p x 1,x 2 +k n 1 p } y 1,y 2 : n N { } 2 k = 1 k p y 1,y 2 +p x1,x 2 inf { k n 1 : n N } = This is a contradiction. Therefore we have u Su and u Tu. Theorem 3.2. Let X be a complete metric space with a metric d and let p be a w-distance on X. Suppose that S and T are two mappings of X into CB p X and ϕ : X X [0, is a mapping such that max { p u 1,u 2,p v1,v 2 } q ϕx,y 3.10 for all x,y X, u 1 Sx, u 2 S 2 x, v 1 Ty, v 2 T 2 y, and some q [0,1, { ϕx,y sup sup min [ px,sx,py,ty ] : x A, y B and 3.3 is satisfied. Then S and T have a common fixed point in X. } : A,B X < 1 q, 3.11

4 322 JEONG SHEOK UME ET AL. By a method similar to that in the proof of Theorem 3.1, the result follows. Theorem 3.3. Let X be a complete metric space with a metric d and let p be a w-distance on X. Suppose that T is a mapping of X into CB p X and ψ : X [0, is a mapping such that p u 1,u 2 q ψx 3.12 for all x X, u 1 Tx, u 2 T 2 x and some q [0,1, { } ψx sup px,tx : x X < 1 q, inf { px,u+px,tx : x X } > 0, 3.13 for every u X with u Tu.ThenT has a fixed point in X. By a method similar to that in the proof of Theorem 3.1, the result follows. Theorem 3.4. Let X be a complete metric space with a metric d and let p be a w-distance on X. Suppose that S and T are self-mapping of X and ϕ : X X [0, is a mapping such that max { p Sx,S 2 x,p Ty,T 2 y } q ϕx,y 3.14 for all x,y X and some q [0,1, { sup } < 1 q, ϕx,y min [ px,sx,py,ty ] : x,y X inf { py,u+px,sx+py,ty : x,y X } > 0, 3.15 for every u X with u Su or u Tu.ThenS and T have a common fixed point in X. By a method similar to that in the proof of Theorem 3.1, the result follows. From Theorem 3.1, we have the following corollary. Corollary 3.5. Let X be a complete metric space with a metric d and let p be a w-distance on X. Suppose that S and T are two mappings of X into CB p X and ϕ : X X [0, is a mapping such that { max sup [ p u 1,u 2 : u1 Sx, u 2 S 2 x ], sup [ p v 1,v 2 : v1 Tx, v 2 T 2 x ]} 3.16 q ϕx,y for all x,y X and some q [0,1, and that 3.3 and 3.11 are satisfied. Then S and T have a common fixed point in X.

5 From Theorem 3.3, we have the following corollaries. FIXED POINT THEOREMS 323 Corollary 3.6. Let X be a complete metric space with a metric d and let p be a w-distance on X. Suppose that T is a mapping of X into CB p X and ψ : X [0, is a mapping such that sup [ p u 1,u 2 : u1 Tx, u 2 T 2 x ] q ψx 3.17 for all x X and some q [0,1, and that 3.13 is satisfied. Then T has a fixed point in X. Corollary 3.7. Let X be a complete metric space with a metric d and let p be a w-distance on X. Suppose that T is a self-mapping of X and ψ : X [0, is a mapping such that p Tx,T 2 x q ψx 3.18 for all x X and some q [0,1, { } ψx sup px,tx : x X < 1 q, inf { px,u+px,tx : x X } > 0, 3.19 for every u X with u Tu.ThenT has a fixed point in X. From Corollary 3.7, we have the following corollaries. Corollary 3.8 see [3]. Let X be a complete metric space with a metric d and let p be a w-distance on X. Suppose that T is a self-mapping of X such that ptx,ty q max { px,y,px,tx,py,ty,px,ty,py,tx } 3.20 for all x,y X and some q [0,1, and that inf { px,u+px,tx : x X } > for every u X with u Tu.ThenT has a unique fixed point in X. By 3.20 andlemma 2.43, we have sup { p T i x,t j x i, j N {0} } < 3.22 for every x X. Thus we may define a function r : X X [0, by rx,y= max { sup [ p T i x,t j x i, j N {0} ],px,y } 3.23 for every x,y X. Clearly, r is a w-distance on X.Letx be a given element of X, then, by using Lemma 2.41, 3.20, and 3.23, we have r Tx,T 2 x = sup { p T i x,t j x i, j N } q sup { p T i x,t j x i, j N {0} } 3.24 = q rx,tx.

6 324 JEONG SHEOK UME ET AL. By 3.21 and 3.23, we obtain inf { rx,u+rx,tx: x X } > for every u X with u Tu. From 3.24, 3.25, and Corollary 3.7, T has a fixed point in X. By3.20 andlemma 2.4, it is clear that the fixed point of T is unique. Corollary 3.9 see [2]. Let X be a complete metric space, let p be a w-distance on X, and let T be a mapping from X into itself. Suppose that there exists q [0,1 such that p Tx,T 2 x q px,tx 3.26 for every x X and that inf { px,y+px,tx : x X } > for every y X with y Ty.ThenT has a fixed point in X. Define ψ : X [0, by ψx = px,tx 3.28 for all x X. Thus the conditions of Corollary 3.7 are satisfied. Hence T has a fixed point in X. From Corollary 3.8, we have the following corollary. Corollary 3.10 see [1]. Let X be a complete metric space with a metric d and let T be a mapping from X into itself. Suppose that T is a quasicontraction, that is, there exists q [0,1 such that dt x,t y q max { dx,y,dx,tx,dy,ty,dx,ty,dy,tx } 3.29 for every x,y X. ThenT has a unique fixed point in X. It is clear that the metric d is a w-distance and inf { dx,y+dx,t x : x X } > for every y X with y Ty. Thus, by Corollary 3.8 or 3.9, T has a unique fixed point in X. Acknowledgment. This work was supported by grant No from the Basic Research Program of the Korea Science & Engineering Foundation. References [1] L. B. Ćirić, A generalization of Banach s contraction principle, Proc. Amer. Math. Soc , [2] O. Kada, T. Suzuki, and W. Takahashi, Nonconvex minimization theorems and fixed point theorems in complete metric spaces, Math. Japon , no. 2,

7 FIXED POINT THEOREMS 325 [3] J. S. Ume, Fixed point theorems related to Ćirić s contraction principle, J. Math. Anal. Appl , no. 2, Jeong Sheok Ume: Department of Applied Mathematics, Changwon National University, Changwon , Korea address: jsume@sarim.changwon.ac.kr Byung Soo Lee: Department of Mathematics, Kyungsung University, Pusan , Korea address: bslee@star.kyungsung.ac.kr Sung Jin Cho: Department of Applied Mathematics, Pukyong National University, Pusan , Korea address: sjcho@dolphin.pknu.ac.kr

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