COMMON STATIONARY POINTS FOR SET-VALUED MAPPINGS

Several theorems on stationary points for set-valued mappings have obtained. These are improvements upon some earlier results due to Fisher.


INTRODUCTION AND PRELIMINARIES.
In this paper, we prove several common stationary point theorems for four set-valued mappings, which are improvements upon some earlier results obtained by Fisher [1], [2], [3].
Let (X,d) be a metric space and CL(X) be the class of all nonempty closed subset of X.For z x and A c_ X, let D(z,A) iny{d(z,y):y e A}.
Then//is called the getera//zed Hadar d/.Cm:e jmct/an for the class CL(X) induced by the metric d.
DEFINITION 1.3.A set-valued mapping S:X-.CL(X) is said to be nettr/y-dem.j//if ct(q(A))<ct(A) for any bounded and $-invariant subset of X with a(A)>0, where a is the Kuratowski's measure of non-compactness.
DEFINITION 1.4.Let F,G,S,T X-.CL(X) be set-valued mappings.For some z X, define the arb/t O(z) of r by O(z) {y X:y z or y =/'(z) for some " }, 'Y being the subsemigroup generated by F,G,S and T in the semigroup of all self-mappings on X with composition operation.
DEFINITION 1.5.A point is said to be a common stionarll point of set-valued mappings F and 2. THE MAIN RESULTS.
Throughout this paper, for any set-valued mapping S:X-,CL(X), we assume that all the powers of $ map X into CL(X).First of all, we prove the following crucial result to be used in the sequel.LEMMA 2.1.Let (X,d) be a compact metric space and S:X-CL(X) be a set-valued mapping such that S' is continuous with respect to the generalized Hausdorff distance function H for some positive integer i.If '4 nF= 1Sk'(X), then S('4) .4.
Then F, G,S and T have a unique common stationary point in X.
Then F,G,S and T have a unique common stationary point z in X.Further, z is the unique common stationary point of F and G.
PROOF.If we put B n= 1(FG)n(X), as in the proof of Theorem 2.1, we have B {z) and z is a unique common stationary point of F,G,S and T. Since any common stationary point of F ad G is a point of B {z}, it follows that z is the unique common stationary point of F and G.This completes the proof.
REMARK.Theorem 2 of Fisher [2] and theorems in Fisher [3] follow as corollaries of our Theorem 2.2.In fact, our theorem can be regarded as an improvement over the above theorems due to Fisher.THEOREM 2.3.Let (X,d) be a complete metric space and F,G,S,T:X-,CL(X) be set-valued mappings such that (2.8) F,G,S,T,F and G are continuous with respect to the distance function H for some positive integers and j.Also, F,G,S and T are nearly-densifying, (2.9) for some z X, the orbit O(zo) is bounded, (2.10) H(FPz, GqF) < m(z,F, FP, Gq, S',Tt) whenever the left-hand side is positive, (2.11) SF FiS" and TtG q GqTt.Then F, G,S and T have a unique common stationary point z in X. PROOF.Let A O(zo).Then as in the proof of Theorem 2.1, is compact.If we define lFin 1Gin B t= (A) and K t= (A), by Lemma 2.1, F(B)= B and G(K)= K. Also, it follows that B and K are compact subsets of X.