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Some results on fixed point theorems for multivalued mappings in complete metric spaces. (English) Zbl 1020.47048

Let (X,d) be a metric space, then a function p:X×X[0,) is called a w-distance on X [O. Kada, T. Suzuki and W. Takahashi, Math. Jap. 44, 381-391 (1996; Zbl 0897.54029)] if: (1) p(x,z)p(x,y)+p(y,z) for all x,y, zX; (2) for any xX, p(x,·):X[0,) is lower semicontinuous; (3) for any ε>0, there exists δ>0 such that p(z,x)δ and p(z,y)δ imply d(x,y)ε. For any xX and AX we denote by p(x,A)={infp(x,y):yA) and by p(A,x)=inf{p(y,x):yA} and CB p (X)={AA is a nonempty closed susbset of X and sup x,yA p(x,y)}<.

Recently, J.-S. Ume [J. Math. Anal. Appl. 225, 630-640 (1998; Zbl 0917.54047)] improved some fixed points theorem in a complete metric space using the concept of w-distance. In the paper under review the authors, using this concept, prove some common fixed point theorems for two multivalued mappings S and T in a complete metric space.

The main result of this paper is the following theorem: 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 f:X×X[0,) is a mapping such that max{p(u 1 ,u 2 ), p(v 1 ,v 2 )}qf(x,y) for all nonempty subsets A,B of X, u 1 SA, u 2 S 2 A, v 1 TB, v 2 T 2 B, xA, yB, and some q[0,1] with sup{sup(f(x,y)/min[p(x,SA), p(y,TB):xA, yB):A,B<X}<1/q, inf{p(y,u)+p(x,Sx)+p(y,Ty):x,yX}>0, for every uX with uSu or uTu, where SA means {Sa:aA}. Then S and T have a common fixed point.

Reviewer: V.Popa (Bacau)
MSC:
47H10Fixed point theorems for nonlinear operators on topological linear spaces
54H25Fixed-point and coincidence theorems in topological spaces
47H04Set-valued operators