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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 $$p:X\times X\to [0,\infty)$$ 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) $$p(x,z)\leq p(x,y)+ p(y,z)$$ for all $$x,y$$, $$z\in X$$; (2) for any $$x\in X$$, $$p(x, .):X\to [0,\infty)$$ is lower semicontinuous; (3) for any $$\varepsilon >0$$, there exists $$\delta >0$$ such that $$p(z,x)\leq \delta$$ and $$p(z,y) \leq\delta$$ imply $$d(x,y) \leq\varepsilon$$. For any $$x\in X$$ and $$A\subset X$$ we denote by $$p(x,A)= \{\inf p(x,y): y\in A)$$ and by $$p(A,x)= \inf\{p(y,x): y\in A\}$$ and $$CB_p(X)= \{A \mid A$$ is a nonempty closed susbset of $$X$$ and $$\sup_{x,y\in A}p(x,y)\}< \infty$$.
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 $$f:X\times X\to [0,\infty)$$ is a mapping such that $$\max\{p (u_1,u_2)$$, $$p(v_1,v_2)\} \leq qf(x,y)$$ for all nonempty subsets $$A,B$$ of $$X$$, $$u_1\in SA$$, $$u_2\in S^2A$$, $$v_1\in TB$$, $$v_2\in T^2B$$, $$x\in A$$, $$y\in B$$, and some $$q\in[0,1]$$ with $$\sup\{\sup (f(x,y)/\min [p (x,SA)$$, $$p(y,TB):x\in A$$, $$y\in B):A, B<X\}<1/q$$, $$\inf\{p(y,u)+ p(x,Sx)+p(y, Ty):x,y\in X\}>0$$, for every $$u\in X$$ with $$u\in Su$$ or $$u\notin Tu$$, where $$SA$$ means $$\cup\{Sa:a\in A\}$$. Then $$S$$ and $$T$$ have a common fixed point.
Reviewer: V.Popa (Bacau)

##### MSC:
 47H10 Fixed-point theorems 54H25 Fixed-point and coincidence theorems (topological aspects) 47H04 Set-valued operators
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