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

Let $\left(X,d\right)$ be a metric space, then a function $p:X×X\to \left[0,\infty \right)$ 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) $p\left(x,z\right)\le p\left(x,y\right)+p\left(y,z\right)$ for all $x,y$, $z\in X$; (2) for any $x\in X$, $p\left(x,·\right):X\to \left[0,\infty \right)$ is lower semicontinuous; (3) for any $\epsilon >0$, there exists $\delta >0$ such that $p\left(z,x\right)\le \delta$ and $p\left(z,y\right)\le \delta$ imply $d\left(x,y\right)\le \epsilon$. For any $x\in X$ and $A\subset X$ we denote by $p\left(x,A\right)=\left\{infp\left(x,y\right):y\in A\right)$ and by $p\left(A,x\right)=inf\left\{p\left(y,x\right):y\in A\right\}$ and $C{B}_{p}\left(X\right)=\left\{A\mid A$ is a nonempty closed susbset of $X$ and ${sup}_{x,y\in A}p\left(x,y\right)\right\}<\infty$.

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 $C{B}_{p}\left(X\right)$ and $f:X×X\to \left[0,\infty \right)$ is a mapping such that $max\left\{p\left({u}_{1},{u}_{2}\right)$, $p\left({v}_{1},{v}_{2}\right)\right\}\le qf\left(x,y\right)$ for all nonempty subsets $A,B$ of $X$, ${u}_{1}\in SA$, ${u}_{2}\in {S}^{2}A$, ${v}_{1}\in TB$, ${v}_{2}\in {T}^{2}B$, $x\in A$, $y\in B$, and some $q\in \left[0,1\right]$ with $sup\left\{sup\left(f\left(x,y\right)/min\left[p\left(x,SA\right)$, $p\left(y,TB\right):x\in A$, $y\in B\right):A,B, $inf\left\{p\left(y,u\right)+p\left(x,Sx\right)+p\left(y,Ty\right):x,y\in X\right\}>0$, for every $u\in X$ with $u\in Su$ or $u\notin Tu$, where $SA$ means $\cup \left\{Sa:a\in A\right\}$. Then $S$ and $T$ have a common fixed point.

Reviewer: V.Popa (Bacau)
##### MSC:
 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 54H25 Fixed-point and coincidence theorems in topological spaces 47H04 Set-valued operators