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Multi-valued fixed point theorems by altering distance between the points. (English) Zbl 1027.47060
Fixed point results involving altering distances were introduced by M. S. Khan, M. Swaleh and S. Sessa [Bull. Aust. Math. Soc. 30, 1-9 (1984; Zbl 0553.54023)]. There exists a vast literature which generalizes the results of this paper. An altering distance is a mapping $$\phi: [0,\infty)\to [0,\infty)$$ which satisfies: (a) $$\phi$$ is increasing continuous and (b) $$\phi(t)= 0$$ if and only if $$t= 0$$. In the present paper, the authors extend the results from [M. El Amrani and A. B. Mbarki, Southwest J. Pure Appl. Math. 1, 16-21 (2000; Zbl 0966.47038)] and M. A. Krasnoselski and P. P. Zabreiko [“Geometrical methods of nonlinear analysis”, Grundlehren 263, Springer-Verlag, Berlin (1984; Zbl 0546.47030), p. 206] to multivalued maps.
Let $$(X,d)$$ be a metric space. We denote by $$CB(X)$$ the set of all nonempty closed bounded subsets of $$(X,d)$$ and by $$H$$ the Hausdorff-Pompeiu metric on $$CB(X)$$, i.e., $H(A,B)= \max\{\sup_{x\in A} d(x,B), \sup_{x\in B} d(x,A)\},$ where $$A,B\in CB(X)$$ and $$d(x,A)= \inf_{y\in A} \{d(x,y)\}$$. We define also $$\delta(A,B)$$ by $$\delta(A,B)= \sup\{d(a, b): a\in A,b\in B\}$$. Let $$F: X\to 2^X$$ be a multi-valued map, then a point $$x\in X$$ is called a fixed point of $$F$$ if $$x\in Fx$$; it is called absolutely fixed if $$Fx= \{x\}$$.
The authors prove the following Theorem: Let $$(X,d)$$ be a complete metric space, $$T: X\to CB(X)$$ a multi-valued map satisfying the following condition: $$\phi(\delta(Tx,Ty))\leq a\phi(dx,y))+ b[\phi(\delta(x, Tx))+ \phi(\delta(y, Ty))]+ c\min\{\phi(d(x, Ty)),\phi(d(y, Tx))\}$$, $$\forall x,y\in X$$, where $$\phi: \mathbb{R}^+\to \mathbb{R}^+$$ is continuous and strictly increasing such that $$\phi(0)= 0$$ and $$a$$, $$b$$, $$c$$ are three positive constants such that $$a+ 2b< 1$$ and $$a+ c< 1$$. Then $$T$$ has a unique absolutely fixed point.
A similar theorem for multi-valued maps with compact values is proved. In the last part, a generalization of Theorem 34.5 of Krasnoselski and Zabreiko [loc. cit.] is proved.

MSC:
 47H10 Fixed-point theorems 47H04 Set-valued operators
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