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Extension and generalization of a common fixed point theorem for set-valued mappings. (Chinese. English summary) Zbl 0922.47054

This paper is a continuation of S. Chang and Y. Peng [Int. J. Math. Math. Sci. 16, No. 1, 87-94 (1993; Zbl 0774.47031)] and the following fixed point theorem is obtained, which generalizes the corresponding results in the paper mentioned above:
Theorem. Let \((X,d)\) be a complete metric space, \(CB(X)\) the family of all nonempty bounded closed subsets of \(X\), \(F:X\to 2^X\) and \(T_n: X\to CB(X)\) be a set-valued mapping and a set-valued mapping sequence satisfying the following conditions:
(i) \(T_n(X)\subset F(X)\) for all \(n\in\mathbb{N}\) and \(F(X)\) is a closed subset;
(ii) for all \(x,y\in X\) and for any \(i, j\in\mathbb{N}\), \(i\neq j\) \[ \begin{split} H(T_i(x), T_j(y))\leq \Phi(\max\{d(F(x), F(y)), d(F(x),T_i(x)),\\ d(F(y),T_j(y),\textstyle{{1\over 2}}[d(F(x), T_j(y))+ d(F(y), T_i(x))]\}),\end{split} \] where \(\Phi: \mathbb{R}^+\to \mathbb{R}^+\) is an increasing function such that \(\Phi(t+0)< t\) for all \(t>0\) and \(\sum^\infty_{n=1} \Phi^n(t)< \infty\) for all \(t>0\).
Then there exists \(x^*\in X\) such that \[ F(x^*)\cap \Biggl(\bigcap^\infty_{n= 1} T_n(x^*)\Biggr)\neq \emptyset. \]

MSC:

47H10 Fixed-point theorems
47H04 Set-valued operators
54H25 Fixed-point and coincidence theorems (topological aspects)
54C60 Set-valued maps in general topology

Citations:

Zbl 0774.47031
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