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Fixed point theorems for metrically weakly inward set-valued mappings. (English) Zbl 0949.47045

The paper is devoted to the existence of a fixed point of set-valued mappings satisfying some metrically inward conditions. The concept of metrically weakly inward set-valued mapping, introduced by the authors is compared to other inward type conditions, in Section 2.
The main results of the paper, Theorem 3.1 and Theorem 3.2 in Section 3, are proved by an argument based on Caristi’s fixed point theorem. Theorem 3.1 shows that any u.s.c. set-valued mapping \(T\) of \(K\) into \(CB(X)\) satisfying both a inward and a contraction condition, does possess a fixed point (\(K\) is a nonempty complete subset of a metric space \((X, d)\), while \(CB(X)\) denotes the family of all nonempty bounded closed subsets of \(X\)).
The statement of Theorem 3.2 is as follows: Let \(K\) be a complete subset of a metric space \((X, d)\) and \(T\) a metrically weakly inward set-valued \(k\)-contraction mapping of \(K\) into \(CB(X)\). Then \(T\) has a fixed point in \(K\). These results and their corollaries improve and generalize various other results in the literature, as shown by means of Example 3.1.

MSC:

47H10 Fixed-point theorems
47H04 Set-valued operators
54C60 Set-valued maps in general topology
54E40 Special maps on metric spaces
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References:

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