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Some fixed points theorems for multivalued mappings. (English) Zbl 0574.54045

Let (x,d) be a complete metric space and let S(X) be the family of all nonempty bounded closed subsets of X. Let \(\psi\) : [0,\(+\infty)\to [0,+\infty)\) be a nondecreasing function, continuous from the right, such that for any \(u>0\), \(\sum^{\infty}_{n=0}\psi^ n(u)<+\infty\), and \(\psi (u)=0\) implies \(u=0\). Let \(D(a,B)=\inf \{d(a,b): b\in B\}\) and \(H(A,B)=\max \{\sup \{D(a,B):\) \(a\in A\}\), sup\(\{\) D(b,A): \(\in B\}\}\), with A,B\(\in S(X)\). (The function H is the Hausdorff metric on S(X).) Main theorem: Let F be a multivalued mapping from X into S(X) for which there exists an \(\epsilon >0\) such that: for all x,y\(\in X\), \(d(x,y)<\epsilon\) implies H(Fx,Fy)\(\leq \psi (d(x,y))\); there exists a \(z\in X\) such that \(D(z,Fz)<\epsilon\); then there exists a point w such that \(w\in Fw\). The author also establishes a corollary in \(\epsilon\)- chained metric spaces, generalizing a theorem of P. Kuhfitting [Pac. J. Math. 65, 399-403 (1976; Zbl 0333.54035)]. A result is also given for Banach spaces.

MSC:

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

Citations:

Zbl 0333.54035
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