Fixed point theorems for a broad class of multimaps
type: Item
instance of: scholarly article
title: Fixed point theorems for a broad class of multimaps
zbMATH Open document ID: 1029.54049
DOI: 10.1016/S0362-546X(02)00146-3
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2003-03-11T00:00:00Z
review text: We say that \(\varphi: \mathbb{R}^+\to [0,1)\) has property \(A\) if there exists an \(M\in (0,\infty)\) such that \(\varphi\) is monotonically increasing on \([M,\infty)\), \(\int^\infty_M(1-\varphi(t)) dt=+\infty\) and \(\sup\{\varphi(t): r\leq t\leq M\}< 1\), \(\forall r\in (0,M)\), \(\varphi\) has property \(B\) if \(\sup\{\varphi(t): r\leq t<\infty\}<1\), \(\forall r\in (0,\infty)\). Let \((X,d)\) be a metric space, \(N(X)\) the collection of all nonempty subsets of \(X\) and \(F: X\to N(X)\). We say that \(\varphi\) has property \(C\) w.r.t. \((F,x_0)\) if it is monotonically increasing in \(\mathbb{R}^+\) and \(\int^\infty_0 (1-\varphi(t)) dt> d(x_0, Fx_0)\). The main result of this paper is the following theorem: Let \(F: X\to C(X)\), where \(C(X)\) is the collection of all nonempty closed subsets of \(X\). Suppose that \((X,d)\) is \(F\)-orbitally complete, the function \(f\) defined on \(X\) as \(f(x)= d(x,Fx)\) is \(F\)-orbitally semicontinuous at any cluster point of any orbit of \(F\) w.r.t. \(x_0\), the function \(\varphi\) has property \(A\) or \(B\) or property \(C\) w.r.t. \((F,x_0)\) and that \(d(y, Fy)\leq \varphi(d(x_0, x)) d(x,y)\) whenever \(x\in 0(F,x_0)\), \(y\in Fx\) and \(x\not\in Fx\). Then there exists an orbit \(\{x_n\}^\infty_0\) of \(F\) w.r.t. \(x_0\) which converges to a fixed point of \(F\). This theorem is a proper generalization of Theorem 2.1 of [\textit{C.-K. Zhong}, \textit{J. Zhu} and \textit{P.-H. Zhao}, Proc. Am. Math. Soc. 128, No. 8, 2439-2444 (2000; Zbl 0948.47058)].
reviewed by: Valeriu Popa
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 54C60
Mathematics Subject Classification ID: 54H10
zbMATH DE Number: 1879990
zbMATH Keywords: multimap
zbMATH Keywords: fixed point
zbMATH Keywords: orbital completeness
zbMATH Keywords: orbital lower semicontinuity
author: S. V. R. Naidu
MaRDI profile type: Publication
cites work: Q3869872
cites work: Multi-valued contraction mappings
cites work: Mapping theorems for gateaux differentiable and accretive operators
cites work: A generalization of Ekerland's variational principle and application to the study of the relation between the weak P.S. condition and coercivity
cites work: An extension of multi-valued contraction mappings and fixed points
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description: scientific article; zbMATH DE number 1879990