Fixed point theory for weakly contractive maps with applications to operator inclusions in Banach spaces relative to the weak topology
type: Item
instance of: scholarly article
title: Fixed point theory for weakly contractive maps with applications to operator inclusions in Banach spaces relative to the weak topology
zbMATH Open document ID: 0911.47057
DOI: 10.4171/ZAA/822
author: Donal O'Regan
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1999-05-04T00:00:00Z
review text: The author presents a new fixed point theory for weakly contractive multivalued maps between Banach spaces. Theorem 2.2. Let \(Q\) be a non-empty, bounded, convex, closed set in a Banach space \(E\). Assume \(F:Q\to C(Q)\) is weakly sequentially upper semicontinuous and \(\alpha\) \(W\)-contractive. Then \(F\) has a fixed point; here \(C(Q)\) denotes the family of non-empty, closed, convex subsets of \(Q\). Theorem 2.3. Let \(Q\) and \(C\) be closed bounded, convex subsets of a Banach space \(E\) and \(Q\subset C\). In addition, let \(U\) be a weakly open subset of \(Q\) with \(0\in U\), \(U^w\) a weakly compact subset of \(Q\) and \(F: U^w\to CK(Q)\) a weakly sequentially upper semicontinuous, \(\alpha\) \(w\)-contractive (here \(0\leq\alpha< 1\)) map, here \(CK(C)\) denotes the family of non-empty, convex, weakly compact subsets of \(C\) and \(\overline{U^w}\) the weak closure of \(U\) in \(E\). Then either: \((\text{A}_1)\) \(F\) has a fixed point; or \((\text{A}_2)\) there is a point \(u\in\partial_QU\) (the weak boundary of \(U\) in \(Q\)) and \(\lambda\in(0,1)\) with \(u\in\lambda Fu\). Theorem 2.4. Let \(E=(E,\|\cdot\|)\) be a separable and reflexive Banach space, \(C\) and \(Q\) are closed, bounded, convex subsets of \(E\) with \(Q\subset C\) and \(0\in Q\). Also, assume \(F: Q\to CK(Q)\) is a weakly sequentially upper semicontinuous (and weakly compact) map. In addition suppose the following: For any \(\Omega_\varepsilon= \{x\in E: d(x,Q)\leq \varepsilon\}\) \((\varepsilon>0)\), if \(\{(x_j,\lambda_j)\}^\infty_{j= 1}\) is a sequence in \(Q\times[0, 1)\) with \(x_j\rightharpoonup x\in\partial\Omega_\varepsilon Q\) and \(\lambda_j\to\lambda\) and if \(x\in\lambda F(x)\), \(0\leq\lambda< 1\), then \(\{\lambda_jF(x_j)\}\subset Q\) for \(j\) sufficiently large (\(\partial\Omega_\varepsilon Q\) is the weak boundary of \(Q\) relative to \(\Omega_\varepsilon\), \(d(x,y)=| x-y|\) and \(\rightharpoonup\) denotes weak convergence). Then \(F\) has a fixed point in \(Q\…
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 47J05
Mathematics Subject Classification ID: 54C60
Mathematics Subject Classification ID: 47H04
zbMATH DE Number: 1182348
zbMATH Keywords: fixed point
zbMATH Keywords: weakly contractive multivalued maps
zbMATH Keywords: nonlinear abstract operator inclusion
reviewed by: Valeriu Popa
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1182348