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Proximal maps, prox maps and coincidence points. (English) Zbl 0726.41035
Let (X,\(\tau\)) be a Hausdorff locally convex linear topological space and \({\mathcal P}\) be a separating family of continuous seminorms generating the topology \(\tau\). The class of the nonempty closed subsets of X is denoted by CL(X) and the generalized Hausdorff pseudometric \(H_ p\) on CL(X), where \(p\in {\mathcal P}\), is defined for sets A, B in CL(X) by \(H_ p(A,B)=\max \{\sup_{a\in A}d_ p(a,B),\sup_{b\in B}d_ p(b,A)\}\), where \(d_ p(a,B)=\inf \{p(a-b):\) \(b\in B\}\). Given a family \({\mathcal B}\subset CL(X)\), \(\tau_ H\) will denote the topology induced on \({\mathcal B}\) by the uniformity generated by the family of pseudometrics \(\{H_ p:\) \(p\in {\mathcal P}\}\). In this paper the authors mainly prove the following result. Let \(p\in {\mathcal P}\) and C be a nonempty convex subset of X. Let g: \(C\to C\) be a continuous, almot p-affine and surjective map which is perfect (i.e., g(A) is closed whenever A is closed and for each \(c\in C\), \(g^{-1}(C)\) is a compact subset of C), let F: \(C\to (KC(X),\tau_ H)\) be continuous, where KC(X) denote the class of the nonempty compact and convex subsets of X, assume either (1) \({\mathcal B}=\{Fx:\) \(x\in C\}\) is \(\tau_ H\)-compact and C is proximally-p-compact with respect to \({\mathcal B}\); or (2) F(C) is relatively compact and C is proximally-p-compact with respect to \({\mathcal B}\), where \({\mathcal B}\) is class of the nonempty closed and convex subsets of cl F(C). Then there exists an \(x\in C\) satisfying \(d_ p(g(x),F(x))=d_ p(F(x),C),\) where \(d_ p(F(x),C)=\inf \{d_ p(a,C):\;a\in F(x)\}.\) There after, they particularized the above result to the setting of a normed linear space X to obtain a coincidence theorem using a boundary condition on F and g.

MSC:
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
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