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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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