Proximal maps, prox maps and coincidence points. (English) Zbl 0726.41035

Beer, Gerald; Pai, Devidas

Numer. Funct. Anal. Optimization 11, No. 5-6, 429-448 (1990).

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.

Reviewer: Zhang Ganglu (Dongying)

MSC:

41A65

Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)

Keywords:

Hausdorff locally convex linear topological space; pseudometrics

References:

[1]	DOI: 10.1017/S0004972700027519 · Zbl 0669.52002
[2]	Beer G., J. Math. Anal. Appl. 38 (1988)
[3]	Castaing C., Lecture Notes in Mathematics (1975)
[4]	Engleking R., General Topology (1977)
[5]	DOI: 10.1007/BF01110225 · Zbl 0185.39503
[6]	DOI: 10.1016/0022-247X(72)90128-X
[7]	Horvath J., Topological Vector Spaces and Distributions 1 (1966) · Zbl 0143.15101
[8]	Klein K., Theory of Correspondences (1984)
[9]	DOI: 10.1016/0021-9045(76)90075-7 · Zbl 0331.41026
[10]	Pai D. V., Optimizing Methods in Statistics (1979)
[11]	Pai D. V., Methods of Functional Analysis in Approximation Theory, ISNM 76 (1986)
[12]	DOI: 10.1080/01630568308816149 · Zbl 0513.41015
[13]	DOI: 10.1016/0022-247X(78)90222-6 · Zbl 0375.47031
[14]	DOI: 10.1016/0022-247X(79)90148-3 · Zbl 0423.47027
[15]	Sehgal V. M., Numer. Funct. Anal. and Optimiz. 69 (1979)
[16]	Sehgal V. M., Proc. Amer. Math. Soc. 102 pp 534– (1988)
[17]	Singer I., Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces (1970) · Zbl 0197.38601
[18]	Smithson R. E., Pacific J. Math. 61 pp 283– (1975) · Zbl 0317.54022