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.


41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
Full Text: DOI


[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.