Recent advances in the Lefschetz fixed point theory for multivalued mappings

Authors : Lech GÓRNIEWICZ

Pages : 116-126

Doi:10.53006/rna.941060

View : 18 | Download : 7

Publication Date : 2021-06-30

Article Type : Research Paper

Abstract :In 1923 S. Lefschetz proved the famous fixed point theorem which is now known as the Lefschetz fixed point theorem insert ignore into journalissuearticles values(comp. [5], [9], [20], [21]. The multivalued case was considered for the first time in 1946 by S. Eilenberg and D. Montgomery insert ignore into journalissuearticles values([10]);. They proved the Lefschetz fixed point theorem for acyclic mappings of compact ANR-spaces insert ignore into journalissuearticles values(absolute neighborhood retracts insert ignore into journalissuearticles values(see [4] or [13]); using Vietoris mapping theorem insert ignore into journalissuearticles values(see [4], [13], [16]); as the main tool. In 1970 Eilenberg, Montgomery`s result was generalized for acyclic mappings of complete ANR-s insert ignore into journalissuearticles values(see [17]);. Next, a class of admissible multivalued mappings was introduced insert ignore into journalissuearticles values([13] or [16]);. Note that the class of admissible mappings is quite large and contains as a special case not only acyclic mappings but also infinite compositions of acyclic mappings. For this class of multivalued mappings several versions of the Lefschetz fixed point theorem were proved insert ignore into journalissuearticles values(comp. [11], [13]-15], [18], [19], [27]);. In 1982 G. Skordev and W. Siegberg insert ignore into journalissuearticles values([26]); introduced the class of multivalued mappings so-called now insert ignore into journalissuearticles values(1 − n);-acyclic mappings. Note that the class insert ignore into journalissuearticles values(1−n);-acyclic mappings contain as a special case n-valued mappings considered in [6], [12], [28]. We recommend [8] for the most important results connected with insert ignore into journalissuearticles values(1 − n);-acyclic mappings. Finally, the Lefschetz fixed point theorem was considered for spheric mappings insert ignore into journalissuearticles values(comp. [3], [2], [7], [23]); and for random multivalued mappings insert ignore into journalissuearticles values(comp. [1], [2], [13]);. Let us remark that the main classes of spaces for which the Lefschetz fixed point theorem was formulated are the class of ANR-spaces insert ignore into journalissuearticles values([4]); and MANR-spaces insert ignore into journalissuearticles values(multi absolute neighborhood retracts insert ignore into journalissuearticles values(see [27]);. The aim of this paper is to recall the most important results concerning the Lefschetz fixed point theorem for multivalued mappings and to prove new versions of this theorem mainly for AANR-spaces insert ignore into journalissuearticles values(approximative absolute neighborhood retracts insert ignore into journalissuearticles values(see [4] or [13]); and for MANR-s. We believe that this article will be useful for analysts applying topological fixed point theory for multivalued mappings in nonlinear analysis, especially in differential inclusions.
Keywords : multivalued mappings, absolute retracts, admissible mappigns 1 n, acyclic mappings, spheric and random mappings, Lefschetz number, fixed point problem