Intersection theorems for generalized weak KKM set‐valued mappings with applications in optimization

Abstract

In this paper, we introduce the concept of generalized weak KKM mapping that is more general than many others encountered in the KKM theory. Then, two previous intersection theorems of the author are extended from weak KKM mappings to generalized weak KKM mappings. Applications of these results to set‐valued equilibrium problems and minimax inequalities are given in the last two sections.

Nonlinear analysis by applying best approximation method in p-vector spaces

“…Here we first give some notion and a brief introduction to the abstract convex spaces, which play an important role in the development of the KKM principle and related applications. Once again, for the corresponding comprehensive discussion on the KKM theory and its various applications to nonlinear analysis and related topics, we refer to Agarwal et al [1], Alghamdi et al [5], Balaj [8], Mauldin [75], Granas and Dugundji [48], Park [91] and [92], Yuan [133], and related comprehensive references therein.…”

“…We all know that the best approximation in nature is related to fixed points for nonself mappings, which tightly link with the classical Leray-Schauder alternative based on the Leray-Schauder continuation theorem by Leray and Schauder [67], which is a remarkable result in nonlinear analysis; and in addition, there exist several continuation theorems, which have many applications to the study of nonlinear functional equations (see Agarwal et al [1], Alghamdi et al [5], Balaj [8], O'Regan and Precup [83]). Historically, it seems that the continuation theorem is based on the idea of obtaining a solution of a given equation, starting from one of the solutions of a simpler equation.…”

“…It is known that the class of p-seminorm spaces (0 < p ≤ 1) is an important generalization of the usual normed spaces with rich topological and geometrical structures, and related study has received a lot of attention (e.g., see Agarwal et al [1], Alghamdi et al [5], Balaj [8], Balachandran [7], Bayoumi [9], Bayoumi et al [10], Bernuées and Pena [12], Ding [31], Ennassik and Taoudi [34], Ennassik et al [33], Gal and Goldstein [40], Gholizadeh et al [41],…”

“…On the other hand, after the Birkhoff and Kellogg theorem given by Birkhoff and Kellogg in 1920s, the study on Birkhoff-Kellogg problem has received a lot of scholars' attention since then. For example, in 1934, one of the fundamental results in nonlinear functional analysis, famously called Leray-Schauder alternative first establoished by Leray and Schauder [67] was also studied via topological degree and some other advanced approaches by Agarwal et al [1], Alghamdi et al [5], Balaj [8], and many others. Thereafter, certain other types of Leray-Schauder alternatives were proved using techniques other than topological degree, see the works given by Granas and Dugundji [48], Furi and Pera [39] in the Banach space setting and applications to the boundary value problems for ordinary differential equations in noncompact problems, a general class of mappings for nonlinear alternative of Leray-Schauder type in normal topological spaces, and some Birkhoff-Kellogg type theorems for general class mappings in TVS or LCS by Agarwal et al [2], Agarwal and O'Regan [3,4], Park [88], O'Regan [81] by using the Leray-Schauder type coincidence theory applying to establish Birkhoff-Kellogg problem, the Furi-Pera type result for a general class of set-valued mappings.…”

“…It is well known that the best approximation is one of very important aspects for the study of nonlinear problems related to the problems on their solvability for partial differential equations, dynamic systems, optimization, mathematical program, operation research; and in particular, the one approach well accepted for the study of nonlinear problems in optimization, complementarity problems, of variational inequality problems, and so on, is strongly based on what is today called Fan's best approximation theorem given by Fan [37] in 1969, which acts as a very powerful tool in nonlinear analysis (see also the book of Singh et al [114] for the related discussion and study on the fixed point theory and best approximation with the KKM-map principle). Among them, the related tools are Rothe type and the principle of Leray-Schauder alterative in topological vector spaces (TVS) and local topological vector spaces (LCS), which are comprehensively studied by Agarwal et al [1], Alghamdi et al [5], Balaj [8], Chang et al [24], Chang et al [25][26][27], Carbone and Conti [20], Ennassik and Taoudi [34], Ennassik et al [33], Isac [53], Granas and Dugundji [48], Kirk and Shahzad [60], Liu [72], Park [91], Rothe [104,105], Shahzad [109][110][111], Xu [126], Yuan [132,133], Zeidler [134] (see also the references therein).…”