Nonlinear analysis in p-vector spaces for single-valued 1-set contractive mappings

Abstract

The goal of this paper is to develop some fundamental and important nonlinear analysis for single-valued mappings under the framework of p-vector spaces, in particular, for locally p-convex spaces for $0 < p \leq 1$ 0 < p ≤ 1 . More precisely, based on the fixed point theorem of single-valued continuous condensing mappings in locally p-convex spaces as the starting point, we first establish best approximation results for (single-valued) continuous condensing mappings, which are then used to develop new results for three classes of nonlinear mappings consisting of 1) condensing; 2) 1-set contractive; and 3) semiclosed 1-set contractive mappings in locally p-convex spaces. Next they are used to establish the general principle for nonlinear alternative, Leray–Schauder alternative, fixed points for nonself mappings with different boundary conditions for nonlinear mappings from locally p-convex spaces, to nonexpansive mappings in uniformly convex Banach spaces, or locally convex spaces with the Opial condition. The results given by this paper not only include the corresponding ones in the existing literature as special cases, but are also expected to be useful tools for the development of new theory in nonlinear functional analysis and applications to the study of related nonlinear problems arising from practice under the general framework of p-vector spaces for $0< p \leq 1$ 0 < p ≤ 1 .Finally, the work presented by this paper focuses on the development of nonlinear analysis for single-valued (instead of set-valued) mappings for locally p-convex spaces. Essentially, it is indeed the continuation of the associated work given recently by Yuan (Fixed Point Theory Algorithms Sci. Eng. 2022:20, 2022); therein, the attention is given to the study of nonlinear analysis for set-valued mappings in locally p-convex spaces for $0 < p \leq 1$ 0 < p ≤ 1 .

Fixed Point Theorems for Upper Semicontinuous Set-Valued Mappings in p-Vector Spaces

“…given in this paper are new, and may easily understand to use for general readers in mathematical community, and see more Yuan [51]- [52] on the study of nonlinear analysis and related applications in both locally p-convex spaces, or p-vector spaces, and references wherein for 0 < p ≤ 1.…”

“…It is known that the class of p-seminorm spaces (0 < p ≤ 1) is an important generalization of usual normed spaces with rich topological and geometrical structures, and related study has received a lot of attention (e.g., see Alghamdi et al [2], Balachandran [4], Bayoumi [5], Ennassik and Taoudi [14], Ennassik et al [15], Gholizadeh et al. [18], Granas and Dugundji [21], Jarchow [22], Kalton [23]- [24], Kalton et al [25], Park [35], Rolewicz [40], Xiao and Lu [46], Xiao and Zhu [47], Yuan [49]- [52], and many others). However, to the best of our knowledge, the corresponding basic tools and associated results in the category of nonlinear functional analysis have not been well developed, thus the goal of this paper is to develop some fundamental fixed point theorems for upper semicontinuous set-valued mappings under the framework of p-vector spaces by including topological vector spaces as a special class.…”

“…Tychonoff [43], Weber [44]- [45], Xiao and Lu [46], Xiao and Zhu [47], Xu [48], Yuan [49]- [52] and many other scholars' contributions, and comprehensive references and related discussion under the general framework of topological vector space, or p-vector spaces for non-self set-valued mappings (0 < p ≤ 1).…”

Fixed Point Theorems for Upper Semicontinuous Set-valued Mappings in p-Vector Spaces

“…It is known that the class of p-seminorm spaces (0 < p ≤ 1) is an important generalization of usual normed spaces with rich topological and geometrical structures, and related study has received a lot of attention (e.g., see Alghamdi et al [2], Balachandran [4], Bayoumi [5], Ennassik and Taoudi [14], Ennassik et al [15], Gholizadeh et al. [18], Granas and Dugundji [21], Jarchow [22], Kalton [23]- [24], Kalton et al [25], Park [35], Rolewicz [40], Xiao and Lu [46], Xiao and Zhu [47], Yuan [49]- [52], and many others). However, to the best of our knowledge, the corresponding basic tools and associated results in the category of nonlinear functional analysis have not been well developed, thus the goal of this paper is to develop some fundamental fixed point theorems for upper semicontinuous set-valued mappings under the framework of p-vector spaces by including topological vector spaces as a special class.…”

Fixed Point Theorems for Upper Semicontinuous Set-valued Mappings in p-Vector Spaces

“…Ben-El-Mechaiekh and Saidi [7], Browder [8], Cellina [11], Chang [12], Ennassik et al [16], Fan [17]- [18], Górniewicz [20], Granas and Dugundji [22], Guo et al [21], Nhu [31], Park [33], Reich [35], Smart [40], Tychonoff [41], Weber [42]- [43], Xiao and Lu [44], Xiao and Zhu [45], Xu [46], Yuan [47]- [50] and many other scholars, and also see the comprehensive references and related discussion under the general framework of topological vector space, or p-vector spaces for non-self set-valued mappings (0 < p ≤ 1).…”