Fixed points for convex continuous mappings in topological vector spaces

Abstract

Abstract. We prove the following result. Let C be a convex compact subset in a topological vector space, and T : C → C a convex continuous mapping. (See Definition 1.1.) Then T has a fixed point. Moreover, continuous mappings that can be approximated by convex continuous mappings also have the fixed point property.

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

“…Proof Apply Theorem 4.4 with p = 1, this completes the proof. Theorem 4.4 improves or unifies corresponding results given by Askoura and Godet-Thobie [6], Cauty [21], Cauty [22], Chen [29], Isac [53], Li [70], Nhu [77], Okon [79], Park [92], Reich [100], Smart [115], Yuan [133], Theorem 3.3 of Ennassik and Taoudi [33], Theorem 3.14 of Gholizadeh et al [41], Xiao and Lu [123], Xiao and Zhu [124,125] under the framework of topological vector spaces.…”

“…A generalization of Schauder's theorem from a normed space to general topological vector spaces is an old conjecture in fixed point theory, which is explained by Problem 54 of the book "The Scottish Book" by Mauldin [75] and stated as Schauder's conjecture: "Every nonempty compact convex set in a topological vector space has the fixed point property, or in its analytic statement, does a continuous function defined on a compact convex subset of a topological vector space to itself have a fixed point?" Recently, this question has been answered by the work of Ennassik and Taoudi [34] by using the p-seminorm methods under locally p-convex spaces; see also related contribution given by Cauty [22], plus the works by Askoura and Godet-Thobie [6], Cauty [21], Chang [23], Chang et al [24], Chen [29], Dobrowolski [32], Gholizadeh et al [41], Isac [53], Li [70], Li et al [69], Liu [72], Nhu [77], Okon [79], Park [90][91][92], Reich [100], Smart [115], Weber [121,122], Xiao and Lu [123], Xiao and Zhu [124,125], Xu [129], Xu et al [130], Yuan [132,133] in both TVS, LCS and related references therein under the general framework of p-vector spaces for nonself set-valued or single-valued mappings (0 < p ≤ 1).…”

Nonlinear Analysis in p-Vector Spaces for Singe-Valued 1-Set Contractive Mappings

“…Cauty [20], Chen [26], Isac [52], Li [68], Nhu [76], Okon [78], Park [91], Reich [99], Smart [114], Yuan [133], Theorem 3.14 of Gholizadeh et al [39], Xiao and Lu [122], Xiao and Zhu [124]- [123] under the framework of LCS for set-valued mappings instead of single-valued functions.…”

Nonlinear Analysis by Applying Best Approximation Method in p-Vector Spaces

“…A generalization of Schauder's theorem from normed space to general topological vector spaces is an old conjecture in fixed point theory which is explained by the Problem 54 of the book "The Scottish Book" by Mauldin [75] as stated as Schauder's conjecture: "Every nonempty compact convex set in a topological vector space has the fixed point property, or in its analytic statement, does a continuous function defined on a compact convex subset of a topological vector space to itself have a fixed point?" Recently, this question has been recently answered by the work of Agarwal et al [1], Alghamdi et al [5], Balaj [8], Górniewicz [46], L. Górniewicz et al [47], Ennassik and Taoudi [33], Ennassik 100 et al [34] by using the p-seminorm methods under p-vector spaces; and also singificant contribution by Cauty [22], plus corresponding contributions by Askoura and Godet-Thobie [6], Cauty [21], Chang [23], Chang et al [24], Chen [28], Dobrowolski [32], Gholizadeh et al [41], Isac [54], Li [70], Li et al [69], Liu [72], Nhu [77], Okon [79], Park [90]- [92], Reich [100], Smart [114], Weber [120]- [121], Xiao and Lu [122], Xiao and Zhu [123]- [124], Xu [128], Xu et al [129], Yuan [131]- [134] and related references therein under the general framework of p-vector spaces for even non-self set-valued mappings (0 < p ≤ 1).…”