Fixed point theorem in subsets of topological vector spaces

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.…”

“…Before ending this section, we would also like to remark that compared with the topological method or related arguments used by Askoura et al [6], Cauty [21,22], Nhu [77], Reich [100], the fixed points given in this section improve or unify the corresponding ones given by Alghamdi et al [5], Darbo [30], Liu [72], Machrafi and Oubbi [73], Sadovskii [106], Silva et al [112], Xiao and Lu [123], and those from the references therein.…”

“…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 by Applying Best Approximation Method in p-Vector Spaces

“…From the definition above, we first observe that the definitions (1) to (6) for set-valued mappings can be given by the similar way with the Hausdorff metric H (we omit their definitions here in details by saving spaces); Secondly, if f is a continuous demicompact mapping, then (I − f ) is closed, where I is the identity mapping on X. it is also clear from definitions that every demicompact map is hemicompact in seminorm spaces, but the converse is not true by the example in p.380 by Tan and Yuan [117]. It is evident that if f is demicompact, then I − f is demiclosed.…”

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

“…From the definition above, we first observe that the definitions (1) to (6) for set-valued mappings can be given by the similar way with the Hausdorff metric H (we omit their definitions here in details by saving spaces); Secondly, if f is a continuous demicompact mapping, then (I − f ) is closed, where I is the identity mapping on X. it is also clear from definitions that every demicompact map is hemicompact in seminorm spaces, but the converse is not true by the example in p.380 by Tan and Yuan [117]. It is evident that if f is demicompact, then I − f is demiclosed.…”