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

Abstract

It is known that the class of p-vector spaces $(0 < p \leq 1)$ ( 0 < p ≤ 1 ) is an important generalization of the usual norm spaces with rich topological and geometrical structure, but most tools and general principles with nature in nonlinearity have not been developed yet. The goal of this paper is to develop some useful tools in nonlinear analysis by applying the best approximation approach for the classes of 1-set contractive set-valued mappings in p-vector spaces. In particular, we first develop general fixed point theorems of compact (single-valued) continuous mappings for closed p-convex subsets, which also provide an answer to Schauder’s conjecture of 1930s in the affirmative way under the setting of topological vector spaces for $0 < p \leq 1$ 0 < p ≤ 1 . Then one best approximation result for upper semicontinuous and 1-set contractive set-valued mappings is established, which is used as a useful tool to establish fixed points of nonself set-valued mappings with either inward or outward set conditions and related various boundary conditions under the framework of locally p-convex spaces for $0 < p \leq 1$ 0 < p ≤ 1 . In addition, based on the framework for the study of nonlinear analysis obtained for set-valued mappings with closed p-convex values in this paper, we conclude that development of nonlinear analysis and related tools for singe-valued mappings in locally p-convex spaces for $0 < p \leq 1$ 0 < p ≤ 1 seems even more important, and can be done by the approach established in this paper.

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

“…Remark 2.5. We like to point out that Lemma 2.4 shows that any set-valued mapping with closed p-convex values in p-spaces for 0 < p < 1 has the zero point as its trivial fixed point, thus it is very important to study fixed point and related principle of nonlinear analysis for single-valued (instead of set-valued) mappings for p-vector spaces (for 0 < p < 1), as pointed out and the discussion given in P.40-41 by Yuan [134], thus the most new results established in this paper is for singl-valued mappings for the three classes of (single-valued) continuous mappings which are:…”

“…Though some new results for set-valued mappings in locally p-convex spaces have been developed (see Gholizadeh et al [39], Park [89], Qiu and Rolewicz [98], Xiao and Zhu [124]- [123], Yuan [134] and others), we still like to emphasize that results obtained for set-valued mappings for p-vector spaces may face some challenging in dealing with true nonlinear problems. One example is that the assumption used for "set-valued mappings with closed p-convex values" seems too strong as it always means that the zero element is a trivial fixed point of the set-valued mappings, and this was also discussed in P.40-41 by Yuan [134] for 0 < p ≤ 1.…”

“…Bernstein [9], Chang et al [22], Granas and Dugundji [46], Isac [52], Park [87], Singh et al [111], Zeidler [135]; and also see work contributed by Agarwal and O'Regan [2]- [3], Furi and Pera [37], Park [87], O'Regan [81], O'Regan and Precup [82]), Poincare [96], Rothe [104]- [103], Yuan [132]- [134], Zeidler [135].…”

“…Rothe [104]- [103], Shahzad [109]- [108], Xu [126], Yuan [132]- [134], Zeidler [135], and references therein.…”

“…Recently, this question has been recently answered by the work of Ennassik and Taoudi [31] by using p-seminorm methods under locally p-convex spaces! See also related work in this direction given by Askoura and Godet-Thobie [5], Cauty [19]- [20], Chang [21], Chang et al [22], Chen [26], Dobrowolski [30], Gholizadeh et al [39], Górniewicz [44], Górniewicz et al [45], Isac [52], Li [68], Li et al [67], Liu [70], Nhu [76], Okon [78], Park [89]- [91], Reich [99], Smart [114], Weber [121]- [120], Xiao and Lu [122], Xiao and Zhu [124]- [123], Xu [129], Xu et al [130], Yuan [132]- [134], and related references therein under the general framework of p-vector spaces, in particular, locally p-convex spaces for non-self mappings with various boundary conditions for 0 < p ≤ 1.…”