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

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arXiv:2210.10286v1 [math.FA] 19 Oct 2022

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

Mappings

George Xianzhi YUAN

Chengdu University, Chengdu 610601, China

Mathematics, Sichuan University, Chengdu 610065, China

Sun Yat-Sen University, Guangzhou 510275, China and

East China University of Science and Technology, Shanghai 200237, China

george yuan99@yahoo.com

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 ≤1. More precisely,

based on the ﬁxed point theorem of single-valued continuous condensing mapping 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 general principle for nonlinear alternative, Leray - Schauder alternative, ﬁxed points for non-self 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 Opial condition. The results given by this paper

not only include the corresponding ones in the existing literature as special cases, but 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 ≤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, is indeed the continuation of the associated

work given recently by Yuan [134] therein, the attention is given to the study of nonlinear analysis for set-valued

mappings in locally p-convex spaces for 0 < p ≤1.

Keywords: Nonlinear analysis, p-convex, Fixed points, Measure of noncompactness, Condensing mapping, 1-set

contractive mapping, Semiclosed mapping, Nonexpansive mapping, Best approximation, Nonlinear alternative,

Leray - Schauder alternative, Demiclosed principle, Opial condition, p-inward and p-outward set, p-vector space,

locally p-convex space, Uniform convex space.

AMS Classiﬁcation: 47H04, 47H10, 46A16, 46A55, 49J27, 49J35, 52A07, 54C60, 54H25, 55M20

Preprint submitted to The Report October 20, 2022

1. Introduction

It is known that the class of p-semi-norm 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

work by Alghamdi et al.[4], Balachandran [6], Bayoumi [7], Bayoumi et al.[8], Bernu´ees and Pena [10], Ding [29],

Ennassik and Taoudi [31], Ennassik et al.[32], Gal and Goldstein [38], Gholizadeh et al.[39], Jarchow [51], Kalton

[53]-[54], Kalton et al.[55], Machraﬁ and Oubbi [72], Park [89], Qiu and Rolewicz [98], Rolewicz [102], Silva et

al.[113], Simons [110], Tabor et al.[115], Tan [116], Wang [119], Xiao and Lu [122], Xiao and Zhu [124]-[123], Yuan

[134], and many others. However, to the best of our knowledge, the corresponding basic tools and associated results

in the category of nonlinear functional analysis for p-vector spaces have not been well developed, in particular for

the three classes of (single-valued) continuous nonlinear mappings which are: 1) condensing; 2) 1-set contractive;

and 3) semiclosed 1-set contractive operators under locally p-convex spaces. Our goal in 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 ≤1. More precisely, based on the ﬁxed point theorem

of single-valued continuous condensing mapping 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, which are 1): condensing; 2): 1-set contractive; and 3): semiclosed 1-

set contractive in locally p-convex spaces. Then these new results are used to establish general principle for nonlinear

alternative, Leray - Schauder alternative, ﬁxed points for non-self 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 Opial condition. The results given by this paper not only include the corresponding

results in the existing literature as special cases, but 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 ≤1.

In addition, we like to point out that 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, is very important,

and also the continuation of the work given recently by Yuan [134] therein, the attention was given to establish new

results on ﬁxed points, principle of nonlinear alternative for nonlinear mappings mainly on set-valued (instead of

single-valued) mappings developed in locally p-convex spaces for 0 < p ≤1. 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.

For the development since 1920s on the development, and in particular, how the ﬁxed points for non-self

mappings, best approximation method and related to the study on some key aspects of nonlinear analysis related to

Birkhoﬀ-Kellogg problems, nonlinear alternative, Leray - Schauder alternative, KKM principle, best approximation,

and related topics, readers can ﬁnd some most important contributions by Birkhoﬀ and Kellogg [11] in 1920’s, Leray

and Schauder [65] in 1934’s, Fan [35] in 1969; plus the related comprehensive references given by Agarwal et al.[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].

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, math-

ematical program, operation research; and in particularly, the one approach well accepted for study of nonlinear

problems in optimization, complementarity problems and of variational inequalities problems and so on, strongly

based on today called Fan’s best approximation theorem given by Fan [33]-[36] in 1969 which acts as a very powerful

tool in nonlinear analysis, and see the book of Singh et al.[111] 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

principle of Leray-Schauder alterative in topological vector spaces (TVS), and local topological vector spaces (LCS)50

which are comprehensively studied by Chang et al.[22], Chang et al.[25]-[23], Carbone and Conti [18], Ennassik and

Taoudi [31], Ennassik et al.[32], Isac [52], Granas and Dugundji [46], Kirk and Shahzad [58], Liu [70], Park [90],

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

On the other hand, since the celebrated so-called KKM principle established in 1929 in [60], was based on the

celebrated Sperner combinatorial lemma and ﬁrst applied to a simple proof of the Brouwer ﬁxed point theorem.

Later it became clear that these three theorems are mutually equivalent and they were regarded as a sort of

mathematical trinity (Park [90]). Since Fan extended the classical KKM theorem to inﬁnite-dimensional spaces in

1961 by Fan [34]-[36], there have been a number of generalizations and applications in numerous areas of nonlinear

analysis, and ﬁxed points in TVS and LCS as developed by Browder [12]-[17] and related references therein. Among

them, Schauder’s ﬁxed point theorem [112] in normed spaces is one of the powerful tools in dealing with nonlinear

problems in analysis. Most notably, it has played a major role in the development of ﬁxed point theory and related

nonlinear analysis and mathematical theory of partial and diﬀerential equations and others. 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 [74] 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 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], G´orniewicz [44], G´orniewicz 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.

The goal of this paper is to establish the general new tools of nonlinear analysis under the framework of gen-

eral locally p-convex space (p-seminorm spaces) for general condensing mappings, 1-set contractive mappings, and

semiclosed mappings (here 0 < p ≤1), and we do wish these new results such as best approximation, theorems

of Birkhoﬀ-Kellogg type, nonlinear alternative, ﬁxed point theorems for non-self (singl-valued) continuous opera-

tors with various boundary conditions, Rothe, Petryshyn type, Altman type, Leray-Schedule types, related others

nonlinear problems would play important roles for the nonlinear analysis of p-seminorm spaces for 0 < p ≤1. In

addition, our results also show that ﬁxed point theorem for condensing continuous mappings for closed p-convex

subsets provide solutions for Schauder’s conjecture since 1930’a in the aﬃrmative way under the general setting

of p-vector spaces (which may not locally convex, see related study given by Ennassik and Taoudi [31], Kalton

[53]-[54], Kalton et al.[55], Jarchow [51], Roloewicz [102] on this direction).

The paper has seven sections. Section 1 is the introduction. Section 2 describes general concepts for the p-convex

subsets of topological vector spaces (0 < p ≤1). In Section 3, then some basic results of KKM principle related

to abstract convex spaces are given. In Section 4, as the application of the KKM principle in abstract convex

spaces which including p-convex vector spaces as a special class (0 < p ≤1) by combining the embedding lemma for

compact p-convex subsets from topological vector spaces into locally p-convex spaces, we provide general ﬁxed point

theorems for condensing continuous mappings for both single-valued version in topological vector spaces; and upper

semi-continuous set-valued version in locally convex spaces deﬁned on closed p-convex subsets for 0 < p ≤1. The

Sections 5, 6 and 7 mainly focus on the study of nonlinear analysis for 1-set contractive (single-valued ) continuous

mappings in locally p-convex vector spaces to establish the general existence theorems for solutions of Birkhoﬀ-

Kellogg (problem) alternative, general principle of nonlinear alterative, and including Leray-Schauder alternative,

Rothe type, Altman type associated with diﬀerent boundary conditions. The Sections 8, 9 and 10 mainly focus

on the study of new results based on semiclosed 1-set contractive (single-valued) continuous mappings related to

nonlinear alternative principles, Birkhoﬀ-Kellogg theorems, Leray-Schauder alternative and non-self operations from

general locally p-convex spaces to uniformly convex Banach spaces for nonexpansive mappings, or locally convex

topological spaces with Opial condition.

For the convenience of our discussion, throughout this paper, we always assume that all p-vector spaces are

Hausdorﬀ for 0 < p ≤1 unless speciﬁed; and we also denote by Nthe set of all positive integers, i.e., N:= {1,2,··· ,}.100

2. Some Basic Results for p-Vector Spaces

For the convenience of self-containing reading dor readers, we recall some notion and deﬁnitions for p-convex

vector spaces below as summarized by Yuan [134], see also Balachandran [6], Bayoumi [7], Jarchow [51], Kalton

[53], Rolewicz [102], Gholizadeh et al.[39], Ennassik and Taoudi [31], Ennassik et al.[32], Xiao and Lu [122], Xiao

and Zhu [124] and references therein for more in details.

Deﬁnition 2.1. A set Ain a vector space Xis said to be p-convex for 0 < p ≤1 if, for any x, y ∈A, 0 ≤s, t ≤1

with sp+tp= 1, we have s1/px+t1/py∈A; and if Ais 1-convex, it is simply called convex (for p= 1) in general

vector spaces; the set Ais said to be absolutely p-convex if s1/px+t1/py∈Afor 0 ≤ |s|,|t| ≤ 1 with |s|p+|t|p≤1.

Deﬁnition 2.2. If Ais a subset of a topological vector space X, the closure of Ais denoted by A, then the p-convex

hull of Aand its closed p-convex hull denoted by Cp(A), and Cp(A), respectively, which is the smallest p-convex

set containing A, and the smallest closed p-convex set containing A, respectively.

Deﬁnition 2.3. Let Abe p-convex and x1,··· , xn∈A, and ti≥0,

i= 1. Then

tixiis called a p-convex

combination of {xi}for i= 1,2,· · · , n. If

|ti|p≤1, then

tixiis called an absolutely p-convex combination.

It is easy to see that

tixi∈Afor a p-convex set A.

Deﬁnition 2.4. A subset Aof a vector space Xis called circled (or balanced) if λA ⊂Aholds for all scalars λ

satisfying |λ| ≤ 1. We say that Ais absorbing if for each x∈X, there is a real number ρx>0 such that λx ∈A

for all λ > 0 with |λ| ≤ ρx.

By the deﬁnition 2.4, it is easy to see that the system of all circled subsets of Xis easily seen to be closed under

the formation of linear combinations, arbitrary unions, and arbitrary intersections. In particular, every set A⊂X

determines a smallest circled subset ˆ

Aof Xin which it is contained: ˆ

Ais called the circled hull of A. It is clear

that ˆ

A=∪|λ|≤1λA holds, so that Ais circled if and only if (in short, iﬀ) ˆ

A=A. We use ˆ

Ato denote for the closed

circled hull of A⊂X.

In addition, if Xis a topological vector space, we use the int(A) to denote the interior of set A⊂Xand if

0∈int(A), then int(A) is also circled, and using ∂A to denote the boundary of Ain Xunless speciﬁed.

Deﬁnition 2.5. A topological vector space is said to be locally p-convex if the origin has a fundamental set of

absolutely p-convex 0-neighborhoods. This topology can be determined by p-seminorms which are deﬁned in the

obvious way (see P.52 of Bayoumi [7], Jarchow [51] or Rolewicz [102]).

Deﬁnition 2.6. Let Xis a vector space and R+is a non-negative part of a real line R. Then a mapping

P:X−→ R+is said to be a p-seminorm if it satisﬁes the requirements for (0 < p ≤1)

(i) P(x)≥0 for all x∈X;

(ii) P(λx) = |λ|pP(x) for all x∈Xand λ∈R;

(iii) P(x+y)≤P(x) + P(y) for all x, y ∈X.

An p-seminorm Pis called a p-norm if x= 0 whenever P(x) = 0, so a vector space with a speciﬁc p-norm is

called an p-normed space, and of course if p= 1, Xis a normed space as discussed beofe (e.g., see Jarchow [51]).

By Lemma 3.2.5 of Balachandran [6], the following proposition gives a necessary and suﬃcient condition for an

p-seminorm to be continuous.

Proposition 2.1. Let Xbe a topological vector space, Pis a p-seminorm on Xand V:= {x∈X:P(x)<1}.

Then Pis continuous if and only if 0 ∈int(V), where int(V) is the interior of V.

Now given an p-seminorm P, the p-seminorm topology determined by P(in short, the p-topology) is the class

of unions of open balls B(x, ǫ) := {y∈X:P(y−x)< ǫ}for x∈Xand ǫ > 0.

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arXiv:2210.10286v1[math.FA]19Oct2022NonlinearAnalysisinp-VectorSpacesforSinge-Valued1-SetContractiveMappingsGeorgeXianzhiYUANChengduUniversity,Chengdu610601,ChinaMathematics,SichuanUniversity,Chengdu610065,ChinaSunYat-SenUniversity,Guangzhou510275,ChinaandEastChinaUniversityofScienceandTechnology,Sh...

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

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