# Further Properties of 1- and 2-Dimensional U- and W-Convexity and Fixed Point of Nonexpansive Mappings in Banach SpacesX and X∗

Authors: Ji Gao
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we further studied properties of the modulus ofn-dimensional U-convexity and the modulus of n-dimensional U-flatness when n = 1 (2-dimensional character) and n = 2 (3-dimensional character). The new properties of these moduli are investigated, and the relationships between these moduli and other geometric parameters of Banach spaces are studied. Some results on fixed point theory for nonexpansive mappings and normal structure in Banach spaces are obtained.


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## Body

## 1. Introduction

LetX be a real Banach space with the dual space X∗. Let BX=x∈X:x≤1 and SX=x∈X:x=1 be the unit ball and the unit sphere of X, respectively. Let X∗ be the dual space of X. Let ∇x⊂SX∗ denote the set of norm 1 supporting functionals of x∈SX.Brodskiĭ and Mil’man [1] introduced the following geometric concepts in 1948.Definition 1.
LetX be a Banach space. A nonempty bounded and convex subset K of X is said to have normal structure if, for every convex subset C of K that contains more than one point, there is a point x0∈C such that(1)supx0−y:y∈C<diamC.
A Banach spaceX is said to have(a)
Normal structure if every bounded convex subset ofX has normal structure.(b)
Weak normal structure if every weakly compact convex setK of X has normal structure.(c)
Uniform normal structure if there exists0<c<1 such that, for every bounded closed convex subset C of K that contains more than one point, there is a point x0∈C such that(2)supx0−y:y∈C<c⋅diamC.Remark 1.
The following facts are known:(a)
Uniform normal structure⇒ normal structure ⇒ weak normal structure(b)
For a reflexive spaces, normal structure⇔ weak normal structure
Kirk [2] proved that if a Banach space X has weak normal structure, then it has weak fixed point property, that is, every nonexpansive mapping from a weakly compact and convex subset of X to itself has a fixed point.
Letℕ be the set of all natural numbers and n∈ℕ.
For two sets of vectorsxii=1n+1⊆X and fii=2n+1⊆X∗, the n+1×n+1 matrix,(3)11⋯1x1,f2x2,f2⋯xn+1,f2⋮⋮⋱⋮x1,fn+1x2,fn+1⋯xn+1,fn+1,is denoted by mx1,x2,…,xn+1;f2,f3,…,fn+1 [3].
Gao and Saejung [3] introduced the concept of volume by the convex hull of x1,x2,…,xn+1 in X of(4)vx1,x2,…,xn+1≔supdetmx1,x2,…,xn+1;f2,f3,…,fn+1,where the supremum is taken over all fi∈∇xi, where i=2,3,…,n+1.Definition 2.
(see [3]). Let νXn=supvx1,x2,…,xn+1:x1,x2,…xn+1∈SX be the upper bound of all n-dimensional volume in X.
The following result was proved [3].Proposition 1.
For a Banach spaceX with dimX>n, we have νXn≥2.Gao and Saejung introduced the concept of the modulus ofn-dimensional U-convexity of X as follows.Definition 3.
(see [3]). Let X be a Banach space. Define(5)UXnε=inf1−1n+1x1+x2+…+xn+1:x1,x2,…,xn+1∈SX,vx1,x2,…,xn+1≥ε,where 0≤ε≤νXn is the modulus of n-dimensional U-convexity of X.
The following results were proved in [3] too.Theorem 1.
(a)
UXnε is an increasing and continuous function in 0,νXn for any n∈ℕ.(b)
IfX is a Banach space with UXn1>0 for some n∈ℕ, then X is super-reflexive.(c)
IfX is a Banach space with UXn1>0 for some n∈ℕ, then X has uniform normal structure
The following results were proved in [4].Theorem 2.
IfX is a Banach space with UXn2n+1/2n>1−1/n+1 for some n∈ℕ and n is even, then X is reflexive.Letn=2, and we have the following.Corollary 1.
IfX is a Banach space with UX25/4>2/3, then X is reflexive.The following results were also proved in [4].Theorem 3.
IfX is a Banach space with UX25/4>2/3 and UX25/2>5/6, then X has uniform normal structure.Gao and Saejung also introduced the concept of the modulus ofn-dimensional U-flatness as follows.Definition 4.
(see [5]). Let X be a Banach space. Define(6)WXnε=sup1−1n+1x1+x2+…+xn+1:x1,x2,…,xn+1∈SX,vx1,x2,…,xn+1≤ε,where 0≤ε≤νXn is the modulus of n-dimensional U-flatness of X.Remark 2.
(a) The name of the modulus ofU-flatness is defined by comparing with the name of the modulus of U-convexity of X. (b) UXnε≤WXnε in 0,νXn.
The following results were proved in [5] too.Theorem 4.
(a)
WXnε is an increasing and continuous function in 0,νXn for any n∈ℕ.(b)
IfX is a Banach space with WX15/3<2/3, or WX27/9<2/3, then X is reflexive.(c)
IfX is a Banach space with WXn2n+1/2n<1−1/n+1, where n∈ℕ, then X is super-reflexive.(d)
Suppose thatX is a Banach space satisfying one of the following conditions:(i)
WXn1<1−1/n+1, for some n∈ℕ with n≥3(ii)
WXn2n+1/2n<1−1/n+1, for n=1 or n=2
Then,X has uniform normal structure.Recently, Gabeleh introduced concepts of pointwise cyclic relatively nonexpansive mapping and weak proximal normal structure for an extension of geometric property of normal structure. Some interesting results are obtained there [6].In this paper, we studied further properties of the modulus ofn-dimensional U-convexity and the modulus of n-dimensional U-flatness when n = 1 (2-dimensional character) and n = 2 (3-dimensional character). The new properties of these moduli are investigated and the relationships between these moduli and other geometric parameters of Banach spaces are studied. Some results on fixed point theory for nonexpansive mappings and normal structure in Banach spaces are obtained.


## 2. Main Results

Lemma 1 (Bishop-Phelps-Bollobás, see [7]).
LetX be a Banach space, and let 0<ε<1. Given z∈BX and h∈SX∗ with 1−z,h<ε2/4, then there exist y∈SX and g∈∇y such that y−z<ε and g−h<ε.Sincez,h is a continuous function in norm for z∈X and norm for h∈X∗, Lemma 1 can be stated as follows.Lemma 2.
LetX be a Banach space, and let 0<ε<1. Given z∈SX and h∈BX∗ with 1−z,h<ε2/4, then there exist y∈SX and g∈∇y such that y−z<ε and g−h<ε.Theorem 5 (see [8]).
LetX be a Banach space. Then, X is not reflexive if and only if, for any 0<δ<1, there are a sequence xn⊆SX and a sequence fn⊆SX∗ such that(a)
xm,fn=δ whenever n≤m(b)
xm,fn=0 whenever n>mTheorem 6.
IfX is a Banach space with UX∗12>3/4, or WX∗12<1/2, then X is reflexive.Proof.
Suppose thatX is not reflexive. Let 0<δ<1 be given.
Letf1∈SX∗,−f1+f2/2∈BX∗, and x1∈SX∗∗,−x2∈SX∗∗. Then, we have(7)mf1,−f1+f22;x1,−x2=11f1,−x2−f1+f22,−x2=11−δδ=2δ.
We also have(8)14≤x1,12f1−f1+f22≤12f1−f1+f22=12f1−f22≤12.
So,(9)12≤1−f1−f1+f2/22≤34.
Sinceδ can be arbitrarily close to 1, by using Lemma 2 and the definition of UX1ε and WX1ε, we have if UX∗12>3/4, or WX∗12<1/2, then X is reflexive.Theorem 7.
IfX is a Banach space with UX∗22/3>2/3, or WX∗22/3<5/9, then X is reflexive.Proof.
Suppose thatX is not reflexive. Let 0<δ<1 be given.
Letf1∈SX∗,−f1+f2/2∈BX∗,f1+f2+f3/3∈BX∗, and x1∈SX∗∗,−x2∈SX∗∗,andx3∈SX∗∗.
Then, we have(10)mf1,−f1+f22,f1+f2+f33;x1,−x2,x3=111f1,−x2−f1+f22,−x2f1+f2+f33,−x2f1,x3−f1+f22,x3f1+f2+f33,x3=111−δδ−23δδ−δδ=23δ2.
We also have(11)13f1−f1+f22+f1+f2+f33=5f1−f2+2f318≤49,13f1−f1+f22+f1+f2+f33=5f1−f2+2f318≥5f1−f2+2f318,x3=13.
We have(12)59≤1−13f1−f1+f22+f1+f2+f33≤23.
Sinceδ can be arbitrarily close to 1, by using Theorem 2.2 and the definition of UX2ε and WX2ε, we have if UX∗22/3>2/3, or WX∗22/3<5/9, then X is reflexive.
The following result refers to a Banach space with weak∗ sequentially compact unit ball of the dual. Notice that this property is satisfied by reflexive or separable Banach spaces and by those that admit an equivalent smooth norm.Lemma 3 (see [9]).
IfX is a Banach space with BX∗ weak∗ sequentially compact and fails to have weak normal structure, then, for any ε>0, there are sequence xn⊆SX and sequence fn⊆SX∗ such that(a)
xi−xj−1<ε, whenever i≠j(b)
xi,fi=1, whenever 1≤i≤∞(c)
xj,fi<ε, whenever i≠j(d)
fi−fj>2−ε, whenever i≠jTheorem 8.
IfX is a Banach space with UX∗12>1/2, or WX∗12<1/2, then X has weak normal structure.Proof.
Suppose thatX does not have weak normal structure. Let ε>0 be given. Since 2≥f2−f1≥f2−f1,x2−x1/1+ε≥2−2ε/1+ε, we have 1−ε/1+ε≤f2−f1/2≤1.
Letf1∈SX∗,f2−f1/2∈BX∗; and x1∈SX∗∗andx2−x1/x2−x1∈SX∗∗. Then, we have(13)mf1,f2−f12;x1,x2−x1x2−x1=11f1,x2−x1x2−x1f2−f12,x2−x1x2−x1=11−1+aεx2−x11+bεx2−x1=2+cε,where a, b, and c are constants and ε is arbitrarily small.
We also have(14)12−dε≤x1−x2x1−x2,12f1−f22≤12f1+f2−f12=12f1+f22≤12,where d is a constant.
So,(15)12≤1−12f1+f2−f12=12+dε.
By using Theorem 2.2 and the definition ofUX1ε and WX1ε, we have if UX∗12>1/2, or WX∗12<1/2, then X has weak normal structure.Theorem 9.
IfX is a Banach space with UX∗29/4>5/6, or WX∗29/4<2/3, then X has weak normal structure.Proof.
Suppose thatX does not have weak normal structure. Let ε>0 be given.
Letf1∈SX∗,f2−f1/2∈BX∗,f3−f2/2∈BX∗; and x1∈SX∗∗,x2−x1/x2−x1∈SX∗∗,andx3−x2/x3−x2∈SX∗∗. Then, similar to the proof of Theorem 8, we have(16)mf1,f2−f12,f3−f22;x1,x2−x1x2−x1,x3−x2x3−x2=111f1,x2−x1x2−x1f2−f12,x2−x1x2−x1f3−f22,x2−x1x2−x1f1,x3−x2x3−x2f2−f12,x3−x2x3−x2f3−f22,x3−x2x3−x2=111−1+aε1+bε−12+cεdε−12+eε1+fε=94+gε,where a, b, c, d, e, f, and g are constants and ε is arbitrarily small.
We also have(17)16−ε≤x1,f1+f36≤13f1+f2−f12+f3−f22=f1+f36≤13,and we have(18)23≤1−f1+f36≤56+ε.
By using Theorem 2.2 and the definition ofUX2ε and WX2ε, we have if UX∗29/4>5/6, or WX∗29/4<2/3, then X has weak normal structure.Definition 5.
(see [10, 11]). Let X and Y be Banach spaces. We say that Y is finitely representable in X if, for any ε>0 and any finite dimensional subspace N⊆Y, there is an isomorphism T:N⟶X such that, for any y∈N, 1−εy≤Ty≤1+εy.
We say thatX is super-reflexive if any space Y which is finitely representable in X is reflexive.Theorem 10.
Suppose thatX is a Banach space satisfying one of the following conditions:(a)
UX∗12>3/4(b)
UX∗22/3>2/3(c)
WX∗12<1/2(d)
WX∗22/3<5/9
Then,X is super-reflexive.Proof.
The proof of this theorem follows from the fact that if a Banach spaceY is finitely representable in X, then UYnε=UXnε and WYnε=WXnε, for any n∈ℕ and for any ε∈νXn.
We consider the uniform normal structure. To discuss this result, let us recall the concept of the “ultra”-technique.
Letℱ be a filter of an index set I, and let xii∈I be a subset in a Hausdorff topological space X, xii∈I is said to converge to x with respect to ℱ, denoted by limℱxi=x, if, for each neighborhood U of x, i∈I:xi∈U∈ℱ. A filter U on I is called an ultrafilter if it is maximal with respect to the ordering of the set inclusion. An ultrafilter is called trivial if it is of the form A:A⊆I,i0∈A for some i0∈I. We will use the fact that if U is an ultrafilter, then(i)
For anyA⊆I, either A⊆U or I−A⊆U(ii)
Ifxii∈I has a cluster point x, then limUxi exists and equals to x
LetXii∈I be a family of Banach spaces and let l∞I,Xi denote the subspace of the product space equipped with the norm xi=supi∈Ixi<∞.Definition 6.
(see [12, 13]). Let U be an ultrafilter on I and let NU=xi∈l∞I,Xi:limUxi=0. The ultraproduct of Xii∈I is the quotient space l∞I,Xi/NU equipped with the quotient norm.
We will usexiU to denote the element of the ultraproduct. It follows from remark (ii) above and the definition of quotient norm that(19)xiU=limUxi.
In the following, we will restrict our index setI to be ℕ, the set of natural numbers, and let Xi=X,i∈ℕ, for some Banach space X. For an ultrafilter U on ℕ, we use XU to denote the ultraproduct. Note that if U is nontrivial, then X can be embedded into XU isometrically.Lemma 4 (see [13]).
Suppose thatU is an ultrafilter on ℕ and X is a Banach space. Then, X∗U≅XU∗ if and only if X is super-reflexive; in this case, the mapping J defined by(20)xiU,JfiU=limUxi,fi,for allxiU∈XU,which is the canonical isometric isomorphism from X∗U onto XU∗.Theorem 11.
LetX be a super-reflexive Banach space. Then, for any nontrivial ultrafilter U on ℕ and for all n∈ℕ and ε>0, we have UXUnε=UXnε, and WXUnε=WXnε.Proof.
The proof is the same as the proof of Theorem 2.17 in [3].Lemma 5 (see [14]).
IfX is a super-reflexive Banach space, then X has uniform normal structure if and only if XU has normal structure.Theorem 12.
Suppose thatX is a Banach space satisfying one of the following conditions:(a)
UX∗12>3/4(b)
UX∗22/3>2/3 and zX∗12>1/2, or UX∗22/3>2/3 and WX∗12<1/2, or UX∗22/3>2/3 and UX∗29/4>5/6, or UX∗22/3>2/3 and WX∗29/4<2/3(c)
WX∗12<1/2(d)
WX∗22/3<5/9 and UX∗12>1/2, or WX∗22/3<5/9 and WX∗12<1/2, or UX∗22/3<5/9 and UX∗29/4>5/6, or WX∗22/3<5/9 and WX∗29/4<2/3
Then,X has uniform normal structure.Proof.
The results (a) and (c) follow directly from Theorem8, Theorems 10 and 11, and Lemma 5. The results (b) and (d) follow directly from Theorems 9, 10, and 11 and Lemma 5.



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