Several Geometric Properties in Banach Spaces and Their Further Application in Orlicz Spaces

Abstract

In this paper, locally nearly uniformly convex $(LNUC)$ is studied in Banach space. Furthermore, the implication relationship between $(LNUC)$ and the Kadec–Klee property $(KK)$, the fixed–point property $(FPP)$ are investigated in Banach space. Finally, the relationship between the uniform Kadec−Klee property $(UKK)$, the coordinate-wise uniform Kadec–Klee property $(UK{K}_C)$, the coordinate-wise Kadec–Klee property $(H_c)$ and ${\delta }_2$ conditions are investigated in Orlicz sequence spaces equipped with the Orlicz norm, meanwhile we get a criteria that Orlicz sequence spaces equipped with the Orlicz norm are $(LNUC)$.

1. Introduction

Let $X$ be a Banach space and $X^{\ast }$ be its dual space. By $U(X)$ and $S(X)$ denoted the closed unit ball and the unit sphere of Banach space $X$, respectively.

By $U(X^{\ast })$ and $S(X^{\ast })$ denoted the closed unit ball and the closed unit sphere of the dual space of $X^{\ast }$, respectively. $w$, $w^{\ast }$ and $‖\cdot ‖$ represent the $w$, $w^{\ast }$ and norm topologies, respectively. By $co(A)$ and $\overline{co}(A)$, we denote the convex hull and the closed convex hull of the set $A$, respectively. By $x_n\stackrel{w}{\to }x$ we denote $\left\{x_n\right\}$ converging to $x$, and $x_n^{\ast }\stackrel{w^{\ast }}{\to }{x}^{\ast }$ denotes $\left\{x_n^{\ast }\right\}$ $weakl{y}^{\ast }$ converging to $x^{\ast }.$ Let us define the following:

$$\begin{gathered} sep(x_n)=\inf \{\parallel x_n-x_m\parallel :m\ne n\}, \\ {B}_{\delta }(0)=\{x:\parallel x\parallel \le \delta \}. \end{gathered}$$

In 1973, M. M. Day [1] introduced the concept of the $(KK)$ property (also called $(H)$ property).

We say that a Banach space $X$ has the Kadec−Klee $(KK)$ property if for any $x\in S(X)$, $\left\{x_n\right\}\subset X$, $\parallel x_n\parallel \to \parallel x\parallel$ and $x_n\stackrel{w}{\to }x$, then $\parallel x_n-x\parallel \to 0$.

In 1980, Huff [2] introduced the concepts of $(UKK)$ spaces and $(NUC)$ spaces.

We say that a Banach space $X$ has the uniform Kadec−Klee $(UKK)$ property if for every $\epsilon >0$, there exists $0<\delta <1$ such that $\left\{x_n\right\}\subset U(X)$, $x_n\stackrel{w}{\to }x$ and $sep(x_n)\ge \epsilon$, then $x\in B_{\delta }(0)$.

A Banach space $X$ is called a Nearly Uniform Convex $(NUC)$ space if for every $\epsilon >0$, there exists $0<\delta =\delta (\epsilon )<1$ such that $\left\{x_n\right\}\subset U(X)$ with $sep(x_n)\ge \epsilon$, then there is an $N\ge 1$ and scalars ${\lambda }_1,\cdots ,{\lambda }_N\ge 0$ with ${\displaystyle \sum _{n=1}^N{\lambda }_n}=1$ such that $\parallel \sum {\lambda }_n{x}_n\parallel \le 1-\delta$.

Huff showed that a Banach space $X$ is $(NUC)$ if and only if $X$ is $(UKK)$ and reflexive.

It is widely recognized that the condition equivalent to nearly uniformly convex $(NUC)$ was separately articulated in [3,4].

A mapping $T:C\to X$ defined on a subset $C$ of a Banach space $X$ is said to be non-expansive if $\parallel Tx-Ty\parallel \le \parallel x-y\parallel$ for all $x,y\in C$.

For the fixed–point theory of nonexpansive mappings is referred to Kirk [5,6], he also proved that if a weakly compact subset $K$ of $X$ that has a normal structure, then any nonexpansive mapping on $K$ has a fixed point.

Many research have focused on $(FPP)$; for example A. Granas [7] covers topology, functional analysis, operator theory, and other related fields, focusing on the classical fixed point theory for continuous mappings, as well as the theoretical achievements of scholars such as Poincaré and Brouwer and their modern extensions. A. Górniewicz, L. [8] systematically elaborates on the relevant achievements and methods of the topological fixed point theory for multivalued mappings, and also incorporates the applications of this theory in fields such as differential inclusions, convex analysis, game theory, and mathematical economics., and D. van Dulst [9] researched equivalent norms and the fixed–point property for non-expansive mappings.

Jean Saint Raymond studied the relationship between the Kadec-Klee property, the Dual Kadec-Klee property and the fixed point property (see [10,11]).

In 1999, P. Foralewski H. and Hudzik [12] introduced a new property the coordinate-wise Kadec–Klee property $(H_c)$.

Let $X$ be a Köthe sequence space. Say that $X$ has the coordinate-wise Kadec–Klee property $(H_c)$ if for every $x\in X$ and every sequence $\left\{x_n\right\}\subset X$ such that $\parallel x_n\parallel \to \parallel x\parallel$ and $x_n(i)\to x(i)$ for all $n\in \mathbb{N}$, then we have $\parallel x-x_n\parallel \to 0$.

In 2003, T. Zhang [13] introduced a new property the coordinate-wise uniform Kadec–Klee property $(UK{K}_c)$ for a Banach space and studied it in the Orlicz sequence space.

Let $X$ Köthe sequence space. Say that $X$ has the coordinate-wise uniform Kadec–Klee property $(UK{K}_c)$ if for every $\epsilon >0$ there exists $\delta >0$ such that $\left\{x_n\right\}\subset U(X)$, $sep(x_n)\ge \epsilon$, $\parallel x_n\parallel \to \parallel x\parallel$ and $x_n(i)\to x(i)$ implies $\parallel x\parallel <1-\delta$.

In 1932, W. Orlicz [14] introduced the Orlicz space, and in 1936, he introduced the Orlicz norm in Orlicz space [15]. In 1955, W. A. Luxemburg [16] introduced the Luxemburg norm in Orlicz space.

One aim of this study is to give the sufficient and necessary condition of $(LNUC)$ in the Orlicz sequence space. We next recall some basic facts about Orlicz spaces.

A mapping $\mathsf{\Phi }$ is called a Orlicz function provided that $\mathsf{\Phi }:R\to R^{+}=[0,+\infty )$ is even, convex on $[0,+\infty )$ and $\mathsf{\Phi }(0)=0$.

Let $p(u)$ stand for the right derivative of $\mathsf{\Phi }$ at $u\in R$ and $q(v)$ is left inverse of $p(v)$. Then $\mathsf{\Phi }(u)={\int }_0^{|u|}p(t)dt$ and $\mathsf{\Psi }(v)={\int }_0^{|v|}q(s)ds$ be a pair of complementary functions in the sense of Young.

A Orlicz function $\mathsf{\Phi }$ is said to be $N-$ function if it also satisfies $\underset{u\to 0}{\lim }{\displaystyle \frac{\mathsf{\Phi }(u)}{u}}=0$ and $\underset{u\to +\infty }{\lim }{\displaystyle \frac{\mathsf{\Phi }(u)}{u}}=+\infty$.

Let $l^o$ denote the space of all real sequence. Given any Orlicz function $\mathsf{\Phi }$, we define on $l^o$ the convex modular $I_{\mathsf{\Phi }}$ by

$$I_{\mathsf{\Phi }}\left(x\right)={\displaystyle \sum _{i=1}^{\infty }\mathsf{\Phi }\left(x(i)\right)},$$

for any $x\in l^o$.

For following details on Orlicz space please see [17].

The Orlicz sequence space $l_{\mathsf{\Phi }}$ is defined as the set

$$l_{\mathsf{\Phi }}=\left\{x=(x(1),x(2),\dots ):I_{\mathsf{\Phi }}(\lambda x)=\sum _{i=1}^{\infty }\mathsf{\Phi }(\lambda |x(i)|)<\infty for some \lambda >0\right\}.$$

$$h_{\mathsf{\Phi }}=\left\{x\in l^o:I_{\mathsf{\Phi }}(\lambda x)<+\infty for any \lambda >0\right\}.$$

The spaces $l_{\mathsf{\Phi }}$ and $h_{\mathsf{\Phi }}$ endowed with the following norm

$$\parallel x\parallel =\inf \left\{\lambda >0:I_{\mathsf{\Phi }}\left({\displaystyle \frac{x}{\lambda }}\right)\le 1\right\},$$

or the Orlicz norm is defined, respectively, by

$$\parallel x{\parallel }^o=\sup \left\{\sum _{i=1}^{\infty }|x(i)y(i)|:I_{\mathsf{\Psi }}(y)=\sum _{i=1}^{\infty }\mathsf{\Psi }(|y(i)|)\le 1\right\}=\underset{k>0}{\inf }{\displaystyle \frac{1}{k}}[1+I_{\mathsf{\Phi }}(kx)],$$

are Banach spaces. It is well known that ${‖x‖}_{\mathsf{\Phi }}^o={\displaystyle \frac{1}{k}}(1+I_{\mathsf{\Phi }}(kx))$ if and only if $k\in K(x)=[k_x^{*},k_x^{**}]$, where

$$\begin{array}{l}{k}_x^{*}=k^{*}(x)=\inf \{k>0:I_{\mathsf{\Phi }}(p(kx))\ge 1\},\\ k_x^{**}=k^{**}(x)=\sup \{k>0:I_{\mathsf{\Phi }}(p(kx))\le 1\}.\end{array}$$

We say that the Orlicz function $\mathsf{\Phi }$ satisfies the ${\delta }_2$-condition ($\mathsf{\Phi }\in {\delta }_2$ for short) if there exist a constant $k\ge 2$ such that $\mathsf{\Phi }(2u)\le k\mathsf{\Phi }(u)$. For every $|u|\le u_0$.

We say that the Orlicz function $\mathsf{\Phi }$ satisfies the ${\delta }_2^o$-condition ($\mathsf{\Phi }\in {\delta }_2^o$ for short) if its complementary function $\mathsf{\Psi }$ satisfies the ${\delta }_2$-condition. $\mathsf{\Phi }\in {\delta }_2^0$ if and only if $\mathsf{\Psi }\in {\delta }_2$.

2. Materials and Methods

In this study, we introduced a new geometric property locally nearly uniformly convex $(LNUC)$ in Banach space. Then the implication relationship between $(LNUC)$ and the Kadec–Klee property $(KK)$, the fixed–point property $(FPP)$ are investigated in Banach space by using the basic theoretical knowledge of Banach space, such as the triangle inequality of norm, weak lower continuous of norm and several relative Lemmas. And the relationship between $(UKK)$, the coordinate-wise uniform Kadec–Klee property $(UK{K}_C)$, the coordinate-wise Kadec–Klee property $(H_c)$, $(LNUC)$ and ${\delta }_2$, ${\delta }_2^o$ conditions are investigated in Orlicz sequence spaces by using the basic theoretical knowledge of Orlicz space and several relative Lemmas.

3. Result

3.1. Preliminaries

Let us recall some lemmas which will be used in the remainder of the paper.

Lemma 1 (see [17]). 

Suppose that $\mathsf{\Phi }\in {\delta }_2$ and $u,u_n\in l_{\mathsf{\Phi }}$. Then For every $\epsilon >0$, there exists $\delta >0$ such that ${‖u‖}_{\mathsf{\Phi }}^O\ge \epsilon \Rightarrow I_{\mathsf{\Phi }}(u)\ge \delta$.

Lemma 2 (see [10]). 

Let $X$ be a reflexive infinite dimensional Banach space having the Kadec–Klee property $(KK)$. If $f:B(X)\to X^{*}\backslash \left\{0\right\}$ is a compact function, then there exist some $\stackrel{\wedge }{x}\in S(X)$ such that $⟨f(\stackrel{\wedge }{x}),\stackrel{\wedge }{x}⟩=‖f(\stackrel{\wedge }{x})‖$.

Lemma 3 (see [10]). 

Let $X$ be a separable reflexive Banach space. Then there exists a linear compact mapping $K:X\to X^{*}$ such that $⟨Kx,x⟩>0$ for all $x\in X$ with $x\ne 0$.

Lemma 4 (see [10]). 

Let $X$ be a separable reflexive Banach space without the Kadec–Klee property. Then there exists a compact function $f:B(X)\to X^{*}$ such that $‖f(x)‖\ge 1$ and $⟨f(x),x⟩<‖f(x)‖$ for all $x\in B(X)$.

Lemma 5 (see [11]). 

Let $X$ be an infinite-dimensional Banach space having the dual Kadec–Klee property. Then there exists a compact function $f:B(X)\to X^{*}$ such that $‖f(x)‖\ge 1$ and $⟨f(x),x⟩<‖f(x)‖$ for all $x\in B(X)$.

Lemma 6 (see [13]). 

Let $\mathsf{\Phi }$ be a $N-$function. Then the following are equivalent:

(1)

$l_{\mathsf{\Phi }}^o$ has the $(UK{K}_c)$ property;

(2)

$l_{\mathsf{\Phi }}^o$ has the $(H_c)$ property;

(3)

$\mathsf{\Phi }\in {\delta }_2$.

3.2. Results and Discussion

We present the main definitions of this section.

Definition 1. 

Let $X$ be a Banach space. Say that $X$ has the locally nearly uniformly convex $(LNUC)$ property if for every $\epsilon >0$, $x\in S(X)$ there exists $0<\delta <1$ such that $\left\{x_n\right\}\subset U(X)$, $sep(x_n)\ge \epsilon$, then there hold $\lambda \in (0,1)$ and $n_0\in N$ for which

$$\begin{array}{c}\parallel \lambda x_n+(1-\lambda )x\parallel <1\end{array}-\delta .$$

(1)

Below, we present the main theorem of this section.

Theorem 1. 

If Banach space $X$ has the $(LNUC)$ property, then $X$ has the $(KK)$.

Proof of Theorem 1. 

Suppose $X$ does not have the $(KK)$ property, then there exists $x\in S(X)$ and $\left\{x_n\right\}\subset S(X)$ such that although $x_n\stackrel{w}{\to }{x}_0$, we still have

$$\underset{n\to +\infty }{\lim }‖x_n-x‖>0.$$

(2)

In other words, the sequence $\left\{x_n\right\}$ is not a Cauchy sequence, thanks to $x_n\stackrel{w}{\to }{x}_0$, i.e.,

$$‖\lambda x_0+(1-\lambda )x‖\ge \delta .$$

(3)

there exists ${\epsilon }_0>0$ and a subsequence $\left\{x_{n_i}\right\}\subset \left\{x_n\right\}$ for which $sep(x_{n_i})\ge {\epsilon }_0$ for convenience, we may assume that $sep(x_n)\ge {\epsilon }_0$. For any $\lambda \in [0,1]$, $\lambda x+(1-\lambda )x_n\stackrel{w}{\to }x$, using the weak lower continuous of norm, we get

$$\underset{n\to +\infty }{\lim }‖\lambda x+(1-\lambda )x_n‖\ge ‖x‖=1.$$

(4)

This contradicts with that $X$ is $(LNUC)$. □

By the Lemma 2–5 and Theorem 1 we can obtain the following theorems.

Theorem 2. 

If Banach space $X$ has the $(LNUC)$ property, then $X$ has the $(FPP)$ property.

Theorem 3. 

The following statements are equivalent:

(1)

$l_{\mathsf{\Phi }}^o$ has the $(UKK)$;

(2)

$l_{\mathsf{\Phi }}^o$ has the $(UK{K}_c)$;

(3)

$l_{\mathsf{\Phi }}^o$ has the $(H_c)$;

(4)

$\mathsf{\Phi }\in {\delta }_2$.

Proof of Theorem 3. 

$(3)\Rightarrow (4)$. By Lemma 6 we get that $\mathsf{\Phi }\in {\delta }_2$.

$(2)\Rightarrow (3)$ is obviously.

$(4)\Rightarrow (1)$. Since $\mathsf{\Phi }(u)$ is convex and $\mathsf{\Phi }(0)=0$, thus $\mathsf{\Phi }(\alpha u)\le \alpha \mathsf{\Phi }(u)$ whenever $\alpha \in [0,1]$ Then for any $0<u_1<u_2$, we have

$$\mathsf{\Phi }(u_1)<\mathsf{\Phi }({\displaystyle \frac{u_1}{u_2}}{u}_2)\le {\displaystyle \frac{u_1}{u_2}}\mathsf{\Phi }(u_2).$$

(5)

$${\displaystyle \frac{\mathsf{\Phi }(u_1)}{u_1}}\le {\displaystyle \frac{\mathsf{\Phi }(u_2)}{u_2}}.$$

(6)

Hence, $\underset{u\to +\infty }{\lim }{\displaystyle \frac{\mathsf{\Phi }(u)}{u}}=A$ exists.

Case I: $A=+\infty$.

• Then

$$k_x^{*}=k^{*}(x)=\inf \{k>0:I_{\mathsf{\Phi }}(p(kx))\ge 1\}<+\infty .$$

(7)

So, for any $x\in L_{\mathsf{\Psi }}$ with $x\ne 0$, we have $k_x^{*}\le k_x^{**}$. Consequently,

$$K(x)=[k_x^{*},k_x^{**}]\ne \varphi .$$

(8)

Case II: $A<+\infty$. In this case $\underset{u\to \infty }{\lim }p(u)=A$. If ${\displaystyle \sum _{i\in \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(x)}}\mathsf{\Psi }(A)<1$, then $k_x^{**}=+\infty$, where $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(x)=\{i\in N:x(i)\ne 0\}$. It is easy to see that if ${\displaystyle \sum _{i\in \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(x)}}\mathsf{\Psi }(A)<1$, then $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(x)$ is a finite set. Hence,

$$\begin{array}{cc}{‖x‖}_{\mathsf{\Phi }}^o& =\underset{k\to +\infty }{\lim }{\displaystyle \frac{1}{k}}(1+I_{\mathsf{\Phi }}(kx))\hfill \\ & =\underset{k\to +\infty }{\lim }{\displaystyle \frac{1}{k}}{\displaystyle \sum _{i\in \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(x)}}\mathsf{\Phi }(kx(i))\hfill \\ & ={\displaystyle \frac{1}{k}}{\displaystyle \sum _{i\in \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(x)}}\mathsf{\Phi }(kx(i))\hfill \\ & =A{\displaystyle \sum _{i\in \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(x)}\left|x(i)\right|}.\hfill \end{array}$$

(9)

We next consider two cases:

(i)

$K(x)=\varphi$, i.e.,

$${‖x‖}_{\mathsf{\Phi }}^o=A{\displaystyle \sum _{i\in \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(x)}\left|x(i)\right|}.$$

(10)

Without loss of generality, we may assume that there exists $m\in N$ satisfy $x(i)=0(i\ge m+1).$

①

If $K(x_n)=\varphi (n\in N)$, then

$${‖x_n‖}_{\mathsf{\Phi }}^o=A{\displaystyle \sum _{i=1}^{\infty }\left|x_n(i)\right|}.$$

(11)

Since $\underset{n\to \infty }{\lim }{x}_n(i)=x(i)$ uniformly for $(i=1,2,\cdots ,m)$. We get that there exists $n_0\in N$ such that

$${‖\sum _{i=1}^m{x}_n{e}_i-\sum _{i=1}^{m}x{e}_i‖}_{\mathsf{\Phi }}^o<{\displaystyle \frac{\epsilon }{4}} whenever n\ge n_0.$$

(12)

Then $sep(\overline{x_n})=0$, put $\overline{x_n}=(x_n(1),\cdots ,x_n(m),0,\cdots )$ and $sep(x_n-\overline{x_n})\ge \epsilon .$ Hence there exists $n_1\in N$ such that

$${‖x_{n_1}-\overline{x_{n_0}}‖}_{\mathsf{\Phi }}^o\ge {\displaystyle \frac{\epsilon }{2}} whenever n\ge n_0.$$

(13)

Hence,

$$\begin{array}{cc}1={\displaystyle \sum _{i=1}^{\infty }A\left|x_{n_1}(i)\right|}& ={\displaystyle \sum _{i=1}^{m}A\left|x_{n_1}(i)\right|}+{\displaystyle \sum _{i=m+1}^{m}A\left|x_{n_1}(i)\right|}\hfill \\ & \ge {‖x‖}_{\mathsf{\Phi }}^o-{\displaystyle \frac{\epsilon }{4}}+{\displaystyle \frac{\epsilon }{2}}\hfill \\ & \ge {‖x‖}_{\mathsf{\Phi }}^o+{\displaystyle \frac{\epsilon }{4}},\hfill \end{array}$$

(14)

i.e., ${‖x‖}_{\mathsf{\Phi }}^o\le 1-{\displaystyle \frac{\epsilon }{4}}$.

②

$K(x_n)\ne \varphi (n\in N)$.

In this case, there exists $k_n\ge 1$ such that

$${‖x_n‖}_{\mathsf{\Phi }}^o={\displaystyle \frac{1}{k_n}}(1+\sum _{i=1}^{\infty }\mathsf{\Phi }(k_n{x}_n(i))).$$

(15)

using ${‖{\overline{x}}_n‖}_{\mathsf{\Phi }}^o=\underset{k>0}{\inf }{\displaystyle \frac{1}{k}}(1+I_{\mathsf{\Phi }}(k{x}_n)),$ we get that

$${\displaystyle \frac{1}{k_n}}(1+I_{\mathsf{\Phi }}(k_n{\overline{x}}_n))\ge {‖{\overline{x}}_n‖}_{\mathsf{\Phi }}^o.$$

(16)

In virtue of that $sep(x_n)\ge 0$ and $sep({\overline{x}}_n)\ge \epsilon$, we get that $sep(x_n-{\overline{x}}_n)\ge \epsilon ,$ i.e., for any $k\in N$, there exists $n_k>n$ such that $I_{\mathsf{\Phi }}(x_{n_k}-{\overline{x}}_{n_k})\ge \delta$. Using $\underset{n\to \infty }{\lim }{x}_n(i)=x(i)$ $(i=1,2,\cdots ,m)$, there exists $n_2\in N$ such that

$${‖\widetilde{x_n}‖}_{\mathsf{\Phi }}^o\ge {‖x‖}_{\mathsf{\Phi }}^o-{\displaystyle \frac{\delta }{2}}(n\ge n_2).$$

(17)

Hence, for $n_2\in N$ there exists a $n_l>n_2$ such that

$$I_{\mathsf{\Phi }}(x_{n_l}-\overline{x_{n_l}}))\ge \delta .$$

(18)

Therefore,

$$\begin{array}{cc}1={‖\widetilde{x_{n_l}}‖}_{\mathsf{\Phi }}^o& ={\displaystyle \frac{1}{k_{n_l}}}(1+I_{\mathsf{\Phi }}(k_{n_l}{x}_{n_l}))\hfill \\ & ={\displaystyle \frac{1}{k_{n_l}}}(1+{\displaystyle \sum _{i=1}^m}\mathsf{\Phi }(k_{n_l}{x}_{n_l}(i))+{\displaystyle \sum _{i=m+1}^{\infty }}\mathsf{\Phi }(k_{n_l}{x}_{n_l}(i)))\hfill \\ & \ge {‖x_{n_l}‖}_{\mathsf{\Phi }}^o+{\displaystyle \sum _{i=m+1}^{\infty }}\mathsf{\Phi }(x_{n_l}(i))\hfill \\ & ={‖x_{n_l}‖}_{\mathsf{\Phi }}^o+I_{\mathsf{\Phi }}(x_{n_l}-\widetilde{x_{n_l}}))\hfill \\ & \ge {‖x‖}_{\mathsf{\Phi }}^o-{\displaystyle \frac{\delta }{2}}+\delta \hfill \\ & ={‖x‖}_{\mathsf{\Phi }}^o+{\displaystyle \frac{\delta }{2}},\hfill \end{array}$$

(19)

i.e., ${‖x‖}_{\mathsf{\Phi }}^o\le 1-{\displaystyle \frac{\delta }{2}}.$

(ii)

$K(x)\ne \varphi .$

In this case, we have

$${\displaystyle \sum _{i\in \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(y)}}\mathsf{\Psi }(A)\ge 1.$$

(20)

If $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(x)$ is finite set, we may assume that $x=(x(1),x(2),\cdots ,x(m),0,\cdots )$ where $x(i)\ne 0(i=1,2,\cdots ,m)$. Using $\underset{n\to \infty }{\lim }{x}_n(i)=x(i)$ $(i=1,2,\cdots ,m)$, we get that ${\displaystyle \sum _{i\in \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(x_n)}}\mathsf{\Psi }(A)\ge 1$ for $n$ large enough, i.e., $K(x_n)\ne \varphi .$

If $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(x)$ is finite set, we can find a $m\in N$ such that

$${\displaystyle \sum _{\begin{array}{l}i\in \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(y)\\ \hfill i\le m\end{array}}}\mathsf{\Psi }(A)\ge 1.$$

(21)

Hence, we also have $K(x_n)\ne \varphi$ for $n$ large enough.

Without loss of generality, in the case of $K(x)\ne \varphi$, we may assume that $K(x_n)\ne \varphi$ for each $n\in N$.

Suppose that $\left\{x_n\right\}\subset S(l_{\mathsf{\Phi }}^o)$, $x_n\stackrel{w}{\to }x$ and $sep(x_n)\ge \epsilon$, using for any $m\in N$, the set $\{x_n(j):1\le j\le m\}$ is compact, we set $sep(x_n)\ge \epsilon$, where

$$x_{n,m}=(0,\cdots 0,x_n(m+1),x_n(m+2),\cdots ).$$

(22)

There holds that

$${‖x_{n,m}‖}_{\mathsf{\Phi }}^o\ge {\displaystyle \frac{\epsilon }{2}}$$

(23)

Since $\mathsf{\Phi }\in {\delta }_2$, by Lemma 1 there exists $\delta >0$ such that

$$I_{\mathsf{\Phi }}(x_{n,m})\ge \delta ,$$

(24)

using $\mathsf{\Phi }\in {\delta }_2$ again, there exists $m\in N$ such that

$${‖\sum _{i=m}^{\infty }x(i)e_i‖}_{\mathsf{\Phi }}^o<{\displaystyle \frac{\delta }{4}}.$$

(25)

In virtue of that $x_n\stackrel{w}{\to }x$ we get

$$x_n(i)\to x(i),$$

(26)

for each $i\in N$.

• Thus,

  $${‖\sum _{i=1}^{m-1}{x}_n(i)e_i‖}_{\mathsf{\Phi }}^o\to {‖\sum _{i=1}^{m-1}x(i)e_i‖}_{\mathsf{\Phi }}^o,$$

  (27)

  where $e_i\in (0,0,\cdots ,0,\stackrel{i-th}{1},0,\cdots )$.

Then there exists $n_0\in N$,

$${‖\sum _{i=1}^{m-1}{x}_n(i)e_i‖}_{\mathsf{\Phi }}^o\ge {‖\sum _{i=1}^{m-1}x(i)e_i‖}_{\mathsf{\Phi }}^o-{\displaystyle \frac{\delta }{4}},(n\ge n_0).$$

(28)

Take $k_n\ge 1$ such that

$$\begin{array}{cc}1={‖x_n‖}_{\mathsf{\Phi }}^o\hfill & ={\displaystyle \frac{1}{k_n}}(1+I_{\mathsf{\Phi }}(k_n{x}_n))\hfill \\ & ={\displaystyle \frac{1}{k_n}}(1+{\displaystyle \sum _{i=1}^{m-1}}\mathsf{\Phi }(k_n{x}_n(i))+{\displaystyle \sum _{i=m}^{\infty }}\mathsf{\Phi }(k_n{x}_n(i)))\hfill \\ & \ge {‖{\displaystyle \sum _{i=1}^{m-1}}{x}_n(i)e_i‖}_{\mathsf{\Phi }}^o+{\displaystyle \frac{1}{k_n}}{\displaystyle \sum _{i=m}^{\infty }}\mathsf{\Phi }(k_n{x}_n(i))\hfill \\ & \ge {‖{\displaystyle \sum _{i=1}^{m-1}}{x}_n(i)e_i‖}_{\mathsf{\Phi }}^o-{\displaystyle \frac{\delta }{4}}+{\displaystyle \sum _{i=m}^{\infty }}\mathsf{\Phi }(x_n(i)).\hfill \end{array}$$

(29)

Hence,

$${‖\sum _{i=1}^{m-1}{x}_n(i)e_i‖}_{\mathsf{\Phi }}^o\le 1-{\displaystyle \frac{3}{4}}\delta .$$

(30)

Therefore,

$$\begin{array}{cc}1={‖x‖}_{\mathsf{\Phi }}^o& \le {‖{\displaystyle \sum _{i=1}^{m-1}}x(i)e_i‖}_{\mathsf{\Phi }}^o+{‖{\displaystyle \sum _{i=m}^{\infty }}x(i)e_i‖}_{\mathsf{\Phi }}^o\hfill \\ & \le 1-{\displaystyle \frac{3\delta }{4}}+{\displaystyle \frac{\delta }{2}}\hfill \\ & =1-{\displaystyle \frac{\delta }{2}}.\hfill \end{array}$$

(31)

Consequently, $l_{\mathsf{\Phi }}^o$ is $(UKK)$.

• $(1)\Rightarrow (2)$. We only need to prove that weak convergence implies coordinate-wise convergence in $l_{\mathsf{\Phi }}^o$ if $\mathsf{\Phi }\in {\delta }_2$. Suppose that $\left\{x_n\right\}\subset S(l_{\mathsf{\Phi }}^o)$, $x_n\stackrel{w}{\to }x$. Take

  $$e_i\in (0,0,\cdots ,0,\stackrel{i-th}{1},0,\cdots )\in {(l_{\mathsf{\Phi }}^o)}^{*}=l_{\mathsf{\Psi }}.$$

  (32)

  Then,

  $$⟨e_i x_n⟩=x_n(i)\to ⟨e_i x⟩=x(i).$$

  (33)

  Hence, $x_n$ converges coordinate-wise to $x$. □

Theorem 4. 

Orlicz sequence space $l_{\mathsf{\Phi }}^o$ equipped with the Orlicz norm has the $(LNUC)$ property if and only if $\mathsf{\Phi }\in {\delta }_2\cap {\delta }_2^o$.

Proof of Theorem 4. 

Necessity. Suppose that $\mathsf{\Phi }\notin {\delta }_2$. Then there exists a sequence $\left\{u_n\right\}\subset R^{+}$ and $u_n\searrow 0$ such that

$$\mathsf{\Phi }((1+{\displaystyle \frac{1}{n}})u_n)\ge 2^n\mathsf{\Phi }(u_n) (n=1,2,\cdots ).$$

(34)

• Take $l_n\in N$ such that

  $${\displaystyle \frac{1}{2^n}}\le l_n\mathsf{\Phi }(u_n)<{\displaystyle \frac{1}{2^{n-1}}} (n=1,2,\cdots ).$$

  (35)

  Put

  

  (36)

  

  (37)

  Then

  $$I_{\mathsf{\Phi }}\left(x_n\right)=I_{\mathsf{\Phi }}\left(x\right)=\sum _{i=1}^{\infty }{l}_i\mathsf{\Phi }\left(u_i\right)\le \sum _{i=1}^{\infty }{\displaystyle \frac{1}{2^{i-1}}}=1.$$

  (38)

  For any $\epsilon \in (0,1)$, there exists $i_0\in N$ such that

  $$1+{\displaystyle \frac{1}{i}}\le {\displaystyle \frac{1}{\epsilon }},$$

  (39)

  whenever $i\ge i_0$. Hence,

  $$\begin{array}{cc}{I}_{\mathsf{\Phi }}\left({\displaystyle \frac{x_n}{\epsilon }}\right)& ={\displaystyle \sum _{i=1}^{\infty }}{l}_i\mathsf{\Phi }\left({\displaystyle \frac{u_i}{\epsilon }}\right)\hfill \\ & \ge {\displaystyle \sum _{i=i_0}^{\infty }}{l}_i\mathsf{\Phi }\left((1+{\displaystyle \frac{1}{i}})u_i\right)\hfill \\ & \ge {\displaystyle \sum _{i=i_0}^{\infty }}{l}_i{2}^i\mathsf{\Phi }\left(u_i\right)\hfill \\ & \ge {\displaystyle \sum _{i=i_0}^{\infty }}{l}_i{\displaystyle \frac{1}{2^{i-1}}}=\infty .\hfill \end{array}$$

  (40)

  i.e.,

  $${‖x_n‖}_{\mathsf{\Phi }}={‖x‖}_{\mathsf{\Phi }}=1.$$

  (41)

  Take $c>0$ such that ${‖\overline{x_n}‖}_{\mathsf{\Phi }}^o={‖x‖}_{\mathsf{\Phi }}^o=1$, where

  $$\overline{x_n}=(c,x_n(1),x_n(2),\cdots ),\overline{x}=(c,x(1),x(2),\cdots ).$$

  (42)

  Then, for any $m,n\in N$ with $m\ne n$,

  $$\begin{array}{cc}{I}_{\mathsf{\Phi }}\left(\overline{x_n}-\overline{x_m}\right)& =l_m\mathsf{\Phi }\left(2u_m\right)+l_n\mathsf{\Phi }\left(2u_n\right)\hfill \\ & \ge l_m{2}^m\mathsf{\Phi }\left(u_m\right)+l_n{2}^n\mathsf{\Phi }\left(u_n\right)\hfill \\ & \ge 1.\hfill \end{array}$$

  (43)

  Thus,

  $${‖\overline{x_n}-\overline{x_m}‖}_{\mathsf{\Phi }}^o\ge {‖x_n-x_m‖}_{\mathsf{\Phi }}\ge 1$$

  (44)

  For any $\lambda \in (0,1)$,

  $${‖\lambda \overline{x_n}+(1-\lambda )\overline{x}‖}_{\mathsf{\Phi }}^o\ge \underset{n\to +\infty }{\lim }{‖(c,x_{(1)},x_{(2)},\cdots x_{(n-1)},0,\cdots )‖}_{\mathsf{\Phi }}^o={‖x‖}_{\mathsf{\Phi }}^o=1,$$

  (45)

  i.e.,

  $$\underset{n\to +\infty }{\lim }{‖\lambda x_n+(1-\lambda )x‖}_{\mathsf{\Phi }}^o=1,$$

  (46)

  a contradiction.

Suppose that $\mathsf{\Phi }\notin {\delta }_2^0$, i.e., $\mathsf{\Psi }\notin {\delta }_2$. We divided $N$ into infinite disjoint subsets

$$N_1,N_2,\cdots , \mathrm{i}.\mathrm{e}., N={\displaystyle \underset{i=1}{\overset{\infty }{\cup }}{N}_i}, {{\rm N}}_i{\displaystyle \cap {{\rm N}}_j=\varphi ,} \mathrm{and} Card({{\rm N}}_i)=\infty .$$

(47)

Based on the counter examples above, we can built a sequence $\left\{z_n\right\}\subset S(l_{\mathsf{\Phi }})$ such that

$$\mathrm{supp}(z_n)=\{i\in N:z_n(i)\ne 0\}\subset N_n \mathrm{and} \underset{n\to +\infty }{\lim }{I}_{\mathsf{\Phi }}(z_n)=0,$$

(48)

Take a supporting function $y_n\subset S(l_{\mathsf{\Phi }}^o)$ of $z_n$ for each $n\in N$ with $\mathrm{supp}(y_n)\subset N_n$. Without loss generality, we can assume that ${\displaystyle \sum _{n=1}^{\infty }}{I}_{\mathsf{\Phi }}\left(z_n\right)\le 1$ (If necessary, we can take a subsequence $\left\{z_{n_i}\right\}\subset \left\{z_n\right\}$ such that ${\displaystyle \sum _{i=1}^{\infty }}{I}_{\mathsf{\Phi }}\left(z_{n_i}\right)\le 1.$) (see [17] proposition 1.84)

• Put

  $$z=(z_{(1)},z_{(2)},\cdots ,z_{(n)},\cdots )\mathrm{where} z(i)=z_n(i)\mathrm{if} i\in N_n.$$

  (49)

  Then ${‖z‖}_{\mathsf{\Phi }}=1$ and for any $\lambda \in (0,1)$

  $$\begin{array}{cc}z(\lambda y_n+(1-\lambda )y_1)& =\lambda z(y_n)+(1-\lambda )z(y_1)\hfill \\ & =\lambda z_n(y_n)+(1-\lambda )z_1(y_1)\hfill \\ & =\lambda +(1-\lambda )\hfill \\ & =1.\hfill \end{array}$$

  (50)

  i.e., ${‖\lambda y_n+(1-\lambda )y_1‖}_{\mathsf{\Phi }}^o=1$, a contradiction. □

By Theorem 4 we can obtain that $l_{\mathsf{\Phi }}^o$ is $(UKK)$. Since $\mathsf{\Phi }\in {\delta }_2\cap {\delta }_2^o$, thus $l_{\mathsf{\Phi }}^o$ is $(LNUC)$.

Corollary 1. 

Orlicz space $l_{\mathsf{\Phi }}^o$ is $(NUC)$ if and only if $l_{\mathsf{\Phi }}^o$ is reflexive.

4. Discussion

Our study focuses on a new geometric property locally nearly uniformly convex $(LNUC)$ and the classical geometric Kadec–Klee property $(KK)$ in Banach space.

We also study $(UKK)$, $(UK{K}_C)$, $(H_c)$, $(LNUC)$ and $(NUC)$ five geometric properties in Orlicz sequence spaces with four theorems and a corollary, these properties indicate that they play a very significant role in some recent trends related to the geometric theory of Banach and Orlicz spaces.

5. Conclusions

In this paper, locally nearly uniformly convex $(LNUC)$ is studied in Banach space. Furthermore, the implication relationship between $(LNUC)$ and the Kadec–Klee property $(KK)$, the fixed–point property $(FPP)$ are investigated in Banach space according to the following principles: (1) if Banach space $X$ has the $(LNUC)$ property, then $X$ has the $(KK)$ property; (2) if Banach space $X$ has the $(LNUC)$ property, then $X$ has the $(FPP)$ property. Finally, the relationship between the coordinate-wise uniform Kadec–Klee property $(UK{K}_C)$, the coordinate-wise Kadec–Klee property $(H_c)$ and ${\delta }_2$ conditions are investigated in Orlicz sequence spaces equipped with the Orlicz norm according to the following principle: Let $X=l_{\mathsf{\Phi }}^o$. Then, the following are equivalent: (i) $X$ has $(UK{K}_c)$ property; (ii) $X$ has the $(H_c)$ property; (iii) $\mathsf{\Phi }\in {\delta }_2$. Meanwhile we get a criteria that Orlicz sequence spaces equipped with the Orlicz norm are $(LNUC)$ according to the following principles: Orlicz sequence space $l_{\mathsf{\Phi }}^o$ equippeds with Orlicz norm has the $(LNUC)$ property if and only if $\mathsf{\Phi }\in {\delta }_2\cap {\delta }_2^o$. We also get a criteria that Orlicz sequence spaces equipped with the Orlicz norm are $(NUC)$ if and only if $l_{\mathsf{\Phi }}^o$ is reflexive.