Demiclosed principle and some fixed-point theorems for generalized nonexpansive map**s in Banach spaces

Abstract

The aim of this paper is to discuss some results concerning the demiclosedness principle of generalized, nonexpansive map**s in uniformly convex spaces. Further, we present some new fixed-point theorems for generalized nonexpansive map**s in different settings of Banach spaces.

1 Introduction

Let $(\mathcal{M},\|.\|)$ be a Banach space and $\mathcal{C}$ a nonempty subset of $\mathcal{M}$. A map** $S: \mathcal{C} \to \mathcal{C}$ is said to be nonexpansive if

$$ \bigl\Vert S(u)-S(v) \bigr\Vert \leq \Vert u-v \Vert , \quad\forall u,v \in \mathcal{C}. $$

A point $u^{\dagger} \in \mathcal{C}$ is said to be a fixed point of S if $S(u^{\dagger})= u^{\dagger}$. In the context of Banach spaces, a nonexpansive map** may not necessarily possess a fixed point. However, it is possible to obtain fixed points for such map**s by enriching the space with certain geometric properties. In 1965, Browder [5] and Göhde [15] separately established that nonexpansive map**s have fixed points in every uniformly convex Banach space. Kirk [20], on the other hand, extended the fixed-point theorem for nonexpansive map**s to the broader category of reflexive Banach spaces with normal structure. Recall that a Banach space $(\mathcal{M},\|.\|)$ is said to have normal structure, if for each bounded, closed, and convex subset $\mathcal{C}$ of $\mathcal{M}$ consisting of more than one point there is a point $u \in \mathcal{C}$ such that

$$ \sup \bigl\{ \Vert v-u \Vert :~ v \in \mathcal{C} \bigr\} < \operatorname{diam}_{ \Vert . \Vert }(\mathcal{C})= \sup \bigl\{ \Vert v-u \Vert :~ u,~v \in \mathcal{C} \bigr\} . $$

In [3], Baillon and Schöneberg weakened the concept of normal structure and introduced the asymptotic normal structure as follows: A Banach space $(\mathcal{M},\|.\|)$ is said to have asymptotic normal structure, if for each bounded, closed, and convex subset $\mathcal{C}$ of $\mathcal{M}$ consisting of more than one point and each sequence $\{u_{n}\}$ in $\mathcal{C}$ satisfying $u_{n} -u_{n+1} \to 0$ as $n\to \infty$, there is a point $u \in \mathcal{C}$ such that

$$ \liminf_{n \to \infty} \Vert u_{n}-u \Vert < \operatorname{diam}_{ \Vert . \Vert }(\mathcal{C}). $$

The relationship between fixed-point theory and the geometry of Banach spaces has been highly productive and significant. In the context of metric fixed-point problems, geometric properties are particularly influential. Nonexpansive map**s are a prominent area of study in metric fixed-point theory. Many authors have since derived generalizations and extensions of nonexpansive map**s and their associated results. The literature contains a considerable body of research on classes of map**s that are more general than the nonexpansive ones. Some of the notable extensions and generalizations of nonexpansive map**s can be found in [1, 2, 4, 6, 7, 11, 13, 21, 22, 24–26]. Some classes of map**s are not necessarily continuous on their domains, unlike nonexpansive map**s. In 2008, Suzuki [27] introduced a new class of nonexpansive-type map**s, referred to as map**s satisfying condition (C), and derived some significant fixed-point results for them. Suzuki [27] also demonstrated that this class of map**s does not necessarily exhibit continuity, unlike nonexpansive map**s. García-Falset et al. [11] explored a broader version of condition (C), called map**s satisfying condition (E). In 2011, Llorens-Fuster and Moreno-Galvez [21] introduced a general class of map**s called (L)-type map**s (or condition (L)), which is contingent on two conditions. First, the existence of an approximate fixed-point sequence (a.f.p.s.) for S in all nonempty, closed, convex, and S-invariant subsets of C. Secondly, the distances between points and their images in the limiting case from the a.f.p.s. For this class of map**s, the nonexpansiveness condition need not hold for all points but only for certain points in the domain. They obtained several fixed-point results for their new class of nonexpansive-type map**s.

It is noted herein that the normal structure condition depends on the distance between all points of set $\mathcal{C}$ and point u, while the asymptotic normal structure condition depends on the limiting distance between sequence $\{u_{n}\}$ and point u. This condition seems similar to the second condition of (L)-type map**s. It looks natural to investigate the fixed-point theorem for (L)-type map**s in the setting of Banach spaces having asymptotic normal structure. In this paper, we present some results concerning the demiclosedness principle of a map** satisfying condition (L-1) in uniform convex spaces. Further, we obtain some fixed-point theorems for (L-1)-type map**s in the setting of Banach spaces having asymptotic normal structure. Moreover, we show that in $\ell _{1}$ and $J_{0}$ (James space), (L-1)-type self-map** of a bounded weak^∗ closed convex subset has a fixed point. In this way, results in [11, 18, 21, 27] have been extended, generalized, and complemented.

2 Preliminaries

Definition 1

[12]. Let $\mathcal{C}$ be a nonempty subset of a Banach space $\mathcal{M}$. A sequence $\{u_{n}\}$ in $\mathcal{C}$ is said to be an approximate fixed-point sequence (in short, a.f.p.s.) for a map** $S: \mathcal{C} \to \mathcal{C}$ if $\lim_{n \to \infty}\|u_{n}-S(u_{n})\|=0$.

Definition 2

[12]. Let $\mathcal{C}$ be a subset of a Banach space $\mathcal{M}$. A map** $G:\mathcal{C}\rightarrow \mathcal{C}$ is said to be demiclosed if for any sequence $\{u_{n}\}$ in $\mathcal{C}$ the following implication holds:

$$ \{u_{n}\} \text{{ converges weakly to }} u \text{{ and }} \lim _{n\rightarrow \infty } \bigl\Vert G(u_{n})-w \bigr\Vert =0 $$

that implies

$$ u \in \mathcal{C} \quad\text{{and}}\quad G(u)=w. $$

Definition 3

[16]. Let $\mathcal{M}$ be a Banach space and $u,v \in \mathcal{M}$. A vector u is orthogonal to v if $\|u\| \leq \|u+\mu v\|$ for all scalars μ. We use to denote $u \bot v$ if u is orthogonal to v.

In general, the relation ⊥ is not symmetric cf. [18].

Definition 4

[18]. Let $\mathcal{M}$ be a Banach space. The relation ⊥ is said to be approximately symmetric if for each $u \in \mathcal{M}$ and $\varepsilon >0$, there exists a closed, linear subspace $\mathcal{Y}=\mathcal{Y}(u,\varepsilon )$ such that the following two conditions hold:

• (i) $\mathcal{Y}$ has finite codimension;

• (ii) $\|z\| \leq \|z+\mu u\|$ for all $z \in \mathcal{Y}$, $\|z\|=1$, and each μ with $\mu \geq \varepsilon$.

Definition 5

[18]. Let $\mathcal{M}$ be a conjugate space, that is, there exists a normed space $\mathcal{Z}$ such that $\mathcal{M}=\mathcal{Z}^{*}$. The relation ⊥ is said to be weak^∗ approximately symmetric if conditions (i) and (ii) in Definition 4 hold along with $\mathcal{Y}$ is weak^∗ closed.

Definition 6

[18]. Let $\mathcal{M}$ be a Banach space. The relation ⊥ is said to be uniformly approximately symmetric (uniformly weak^∗ approximately symmetric) if it is approximately symmetric (weak^∗ approximately symmetric) and condition (ii) in Definition 4 is replaced by the following stronger condition:

• (iii) there exists $\delta =\delta (u,\varepsilon )>0$ such that $\|z\| \leq \|z+\mu u\|-\delta$, for all $z \in Y$, $\|z\|=1$, and each μ with $\mu \geq \varepsilon$.

In the spaces $\ell _{p}$, $p \in (1,\infty )$, the relation ⊥ is uniformly approximately symmetric. In spaces $\ell _{1}$ and $J_{0}$ (James space [17]) the relation ⊥ is uniformly weak^∗ approximately symmetric. However, in both spaces $L_{p}$, $p\neq 2$ and $c_{0}$, the relation ⊥ fails to be uniformly approximately symmetric.

Lemma 1

(Goebel–Karlovitz) [14]. Let $\mathcal{C}$ be a subset of a reflexive Banach space $\mathcal{M}$, and suppose $\mathcal{C}$ is minimally invariant with respect to being nonempty, bounded, closed, convex, and S-invariant for some nonexpansive map** S. Let $\{x_{n}\}$ be a sequence in $\mathcal{C}$ that satisfies $\lim_{n \to \infty}\|u_{n}-S(u_{n})\|=0$. Then, for each $u \in \mathcal{C}$, $\lim_{n \to \infty}\|u_{n}-u\| = \operatorname{diam}(\mathcal{C})$.

Theorem 1

[14] Let $\mathcal{M}$ be a uniformly convex Banach space. Then, for any $d>0$, $\varepsilon >0$ and $u,v \in X$ with $\|u\|\leq d, \|v\|\leq d, \|u-v\|\geq \varepsilon$, there exists a $\delta >0$ such that

$$ \biggl\Vert \frac{1}{2}(u+v) \biggr\Vert \leq \biggl[ 1-\delta \biggl( \frac{\varepsilon}{d} \biggr) \biggr]d. $$

Theorem 2

[23]. Let $\mathcal{M}$ be a Banach space. The following conditions are equivalent:

• (i) $\mathcal{M}$ is strictly convex;

• (ii) If $u,v \in \mathcal{M}$ and $\|u+v\|= \|u\|+\|v\|$, then $u=0$ or $v=0$ or $v= cu$ for some $c >0$.

Theorem 3

[3]. Let $\beta \geq 1$ and let $\mathcal{M}_{\beta}$ be the real space $\ell _{2}$ renormed according to

$$ \vert u \vert _{\beta}= \max \bigl\{ \Vert u \Vert _{2}, \beta \Vert u \Vert _{\infty} \bigr\} , $$

where $\|u\|_{\infty}$ denotes the $\ell _{\infty}$-norm and $\|u\|_{2}$ the $\ell _{2}$ norm. Then,

• (1) $\mathcal{M}_{\beta}$ has normal structure if and only if $\beta < \sqrt{2}$; and

• (2) $\mathcal{M}_{\beta}$ has asymptotic normal structure if and only if $\beta < 2$.

Lemma 2

[3]. Let $\beta \geq 1$, $x,y,z \in \mathcal{M}_{\beta}$ and $\alpha \in [0,1]$. Then,

$$ \bigl\Vert x- \bigl((1-\alpha )y+\alpha z \bigr) \bigr\Vert _{2}^{2} + \alpha (1-\alpha ) \Vert y-z \Vert _{2}^{2}=(1- \alpha ) \Vert x-y \Vert _{2}^{2}+\alpha \Vert x-z \Vert _{2}^{2}. $$

Lemma 3

[3, 9, 10]. Let $\beta \geq 1$ and $\mathcal{C}$ be a bounded, closed, and convex subset of $\mathcal{M}_{\beta}$. Let $\{u_{n}\}$ be a sequence in $\mathcal{C}$. Then, there exists a unique point $w \in \mathcal{C}$ that satisfies the following conditions:

• (i) $\limsup_{n \to \infty}\|u_{n}-w\|_{2}^{2} +\|w-u\|_{2}^{2} \leq \limsup_{n \to \infty} \|u_{n}-u\|_{2}^{2}$ for all $u \in \mathcal{C}$; and

• (ii) $2 \limsup_{n \to \infty}\|u_{n}-w\|_{2}^{2} \leq \limsup_{p \to \infty} \{\limsup_{n \to \infty} \|u_{n}-u_{p} \|_{2}^{2}\}$.

Lemma 4

[3]. Let $1 \leq \beta \leq 2$ and $\mathcal{C}$ be a bounded, closed, and convex subset of $\mathcal{M}_{\beta}$ with $d=\operatorname{diam}_{|.|_{\beta}}(\mathcal{C})$. Let $\{u_{n}\}$ be a sequence in $\mathcal{C}$ such that $u_{n}-u_{n+1} \to 0$ as $n \to \infty$ and $\lim_{n \to \infty} |u_{n}-u|_{\beta}=d$ for all $u \in \mathcal{C}$, let $w \in \mathcal{C}$ be the $\|.\|_{2}$-asymptotic-center of $\{u_{n}\}$ in $\mathcal{C}$. Then, $\limsup_{n \to \infty}\|u_{n}-w\|_{2}^{2} \geq 2 ( \frac{d}{\beta} )^{2}$.

Lemma 5

[3]. Let $\mathcal{C}$ be a bounded, closed, and convex subset of $\mathcal{M}_{2}$ and let $\{v_{n}\}$ be a sequence in $\mathcal{C}$ such that $\lim_{n \to \infty} |v_{n}-u|_{2} =d=\operatorname{diam}_{|.|_{\beta}}( \mathcal{C})$ for all $u \in \mathcal{C}$. Then, $\lim_{n \to \infty} \|v_{n}-u\|_{\infty} =\frac{d}{2}$ for all $u \in \mathcal{C}$.

Lemma 6

[3]. Let $\{x_{n}\}$ and $\{y_{n}\}$ be two sequences in $\ell _{2}$. Suppose that $d>0$, $y_{n} \rightharpoonup 0$ as $n \to \infty$, $\lim_{n \to \infty}\|y_{n}\|_{\infty}=\frac{d}{2}$ and $\lim_{n \to \infty}\|x_{n}\|_{\infty}=\frac{d}{2}$, $\|x_{n}-y_{p}\| \leq \frac{d}{2}$ for all $n,p$ and

$$ \limsup_{n \to \infty} \Bigl\{ \limsup_{p \to \infty} \Vert x_{n}+y_{p} \Vert _{2}^{2} \Bigr\} =d^{2}. $$

Then,

$$ \limsup_{n \to \infty} \Bigl\{ \limsup_{p \to \infty} \Vert x_{n}+y_{p} \Vert _{\infty} \Bigr\} < d. $$

Let $\{u_{n}\}$ be a bounded sequence in Banach space $\mathcal{M}$, and $\mathcal{C}$ be a nonempty subset of $\mathcal{M}$. The asymptotic radius of $\{u_{n}\}$ at a point x in $\mathcal{M}$ is defined by

$$ r \bigl(x,\{u_{n}\} \bigr):=\limsup_{n \to \infty} \Vert u_{n}-x \Vert . $$

The asymptotic radius of $\{u_{n}\}$ with respect to $\mathcal{C}$ is defined by

$$ r \bigl(\mathcal{C},\{u_{n}\} \bigr):=\inf \bigl\{ r \bigl(x, \{u_{n}\} \bigr):x \in \mathcal{C} \bigr\} . $$

The asymptotic center of $\{u_{n}\}$ with respect to $\mathcal{C}$ is defined as

$$ A \bigl(\mathcal{C},\{u_{n}\} \bigr):= \{ x \in \mathcal{C}: r \bigl(x, \{u_{n}\} \bigr)=r \bigl( \mathcal{C},\{u_{n}\} \bigr). $$

Definition 7

[27]. Let $\mathcal{M}$ be a Banach space and $\mathcal{C}$ a nonempty subset of $\mathcal{M}$. A map** $S:\mathcal{C} \to \mathcal{C}$ is said to satisfy condition (C) if

$$\begin{aligned} \frac{1}{2} \bigl\Vert u-S(u) \bigr\Vert \leq \Vert u-v \Vert \quad\text{{implies }} \bigl\Vert S(u) - S(v) \bigr\Vert \leq \Vert u-v \Vert \quad \forall u,v \in \mathcal{C}. \end{aligned}$$

Definition 8

[11]. Let $\mathcal{C}$ be a nonempty subset of a Banach space $\mathcal{M}$. A map** $S:\mathcal{C} \to \mathcal{C}$ is said to fulfill condition $(E_{\mu})$ if there exists $\mu \geq 1$ such that

$$ \bigl\Vert u-S(v) \bigr\Vert \leq \mu \bigl\Vert u-S(u) \bigr\Vert + \Vert u-v \Vert \quad \forall u,v \in \mathcal{C}. $$

We say that S satisfies condition (E) if it satisfies $(E_{\mu})$ for some $\mu \geq 1$.

3 Class of map**s satisfying condition (L)

Llorens-Fuster and Moreno-Gálvez [21] introduced the following class of nonlinear map**s:

Definition 9

Let $\mathcal{C}$ be a nonempty subset of a Banach space $(\mathcal{M},\|.\|)$. We say that a map** $S:\mathcal{C} \to \mathcal{C}$ satisfies condition (L), (or it is an (L)-type map**), if the following two conditions hold:

• (1) If a set $\mathcal{D} \subset \mathcal{C}$ is nonempty, closed, convex, and S-invariant, (i.e., $S(\mathcal{D}) \subset \mathcal{D}$), then there exists an a.f.p.s. for S in $\mathcal{D}$.

• (2) For any a.f.p.s. $\{u_{n}\}$ of S in $\mathcal{C}$ and each $u \in \mathcal{C}$

  $$ \limsup_{n \to \infty} \bigl\Vert u_{n}-S(u) \bigr\Vert \leq \limsup_{n \to \infty} \Vert u_{n}-u \Vert . $$

In [21] it is shown that the above two conditions in the definition of (L)-type map**s are independent in nature.

It is proved in [21] that the class of (L)-type map**s contains strictly the following classes:

• (A) nonexpansive map**s;

• (B) Suzuki generalized nonexpansive map**s (cf. [27]);

• (C) generalized nonexpansve in many cases, see [21];

• (D) The class of map**s satisfying condition (E) that in turn satisfy condition (1) in the Definition 9 (cf. [11]).

Now, we consider a subclass of class of (L)-type map**s.

Definition 10

Let $\mathcal{C}$ be a nonempty subset of a Banach space $(\mathcal{M},\|.\|)$ and a map** $S:\mathcal{C} \to \mathcal{C}$ satisfies condition (L-1), (or it is an (L-1)-type map**), if the following two conditions hold:

• (1) If a set $\mathcal{D} \subset \mathcal{C}$ is nonempty, closed, convex, and S-invariant, (i.e., $S(\mathcal{D}) \subset \mathcal{D}$), then there exists an a.f.p.s. for S in $\mathcal{D}$.

• (2) For any a.f.p.s. $\{u_{n}\}$ of S in $\mathcal{C}$, there exists a sequence $\{c_{n}\}$ in $[0,\infty )$ such that $c_{n} \to 0$ as $n \to \infty$ and each $u \in \mathcal{C}$, we have

  $$ \bigl\Vert u_{n}-S(u) \bigr\Vert \leq \Vert u_{n}-u \Vert +c_{n}. $$

  (3.1)

Example 1

Let $(\ell ^{2},\|\cdot \|_{2} )$ be the Banach space of square-summable sequences endowed with its standard norm. Assume that $B [0_{\mathcal{M}}, 1 ]$ is a unit ball centered at $0_{\mathcal{M}}$ (zero element). Suppose that $S: B [0_{\mathcal{M}}, 1 ] \rightarrow B [0_{ \mathcal{M}}, 1 ]$ is the map** given by the following definition:

$$ S(u)= \textstyle\begin{cases} \frac{1}{2} \frac{u}{ \Vert u \Vert } & u \in B [0_{\mathcal{M}}, 1 ] \backslash B [0_{\mathcal{M}}, \frac{1}{2} ], \\ 0_{\mathcal{M}} & u \in B [0_{\mathcal{M}}, \frac{1}{2} ].\end{cases} $$

In fact, the unique fixed point of S is $0_{\mathcal{M}}$. We can have a.f.p.s. $\{u_{n}\}$ given by $u_{n} \equiv 0$. Suppose $\{c_{n}\}= \{ \frac{1}{n} \}$ in $[0,\infty )$, then $c_{n} \to 0$ as $n \to \infty$. Then, if $u \in B [0_{\mathcal{M}}, \frac{1}{2} ]$

$$ \bigl\Vert u_{n}-S(u) \bigr\Vert = \bigl\Vert 0_{\mathcal{M}}-S(u) \bigr\Vert \leq \Vert 0_{\mathcal{M}}-u \Vert \leq \Vert u_{n}-u \Vert +c_{n} . $$

Again, if $u \in B [0_{\mathcal{M}}, 1 ] \backslash B [0_{ \mathcal{M}}, \frac{1}{2} ]$

$$ \bigl\Vert 0_{\mathcal{M}}-S(u) \bigr\Vert = \biggl\Vert \frac{1}{2} \frac{u}{ \Vert u \Vert } \biggr\Vert = \frac{1}{2} \leq \Vert u \Vert = \Vert 0_{ \mathcal{M}}-u \Vert \leq \Vert u_{n}-u \Vert +c_{n}. $$

On the other hand, $u \in B [0_{\mathcal{M}}, 1 ]$ with $\|u\|=\frac{1}{2}$ and $v:=\frac{3}{2} u$, The map** S is not nonexpansive.

Proposition 1

Let $S:\mathcal{C} \to \mathcal{C}$ be a map** satisfying condition (L-1), then S is a map** satisfying condition (L).

Proof

The first conditions in both map**s are the same. Hence, we only compare the second conditions. Since map** S is a map** satisfying condition (L-1), then for any a.f.p.s. $\{u_{n}\}$ of S in $\mathcal{C}$, there exists a sequence $\{c_{n}\}$ in $[0,\infty )$ such that $c_{n} \to 0$ as $n \to \infty$ and each $u \in \mathcal{C}$

$$ \bigl\Vert u_{n}-S(u) \bigr\Vert \leq \Vert u_{n}-u \Vert +c_{n}. $$

Taking lim sup on both sides, we obtain the desired result. □

In the next theorem, we present the structure of the fixed-point set of class of (L-1)-type map**s.

Theorem 4

Let $\mathcal{C}$ be a nonempty, closed subset of a Banach space $\mathcal{M}$ and $S:\mathcal{C} \to \mathcal{C}$ a map** satisfying condition (L-1) with $F(S) \neq \emptyset$. Then, the following implications hold:

• (i) $F(S)$ is closed in $\mathcal{C}$;

• (ii) If $\mathcal{C}$ is convex and $\mathcal{M}$ is strictly convex then $F(S)$ is convex.

Proof

• (i) Let $\{w_{n}\}\subseteq F(S)$ such that $w_{n} \rightarrow w \in \mathcal{C}$ as $n \rightarrow \infty$. Thus, $S(w_{n}) = w_{n}$ and $\{w_{n}\}$ is an a.f.p.s. for S in $\mathcal{C}$. Since S is a (L-1)-type map**, we have

  $$ \bigl\Vert w_{n}-S(w) \bigr\Vert \leq \Vert w_{n}-w \Vert +c_{n}, $$

  making $n \to \infty$, which implies that $S(w)=w$ and $F(S)$ is closed.

• (ii) See [8, Theorem 1].

 □

4 Demiclosedness principle in uniformly convex spaces

In this section, we present some results concerning the demiclosedness principle of a map** satisfying condition (L-1).

Lemma 7

Suppose $\mathcal{C}$ is a bounded convex subset of a uniformly convex Banach space $\mathcal{M}$ and $S:\mathcal{C}\rightarrow \mathcal{M}$ is a map** satisfying condition (L-1). If $\{u_{n}\}$ and $\{v_{n}\}$ are approximate fixed-point sequences, then $\{w_{n}\}=\{\frac{1}{2}(u_{n}+v_{n})\}$ is an approximate fixed-point sequence too.

Proof

Suppose the assertion of the lemma is false. Then, there exist sequences $\{u_{n}\}$ and $\{v_{n}\}$ satisfying $\lim_{n\rightarrow \infty }\Vert u_{n}-S(u_{n})\Vert =0$ and $\lim_{n\rightarrow \infty }\Vert v_{n}-S(v_{n})\Vert =0$ such that $\Vert w_{n}-S(w_{n})\Vert \geq \varepsilon$ for some $\varepsilon >0$ and every $n \in \mathbb{N}$. We can assume by passing to a subsequence that

$$ \lim_{n\rightarrow \infty } \Vert u_{n}-v_{n} \Vert =2r>0. $$

It follows that

$$ \lim_{n\rightarrow \infty } \Vert u_{n}-w_{n} \Vert = \lim_{n\rightarrow \infty } \Vert v_{n}-w_{n} \Vert =r. $$

By the definition of map** S, for a.f.p.s. $\{u_{n}\}$ of S in $\mathcal{C}$, there exists a sequence $\{c_{n,1}\}$ in $[0,\infty )$ such that $c_{n,1} \to 0$ as $n \to \infty$, we have

$$\begin{aligned} \bigl\Vert u_{n}-S(w_{n}) \bigr\Vert \leq & \Vert u_{n}-w_{n} \Vert +c_{n,1}. \end{aligned}$$

(4.1)

Similarly,

$$ \bigl\Vert v_{n}-S(w_{n}) \bigr\Vert \leq \Vert v_{n}-w_{n} \Vert +c_{n,2}, $$

where $c_{n,2} \to 0$ as $n \to \infty$. Choose $s>0$ such that $s<\frac{\varepsilon }{r}$. Hence, for sufficiently large n, we have

$$ s< \frac{\varepsilon }{c_{n,1} + \Vert u_{n}-w_{n} \Vert } $$

(4.2)

and

$$ s< \frac{\varepsilon }{c_{n,2} + \Vert v_{n}-w_{n} \Vert }. $$

Now,

$$ \biggl\Vert u_{n}-\frac{1}{2} \bigl(w_{n}+S(w_{n}) \bigr) \biggr\Vert = \biggl\Vert \frac{u_{n}-S(w_{n})+\frac{(u_{n}-v_{n})}{2}}{2} \biggr\Vert $$

and it can be seen that

$$ \bigl\Vert u_{n}-S(w_{n}) \bigr\Vert \leq \Vert u_{n}-w_{n} \Vert +c_{n,1}. $$

Now,

$$ \Vert u_{n}-w_{n} \Vert = \biggl\Vert u_{n} -\frac{1}{2}(u_{n}+v_{n}) \biggr\Vert = \frac{1}{2} \Vert u_{n}-v_{n} \Vert . $$

Thus,

$$ \biggl\Vert \frac{(u_{n}-v_{n})}{2} \biggr\Vert \leq \Vert u_{n}-w_{n} \Vert +c_{n,1} $$

and $\Vert w_{n}-S(w_{n})\Vert \geq \varepsilon$. By the uniform convexity of $\mathcal{M}$ (see Theorem 1), we have

$$\begin{aligned} \biggl\Vert u_{n}-\frac{1}{2} \bigl(w_{n}+S(w_{n}) \bigr) \biggr\Vert \leq & \biggl(1- \delta \biggl( \frac{\varepsilon }{c_{n,1} + \Vert u_{n}-w_{n} \Vert } \biggr) \biggr) \bigl(c_{n,1} + \Vert u_{n}-w_{n} \Vert \bigr). \end{aligned}$$

It is noted that the modulus of convexity, $\delta (\varepsilon )$, is a nondecreasing function of ε, it follows that

$$ \biggl\Vert u_{n}-\frac{1}{2} \bigl(w_{n}+S(w_{n}) \bigr) \biggr\Vert \leq \bigl(1- \delta (s) \bigr) \bigl(c_{n,1}+ \Vert u_{n}-w_{n} \Vert \bigr). $$

(4.3)

Similarly,

$$\begin{aligned} \biggl\Vert v_{n}-\frac{1}{2} \bigl(w_{n}+S(w_{n}) \bigr) \biggr\Vert \leq & \biggl( 1- \delta \biggl( \frac{\varepsilon }{c_{n,2} + \Vert v_{n}-w_{n} \Vert } \biggr) \biggr) \\ &{}\times \bigl(c_{n,2} + \Vert v_{n}-w_{n} \Vert \bigr) \\ \leq & \bigl(1-\delta (s) \bigr) \bigl(c_{n,2} + \Vert v_{n}-w_{n} \Vert \bigr). \end{aligned}$$

(4.4)

By the triangle inequality, (4.3), and (4.4), we obtain

$$\begin{aligned} \Vert u_{n}-v_{n} \Vert \leq &\biggl\Vert u_{n}- \frac{1}{2}(w_{n}+S(w_{n}) \biggr\Vert + \biggl\Vert v_{n}-\frac{1}{2}(w_{n}+S(w_{n})\biggr\Vert \\ \leq & \bigl(1-\delta (s) \bigr) \bigl\{ \bigl(c_{n,1} + \Vert u_{n}-w_{n} \Vert \bigr)+ \bigl(c_{n,2}+ \Vert v_{n}-w_{n} \Vert \bigr) \bigr\} . \end{aligned}$$

Letting $n\rightarrow \infty$, we obtain $2r\leq 2r(1-\delta (s))$, a contradiction and this completes the proof. □

Proposition 2

Suppose $\mathcal{C}$ is a bounded, closed, and convex subset of a uniformly convex space. Let $S:\mathcal{C} \to \mathcal{C}$ be a map** satisfying condition (L-1). Then, S has a fixed point.

Proof

See [21, Theorem 4.4]. □

Theorem 5

(Demiclosedness principle). Suppose $\mathcal{C}$ is a closed, convex subset of a uniformly convex space. Let $S:\mathcal{C} \to \mathcal{C}$ be a map** satisfying condition (L-1). Then, the map** $G=I-S$ is demiclosed on $\mathcal{C}$.

Proof

Let $\{u_{n}\}$ be a sequence in $\mathcal{C}$ such that $\{u_{n}\}$ converges weakly to $u^{\dagger}$ and $\lim_{n\rightarrow \infty }\|u_{n}-S(u_{n})-w\|=0$. Without loss of generality, we assume $w=0$, as limits are preserved under the translation. Define $\mathcal{C}_{n}= \overline{\operatorname{conv}}\{u_{n},u_{n+1},\dots \}$, using Proposition 2 on set $\mathcal{C}_{n}$, there exists $y_{n} \in \mathcal{C}_{n}$ such that $S(y_{n})=y_{n}$. Since any weak subsequential limit of $y_{n}$ lies in $\bigcap_{n=1}^{\infty}\mathcal{C}_{n}=\{u^{\dagger}\}$, it implies that $y_{n}$ converges weakly to $u^{\dagger}$. Therefore, $u^{\dagger}$ is in the weak closure of the fixed-point set $F(S)$. Since $\mathcal{M}$ is uniformly convex, $\mathcal{M}$ is both reflexive and strictly convex. From Theorem 4, fixed-point set $F(S)$ is closed and convex, so weakly closed and $u^{\dagger} \in F(S)$. This completes the proof. □

5 Some fixed-point theorems

In this section, we present some fixed-point results for the class of map**s satisfying condition (L-1).

Theorem 6

Suppose $\mathcal{C}$ is a closed, convex subset of a uniformly convex space. Let $S:\mathcal{C} \to \mathcal{C}$ be a map** satisfying condition (L-1). If $\{u_{n}\}$ is an a.f.p.s. for S such that it converges weakly to $u^{\dagger} \in \mathcal{C}$, then $u^{\dagger}$ is a fixed point of S.

Proof

It can be easily seen from Theorem 5 that map** $I-S$ is demiclosed at 0. From the demiclosedness principle it follows that $u^{\dagger}$ is a fixed point of S. □

Remark 1

The above theorem should be compared with [21, Theorem 4.6] that asserts the same conclusion in view of the Opial property.

Theorem 7

Let $\mathcal{C}$ be a nonempty bounded, closed, and convex subset of $\mathcal{M}_{2}$ and $S:\mathcal{C} \to \mathcal{C}$ a map** satisfying condition (L-1). Assume the following conditions hold:

• (1) If $\mathcal{D}$ is minimal with respect to S, and there is an a.f.p.s. $\{u_{n}\}$ in $\mathcal{D}$, then $u_{n}-u_{n+1} \to 0$ as $n \to \infty$;

• (2) If $\mathcal{D}$ is minimal with respect to S, and $\{u_{n}\}$ is an a.f.p.s. in $\mathcal{D}$, then $|u_{n}-u|_{2} \to d=\operatorname{diam}_{|.|_{\beta}}(\mathcal{C})$ for all $u \in \mathcal{D}$.

Then, S has a fixed point.

Proof

By the application of Zorn’s lemma there is a nonempty, bounded, closed, convex, and S-invariant subset $\mathcal{D}$ of $\mathcal{C}$ with no proper subsets, so $\mathcal{D}$ is minimal with respect to S. Let $d=\operatorname{diam}_{|.|_{\beta}}(\mathcal{C})$ and assume, for a contradiction, that $d>0$. Let $\{u_{n}\}$ be an a.f.p.s. in $\mathcal{D}$ such that $u_{n}-u_{n+1} \to 0$ as $n \to \infty$. Let $w \in \mathcal{D}$ denote the $\|.\|_{2}$-asymptotic center of $\{u_{n}\}$ in $\mathcal{D}$. By Lemma 4, we have

$$ \limsup_{n\rightarrow \infty } \Vert u_{n}-w \Vert _{2}^{2} \geq \frac{d^{2}}{2}. $$

(5.1)

Without loss of generality, we may assume there is a subsequence $\{u_{n_{k}}\}$ of $\{u_{n}\}$ with $u_{n_{k}} \rightharpoonup u \in \mathcal{D}$ and

$$ \lim_{k\rightarrow \infty } \Vert u_{n_{k}}-w \Vert _{2}^{2}= \limsup_{n\rightarrow \infty } \Vert u_{n}-w \Vert _{2}^{2}. $$

Again, take a subsequence $\{u_{m_{k}}\}$ of $\{u_{n}\}$ with $u_{n_{k}} \rightharpoonup v \in \mathcal{D}$ and

$$ \lim_{k\rightarrow \infty } \Vert u_{m_{k}}-u \Vert _{2}^{2}= \limsup_{n\rightarrow \infty } \Vert u_{n}-u \Vert _{2}^{2}. $$

By (5.1) and Lemma 3(i), we obtain

$$\begin{aligned} d^{2} \geq & \lim_{k\rightarrow \infty } \Bigl\{ \lim _{p \rightarrow \infty } \Vert u_{n_{k}}-u_{m_{p}} \Vert _{2}^{2} \Bigr\} \\ =& \limsup_{n\rightarrow \infty } \Vert u_{n}-u \Vert _{2}^{2} + \limsup_{n\rightarrow \infty } \Vert u_{n}-w \Vert _{2}^{2}- \Vert w-u \Vert _{2}^{2} \\ \geq & 2 \limsup_{n\rightarrow \infty } \Vert u_{n}-w \Vert _{2}^{2} \geq d^{2}. \end{aligned}$$

From the above inequalities, we have the following:

$$ \limsup_{n\rightarrow \infty } \Vert u_{n}-w \Vert _{2}^{2} = \frac{d^{2}}{2} $$

(5.2)

and

$$ \lim_{k\rightarrow \infty } \Bigl\{ \lim_{p\rightarrow \infty } \Vert u_{n_{k}}-u_{m_{p}} \Vert _{2}^{2} \Bigr\} = d^{2}. $$

(5.3)

Now, we show that

$$ \limsup_{k\rightarrow \infty } \biggl\{ \limsup_{p \rightarrow \infty } \biggl\Vert \frac{1}{2}(u_{n_{k}}+u_{m_{p}})-u \biggr\Vert _{\infty} \biggr\} =\frac{d}{2} \quad\text{{for all }}u \in \mathcal{C}. $$

Take $\Gamma _{k} =u_{n_{k}}$ and $\Delta _{k}=u_{m_{k}}$. From Lemma 2, for $k,p \in \mathbb{N}$, we have

$$\begin{aligned} & \biggl\Vert S \biggl(\frac{1}{2}(\Gamma _{k}+\Delta _{p}) \biggr)- \frac{1}{2}(\Gamma _{k}+\Delta _{p}) \biggr\Vert _{2}^{2} + \frac{1}{4} \Vert \Gamma _{k}-\Delta _{p} \Vert _{2}^{2} \\ &\quad= \frac{1}{2} \biggl\Vert S \biggl(\frac{1}{2}(\Gamma _{k}+\Delta _{p}) \biggr)-\Gamma _{k} \biggr\Vert _{2}^{2}\frac{1}{2} \biggl\Vert S \biggl( \frac{1}{2}(\Gamma _{k} +\Delta _{p}) \biggr)-\Delta _{p} \biggr\Vert _{2}^{2}. \end{aligned}$$

(5.4)

Now,

$$ \biggl\Vert S \biggl(\frac{1}{2}(\Gamma _{k}+\Delta _{p}) \biggr)-\Gamma _{k} \biggr\Vert _{2} \leq \biggl\vert S \biggl(\frac{1}{2}(\Gamma _{k}+\Delta _{p}) \biggr)-\Gamma _{k} \biggr\vert _{2}. $$

Since $\{\Gamma _{k}\}$ is a.f.p.s for S, from the definition of condition (L-1), we have

$$ \biggl\vert S \biggl(\frac{1}{2}(\Gamma _{k}+\Delta _{p}) \biggr)-\Gamma _{k} \biggr\vert _{2} \leq \biggl\vert \biggl(\frac{1}{2}(\Gamma _{k}+\Delta _{p}) \biggr)-\Gamma _{k} \biggr\vert _{2}+c_{n,1}, $$

where $c_{n,1} \to 0$ as $n \to \infty$. From the above inequality, we obtain

$$\begin{aligned} \biggl\Vert S \biggl(\frac{1}{2}(\Gamma _{k}+ \Delta _{p}) \biggr)-\Gamma _{k} \biggr\Vert _{2} \leq & \frac{1}{2} \vert \Gamma _{k}-\Delta _{p} \vert _{2} +c_{n,1} \\ \leq & \frac{d}{2} +c_{n,1} \end{aligned}$$

(5.5)

and, similarly,

$$ \biggl\Vert S \biggl(\frac{1}{2}(\Gamma _{k}+ \Delta _{p}) \biggr)-\Delta _{p} \biggr\Vert _{2}^{2} \leq \frac{d}{2} +c_{n,2}, $$

(5.6)

where $c_{n,2} \to 0$ as $n \to \infty$. Using (5.5) and (5.6) in (5.4), we have

$$\begin{aligned} \biggl\Vert S \biggl(\frac{1}{2}(\Gamma _{k}+\Delta _{p}) \biggr)- \frac{1}{2}(\Gamma _{k}+\Delta _{p}) \biggr\Vert _{2}^{2} + \frac{1}{4} \Vert u_{k}-v_{p} \Vert _{2}^{2} \leq \frac{1}{2} \biggl(\frac{d}{2} +c_{n,1} \biggr)^{2}+ \frac{1}{2} \biggl(\frac{d}{2} +c_{n,2} \biggr)^{2}. \end{aligned}$$

Since $\lim_{k\rightarrow \infty } \{\lim_{p\rightarrow \infty } \|\Gamma _{k}-\Delta _{p}\|_{2}^{2}\}= d^{2}$, from the above inequality, we obtain

$$ \limsup_{k\rightarrow \infty } \biggl\{ \limsup_{p \rightarrow \infty } \biggl\Vert S \biggl(\frac{1}{2}(\Gamma _{k}+\Delta _{p}) \biggr)-\frac{1}{2}(\Gamma _{k}+\Delta _{p}) \biggr\Vert _{2}^{2} \biggr\} =0. $$

Thus,

$$ \limsup_{k\rightarrow \infty } \biggl\{ \limsup _{p \rightarrow \infty } \biggl\Vert S \biggl(\frac{1}{2}(\Gamma _{k}+\Delta _{p}) \biggr)-\frac{1}{2}(\Gamma _{k}+\Delta _{p}) \biggr\Vert _{\infty} \biggr\} =0. $$

(5.7)

Assume there exists $u \in \mathcal{C}$ such that

$$ \limsup_{k\rightarrow \infty } \biggl\{ \limsup _{p \rightarrow \infty } \biggl\Vert \frac{1}{2}(\Gamma _{k}+ \Delta _{p})-u \biggr\Vert _{\infty} \biggr\} < \Omega < \frac{d}{2}. $$

(5.8)

From (5.7) and (5.8), we can choose subsequences $\{\Gamma _{k_{q}}\}$ and $\{\Gamma _{p_{q}}\}$ such that for $q \in \mathbb{N}$:

$$ \bigl\vert S(z_{q})-z_{q} \bigr\vert _{2} \leq \frac{2}{q} \quad\text{{and}}\quad \Vert z_{q}-u \Vert _{ \infty} \leq \Omega, \quad\text{{where }} z_{q}= \frac{1}{2}(\Gamma _{k_{q}}+ \Delta _{p_{q}}). $$

Therefore, $\lim_{q \rightarrow \infty }|S(z_{q})-z_{q}|_{2}=0$ and $\limsup_{q \rightarrow \infty } \|z_{q}-u\|_{\infty} \leq \Omega <\frac{d}{2}$, which contradicts Lemma 5 by assumption (2). Hence,

$$ \limsup_{k\rightarrow \infty } \biggl\{ \limsup _{p \rightarrow \infty } \biggl\Vert \frac{1}{2}(u_{n_{k}}+u_{m_{p}})-u \biggr\Vert _{\infty} \biggr\} =\frac{d}{2} \quad\text{{for all }}u \in \mathcal{C}. $$

(5.9)

In particular, it yields,

$$ \limsup_{k\rightarrow \infty } \biggl\{ \limsup_{p \rightarrow \infty } \biggl\Vert \frac{1}{2}(u_{n_{k}}+u_{m_{p}})- \frac{1}{2}(u+v) \biggr\Vert _{2}^{2} \biggr\} \geq \frac{d^{2}}{4}. $$

From Lemma 2, it follows that

$$\begin{aligned} \frac{d^{2}}{4} \leq & \limsup_{k\rightarrow \infty } \biggl\{ \limsup_{p\rightarrow \infty } \biggl\Vert \frac{1}{2}(u_{n_{k}}+u_{m_{p}})- \frac{1}{2}(u+v) \biggr\Vert _{2}^{2} \biggr\} \\ =& \limsup_{k\rightarrow \infty } \biggl\{ \limsup_{p \rightarrow \infty } \biggl\Vert \frac{1}{2}(u_{n_{k}}-u)+\frac{1}{2}(u_{m_{p}}-v) \biggr\Vert _{2}^{2} \biggr\} \\ =& \frac{1}{4} \lim_{k\rightarrow \infty } \Vert u_{n_{k}}-u \Vert _{2}^{2}+ \frac{1}{4} \lim_{p\rightarrow \infty } \Vert u_{m_{p}}-v \Vert _{2}^{2}. \end{aligned}$$

(5.10)

Since $u_{n_{k}} \rightharpoonup u \in \mathcal{D}$ as $k \to \infty$, then for each $k \in \mathbb{N}$,

$$\begin{aligned} \Vert u_{n_{k}}-w \Vert _{2}^{2} =& \Vert u_{n_{k}}-u+u-w \Vert _{2}^{2} \\ =& \Vert u_{n_{k}}-u \Vert _{2}^{2} +2 \langle u_{n_{k}}-u,u-w \rangle + \Vert u-w \Vert _{2}^{2}. \end{aligned}$$

From (5.2), we have

$$ \lim_{k\rightarrow \infty } \Vert u_{n_{k}}-u \Vert _{2}^{2}= \frac{d^{2}}{2}- \Vert w-u \Vert _{2}^{2}. $$

(5.11)

Similarly,

$$ \lim_{p \rightarrow \infty } \Vert u_{m_{p}}-v \Vert _{2}^{2}= \frac{d^{2}}{2}- \Vert w-v \Vert _{2}^{2}. $$

(5.12)

Using (5.11) and (5.12) in (5.10) it follows that

$$\begin{aligned} \frac{d^{2}}{4} \leq & \frac{1}{4} \biggl(\frac{d^{2}}{2}- \Vert w-u \Vert _{2}^{2} \biggr)+\frac{1}{4} \biggl( \frac{d^{2}}{2}- \Vert w-v \Vert _{2}^{2} \biggr) \\ \leq & \frac{d^{2}}{4}- \frac{1}{4} \bigl( \Vert w-u \Vert _{2}^{2}+ \Vert w-v \Vert _{2}^{2} \bigr) \end{aligned}$$

and it proves that $u=v=w$. Take $\sigma _{k}=u_{n_{k}}-w$ and $\varrho _{k}=u_{m_{k}}-w$. Since $\{u_{m_{k}}\}$ converges weakly to $v=w$,

$$ \varrho _{k} \rightharpoonup 0 \quad\text{{as }} k \to \infty. $$

(5.13)

Since $|u_{m_{k}}-w|_{2} \to d$ and $|u_{n_{k}}-w|_{2} \to d$ as $k \to \infty$, from Lemma 5, the following hold:

$$ \Vert \varrho _{k} \Vert _{\infty} \to \frac{d}{2} \quad\text{{and}}\quad \Vert \sigma _{k} \Vert _{\infty} \to \frac{d}{2}\quad \text{{as }} k \to \infty. $$

(5.14)

By the definition of d, the following condition is satisfied:

$$ \text{{for each }} k, p \in \mathbb{N},\quad \Vert \sigma _{k}-\varrho _{k} \Vert _{\infty} \leq \frac{d}{2}. $$

(5.15)

From (5.3), we have

$$\begin{aligned} \lim_{k\rightarrow \infty } \Bigl\{ \lim_{p\rightarrow \infty } \Vert u_{n_{k}}-u_{m_{p}} \Vert _{2}^{2} \Bigr\} =& \lim_{k \rightarrow \infty } \Bigl\{ \lim_{p\rightarrow \infty } \bigl\Vert (u_{n_{k}}-w)-(u_{m_{p}}-w) \bigr\Vert _{2}^{2} \Bigr\} \\ =&\lim_{k\rightarrow \infty } \Bigl\{ \lim_{p \rightarrow \infty } \Vert \sigma _{k}-\varrho _{p} \Vert _{2}^{2} \Bigr\} = d^{2}. \end{aligned}$$

(5.16)

From (5.9), we obtain $\frac{1}{2} \limsup_{k\rightarrow \infty } \{\limsup_{p\rightarrow \infty } \Vert (u_{n_{k}}-w)+(u_{m_{p}}-w) \Vert _{\infty} \}=\frac{d}{2}$ and it follows that

$$ \limsup_{k\rightarrow \infty } \Bigl\{ \limsup _{p \rightarrow \infty } \Vert \sigma _{k}+\varrho _{p} \Vert _{ \infty} \Bigr\} =d. $$

(5.17)

From Lemma 2, for all $k,p \in \mathbb{N}$, we have

$$ \Vert \sigma _{k}+\varrho _{p} \Vert _{2}^{2} =2 \Vert \sigma _{k} \Vert _{2}^{2} + \Vert \varrho _{p} \Vert _{2}^{2} - \Vert \sigma _{k} -\varrho _{p} \Vert _{2}^{2}. $$

In view of (5.13), (5.14), (5.15), (5.16), (5.17), and Lemma 6, this implies

$$ \lim_{k\rightarrow \infty } \Bigl\{ \lim_{p\rightarrow \infty } \Vert \sigma _{k} + \varrho _{p} \Vert _{2}^{2} \Bigr\} = d^{2}, $$

which is impossible. This completes the proof. □

Theorem 8

Let $\mathcal{C}$ be a nonempty, bounded, closed (resp., weak^∗ closed), and convex subset of a reflexive Banach space (resp., the conjugate of a separable Banach space) $\mathcal{M}$. Let $S:\mathcal{C} \to \mathcal{C}$ be a map** satisfying condition (L-1). Suppose that the relation ⊥ is uniformly approximately symmetric (resp., uniformly weak^∗ approximately symmetric) in $\mathcal{M}$, then $F(S) \neq \emptyset$.

Proof

By the application of Zorn’s lemma there exists a nonempty, bounded, closed, convex, and S-invariant subset $\mathcal{D}$ of $\mathcal{C}$ with no proper subsets, so $\mathcal{D}$ is minimal with respect to S. Since S satisfies condition (L-1), there exists an a.f.p.s. $\{u_{n}\}$ for S in $\mathcal{D}$. By the reflexiveness of $\mathcal{M}$, there exists a subsequence $\{u_{n_{k}}\}$ of $\{u_{n}\}$ such that $u_{n_{k}}$ converges weakly to $u^{\dagger}$. After possible extraction of a subsequence, if necessary, we assume that $\lim_{k \to \infty}\|x_{n_{k}}-x^{\dagger}\|=\Theta$. Take $v=u^{\dagger}-S(u^{\dagger})$. If $\Theta =0$ or $v=0$, then $S(u^{\dagger})=u^{\dagger}$ and the proof is completed. Therefore, we assume that $\Theta >0$ and $v \neq 0$. Following largely the same argument in [18, Theorem 1] let $\varepsilon =\frac{1}{2\Theta}$. By the assumptions, there exists a closed (resp., weak^∗ closed) linear subspace $\mathcal{Y}$ such that conditions (i) in Definition 4 and (iii) in Definition 6 are satisfied. This implies that there exists a $\delta >0$ such that

$$ \vert \mu \vert \leq \Vert v+\mu u \Vert - \vert \mu \vert \delta $$

(5.18)

for every $u \in \mathcal{Y}$, $\|u\|=1$ and each μ with $|\mu | \leq 2\Theta$. Further, the subspace spanned by $\mathcal{Y}$ and v has a finite-dimensional complement $\mathcal{Z}$. Therefore, for each $k \in \mathbb{N}$, $\sigma _{n_{k}} \in \mathcal{Y}$ and $\varrho _{n_{k}} \in \mathcal{Z}$, we have

$$ u_{n_{k}}-u^{\dagger}=\mu _{n_{k}} v+\sigma _{n_{k}}+\varrho _{n_{k}}. $$

(5.19)

Since $\mathcal{Z}$ is a finite-dimensional space and noting the convergence of $u_{n_{k}}-u^{\dagger}$, it follows that $\mu _{n_{k}} \to 0$ and $\|\varrho _{n_{k}}\| \to 0$ as $k \to \infty$. Thus, $\|\sigma _{n_{k}}\| \to \Theta$ and for sufficiently large k, $\frac{\|\sigma _{n_{k}}\|}{(1+\mu _{n_{k}})} \leq 2 \Theta$. From (5.18) and (5.19), we have

$$\begin{aligned} \bigl\Vert u_{n_{k}}-S \bigl(u^{\dagger} \bigr) \bigr\Vert =& \bigl\Vert u_{n_{k}}-u^{\dagger}+u^{\dagger}-S \bigl(u^{ \dagger} \bigr) \bigr\Vert = \bigl\Vert (1+\mu _{n_{k}}) y+ \sigma _{n_{k}}+\varrho _{n_{k}} \bigr\Vert \\ \geq & \bigl\Vert (1+\mu _{n_{k}}) v+\sigma _{n_{k}} \bigr\Vert - \Vert \varrho _{n_{k}} \Vert \\ \geq & \vert 1+\mu _{n_{k}} \vert \biggl\Vert v+ \biggl( \frac{ \Vert \sigma _{n_{k}} \Vert }{(1+\mu _{n_{k}})} \biggr) \frac{\sigma _{n_{k}}}{ \Vert \sigma _{n_{k}} \Vert } \biggr\Vert - \Vert \varrho _{n_{k}} \Vert \\ \geq & \Vert \sigma _{n_{k}} \Vert (1+\delta )- \Vert \varrho _{n_{k}} \Vert . \end{aligned}$$

(5.20)

Since the map** S satisfies condition (L-1), we have

$$ \bigl\Vert u_{n_{k}}-S \bigl(u^{\dagger} \bigr) \bigr\Vert \leq \bigl\Vert u_{n_{k}}-u^{\dagger} \bigr\Vert + c_{k}, $$

(5.21)

where $c_{k} \to 0$ as $k \to \infty$. Making $k \to \infty$, $\|u_{n_{k}}-S(u^{\dagger})\| \to \Theta$. From (5.20), noting that $\|u_{n_{k}}\| \to \Theta$ and $\|v_{n_{k}}\| \to 0$ as $k \to \infty$ we obtain the following inequality

$$ \Theta \geq (1+\delta ) \Theta, $$

which is a contradiction. Therefore, $\Theta =0$, and this completes the proof. □

Corollary 1

Let $\mathcal{C}$ be a convex, bounded, and weak^∗ closed subset of $\ell _{1}$ or the James space $J_{0}$. Let $S:\mathcal{C} \to \mathcal{C}$ be a map** satisfying condition (L-1). Then, S has a fixed point in $\mathcal{C}$.

We conclude the paper by posing the following interesting problem.

Kassay [19] showed that the converse of the above theorem is also true. More precisely, a reflexive Banach space having normal structure can be characterized by the fixed-point property for Jaggi-nonexpansive map**s.

5.1 Problem

Can a reflexive Banach space having asymptotic normal structure be characterized by the fixed-point property for map** satisfying condition (L-1)?