The Existence and Approximation Fixed Point

ISSN 0975-5748 (online); 0974-875X (print)

Published by RGN Publications http://www.rgnpublications.com

DOI: 10.26713/jims.v11i3-4.419

Research Article

The Existence and Approximation Fixed Point

Theorems for Monotone Nonspreading Mappings in Ordered Banach Spaces

Khanitin Muangchoo-in^1,2 and Poom Kumam^2,3,*

1Department of Exercise and Sports Science, Faculty of Sports and Health Science, 239, Moo. 4, Muang Chaiyaphum, Chaiyaphum 36000, Thailand

2KMUTTFixed Point Research Laboratory, KMUTT-Fixed Point Theory and Applications Research Group, Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand

3Center of Excellence in Theoretical and Computational

Science (TaCS-CoE), SCL 802 Fixed Point Laboratory, Science Laboratory Building, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand

*Corresponding author: [email protected]

Abstract. In this paper, we proved some existence theorems of fixed points for monotone nonspreading mappings T in a Banach space E with the partial order ≤. In order to finding a fixed point of such a mapping T , moreover we proved the convergence theorem of Mann iterative schemes under the condition P^∞

n=1βn(1 −βn) = ∞, which containβn=_n+1¹ as a special case.

Keywords. Ordered Banach space; Fixed point; Monotone nonspreading mapping; Mann iterative scheme

MSC. 47H07; 47H10

Received: June 6, 2016 Accepted: October 20, 2018

Copyright © 2019 Khanitin Muangchoo-in and Poom Kumam. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Let E be a Banach space and E^∗ be the dual space of E. For all x ∈ E and f ∈ E^∗, let the value of f at x be denoted by 〈x, f 〉. The normalized duality mapping J : E → 2^E∗ is defined by

J(x) = {f ∈ E^∗: 〈x, f 〉 = kxk², kf k = kxk}

for all x ∈ E. A single-valued normalized duality mapping is denoted by j, which means a mapping j : E → E^∗ such that, for each u ∈ E, j(u) ∈ E^∗satisfying the following:

〈 j(u), u〉 = k j(u)kkuk, k j(u)k = kuk.

In 2008, Kohsaka and Takahashi [3] also introduced the class of mappings called the class of nonspreading mappings to study the resolvent of a maximal monotone operator in Banach spaces. Let E be a smooth, strictly convex and reflexive Banach space and K be a nonempty closed convex subset of E.

Definition 1.1. A mapping T : K → K is said to be nonspreading if φ(Tx, T y) + φ(T y, Tx) ≤ φ(Tx, y) + φ(T y, x)

for all x, y ∈ C, where

φ(x, y) = kxk²− 2〈x, j(y)〉 + kyk².

The set of fixef points of a mapping T : K → K is defined by F(T) := {x ∈ C : T x = x}.

In 1954, Mann [5] introduced the following iteration to finding a fixed point, which is referred to as the Mann iteration,

x_n+1= βnx_n+ (1 − βn)T x_n

for each n ≥ 1, where βn∈ [0, 1] is a sequence with some condition. However, there are not many convergence theorems of such a iteration in a order Banach space (E, ≤). Motivated by the above results, we consider the weak convergence of the Mann iterative scheme for a monotone nonspreading mapping T under the condition

X∞ n=1

βn(1 − βn) = ∞

which contain βn= _n+1¹ as a special case. By motivation of Mann iteration for a monotone nonexpansive mapping of Dehaish and Khamsi [2].

2. Preliminaries

Let P be a closed convex cone of a real Banach space E. A partial order “≤” with respect to P in E is defined as follows:

x ≤ y (x ≤ y) if and on if y − x ∈ P (y − x ∈ ˚P) for all x, y ∈ E, where ˚P is the interior of P .

Throughout this paper, let E be a Banach space with the norm “k · k” and the partial order

that the order intervals are closed and convex. An order interval [x, y] for all x, y ∈ E is given by

[x, y] = {z ∈ E : x ≤ z ≤ y}. (1)

Then the convexity of the order interval [x, y] implies that

x ≤ tx + (1 − t)y ≤ y for all x, y ∈ E with x ≤ y. (2)

Definition 2.1. Let K be a nonempty closed and convex subset of a Banach space E. A mapping T : K → K is said to be:

(1) monotone if T x ≤ T y for all x, y ∈ K with x ≤ y;

(2) monotone nonspreading if T is monotone and kT x − T yk²≤ kx − yk²+ 2〈x − T x, J(y − T y)〉

for all x, y ∈ K with x ≤ y and J is normalized duality mapping.

A Banach space E is said to be:

(1) strictly convex if k^x+y₂ k < 1 for all x, y ∈ E with kxk = kyk = 1 and x 6= y;

(2) uniformly convex if, for allε ∈ (0,2], there exists δ > 0 such that ^kx+yk₂ < 1−δ for all x, y ∈ E with kxk = kyk = 1 and kx − yk ≥ ε.

The following inequality was showed by Xu [9] in a uniformly convex Banach space E, which is known as Xu’s inequality.

Lemma 2.2 (Xu [9, Theorem 2]). For any real numbers q > 1 and r > 0, a Banach space E is uniformly convex if and only if there exists a continuous strictly increasing convex function g : [0, +∞) → [0,+∞) with g(0) = 0 such that

ktx + (1 − t)yk^q≤ tkxk^q+ (1 − t)kyk^q− ω(q, t)g(kx − yk) (3) for all x, y ∈ Br(0) = {x ∈ E;kxk ≤ r} and t ∈ [0,1], where ω(q, t) = t^q(1 − t) + t(1 − t)^q. In particular, take q = 2 and t =¹₂,

°

°

° x + y

2

°

°

°

2

≤1

2kxk²+1

2kyk²−1

4g(kx − yk). (4)

The following conclusion is well known:

Lemma 2.3 (Takahashi [8, Theorem 1.3.11]). Let K be a nonempty closed convex subset of a reflexive Banach space E. Assume thatϕ : K → R is a proper convex lower semi-continuous and coercive function. Then the functionϕ attains its minimum on K, that is, there exists x ∈ K such that

ϕ(x) = inf

y∈Kϕ(y).

Theorem 2.4. Let {x_n} be a bounded above monotone nondecreasing sequence. Then {x_n} converges to the supemum of {x_n: n ∈N}.

Lemma 2.5. Let K be a nonempty closed convex subset of a uniformly convex Banach space (E, ≤) and T : K → K be a monotone nonspreading mapping. If x ∈ K such that x_n+1= T xn, the sequence {T x_n}^∞

n=1 is bounded. Then lim sup

n→∞ kxn− T xnk → 0.

Proof. From Theorem 2.4 {x_n} is bounded and monotone increasing then there exists M > 0 such that kxnk ≤ M so,

lim sup

n→∞ kxn− Mk ≤ 0 by analogy we obtain

lim sup

n→∞ kx_n+1− Mk ≤ 0 so,

lim sup

n→∞ kxn− x_n+1k ≤ lim sup

n→∞ [kxn− Mk + kx_n+1− Mk]

≤ lim sup

n→∞ kxn− Mk + lim sup

n→∞ kx_n+1− Mk

= 0, therefore, we can conclude that

lim sup

n→∞ kxn− x_n+1k → 0. (5)

Theorem 2.6. Let {x_n} be a bounded above monotone nonincreasing sequence, then {x_n} converges to the infimum of {xn: n ∈N}.

Lemma 2.7. Let K be a nonempty closed convex subset of an uniformly convex Banach space (E, ≤) and T : K → K be a monotone nonspreading mapping. If x ∈ K such that x_n+1= T xn, the sequence {T x_n}^∞

n=1is bounded. Then lim sup

n→∞ kxn− T xnk → 0.

Proof. From Theorem 2.6 {x_n} is bounded and monotone decreasing then there exists M > 0 such that kxnk ≤ M so,

lim inf

n→∞ kxn− Mk ≤ 0 by analogy we get

lim inf

n→∞ kx_n+1− Mk ≤ 0 so,

lim inf

n→∞ kxn− x_n+1k ≥ lim inf

n→∞ [kxn− Mk + kx_n+1− Mk]

≥ lim inf

n→∞ kxn− Mk + lim inf

n→∞ kx_n+1− Mk

= 0.

Therefore, we can conclude that lim inf

n→∞ kxn− x_n+1k → 0

on the other hand, we can conclude that lim inf

n→∞ kxn− x_n+1k = lim sup

n→∞ kxn− x_n+1k → 0. (6)

Definition 2.8. Let E be a smooth Banach space and define the functionalφ : E × E → R by φ(x, y) = kxk²− 2〈x, J y〉 + |y|²

for x, y ∈ E from the definition of φ, we have (kxk − kyk)²≤ φ(x, y) ≤ (kxk + kyk)²,

kxk²− 2kxkkyk + kyk²≤ kxk²− 2〈x, J y〉 + kyk²≤ kxk²+ 2kxkkyk + kyk² and, we can conclude that

2〈x, J y〉 ≤ 2kxkkyk. (7)

3. Main Results

3.1 Existence of Fixed Points

In this section, we prove the existence theorem of fixed points of a monotone nonspreading mapping in an uniformly convex Banach space (E, ≤).

Theorem 3.1. Let K be a nonempty and closed convex subset of an uniformly convex Banach space (E, ≤) and T : K → K be a monotone nonspreading mapping. Assume that there exists x ∈ K such that x ≤ T x, the sequence {T xn}^∞

n=1is bounded and T x_n≤ y for some y ∈ K and all n ≥ 1.

ThenF(T) 6= ; and x ≤ y^∗ for some y^∗∈ F(T).

Proof. Let x1= x, and x_n+1= T xn= Tⁿx. So, we have x1= x ≤ T x = x2, and so, we get x₂= T x1= T x ≤ T x2= T²x = x3.

By analogy, we must have

x = x1≤ x2≤ x3≤ · · · ≤ xn≤ x_n+1≤ · · · .

Let K_n= {z ∈ K; xn≤ z} for all n ≥ 1. Clearly, for each n ≥ 1, Kn is closed convex (K_n∈ K ) and K_n is nonempty too ( y ∈ Kn). Let K^∗= ^∞T

n=1

K_n. Then K^∗ is a nonempty closed convex subset of K . Since {x_n} is bounded, we can define a functionϕ : K^∗→ [0, +∞) as follows:

ϕ(z) = limsup

n→∞ kxn− zk²

for all z ∈ K^∗. From Lemma 2.3, it follows that there exists y^∗∈ K such that ϕ(y^∗) = inf

z∈K^∗ϕ(z). (8)

Now, we show y^∗= T y^∗. In fact, by the definition of K^∗, we obtain x₁≤ x2≤ x3≤ · · · ≤ xn≤ x_n+1≤ · · · ≤ y^∗.

Then, we have x_n+1= T xn≤ T y^∗by the monotonicity of T and hence, for each n ≥ 1, xn≤ T y^∗. So we have T y^∗∈ K^∗. From the convexity of K^∗, it follows that ^{y∗+T y}₂ ^∗ ∈ K^∗and so, by equation (8), we have

ϕ(y^∗) ≤ ϕ³y^∗+ T y^∗ 2

´

and ϕ(y^∗) ≤ ϕ(T y^∗). (9)

By the way, from equations (5) and (7), we obtain ϕ(T y^∗) = limsup

n→∞ kx_n+1− T y^∗k²

= lim sup

n→∞ kT xn− T y^∗k²

≤ lim sup

n→∞ [kxn− y^∗k²+ 2〈xn− T xn, J( y^∗− T y^∗)〉]

≤ lim sup

n→∞ [kxn− y^∗k²+ 2kxn− T xnkky^∗− T y^∗k]

≤ lim sup

n→∞ kxn− y^∗k²+ lim sup

n→∞ 2kxn− T xnkky^∗− T y^∗k

≤ lim sup

n→∞ kxn− y^∗k² (10)

= ϕ(y^∗).

Combining equations (9) and (10), we have

ϕ(T y^∗) = ϕ(y^∗). (11)

It follows from Lemma 2.7 (q = 2 and t =¹₂) and equation (11) that ϕ³y^∗+ T y^∗

2

´

= lim sup

n→∞

°

°

°x_n−y^∗+ T y^∗ 2

°

°

°

2

= lim sup

n→∞

°

°

°

x_n− y^∗

2 +x_n− T y^∗ 2

°

°

°

2

≤ lim sup

n→∞

³1

2kxn− y^∗k²+1

2kxn− T y^∗k²−1

4g(ky^∗− T y^∗k)

´

≤1

2ϕ(y^∗) +1

2ϕ(T y^∗) −1

4g(ky^∗− T y^∗k)

= ϕ(y^∗) −1

4g(ky^∗− T y^∗k).

Noticing equation (9), we have g(ky^∗− T y^∗k) ≤ ϕ(y^∗) − ϕ

³y^∗+ T y^∗ 2

´

≤ 0

and from Lemma 2.2, we have g(ky^∗− T y^∗k) = 0. Thus we have y^∗= T y^∗ by the property of g.

This yields the desired conclusion.

Theorem 3.2. Let K be a nonempty and closed convex subset of an uniformly convex Banach space (E, ≤) and T : K → K be a monotone nonspreading mapping. Assume that there exists x ∈ K such thatT x ≤ x, the sequence {T xn}^∞

n=1is bounded and y ≤ T xn for some y ∈ K and all n ≥ 1.

ThenF(T) 6= ; and y^∗≤ x for some y^∗∈ F(T).

Proof. Let x₁= x and x_n+1= T xn= T xn. Then T x = x2≤ x1= x, and so, T x₂= T²x = x3≤ x2= T x1= T x .

By analogy, we have

· · · ≤ x_n+1≤ xn≤ · · · ≤ x3≤ x2≤ x = x1.

Let K_n= {z ∈ K; z ≤ xn}for all n ≥ 1. Clearly, for each n ≥ 1, Kn is closed convex (K_n∈ K ) and K_n is nonempty too ( y ∈ Kn). Let K^∗= ^∞T

n=1

K_n. Then K^∗ is a nonempty closed convex subset of

K . Since {x_n} is bounded, we can define a functionϕ : K^∗→ [0, +∞) as follows:

ϕ(z) = limsup

n→∞ kxn− zk²

for all z ∈ K^∗. From Lemma 2.2, it follows that there exists y^∗∈ K such that ϕ(y^∗) = inf

z∈K^∗ϕ(z). (12)

Now, we show y^∗= T y^∗. In fact, by the definition of K^∗, we obtain y^∗≤ · · · ≤ x_n+1≤ xn≤ · · · ≤ x3≤ x2≤ x1.

Then, we have T y^∗≤ x_n+1= T xn by the monotonicity of T and hence, for each n ≥ 1, T y^∗≤ xn. So we have T y^∗∈ K^∗. From the convexity of K^∗, it follows that ^{y∗+T y}₂ ^∗ ∈ K^∗and so, by equation (12), we have

ϕ(y^∗) ≤ ϕ³y^∗+ T y^∗ 2

´

and ϕ(y^∗) ≤ ϕ(T y^∗). (13)

On the other hand, by using equations (6) and (7) we get ϕ(T y^∗) = limsup

n→∞ kx_n+1− T y^∗k²= lim sup

n→∞ kT xn− T y^∗k²

≤ lim sup

n→∞ [kxn− y^∗k²+ 2〈xn− T xn, J( y^∗− T y^∗)〉]

≤ lim sup

n→∞ [kxn− y^∗k²+ 2kxn− T xnkky^∗− T y^∗k]

≤ lim sup

n→∞ kxn− y^∗k²+ lim sup

n→∞ 2kxn− T xnkky^∗− T y^∗k

≤ lim sup

n→∞ kxn− y^∗k²

= ϕ(y^∗). (14)

Combining equations (13) and (14), we have

ϕ(T y^∗) = ϕ(y^∗). (15)

It follows from Lemma 2.7 (q = 2 and t =¹₂) and (15) that ϕ³y^∗+ T y^∗

2

´

= lim sup

n→∞

°

°

°x_n−y^∗+ T y^∗ 2

°

°

°

2

= lim sup

n→∞

°

°

°

x_n− y^∗

2 +x_n− T y^∗ 2

°

°

°

2

≤ lim sup

n→∞

³1

2kxn− y^∗k²+1

2kxn− T y^∗k²−1

4g(ky^∗− T y^∗k)

´

≤1

2ϕ(y^∗) +1

2ϕ(T y^∗) −1

4g(ky^∗− T y^∗k)

= ϕ(y^∗) −1

4g(ky^∗− T y^∗k).

Noticing equation(13), we have

g(ky^∗− T y^∗k) ≤ ϕ(y^∗) − ϕ³y^∗+ T y^∗ 2

´

≤ 0

and from Lemma 2.2, we have g(ky^∗− T y^∗k) = 0. Thus we have y^∗= T y^∗by the property of g.

This yields the desired conclusion. This completes the proof.

3.2 The Convergence of the Mann Iteration

In this section, for a monotone nonspreading mapping T , we consider the Mann iteration sequence defined by

x_n+1= βnx_n+ (1 − βn)T x_n (16)

for each n ≥ 1, where {βn} in (0, 1) satisfies the following condition:

X∞ n=1

βn(1 − βn) = ∞.

Clearly, the above condition containsβn=_n+1¹ as a special case.

Lemma 3.3 (Dehaish and Khamsi [2, Lemma 3.1]). Let K be a nonempty and closed convex subset of a Banach space (E, ≤) and T : K → K be a monotone mapping. Assume that the sequence {x_n} is defined by equation(16) and x₁≤ T x1 (or T x₁≤ x1). If F(T) 6= ; and p ≤ x1 (or x₁≤ p) for some p ∈ F(T), then

(1) {x_n} is bounded and x_n≤ x_n+1≤ T xn(or T x_n≤ x_n+1≤ xn);

(2) x_n≤ x (or x ≤ xn) for all n ≥ 1 provided {xn} weakly converges to a point x ∈ K .

Lemma 3.4. Let K be a nonempty and closed convex subset of a Banach space (E, ≤) and T : K → K be a monotone nonspreading mapping. Assume that the sequence {xn} is defined by equation (16) and x₁≤ T x1 (or T x₁≤ x1). If F(T) 6= ; and p ≤ x1 (or x₁≤ p) for some p ∈ F(T), then

n→∞lim kxn− pk exists.

Proof. Assume that p ≤ xn for any n ≥ 1. Since T is monotone, then we obtain p = T p ≤ T x1. Since the order interval [p, →) is convex, assume p ≤ x2 since T is monotone, then we have T p ≤ T x2. By induction we will show that p ≤ xnfor any n ≥ 1, as claimed. Since T is monotone nonspreading, from equation (7), we have p = T p and we get

kT xn− pk²= kT xn− T pk²

≤ kxn− pk²+ 2〈xn− T xn, J(p − T p)〉

≤ kxn− pk²+ 2kxnkkp − T pk

≤ kxn− pk² it follows that,

kT xn− pk ≤ kxn− pk.

Since equation (16), which implies

kx_n+1− pk ≤ tnkT xn− pk + k(1 − tn)kxn− pk

≤ tnkxn− pk + k(1 − tn)kxn− pk

≤ kxn− pk

for any n ≥ 1. This means that kxn− pk is a monotone sequence, which implies that lim

n→∞kxn− pk exists.

Theorem 3.5. Let K be a nonempty and closed convex subset of an uniformly convex Banach space (E, ≤) and T : K → K be a monotone nonspreading mapping. Assume that the sequence {x_n} is defined by equation(16) and x1≤ T x1 (or T x1≤ x1). If F(T) 6= ; and p ≤ x1 (or x1≤ p) for some p ∈ F(T), then

n→∞lim kxn− T xnk = 0.

Proof. It follows from Lemma 3.4 that p ≤ x1≤ xn(or xn≤ x1≤ p)

for all n ≥ 1. Then it follows from the nonspreadingness of T, p = T p and an application of Lemma 2.7 (q = 2 and t = βn) that

kx_n+1− pk²= kβn(x_n− p) + (1 − βn)(T x_n− T p)k²

≤ βnkxn− pk²+ (1 − βn)kT xn− T pk²− βn(1 − βn)g(kxn− T xnk)

≤ βnkxn− pk²+ (1 − βn)kxn− pk²+ 2kxn− T xnkkp − T pk − βn(1 − βn)g(kxn− T xnk)

≤ βnkxn− pk²+ (1 − βn)kxn− pk²− βn(1 − βn)g(kxn− T xnk)

≤ kxn− pk²− βn(1 − βn)g(kxn− T xnk) and so

βn(1 − βn)g(kxn− T xnk) ≤ kxn− pk²− kx_n+1− pk². Therefore, we have

X∞ n=1

βn(1 − βn)g(kxn− T xnk) ≤ kx1− pk²< +∞. (17) Now, we claim that there exists a subsequence {x_nk}such that

k→∞lim g(kxn_k− T xn_kk) = 0. (18)

Suppose that the conclusion is not true. Then, for all subsequence {x_nk}such that lim

k→∞g(kxnk− T x_nkk) > 0, we have

lim inf

n→∞ g(kxn− T xnk) > 0.

Thus there exists a positive number a and a positive integer N such that g(kxn− T xnk) > a > 0 for all n > N. Consequently, we have

βn(1 − βn)g(kxn− T xnk) ≥ aβn(1 − βn) and hence, by the condition ^∞P

n=1βn(1 − βn) = +∞, we obtain X∞

n=1

βn(1 − βn)g(kxn− T xnk) = +∞.

This contradicts equation (17). So equation (18) holds and hence, by the property of g(0) = 0, we have

k→∞limkxnk− T xnkk = 0.

Otherwise, we obtain

kx_n+1− T x_n+1k = kβn(x_n− T xn) + (T xn− T x_n+1)k

≤ βnkxn− T xnk + kx_n+1− xnk

= βnkxn− T xnk + (1 − βn)kxn− T xnk

= kxn− T xnk.

Therefore, the sequence {kxn− T xnk} is monotonically nonincreasing and hence it follows that

n→∞lim kxn− T xnk exists. This yields the desired conclusion.

Recall that a Banach space E is said to satisfy Opial’s condition ([7]) if a sequence {x_n}with {x_n}weakly converges to a point x ∈ E implies

lim sup

n→∞ kxn− xk < lim sup

n→∞ kxn− yk for all y ∈ E with y 6= x.

Next, we show the weak convergence of the sequence {xn}defined by equation (16). The proof is similar to ones of Dehaish and Khamsi [2], but, for more details, we give the proof.

Theorem 3.6. Let K be a nonempty and closed convex subset of an uniformly convex Banach space (E, ≤) and T : K → K be a monotone nonspreading mapping. Assume that E satisfies Opial’s condition and the sequence {x_n} is defined by equation(16) with x₁≤ T x1 (or T x₁≤ x1).

If F(T) 6= ; and p ≤ x1 (or x₁≤ p) for some p ∈ F(T), then {xn} weakly converges to a fixed point x^∗ ofT .

Proof. It follows from Lemma 3.4 and Theorem 3.5 that {xn} is bounded and

n→∞lim kxn− T xnk = 0.

Then, there exists a subsequence {xn_k} ⊂ {xn}such that {xn_k}weakly converges to a point x^∗∈ K.

Following Lemma 3.4, we have x₁≤ xnk≤ x^∗ (or x^∗≤ xnk ≤ x1) for all k ≥ 1. In particular, we have

k→∞limkxnk− T xnkk = 0.

Now, we claim that x^∗ = T x^∗. In fact, assume that this is not true. Then, from the nonspreadingness of T and Opial’s condition, it follows that

lim sup

k→∞

kxnk− x^∗k < lim sup

k→∞

kxnk− T x^∗k

≤ lim sup

k→∞ (kxnk− T xnkk + kT xnk− T x^∗k)

≤ lim sup

k→∞ (kT xnk− T x^∗k). (19)

Consider equation (19), lim sup

k→∞ kT xnk− T x^∗k²≤ lim sup

k→∞ [kxnk− x^∗k²+ 2kxnk− T xnkkkx^∗− T x^∗k]

≤ lim sup

k→∞

kxn_k− x^∗k²+ lim sup

k→∞ 2kxn_k− T xn_kkkx^∗− T x^∗k

≤ lim sup

k→∞

kxnk− x^∗k². (20)

So, we get

From equations (19) and (21), we obtain lim sup

k→∞

kxn_k− x^∗k ≤ lim sup

k→∞

kxn_k− x^∗k

which is a contradiction. Thus, by Lemma 3.4, it follows that the limit lim

n→∞kxn− x^∗k exists. Now, we show that {xn} weakly converges to the point x^∗. Suppose that this is not true. Then There exists a subsequence {x_nj} to converge weakly to a point z ∈ K and z 6= x^∗. Similarly, it follows that z = T z and lim

n→∞kxn− zk exists. It follows from Opial’s condition that

n→∞lim kxn− zk < lim

n→∞kxn− x^∗k = lim sup

i→∞

kxn_i− x^∗k < lim

n→∞kxn− zk.

This is a contradiction and hence x^∗= z. This completes the proof.

4. Conclusions

We prove some existence theorems of fixed point for monotone nonspreading mapping in a Banach space E with the partial order ≤.

Theorem 4.1. Let K be a nonempty and closed convex subset of an uniformly convex Banach space (E, ≤) and T : K → K be a monotone nonspreading mapping. Assume that there exists x ∈ K such that x ≤ T x, the sequence {T xn}^∞_n=1is bounded and T x_n≤ y for some y ∈ K and all n ≥ 1.

ThenF(T) 6= ; and x ≤ y^∗ for some y^∗∈ F(T).

Theorem 4.2. Let K be a nonempty and closed convex subset of an uniformly convex Banach space (E, ≤) and T : K → K be a monotone nonspreading mapping. Assume that there exists x ∈ K such thatT x ≤ x, the sequence {T xn}^∞

n=1is bounded and y ≤ T xn for some y ∈ K and all n ≥ 1.

ThenF(T) 6= ; and y^∗≤ x for some y^∗∈ F(T).

In part of convergence theorem, we prove a weak convergence theorem for monotone nonspreading in order Banach space (E, ≤).

Theorem 4.3. Let K be a nonempty and closed convex subset of an uniformly convex Banach space (E, ≤) and T : K → K be a monotone nonspreading mapping. Assume that E satisfies Opial’s condition and the sequence {x_n} is defined by equation(16) with x₁≤ T x1 (or T x₁≤ x1).

If F(T) 6= ; and p ≤ x1 (or x₁≤ p) for some p ∈ F(T), then {xn} weakly converges to a fixed point x^∗ ofT .

And we can get some results if we reduce some conditions for prove some existence theorems of fixed point by using a monotone nonexpansive mapping T in a Banach space E with the partial order “≤”, in Theorem 4.1 and 4.2, we have following corollaries respectively.

Corollary 4.4. Let K be a nonempty and closed convex subset of an uniformly convex Banach space (E, ≤) and T : K → K be a monotone nonexpansive mapping. Assume that there exists x ∈ K such that x ≤ T x, the sequence {T xn}^∞

n=1is bounded and T x_n≤ y for some y ∈ K and all n ≥ 1.

ThenF(T) 6= ; and x ≤ y^∗ for some y^∗∈ F(T).

Corollary 4.5. Let K be a nonempty and closed convex subset of an uniformly convex Banach space (E, ≤) and T : K → K be a monotone nonexpansive mapping. Assume that there exists x ∈ K

such thatT x ≤ x, the sequence {T xn}^∞

n=1is bounded and y ≤ T xn for some y ∈ K and all n ≥ 1.

ThenF(T) 6= ; and y^∗≤ x for some y^∗∈ F(T).

And if we consider the convergence of Mann iteration for a monotone nonexpansive mapping T , in Theorem 3.5 and 4.3 by using Dehaish and Khamsi [2, Lemmas 3.1 and 3.2], we have following corollary:

Corollary 4.6. Let K be a nonempty and closed convex subset of an uniformly convex Banach space (E, ≤) and T : K → K be a monotone nonexpansive mapping. Assume that the sequence {xn} is defined by equation (16) and x₁≤ T x1 (or T x₁≤ x1). If F(T) 6= ; and p ≤ x1 (or x₁≤ p) for some p ∈ F(T), then lim

n→∞kxn− T xnk = 0.

Corollary 4.7. Let K be a nonempty and closed convex subset of an uniformly convex Banach space (E, ≤) and T : K → K be a monotone nonexpansive mapping. Assume that E satisfies Opial’s condition and the sequence {x_n} is defined by equation(16) with x₁≤ T x1 (or T x₁≤ x1).

If F(T) 6= ; and p ≤ x1 (or x₁≤ p) for some p ∈ F(T), then {xn} weakly converges to a fixed point x^∗ ofT .

Acknowledgement

The author would like to thank Theoretical and Computational Science (TaCS) Center under Computational and Applied Science for Smart Innovation Research Cluster (CLASSIC), Faculty of Science, KMUTT for financial support.

Competing Interests

The authors declare that they have no competing interests.

Authors’ Contributions

All the authors contributed significantly in writing this article. The authors read and approved the final manuscript.

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