The 22nd Annual Meeting in Mathematics (AMM 2017)
Department of Mathematics, Faculty of Science
Chiang Mai University, Chiang Mai, Thailand
Convergence theorem of the Mann iteration for
a class of generalized monotone nonexpansive
mappings
Chayanit Klangpraphan‡ and Weerayuth Nilsrakoo†
Department of Mathematics, Statistics and Computer, Faculty of Science
Ubon Ratchathani University,Ubon Ratchathani 34190, Thailand
Abstract
In this paper, we concerned with convergence of the Mann iteration for finding an order
fixed point of a monotone mapping that satisfies condition (Cλ ) in an ordered Banach space.
As consequence, several convergence theorems for monotone nonexpansive mappings are
deduced. Our results not only include the recent ones announced by Dehaish and Khamsi [8]
and Song et al. [10] as special cases but also are established the weaker assumptions.
Keywords: convergence, Mann iteration, monotone mapping, nonexpansive mapping,
ordered Banach space.
2010 MSC: Primary 47H10; Secondary 47J05, 47H17, 49J40.
1
Introduction
Throughout this paper, X denotes a Banach space and N is the set of natural numbers. Let
C be a subset of X and T : C → C be a mapping. A point z ∈ X is said to be a fixed point of
T if T z = z. The set of fixed points of T is denoted by F (T ). The mapping T : C → C is said
to be nonexpansive if ∥T x − T y∥ ≤ ∥x − y∥ for all x, y ∈ C.
The convergence theorem of nonexpansive mappings have been considered by many researchers in recent years. In 1953, Mann [3] introduced the following iteration to find a fixed
point of a nonexpansive mapping T , which referred as the Mann iteration,
xn+1 = βn T (xn ) + (1 − βn )xn
(1.1)
for each n ≥ 1 and x1 ∈ C where {βn } is a sequence in [0, 1].
In 2008, Suzuki [6] introduced some condition on mappings, condition (C), that weaker than
nonexpansiveness and obtained the convergence theorems for mappings that satisfy such condition. In 2011, Garcia et al. [7] defined a kind of generalization of condition (C) as condition
(Cλ ).
On the other hand, fixed point theory in partially ordered metric spaces has been initiated
by Ran and Reurings [5] for finding applications to metrix equation. Recently, Dehaish and
∗
This research was financially supported by The Science Achievement Scholarship of Thailand.
Corresponding author.
‡
Speaker.
E-mail address: [email protected] (C. Klangpraphan), [email protected]
(W. Nilsrakoo).
†
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Khamsi [8] introduced the concept of a monotone nonexpansive mapping in a Banach space X
endowed with the partial order ≼. Let C be a subset of X and a mapping T : C → C is called
1. monotone if T x ≼ T y for all x, y ∈ C with x ≼ y;
2. monotone nonexpansive if T is monotone and
∥T x − T y∥ ≤ ∥x − y∥
for all x, y ∈ C with x ≼ y.
They used the Mann iteration for finding some order fixed points of monotone nonexpansive
mappings in X and obtained the weak convergence of Mann iteration that provided βn ∈
1
[a, b] ⊂ (0, 1). But their results do not entail βn =
. Very recently, Song et al. [10]
n+1
proved some convergence theorems of the Mann iteration for finding an order fixed point of
monotone nonexpansive mapping in a uniformly convex Banach space under the condition
∞
∑
βn (1 − βn ) = ∞.
n=1
In [9], monotone (Cλ )−conditions are defined in L1 [a, b]. They presented existence of fixed
point of such mapping.
In this paper, we mainly consider the Mann iteration for monotone mappings that satisfy
condition (Cλ ) in an ordered Banach space X with the partial order ≼. In Section 3, we prove
some weak and strong convergence theorems of the Mann iteration for monotone mappings that
satisfy condition (Cλ ) in a uniformly convex Banach space which are extended and improved [10].
Moreover, convergence results without uniformly convexity are presented.
2
Preliminaries
Throughout this paper, let X be an ordered Banach space with the norm ∥ · ∥ and the partial
order ≼. Let → and ⇀ denote as the strong and weak convergences, respectively. In this paper,
we assume that the order intervals are closed and convex.
A space X is said to be uniformly convex if, for all ϵ ∈ (0, 2], there exists δ > 0
∥x + y∥
such that
< 1 − δ for all x, y ∈ X with ∥x∥ = ∥y∥ = 1 and ∥x − y∥ ≥ ϵ.
2
Definition 2.1. Let C be a nonempty subset of X and T : C → C be a monotone mapping. For
λ ∈ [0, 1), T is said to satisfy condition (Cλ ) if for all x, y ∈ C with x ≼ y and λ∥x−T x∥ ≤ ∥x−y∥
implies
∥T x − T y∥ ≤ ∥x − y∥ .
(2.1)
Remark 2.2.
1. If 0 ≤ λ1 < λ2 < 1, then the condition (Cλ1 ) implies condition (Cλ2 ) but the
converse fails. For example, let T : [0, 1] → [0, 1] defined by
x



2
x ̸= 1,
Tx =


1+λ


x=1
2+λ
where λ ∈ (0, 1). Then the mapping T satisfies condition (Cλ ) but it fails condition
(Cλ′ ) whenever 0 ≤ λ′ < λ ( see [7, Example 5] ).
2. T is monotone nonexpansive if and only if a monotone mapping T satisfies condition (C0 ).
The next example is a direct generalization of a monotone nonexpansive mapping.
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Example 2.3 ( [9],Example 1). Let C = {f ∈ Lp [0, 3] : f (x) = a}, where a ∈ [0, 3].
For f, g ∈ C, consider
{ the partial relation f ≼ g iff f (x) ≼ g(x) ∀x ∈ [0, 3]. Let T : C → C be
1,
f = 3,
defined by T (f ) =
0,
f ̸= 3.
Then T satisfies condition (C 1 ).
2
Lemma 2.4. Let C be a nonempty subset of X and T : C → C be a monotone mapping.
Assume that T satisfies condition (Cλ ). Then, for x, y ∈ C with x ≼ y, the following hold:
1. ∥T x − T 2 x∥ ≤ ∥x − T x∥.
2. Either λ∥x − T x∥ ≤ ∥x − y∥ or (1 − λ)∥T x − T 2 x∥ ≤ ∥T x − y∥.
3. Either ∥T x − T y∥ ≤ ∥x − y∥ or ∥T 2 x − T y∥ ≤ ∥T x − y∥.
Proof. It follows from [6, Lemma 5].
Lemma 2.5. Let C be nonempty subset of X. Assume that a monotone mapping T : C → C
satisfies condition (Cλ ). Then ∥x − T y∥ ≤ 3∥T x − x∥ + ∥x − y∥ for all x, y ∈ C with x ≼ y.
Proof. It follows from [6, Lemma 7].
Remark 2.6. If a monotone mapping T satisfies condition (Cλ ), then T is monotone quasinonexpansive, i.e., ∥T x − z∥ ≤ ∥x − z∥ for all x ∈ C and z ∈ F (T ) with x ≼ z.
Lemma 2.7 ( [4], Lemma 2). Let {zn } and {wn } be two sequences of X and {βn } be a sequence
in [0, 1]. Suppose that
zn+1 = (1 − βn )zn + βn wn f or all n ≥ 1.
If the following properties are satisfied:
(i)
∞
∑
βn = ∞ and lim sup βn < 1;
n→∞
n=1
(ii) lim ∥zn ∥ = d and lim sup ∥wn ∥ ≤ d;
n→∞
{
(iii)
n
∑
n→∞
}
βi wi
is bounded.
i=1
Then d = 0.
3
Main Results
From now on, we consider the Mann iteration process as follows: let x1 ∈ C
xn+1 = βn T (xn ) + (1 − βn )xn , n ∈ N
(3.1)
where {βn } is a sequence in [0, 1] and we denote
F≼ (T ) = {p ∈ F (T ) : p ≼ x1 } and F≽ (T ) = {p ∈ F (T ) : x1 ≼ p}.
The following lemma is showed by Dehaish and Khamsi [8].
Lemma 3.1 ( [8], Lemma 3.1). Let C be a nonempty subset of X. Let T : C → C be a
monotone mapping. Assume that the sequence {xn } defined by (3.1) with x1 ≼ T x1 . Then
(i) xn ≼ xn+1 ≼ T xn ≼ T xn+1 ;
(ii) xn ≼ x for all n ∈ N provided {xn } weakly converges to a point x ∈ C.
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Lemma 3.2. Let C be a nonempty closed convex subset of a Banach space X and T : C → C
be a monotone quasi-nonexpansive mapping. Assume that the sequence {xn } defined by (3.1)
with x1 ≼ T x1 and F≽ (T ) ̸= ∅. Then
(i) the sequence {xn } is bounded ;
(ii) lim ∥xn −z∥ and lim d(xn , F≽ (T )) exist provided d(xn , F≽ (T )) denotes the distance from
n→∞
n→∞
x to F≽ (T ).
Proof. Let z ∈ F≽ (T ). Since x1 ≼ z and T is monotone, T x1 ≼ z. By Lemma 3.1 (i), we get
xn ≼ xn+1 ≼ T xn ≼ z.
Since T is monotone quasi-nonexpansive, we get
∥xn+1 − z∥ = ∥(1 − βn )(xn − z) + βn (T xn − z)∥
≤ (1 − βn )∥xn − z∥ + βn ∥T xn − z∥
≤ (1 − βn )∥xn − z∥ + βn ∥xn − z∥
= ∥xn − z∥.
(3.2)
Then {∥xn − z∥} is nonincreasing and bounded. Hence (i) and (ii) follow. This completes the
proof.
3.1
Convergence results in uniformly convex Banach spaces
In this subsection, we present some convergence theorems for monotone mappings T that satisfies condition (Cλ ) using the Mann iteration process (3.1). In the sequel, we also need the
following lemma.
Lemma 3.3 ( [2], Theorem 2). Let r > 0 be a fixed real number. If X is a uniformly convex
Banach space, then there exists a continuous strictly increasing convex function
g : [0, +∞) → [0, +∞) with g(0) = 0 such that
∥ηx + (1 − η)y∥2 ≤ η∥x∥2 + (1 − η)∥y∥2 − η(1 − η)g(∥x − y∥)
for all x, y ∈ Br (0) := {u ∈ X : ∥u∥ ≤ r} and η ∈ [0, 1].
Lemma 3.4. Let C be a nonempty closed convex subset of a uniformly convex Banach space
X and T : C → C be a monotone quasi-nonexpansive mapping. Assume that the sequence {xn }
∞
∑
defined by (3.1) with x1 ≼ T x1 and F≽ (T ) ̸= ∅. If
βn (1 − βn ) = ∞, then
n=1
lim inf ∥T xn − xn ∥ = 0.
n→∞
Proof. Let z ∈ F≽ (T ). By Lemma 3.2, we have {∥xn − z∥} and {∥T xn − z∥} are bounded. We
may assume that such sequences belong to Br (0) where r > 0. By Lemma 3.3, we get
∥xn+1 − z∥2 = ∥(1 − βn )(xn − z) + βn (T xn − z)∥2
≤ (1 − βn )∥xn − z∥2 + βn ∥T xn − z∥2 − βn (1 − βn )g(∥xn − T xn ∥)
≤ ∥xn − z∥2 − βn (1 − βn )g(∥xn − T xn ∥)
and so
βn (1 − βn )g(∥xn − T xn ∥) ≤ ∥xn − z∥2 − ∥xn+1 − z∥2 .
Therefore, we get
∞
∑
(3.3)
βn (1 − βn )g(∥xn − T xn ∥) < ∞.
n=1
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Since
∞
∑
βn (1 − βn ) = ∞ and g is continuous strictly increasing,
n=1
lim inf ∥xn − T xn ∥ = 0.
n→∞
This completes the proof.
Recall that X is said to satisfy Opial property if there exists sequence {xn } in X
such that xn ⇀ x. Then,
lim inf ∥xn − x∥ < lim inf ∥xn − y∥
n→∞
n→∞
(3.4)
for all y ∈ X with x ̸= y.
Next, we present weak convergence theorem as follows.
Theorem 3.5. Let X be a uniformly convex Banach space with the Opial property. Let C be
a nonempty closed convex subset of X and a monotone mapping T : C → C satisfy condition
(Cλ ). Assume that the sequence {xn } defined by (3.1) with x1 ≼ T x1 and F≽ (T ) ̸= ∅.
∞
∑
If
βn (1 − βn ) = ∞ and βn ≥ λ, then {xn } converges weakly to a fixed point of T .
n=1
Proof. Notice that
λ∥xn − T xn ∥ =
λ
∥xn+1 − xn ∥ ≤ ∥xn+1 − xn ∥.
βn
Since T satisfies condition (Cλ ),
∥T xn − T xn+1 ∥ ≤ ∥xn − xn+1 ∥.
Then
∥xn+1 − T xn+1 ∥ ≤ (1 − βn )∥xn − T xn ∥ + ∥T xn − T xn+1 ∥
≤ (1 − βn )∥xn − T xn ∥ + ∥xn − xn+1 ∥
= ∥xn − T xn ∥.
It means that
lim ∥xn − T xn ∥ exists.
n→∞
It follows from Lemmas 3.2 (i) and 3.4 that the sequence {xn } is bounded and
lim ∥xn − T xn ∥ = 0.
n→∞
By Lemma 3.1(i), there exists {xnj } of {xn } such that xnj ⇀ z ∈ C. By Lemma 3.1(ii),
x1 ≼ xnj ≼ z. It follows from Lemma 2.5 that
∥xnj − T z∥ ≤ 3∥xnj − T xnj ∥ + ∥xnj − z∥.
This implies that
lim inf ∥xnj − T z∥ ≤ lim inf ∥xnj − z∥.
j→∞
j→∞
Suppose that T z ̸= z. By the Opial property, we have z ∈ F≽ (T ).
Next we will show that xn ⇀ z. Suppose that {xn } has two subsequences {xnj } and {xnk }
such that xnj ⇀ z, xnk ⇀ w. By similar argument as for z ∈ F≽ (T ), we have w ∈ F≽ (T ).
By Lemma 3.2(ii), lim ∥xn − z∥ and lim ∥xn − w∥ exist. By the Opial property, we have
n→∞
n→∞
lim ∥xn − z∥ = lim ∥xnj − z∥ < lim ∥xnj − w∥
n→∞
j→∞
j→∞
= lim ∥xn − w∥ = lim ∥xnk − w∥
n→∞
k→∞
≤ lim ∥xnk − z∥ = lim ∥xn − z∥
k→∞
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lim ∥xn − z∥ < lim ∥xn − z∥.
n→∞
n→∞
which is contraction. Thus {xn } converges weakly to z ∈ F≽ (T ).
If λ = 0 in Theorem 3.5, then T is monotone nonexpansive. The weak convergence results
are presented as follow corollary.
Corollary 3.6 ( [10], Theorem 3.10). Let X be a uniformly convex Banach space with the
Opial property. Let C be a nonempty closed convex subset of X and a mapping T : C → C be
monotone nonexpansive. Assume that the sequence {xn } defined by (3.1) with x1 ≼ T x1 and
∞
∑
βn (1 − βn ) = ∞, then {xn } converges weakly to a fixed point of T .
F≽ (T ) ̸= ∅. If
n=1
Recall that a mapping T : C → C satisfy condition (I) [1] with respect to K if there exists
a nondecreasing function f : [0, ∞) → [0, ∞) with f (r) > 0 for all r ∈ (0, ∞) such that
f (d(x, K)) ≤ ∥x − T x∥ for all x ∈ C, where d(x, K) denotes the distance of x from K.
Next, we present the strong convergence of T by assume that T satisfies condition (I).
Theorem 3.7. Let C be a nonempty closed convex subset of a uniformly convex Banach space
X and a monotone mapping T : C → C satisfy condition (Cλ ). Assume that the sequence {xn }
defined by (3.1) with x1 ≼ T x1 and F≽ (T ) ̸= ∅. If T satisfies condition (I) with respect to F≽ (T )
∞
∑
and
βn (1 − βn ) = ∞, then {xn } converges strongly to a fixed point of T .
n=1
Proof. Since T satisfies condition (I) with respect to F≽ (T ),
f (d(xn , F≽ (T ))) ≤ ∥xn − T xn ∥,
where f : [0, ∞) → [0, ∞) is nondecreasing function with f (0) = 0 and f (r) > 0 for all
r ∈ (0, ∞). It follows from Lemmas 3.2(ii) and 3.4, that
lim d(xn , F≽ (T )) = 0.
n→∞
Let {xnj } be a subsequence {xn } and {zj } be a sequence in F≽ (T ) such that ∥xnj − zj ∥ ≤
for all j ≥ 1. Then, by (3.2),
∥xnj+1 − zj ∥ ≤ ∥xnj − zj ∥ ≤
1
2j
1
2j
(3.5)
and so
∥zj+1 − zj ∥ ≤ ∥zj+1 − xnj+1 ∥ + ∥xnj+1 − zj ∥
1
1
≤ j+1 + j
2
2
1
< j−1 .
2
Then {zj } is Cauchy in F≽ (T ). Since F≽ (T ) is closed, {zj } converges to some z ∈ F≽ (T ).
Moreover,
∥xnj − z∥ ≤ ∥xnj − zj ∥ + ∥zj − z∥
1
≤ j + ∥zj − z∥ → 0 as j → ∞.
2
Since lim ∥xn − z∥ exists, we get
n→∞
lim ∥xn − z∥ = 0.
n→∞
It means that {xn } converges strongly to z of T . This completes the proof.
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If λ = 0 in Theorem 3.7, then T is monotone nonexpansive. The strong convergence results
are presented as follow corollary.
Corollary 3.8. Let C be a nonempty closed convex subset of a uniformly convex Banach space
X and a mapping T : C → C be monotone nonexpansive. Assume that the sequence {xn }
defined by (3.1) with x1 ≼ T x1 and F≽ (T ) ̸= ∅. If T satisfies condition (I) with respect to
∞
∑
F≽ (T ) and
βn (1 − βn ) = ∞, then {xn } converges strongly to a fixed point of T .
n=1
3.2
Convergence results without uniformly convexity
In this section, we consider the convergence theorem of T that satisfies condition (Cλ ) in general
Banach space.
Theorem 3.9. Let C be a closed convex subset of X. Let a monotone mapping T : C → C
satisfy condition (Cλ ). Assume that the sequence {xn } defined by (3.1) with x1 ≼ T x1 and
∞
∑
F≽ (T ) ̸= ∅. If
βn = ∞ and λ ≤ lim inf βn ≤ lim sup βn < 1, then
n→∞
n=1
n→∞
lim ∥xn − T xn ∥ = 0.
n→∞
Proof. We will apply Lemma 2.7. From the iteration (3.1), we have
zn+1 = (1 − βn )zn + βn wn
where zn = xn − T xn and wn = βn−1 (T xn − T xn+1 ). As the proof of Lemma 3.4,
lim ∥zn ∥ = lim ∥xn − T xn ∥ := d
n→∞
n→∞
Moreover,
lim sup ∥wn ∥ = lim sup ∥βn−1 (T xn − T xn+1 )∥
n→∞
n→∞
≤ lim sup βn−1 ∥xn − xn+1 ∥
n→∞
= lim sup ∥xn − T xn ∥ = d
n→∞
Finally, let z ∈ F≽ (T ). By Remark 2.6 and xn ≼ z for all n ≥ 1, we have
n
n
∑
∑
βi wi = βi (βi−1 (T xi − T xi+1 ))
i=1
i=1
= ∥T x1 − T xn+1 ∥
≤ ∥T x1 − z∥ + ∥z − T xn+1 ∥
By Lemma 3.2, we have that
{ n
∑
≤ ∥x1 − z∥ + ∥z − xn+1 ∥.
}
βi wi
is bounded. It follows then that d = 0. This completes
i=1
the proof.
We now have the following convergence theorems without uniformly convexity.
Theorem 3.10. Let X be a reflexive Banach space. Let C be a nonempty closed convex subset
of X. Let a monotone mapping T : C → C satisfy condition (Cλ ). Assume that the sequence
∞
∑
{xn } defined by (3.1) with x1 ≼ T x1 and F≽ (T ) ̸= ∅. If X satisfies Opial property,
βn = ∞
n=1
and λ ≤ lim inf βn ≤ lim sup βn < 1, then {xn } converges weakly to a fixed point of T .
n→∞
n→∞
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Theorem 3.11. Let C be a closed convex subset of X. Let a monotone mapping T : C → C
satisfy condition (Cλ ). Assume that the sequence {xn } defined by (3.1) with x1 ≼ T x1 and
∞
∑
F≽ (T ) ̸= ∅. If T satisfies condition (I) with respect to F≽ (T ),
βn = ∞ and λ ≤ lim inf βn ≤
n→∞
n=1
lim sup βn < 1, then {xn } converges strongly to a fixed point of T .
n→∞
If λ = 0 in Theorem 3.10 and Theorem 3.11, then the convergence results are presented as
follows.
Corollary 3.12. Let X be a reflexive Banach space. Let C be a nonempty closed convex subset
of X. Let T : C → C be monotone nonexpansive. Assume that the sequence {xn } defined by (3.1)
∞
∑
with x1 ≼ T x1 and F≽ (T ) ̸= ∅. If X satisfies Opial property,
βn = ∞ and lim sup βn < 1,
n→∞
n=1
then {xn } converges weakly to a fixed point of T .
Corollary 3.13. Let C be a closed convex subset of X. Let T : C → C be monotone nonexpansive. Assume that the sequence {xn } defined by (3.1) with x1 ≼ T x1 and F≽ (T ) ̸= ∅ .
∞
∑
If T satisfies condition (I) with respect to F≽ (T ),
βn = ∞ and lim sup βn < 1, then {xn }
n=1
n→∞
converges strongly to a fixed point of T .
Remark 3.14. By using the same ideas and techniques, we can also discuss the convergence
results for a monotone mapping T that satisfies condition (Cλ ) with F≼ (T ) ̸= ∅.
Acknowledgment. The authors are grateful to the referees for their careful reading of the
manuscript and their useful comments.
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