# Necessary and Sufficient Condition for Mann Iteration Converges to a Fixed Point of Lipschitzian Mappings

Authors: Chang-He Xiang; Jiang-Hua Zhang; Zhe Chen
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

Suppose thatE is a real normed linear space, C is a nonempty convex subset of E, T:C→C is a Lipschitzian mapping, and x*∈C is a fixed point of T. For given x0∈C, suppose that the sequence {xn}⊂C is the Mann iterative sequence defined by xn+1=(1-αn)xn+αnTxn,n≥0, where {αn} is a sequence in [0, 1], ∑n=0∞αn2<∞, ∑n=0∞αn=∞. We prove that the sequence {xn} strongly converges to x* if and only if there exists a strictly increasing function Φ:[0,∞)→[0,∞) with Φ(0)=0 such that limsupn→∞infj(xn-x*)∈J(xn-x*){〈Txn-x*,j(xn-x*)〉-∥xn-x*∥2+Φ(∥xn-x*∥)}≤0.


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## Body

## 1. Introduction

LetE be an arbitrary real normed linear space with dual space E*, and let C be a nonempty subset of E. We denote by J the normalized duality mapping from E to 2E* defined by
(1.1)J(x)={f∈E*:〈x,f〉=∥x∥2=∥f∥2},∀x∈E,
where 〈·,·〉 denotes the generalized duality pairing.A mappingT:C→E is called strongly pseudocontractive if there exists a constant k∈(0,1) such that, for all x,y∈C, there exists j(x-y)∈J(x-y) satisfying
(1.2)〈Tx-Ty,j(x-y)〉≤(1-k)∥x-y∥2.T is called ϕ-strongly pseudocontractive if there exists a strictly increasing function ϕ:[0,∞)→[0,∞) with ϕ(0)=0 such that, for all x,y∈C, there exists j(x-y)∈J(x-y) satisfying
(1.3)〈Tx-Ty,j(x-y)〉≤∥x-y∥2-ϕ(∥x-y∥)∥x-y∥.T is called generalizedΦ-pseudocontractive (see, e.g., [1]) if there exists a strictly increasing function Φ:[0,∞)→[0,∞) with Φ(0)=0 such that
(1.4)〈Tx-Ty,j(x-y)〉≤∥x-y∥2-Φ(∥x-y∥)
holds for all x,y∈C and for some j(x-y)∈J(x-y).LetF(T)={x∈C:Tx=x} denote the fixed point set of T. If F(T)≠∅, and (1.3) and (1.4) hold for all x∈C and y∈F(T), then the corresponding mapping T is called ϕ-hemicontractive and generalized Φ-hemicontractive, respectively. It is well known that these kinds of mappings play important roles in nonlinear analysis.ϕ-hemicontractive (resp., generalized Φ-hemicontractive) mapping is also called uniformly pseudocontractive (resp., uniformly hemicontractive) mapping in [2, 3]. It is easy to see that if T is generalized Φ-hemicontractive mapping, then F(T) is singleton.It is known (see, e.g., [4, 5]) that the class of strongly pseudocontractive mappings is a proper subset of the class of ϕ-strongly pseudocontractive mappings. By taking Φ(s)=sϕ(s), where ϕ:[0,∞)→[0,∞) is a strictly increasing function with ϕ(0)=0, we know that the class of ϕ-strongly pseudocontractive mappings is a subset of the class of generalized Φ-pseudocontractive mappings. Similarly, the class of ϕ-hemicontractive mappings is a subset of the class of generalized Φ-hemicontractive mappings. The example in [6] demonstrates that the class of Lipschitzian ϕ-hemicontractive mappings is a proper subset of the class of Lipschitzian generalized Φ-hemicontractive mappings.It is well known (see, e.g., [7]) that if C is a nonempty closed convex subset of a real Banach space E and T:C→C is a continuous strongly pseudocontractive mapping, then T has a unique fixed point p∈C. In 2009, it has been proved in [8] that if C is a nonempty closed convex subset of a real Banach space E and T:C→C is a continuous generalized Φ-pseudocontractive mappings, then T has a unique fixed point p∈C.Many results have been proved on convergence or stability of Ishikawa iterative sequences (with errors) or Mann iterative sequences (with errors) for Lipschitzianϕ-hemicontractive mappings or Lipschitzian generalized Φ-hemicontractive mapping (see, e.g., [4–6, 9–12] and the references therein). In 2010, Xiang et al. [6] proved the following result.Theorem XCZ (see [6, Theorem 3.2]). LetEbe a real normed linear space, letCbe a nonempty convex subset ofE,and letT:C→Cbe a Lipschitzian generalizedΦ-hemicontractive mapping. For givenx0∈C, suppose that the sequence{xn}⊂Cis the Mann iterative sequence defined by(1.5)xn+1=(1-βn)xn+βnTxn,n≥0,where{βn}is a sequence in[0,1]satisfying the following conditions:(1)
∑n=0∞‍βn=∞,(2)
∑n=0∞‍βn2<∞.Then{xn}converges strongly to the unique fixed point ofTinC.The main purpose of this paper is to give necessary and sufficient condition for the Mann iterative sequence which converges to a fixed point of general Lipschitzian mappings in an arbitrary real normed linear space. As an immediate consequence, we will obtain necessary and sufficient condition for the Mann iterative sequence which converges to a solution of a general Lipschitzian operator equationTx=f.


## 2. Preliminaries

The following lemmas will be used in the proof of our main results.Lemma 2.1 (see, e.g., [12]).
LetE be a real normed linear space. Then for all x,y∈E, we have
(2.1)∥x+y∥2≤∥x∥2+2〈y,j(x+y)〉,∀j(x+y)∈J(x+y).Lemma 2.2 (see, e.g., [13]).
Let{an}, {bn}, {cn} be three nonnegative sequences satisfying the following condition:
(2.2)an+1≤(1+bn)an+cn,∀n≥n0,
where n0 is some nonnegative integer, ∑n=n0∞‍bn<∞, and ∑n=n0∞‍cn<∞. Then the limit limn→∞an exists.Lemma 2.3.
Suppose thatφ:[0,∞)→[0,∞) is a strictly increasing function with φ(0)=0 and there exists a natural number n0 such that an, bn, εn, and αn are nonnegative real numbers for all n≥n0 satisfying the following conditions:  (i)
an+1≤(1+bn)an-αnφ(an+1)+αnεn,foralln≥n0,(ii)
∑n=n0∞‍bn<∞,limn→∞εn=0,(iii)
∑n=n0∞‍αn=∞. Then limn→∞an=0.Proof.
Without loss of generality, letlimn→∞infan=a. Now, we will show that a=0. Consider its contrary: a>0 or a=∞. For any given r∈(0,a), there exists a nonnegative integer n1≥n0 such that an≥r>0 and εn<1/2φ(r)≤1/2φ(an+1) for all n≥n1. By condition (i), we have
(2.3)an+1≤(1+bn)an-αnφ(an+1)+αn·12φ(an+1)=(1+bn)an-12αnφ(an+1)≤(1+bn)an,∀n≥n1.
Using Lemma 2.2 and condition (ii), we obtain that limn→∞an exists and {an} is bounded. Suppose that an≤M(foralln≥n1), where M is a nonnegative constant. It follows that
(2.4)an+1≤(1+bn)an-12αnφ(an+1)≤an-12αnφ(r)+Mbn(∀n≥n1).
Thus,
(2.5)∞=12φ(r)∑n=n1∞‍αn≤an1+M∑n=n1∞‍bn<∞,
which is a contradiction. Therefore,
(2.6)liminfn→∞an=0.
By condition (ii), for all ε>0, there exists a nonnegative integer n2≥n0 such that
(2.7)εn<φ(ε)(∀n≥n2),∑n=n2∞‍bn<ln2.
By (2.6), there exists a natural number N≥n2 such that aN<ε. Now, we prove the following inequality (2.8) holds for all k≥N:
(2.8)ak≤ε·exp(∑n=Nk-1‍bn).
It is obvious that (2.8) holds for k=N. Assuming (2.8) holds for some k≥N, we prove that (2.8) holds for k+1. Suppose this is not true, that is, ak+1>ε·exp(∑n=Nk‍bn). Then ak+1≥ε and so φ(ak+1)≥φ(ε). Noting that 1+bk≤exp(bk), it follows from condition (i), (2.7), and (2.8) that
(2.9)ak+1≤(1+bk)ak-αkφ(ak+1)+αkεk≤(1+bk)ak-αkφ(ε)+αkφ(ε)≤ε·(1+bk)exp(∑n=Nk-1‍bn)≤ε·exp(∑n=Nk‍bn),
which is a contradiction. This implies that (2.8) holds for k+1. By induction, (2.8) holds for all k≥N. From (2.7), and (2.8), we have
(2.10)limsupk→∞ak≤ε·exp(∑n=N∞‍bn)<2ε.
Taking ε→0, we obtain limn→∞supak=0. By (2.6), we have limn→∞an=0. This completes the proof.Remark 2.4.
Lemma2.3 is different from Lemma 3 in [14], which requires that bn=0 for all n≥0. It is also different from Lemma 2.3 in [6], which requires that ∑n=n0∞‍αnεn<∞.


## 3. Main Results

Theorem 3.1.
LetE be a real normed linear space, C be a nonempty convex subset of E, let T:C→C be a Lipschitzian mapping, and let x*∈C be a fixed point of T. For given x0∈C, suppose that the sequence {xn}⊂C is the Mann iterative sequence defined by
(3.1)xn+1=(1-αn)xn+αnTxn,n≥0,
where {αn} is a sequence in [0,1] satisfying the following conditions:  (i)
∑n=0∞‍αn2<∞,(ii)
∑n=0∞‍αn=∞. Then {xn} converges strongly to x* if and only if there exists a strictly increasing function Φ:[0,∞)→[0,∞) with Φ(0)=0 such that
(3.2)limsupn→∞infj(xn-x*)∈J(xn-x*){〈Txn-x*,j(xn-x*)〉-∥xn-x*∥2+Φ(∥xn-x*∥)}≤0.Proof.
First, we prove the sufficiency of Theorem3.1.
Suppose there exists a strictly increasing functionΦ:[0,∞)→[0,∞) with Φ(0)=0 such that (3.2) holds. Let
(3.3)γn=infj(xn-x*)∈J(xn-x*){〈Txn-x*,j(xn-x*)〉-∥xn-x*∥2+Φ(∥xn-x*∥)}.
Then there exists j(xn-x*)∈J(xn-x*) such that
(3.4)〈Txn-x*,j(xn-x*)〉-∥xn-x*∥2+Φ(∥xn-x*∥)<γn+1n,∀n≥1.
By (3.2), we obtain  limn→∞supγn≤0. Taking εn=1/(n+1)+max{γn+1,0}(foralln≥0), then
(3.5)limn→∞εn=0.
From (3.1) and (3.4), by using Lemma 2.1, we obtain
(3.6)∥xn+1-x*∥2=∥(1-αn)(xn-x*)+αn(Txn-x*)∥2≤(1-αn)2∥xn-x*∥2+2αn〈Txn-x*,j(xn+1-x*)〉≤(1-αn)2∥xn-x*∥2+2αn〈Txn+1-x*,j(xn+1-x*)〉+2αn〈Txn-Txn+1,j(xn+1-x*)〉≤(1-αn)2∥xn-x*∥2+2αn[∥xn+1-x*∥2-Φ(∥xn+1-x*∥)+γn+1+1n+1]+2Lαn∥xn-xn+1∥·∥xn+1-x*∥≤(1-αn)2∥xn-x*∥2+2αn[∥xn+1-x*∥2-Φ(∥xn+1-x*∥)+εn]+2Lαn∥xn-xn+1∥·∥xn+1-x*∥,
where L is the Lipschitzian constant of T. It follows from (3.1) that
(3.7)∥xn-xn+1∥=∥αn(xn-Txn)∥≤αn(∥xn-x*∥+∥Tx*-Txn∥)≤αn(1+L)∥xn-x*∥.
Substituting (3.7) into (3.6), we have
(3.8)∥xn+1-x*∥2≤(1-αn)2∥xn-x*∥2+2αn∥xn+1-x*∥2-2αnΦ(∥xn+1-x*∥)=+2αnεn+2L(1+L)αn2∥xn-x*∥·∥xn+1-x*∥≤(1-αn)2∥xn-x*∥2+2αn∥xn+1-x*∥2-2αnΦ(∥xn+1-x*∥)=+2αnεn+L(1+L)αn2(∥xn-x*∥2+∥xn+1-x*∥2).
Setting an=∥xn-x*∥2(foralln≥0),φ(s)=2Φ(s), it follows from (3.8) that
(3.9)an+1≤(1-αn)2an+2αnan+1-αnφ(an+1)+2αnεn=+L(1+L)αn2(an+an+1)=[1-2αn+αn2+L(1+L)αn2]an+[2αn+L(1+L)αn2]an+1=-αnφ(an+1)+2αnεn.
It follows from condition (i) that limn→∞[2αn+L(1+L)αn2]=0. Thus, there exists a natural number n0 such that 2αn+L(1+L)αn2≤1/2 for all n≥n0. Let
(3.10)bn=1-2αn+αn2+L(1+L)αn21-2αn-L(1+L)αn2-1=αn2+2L(1+L)αn21-2αn-L(1+L)αn2,∀n≥n0.
Since 1/2≤1-2αn-L(1+L)αn2≤1 for all n≥n0, by (3.9) and (3.10), we have
(3.11)an+1≤(1+bn)an-αnφ(an+1)+4αnεn,∀n≥n0,0≤bn≤2[1+2L(1+L)]αn2,∀n≥n0.
It follows from condition (i) that ∑n=n0∞‍bn<∞. Therefore, by (3.5), condition (ii), and Lemma 2.3, we obtain that limn→∞an=limn→∞∥xn-x*∥2=0. That is, {xn} converges strongly to x*.
Finally, we prove the necessity of Theorem3.1.
Assume that{xn} converges strongly to x*. Let L be the Lipschitzian constant of T. For all j(xn-x*)∈J(xn-x*), we have
(3.12)|〈Txn-x*,j(xn-x*)〉|≤L∥xn-x*∥2.
Taking Φ(s)=s, then Φ:[0,∞)→[0,∞) is a strictly increasing function with Φ(0)=0, and limn→∞Φ(∥xn-x*∥)=0. From (3.12), we obtain
(3.13)limn→∞infj(xn-x*)∈J(xn-x*){〈Txn-x*,j(xn-x*)〉-∥xn-x*∥2+Φ(∥xn-x*∥)}=0,
which implies (3.2) holds. This completes the proof of Theorem 3.1.Remark 3.2.
IfT:C→C is a generalized Φ-hemicontractive mapping, then (3.2) holds. By Theorem 3.1, we obtain Theorem XCZ.Theorem 3.3.
LetE be a real Banach space, let C be a nonempty closed convex subset of E, and let T:C→C be a Lipschitzian generalized Φ-pseudocontractive mapping. For given x0∈C, suppose that the sequence {xn}⊂C is the Mann iterative sequence defined by
(3.14)xn+1=(1-αn)xn+αnTxn,n≥0,
where {αn} is a sequence in [0,1] satisfying the following conditions:  (1)
∑n=0∞‍αn=∞,(2)
∑n=0∞‍αn2<∞. Then {xn} converges strongly to the unique fixed point of T in C.Proof.
By Theorem 2.1 in [8], T has a unique fixed point x* in C. By Theorem 3.1, {xn} converges strongly to x*. This completes the proof of Theorem 3.3.Theorem 3.4.
LetE be a real normed linear space, let S:E→E be a Lipschitzian operator, and let f∈E and x* be a solution of the equation Sx=f. For given x0∈E, suppose that the sequence {xn} is the Mann iterative sequence defined by
(3.15)xn+1=(1-αn)xn+αn(f+xn-Sxn),n≥0,
where {αn} is a sequences in [0,1] satisfying the following conditions:  (i)
∑n=0∞‍αn2<∞,(ii)
∑n=0∞‍αn=∞. Then {xn} converges strongly to x* if and only if there exists a strictly increasing function Φ:[0,∞)→[0,∞) with Φ(0)=0 such that
(3.16)liminfn→∞supj(xn-x*)∈J(xn-x*){〈Sxn-Sx*,j(xn-x*)〉-Φ(∥xn-x*∥)}≥0.Proof.
DefineT:E→E by Tx=f+x-Sx. Since Sx*=f, we have Tx*=x*. From (3.15), we obtain xn+1=(1-αn)xn+αnTxn,n≥0. Since
(3.17)〈Sxn-Sx*,j(xn-x*)〉-Φ(∥xn-x*∥)=-{〈Txn-x*,j(xn-x*)〉-∥xn-x*∥2+Φ(∥xn-x*∥)}.
Therefore,
(3.18)liminfn→∞supj(xn-x*)∈J(xn-x*){〈Sxn-Sx*,j(xn-x*)〉-Φ(∥xn-x*∥)}=-limsupn→∞infj(xn-x*)∈J(xn-x*){〈Txn-x*,j(xn-x*)〉-∥xn-x*∥+Φ(∥xn-x*∥)}.
The condition (3.16) is equivalent to condition (3.2). Since S is a Lipschitzian operator, T is a Lipschitzian mapping. By Theorem 3.1, Theorem 3.4 holds. This completes the proof of Theorem 3.4.



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