# A General Iterative Algorithm with Strongly Positive Operators for Strict Pseudo-Contractions

Authors: Wei Xu; Yuanheng Wang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper deals with a new iterative algorithm{xn} with a strongly positive operator A for a k-strict pseudo-contraction T and a non-self-Lipschitzian mapping S in Hilbert spaces. Under certain appropriate conditions, the sequence {xn} converges strongly to a fixed point of T, which solves some variational inequality. The results here improve and extend some recent related results.


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## Body

## 1. Introduction

LetC be a closed convex subset of Hilbert space H with inner product 〈·,·〉 and norm ∥·∥, T:C→H be a nonlinear mapping. The fixed point set of T is denoted by Fix(T); that is, Fix(T)={x∈C,Tx=x}. Fixed point problem is very general in the sense that it includes, as spacial cases, optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and others.Recall that a mappingT:C↦C is said to be nonexpansive if ∥Tx-Ty∥≤∥x-y∥ for all x,y∈C. A mapping T:H↦H is said to be strongly positive, if there exists a constant γ>0 such that 〈Ax,x〉≥γ∥x∥2 for all x∈H. In 2000, Moudafi [1] investigated the fixed point problem of nonexpansive mapping with viscosity approximation method. Let f be a contraction on H; that is, there exists a constant α∈(0,1) such that ∥f(x)-f(y)∥≤α∥x-y∥ for all x,y∈C; define a sequence {xn} by
(1)xn+1=αnf(xn)+(1-αn)Txn,∀n≥0,
where x0 is an arbitrary starting point in H and {αn} is a sequence in (0,1). In 2004 Xu [2] proved that if the parameter {αn} satisfies some approximate conditions, the sequence {xn} generated by (1) converges strongly to not only a fixed point of T but also the unique solution x* of the variational inequality
(2)〈(I-f)x*,x-x*〉≥0,∀x∈Fix(T).In 2010, Tian [3] considered a general hybrid steepest-descent method:
(3)xn+1=αnγf(xn)+(I-μαnF)Txn,∀n≥0,
where F is a Lipschitzian and strongly monotone operator. Under certain conditions, he proved that the sequence {xn} generated by (3) converges strongly to the unique solution x* of the variational inequality
(4)〈(γf-μF)x*,x-x*〉≤0,∀x∈Fix(T).
On the other hand, Marino and Xu [4] introduced the following iterative scheme:
(5)xn+1=αnγf(xn)+(I-αnA)Txn,∀n≥0,
where A is a strongly positive bounded linear operator. It was proven that under certain conditions on the parameters, the sequence {xn} generated by (5) converges strongly to the unique solution x* of the variational inequality
(6)〈(γf-A)x*,x-x*〉≤0,∀x∈Fix(T).It is well known that a typical convex minimization is that of minimizing a quadratic function on the sets of the fixed points of a nonexpansive mapping:(7)Minx∈Fix(T)12〈Ax,x〉-〈x,b〉,
where b is a given point of H. The solution x* is also the optimality condition for the minimization problem
(8)Minx∈Fix(T)12〈Ax,x〉-h(x),
where h(x) is a potential function for γf; that is, h′(x)=γf(x),∀x∈H. Some authors investigated each iterative method for nonexpansive mappings for solving convex minimization problems and got some convergence results; see, for example [5–7].In 2011, Ceng et al. [8] introduced a general iterative algorithm with strongly positive operators for nonexpansive mappings:
(9)yn=(I-μαnF)Txn+αnγf(xn),xn+1=(I-βnA)Txn+βnyn,∀n≥0
and proved that under certain conditions on the parameters the sequence {xn} generated by (9) converges strongly to a fixed point x* of T, which also solves the variational inequality
(10)〈(I-A)x*,x-x*〉≤0,∀x∈Fix(T).Recently the problems of the approximation of the common fixed points of nonexpansive mappings were extended to the case of a family of finite or infinite pseudo-contractions; see, for example, [9–11].Motivated and inspired by the above research works, we consider some fixed point problems with non-self mappings and introduce a new general iterative algorithm with strongly positive operators fork-strict pseudo-contractions which is a wider map class then the nonexpansive map class
(11)yn=PC[αnτSxn+(I-μαnF)Txn],xn+1=(I-(γn+δn)A)yn+γnxn+δnTxn,(I-(γn+δn)A)yn+γnxn+δnTxn∀n≥0,
where PC:H→C is the metric projection, S:C↦H is a non-self-Lipschitzian mapping, T:C↦C is a k-strict pseudo-contraction, and A:C↦C is a strongly positive bounded linear operator. Under certain conditions on the parameters, we prove that the sequence {xn} generated by (11) converges strongly to a fixed point x* of T, which solves the variational inequality
(12)〈(I-A)x*,z-x*〉≤0,∀z∈Fix(T).


## 2. Preliminaries

In this section, we recall some useful definitions and lemmas for the proof of the main results.Definition 1.
A mappingT:C↦C is said to be L-Lipschitzian, if there exists a constant L>0 such that
(13)∥f(x)-f(y)∥≤L∥x-y∥∀x,y∈C.
A mapping T:C↦C is said to be k-strict pseudo-contraction, if there exists a constant k∈[0,1)  such that
(14)∥Tx-Ty∥2≤∥x-y∥2+k∥(I-T)x-(I-T)y∥2,∥x-y∥2+k∥(I-T)x-x∥2∀x,y∈C.
It is clear that a Lipschitzian map is a contractive map when0<L<1 and is a nonexpansive map when L=1. If k=0; then a k-strict pseudo-contraction map is a nonexpansive map.Definition 2.
A mappingPC:H↦C is said to be the metric projection, if for any x∈H, there exists a unique nearest point in C denoted by PCx such that
(15)∥x-PCx∥≤∥x-y∥,∀y∈C.
And it is well known that if C is a nonempty closed convex subset of H, then the PC exists (e.g., see [12]).Lemma 3 (see [13]).
Letx∈H and z∈C be any points. There holds
(16)〈PCx-PCy,x-y〉≥∥PCx-PCy∥2,∀x,y∈H.
And z=PCx if and only if there holds
(17)〈x-z,y-z〉≤0,∀y∈C,
and if and only if there holds the relation
(18)∥x-z∥2≤∥x-y∥2-∥y-z∥2,∀y∈C.Lemma 4 (see [9], Demiclosedness princple).
LetC be a nonempty closed convex subset of a real Hilbert space H and let T:C↦C be a nonexpansive mapping with F(T)≠∅. If {xn} is a sequence in C weakly converging to x and if {(I-T)xn} converges strongly to y, then (I-T)x=y; in particular if y=0, then x∈F(T).Lemma 5 (see [14]).
Letλ be a number in [0,1] and μ≥0. Let F:H↦H be a t-Lipschitzian and η-strongly monotone operator on a Hilbert space. Associate with a nonexpansive mapping T:H↦H and define the mapping Tλ:H↦H by
(19)Tλx∶=Tx-λμF(Tx),∀x∈H.
Then Tλ is a contraction provided μ≤2η/t2; that is,
(20)∥Tλx-Tλy∥≤[1-λμ(η-μt22)]∥x-y∥,∀x,y∈H.Lemma 6 (see [4]).
Assume thatA is a strongly positive bounded linear operator on a Hilbert space H with coefficient γ->0 and 0<ρ≤∥A∥-1; then ∥I-ρA∥≤1-ργ-.Lemma 7 (see [15]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T:C↦C be a k-strict pseudo-contractive mapping. Let γ and δ be two nonnegative real numbers such that (γ+δ)k≤γ; then
(21)∥γ(x-y)+δ(Tx-Ty)∥≤(γ+δ)∥x-y∥,∀x,y∈C.Lemma 8 (see [16]).
LetH be a Hilbert space and C a nonempty convex subset of H. Let T:C↦H be a k-strict pseudo-contractive mapping. Define a mapping Jx=δx+(1-δ)Tx for all x∈C. Then as δ∈[k,1), J is a nonexpansive mapping such that F(J)=F(T).Lemma 9 (see [17]).
Let{αn} be a sequence of nonnegative real numbers satisfying the following relation: αn+1≤(1-γn)αn+δn, where (i) {γn}⊂(0,1), ∑n=1∞γn=∞; (ii) limsupn→∞(δn/γn)=0 or ∑n=1∞|δn|<∞; then limn→∞αn=0.


## 3. Main Results

In this section, we prove the strong convergence results on the iterative algorithm fork-strict pseudo-contractions.Theorem 10.
LetC be a nonempty closed convex subset of a real Hilbert space H, S:C↦H a non-self-L-Lipschitzian mapping, and T:C↦C a k-strict pseudo-contractive mapping such that Fix(T)≠∅. Let F:C↦H be a t-Lipschitzian and η-strongly monotone mapping and A:C↦C a γ--strongly positive bounded linear operator. For a given x0∈C, let the sequences {xn} and {yn} generated by (11), where {αn}, {γn}, {δn}∈[0,1], satisfy the following conditions:(i)
[1-μ(η-μt2/2)]((1+k)/(1-k))≤1,
μ(η-μt2/2)-τL>0,
γ-∈(1,2);(ii)
limn→∞γn=0, limn→∞δn=0, ∑n=0∞γn=∞, ∑n=0∞δn=∞, (γn+δn)k≤γn;(iii)
limn→∞(αn/(γn+δn))=0, ∑n=1∞|αn-αn-1|<∞, ∑n=1∞|γn-γn-1|<∞, ∑n=1∞|δn-δn-1|<∞.Then the sequence {xn} converges strongly to a fixed point x* of T, which solves the variational inequality
(22)〈(I-A)x*,z-x*〉≤0,∀z∈Fix(T).Proof.
The proof is divided into five steps.
Step  1. We first show that the sequences {xn}, {yn} are bounded. Take p∈Fix(T), own to T:C↦C be a k-strict pseudo-contractive mapping, and define Jx=kx+(1-k)Tx. By Lemma 8  J is nonexpansive and Fix(J)=Fix(T); therefore Tx=(1/(1-k))(Jx-kx):
(23)∥Txn-Tp∥=∥11-k(Jxn-kxn)-11-k(Jp-kp)∥=11-k∥(Jxn-Jp)-k(xn-p)∥≤1+k1-k∥xn-p∥.
Thus we immediately get thatT is a (1+k)/(1-k)-Lipschitzian mapping. Then we estimate ∥yn-p∥:
(24)∥yn-p∥=∥PC[αnτSxn+(I-μαnF)Txn]-PCp∥≤∥αnτSxn+(I-μαnF)Txn-p∥=∥αn(τSxn-μFp)+(I-μαnF)Txn-(I-μαnF)Tp∥≤[1-μ(η-μt22)]1+k1-k∥xn-p∥+αn∥τSxn-μFp∥≤∥xn-p∥+αn∥τSxn-μFp∥.
On the other hand, notice thatlimn→∞γn=0, limn→∞δn=0; without loss of generality, we may assume that γn+δn≤∥A∥-1; thus
(25)∥xn+1-p∥=∥(I-(γn+δn)A)yn+γnxn+δnTxn-p∥≤∥(I-(γn+δn)A)yn-(I-(γn+δn)A)p∥+∥γn(xn-p)+δn(Txn-Tp)∥+∥(γn+δn)(I-A)p∥≤[1-(γn+δn)γ-]∥yn-p∥+(γn+δn)∥xn-p∥+(γn+δn)∥(I-A)∥·∥p∥.
Together with (24), we have
(26)∥xn+1-p∥≤[1-(γn+δn)(γ--1)]∥xn-p∥+αn∥τSxn-μFp∥+(γn+δn)∥(I-A)∥·∥p∥=[1-(γn+δn)(γ--1)]∥xn-p∥+(γn+δn)(γ--1)[1(γn+δn)(γ--1)×(αn∥τSxn-μFp∥+(γn+δn)∥(I-A)∥·∥p∥{1(γn+δn)(γ--1)}]≤max{{αnγn+δn}∥xn-p∥,1γ--1(αnγn+δn∥τSxn-μFp∥+∥(I-A)∥·∥p∥{αnγn+δn})}.
By conditions (ii) and (iii), we get that {xn} is bounded, and so are {yn},{Sxn},{Txn},{FTxn}.
Step  2. Now we prove that ∥xn+1-xn∥→0 as n→∞. Denote ξ=μ(η-(μt2/2)):
(27)∥yn-yn-1∥=∥PC[αnτSxn+(I-μαnF)Txn]-PC[αn-1τSxn-1+(I-μαn-1F)Txn-1]∥≤∥αnτSxn+(I-μαnF)Txn-[αn-1τSxn-1+(I-μαn-1F)Txn-1]∥=∥αnτ(Sxn-Sxn-1)+τ(αn-αn-1)Sxn-1+(I-μαnF)Txn-(I-μαnF)Txn-1-μ(αn-αn-1)FTxn-1∥≤αnτL∥xn-xn-1∥+(1-αnξ)∥xn-xn-1∥+|αn-αn-1|(τ∥Sxn-1∥+μ∥FTxn-1∥)=[1-αn(ξ-τL)]∥xn-xn-1∥+|αn-αn-1|M≤∥xn-xn-1∥+|αn-αn-1|M,
where M is a constant such that
(28)Sup{τ∥Sxn-1∥+μ∥FT(xn-1)∥+∥xn-1∥+∥Txn-1∥+2∥Ayn-1∥}≤M,(29)∥xn+1-xn∥=∥(I-(γn+δn)A)yn+γnxn+δnTxn-[(I-(γn-1+δn-1)A)yn-1+γn-1xn-1+δn-1Txn-1]∥=∥(I-(γn+δn)A)yn-(I-(γn+δn)A)yn-1+(γn-1-γn+δn-1-δn)Ayn-1+γn(xn-xn-1)+δn(Txn-Txn-1)+(γn-γn-1)xn-1+(δn-δn-1)Txn-1∥≤[1-(γn+δn)γ-]∥yn-yn-1∥+(γn+δn)∥xn-xn-1∥+(|γn-γn-1|+|δn-δn-1|)∥Ayn-1∥+|γn-γn-1|∥xn-1∥+|δn-δn-1|∥Txn-1∥≤[1-(γn+δn)γ-]∥xn-xn-1∥+|αn-αn-1|M+(γn+δn)∥xn-xn-1∥+|γn-γn-1|(∥xn-1∥+∥Ayn-1∥)+|δn-δn-1|(∥Txn-1∥+∥Ayn-1∥)≤[1-(γn+δn)(γ--1)]∥xn-xn-1∥+M(|αn-αn-1|+|γn-γn-1|+|δn-δn-1|).
By the conditions (i), (ii), and (iii) and Lemma 9, we get ∥xn+1-xn∥→0 as n→∞.
Step  3. Now we prove that ∥xn+1-Txn∥→0 as n→∞:
(30)∥xn+1-Txn∥=∥(I-(γn+δn)A)yn+γnxn+δnTxn-Txn∥=∥(I-(γn+δn)A)yn-(I-(γn+δn)A)Txn+(I-(γn+δn)A)Txn-(1-(γn+δn))Txn+γn(xn-Txn)+δn(Txn-Txn)∥≤[1-(γn+δn)γ-]∥yn-Txn∥+(γn+δn)∥Txn-ATxn∥+γn∥xn-Txn∥.
On the other hand,
(31)∥yn-Txn∥=∥PC[αnτSxn+(I-μαnF)Txn]-PCTxn∥≤∥αnτSxn+(I-μαnF)Txn-Txn∥=αn∥τSxn-μFTxn∥.
Thus we have ∥xn+1-Txn∥→0 as n→∞. Observe that
(32)∥xn-Txn∥≤∥xn-xn+1∥+∥xn+1-Txn∥;
we immediately get ∥xn-Txn∥→0 as n→∞.
Step  4. Now we show that limsupn→∞〈(I-A)x*,xn-x*〉≤0, where x*∈Fix(T) is the unique solution of the variational inequality. Take a subsequence {xnk} of {xn} such that
(33)limsupn→∞〈(I-A)x*,xn-x*〉=limk→∞〈(I-A)x*,xnk-x*〉.
Observe that the sequence{xn} is bounded; without loss of generality we may assume that xnk⇀x′. By Lemma 4, we get x′∈Fix(T). Therefore by Lemma 3, we have
(34)limsupn→∞〈(I-A)x*,xn-x*〉=〈(I-A)x*,x′-x*〉≤0.
Step  5. Next we prove that ∥xn+1-x*∥→0 as n→∞:
(35)∥xn+1-x*∥2=∥(I-(γn+δn)A)yn-(I-(γn+δn)A)x*+γn(xn-x*)+δn(Txn-Tx*)+(γn+δn)(I-A)x*∥2≤∥(I-(γn+δn)A)(yn-x*)∥2+2〈γn(xn-x*)+δn(Txn-Tx*),xn+1-x*〉+2〈(γn+δn)(I-A)x*,xn+1-x*〉≤[1-(γn+δn)γ-]2∥yn-x*∥2+2∥γn(xn-x*)+δn(Txn-Tx*)∥·∥xn+1-x*∥+2(γn+δn)〈(I-A)x*,xn+1-x*〉≤∥1-(γn+δn)γ-∥2∥yn-x*∥2+2(γn+δn)∥xn-x*∥·∥xn+1-x*∥+2(γn+δn)〈(I-A)x*,xn+1-x*〉.
Notice that(36)∥yn-x*∥2≤∥xn-x*∥2+2αn∥xn-x*∥·∥τSxn-μFx*∥+αn2∥τSxn-μFx*∥2;
thus
(37)∥xn+1-x*∥2≤[1-(γn+δn)γ-]2×[∥xn-x*∥2+2αn∥xn-x*∥·∥τSxn-μFx*∥+αn2∥τSxn-μFx*∥2]×(γn+δn)(∥xn-x*∥2+∥xn+1-x*∥2)+2(γn+δn)〈(I-A)x*,xn+1-x*〉.[1-(γn+δn)]∥xn+1-x*∥2≤[1-(γn+δn)γ-]2∥xn-x*∥2+(γn+δn)∥xn-x*∥2+[2αn∥xn-x*∥·∥τSxn-μFx*∥+αn2∥τSxn-μFx*∥2+2(γn+δn)〈(I-A)x*,xn+1-x*〉],∥xn+1-x*∥2=[1-2(γn+δn)(γ--1)1-(γn+δn)]∥xn-x*∥2+11-(γn+δn)×[(γn+δn)2γ-2∥xn-x*∥2+2αn∥xn-x*∥·∥τSxn-μFx*∥+αn2∥τSxn-μFx*∥2+2(γn+δn)〈(I-A)x*,xn+1-x*〉]=[1-2(γn+δn)(γ--1)1-(γn+δn)]∥xn-x*∥2+2(γn+δn)(γ--1)1-(γn+δn)×{12(γn+δn)(γ--1)×[(γn+δn)2γ-2∥xn-x*∥2+2αn∥xn-x*∥·∥τSxn-μFx*∥+αn2∥τSxn-μFx*∥2+2(γn+δn)×〈(I-A)x*,xn+1-x*〉(γn+δn)2γ-2∥xn-x*∥2]{12(γn+δn)(γ--1)}}.
By the conditions (ii), (iii) and Lemma 9, we conclude that ∥xn-x*∥→0 as n→∞, which solves the variational inequality 〈(I-A)x*,z-x*〉≤0,forallz∈Fix(T). This completes the proof.Remark 11.
The iterative algorithm in Theorem10 here is a new approximating method, and Lemma 7 plays a key role in the proof of the main results which makes the proof simple.Remark 12.
The results in this paper improve and extend some recent related results. For example, Theorem10 here improves and extends Theorem  3.2 in [8] in the following ways:(i)
the nonexpansive mappingT:C↦C in [8] is extended to the case of k-strict pseudo-contractions T:C↦C;(ii)
the self-contractionf:C↦C in [8] is extended to the case of a (possiblly non-self) Lipschitzian mapping S:C↦H.



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