# Approximate Iteration Algorithm with Error Estimate for Fixed Point of Nonexpansive Mappings

Authors: Yongfu Su
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The purpose of this article is to present a general viscosity iteration process{xn} which defined by xn+1=(I-αnA)Txn+βnγf(xn)+(αn-βn)xn and to study the convergence of {xn}, where T is a nonexpansive mapping and A is a strongly positive linear operator, if {αn}, {βn} satisfy appropriate conditions, then iteration sequence {xn} converges strongly to the unique solution x*∈f(T) of variational inequality 〈(A−γf)x*,x−x*〉≥0, for all x∈f(T). Meanwhile, a approximate iteration algorithm is presented which is used to calculate the fixed point of nonexpansive mapping and solution of variational inequality, the error estimate is also given. The results presented in this paper extend, generalize, and improve the results of Xu, G. Marino and Xu and some others.


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## Body

## 1. Introduction

Iteration methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [1–4] and the references therein. A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space H:
(1.1)minx∈F(T)12〈Ax,x〉-〈x,b〉,
where F(T) is the fixed points set of a nonexpansive mapping T on H, b is a given point in H, and A:H→H is strongly positive operator, that is, there exists a constant δ>0 with the property
(1.2)〈Ax,x〉≥δ∥x∥2,∀x∈H.Recall thatT:H→H is nonexpansive if ∥Tx-Ty∥≤∥x-y∥ for all x,y∈H. Throughout the rest of this paper, we denote by F(T) the fixed points set of T and assume that F(T) is nonempty. It is well known that F(T) is closed convex (cf. [5]). In [4] (see also [2]), it is proved that the sequence {xn} defined by the iteration method below, with the initial guess x0 chosen arbitrarily,
(1.3)xn+1=(I-αnA)Txn+αnb,n≥0,
converges strongly to the unique solution of minimization problem (1.1) provided the sequence {αn} satisfies certain conditions.On the other hand, Moudafi [6] introduced the viscosity approximation method for nonexpansive mappings (see [7] for further developments in both Hilbert and Banach spaces). Let f be a contraction on H. Starting with an arbitrary initial guess x0∈H, define a sequence {xn} recursively by
(1.4)xn+1=(I-αnA)Txn+αnf(xn),n≥0,
where {αn} is a sequence in (0,1). It is proved [6, 7] that under certain appropriate conditions imposed on {αn}, the sequence {xn} generated by (1.4) converges strongly to the unique solution x* in F(T) of the variational inequality
(1.5)〈(I-f)x*,x-x*〉≥0,x∈F(T).Recently (2006), Marino and Xu [2] combine the iteration method (1.3) with the viscosity approximation method (1.4) and consider the following general iteration method:
(1.6)xn+1=(I-αnA)Txn+αnγf(xn),n≥0,
they have proved that if the sequence {αn} of parameters satisfies appropriate conditions, then the sequence {xn} generated by (1.6) converges strongly to the unique solution of the variational inequality
(1.7)〈(A-γf)x*,x-x*〉≥0,x∈F(T),
which is the optimality condition for the minimization problem
(1.8)minx∈F(T)12〈Ax,x〉-h(x),
where h is a potential function for γf (i.e., h′(x)=γf(x) for x∈H).The purpose of this paper is to present a general viscosity iteration process{xn} which is defined by
(1.9)xn+1=(I-αnA)Txn+βnγf(xn)+(αn-βn)xn
and to study the convergence of {xn}, where T is a nonexpansive mapping and A is a strongly positive linear operator, if {αn}, {βn} satisfy appropriate conditions, then iteration sequence {xn} converges strongly to the unique solution x*∈F(T) of variational inequality (1.7). Meanwhile, an approximate iteration algorithm
(1.10)xn+1=(I-sA)Txn+tγf(xn)+(s-t)xn
is presented which is used to calculate the fixed point of nonexpansive mapping and solution of variational inequality; the convergence rate estimate is also given. The results presented in this paper extend, generalize and improve the results of Xu [7], Marino and Xu [2], and some others.


## 2. Preliminaries

This section collects some lemmas which will be used in the proofs for the main results in the next section. Some of them are known; others are not hard to derive.Lemma 2.1 (see [3]).
Assume that{an} is a sequence of nonnegative real numbers such that
(2.1)an+1≤(1-λn)an+δn,
where {λn} is a sequence in (0,1) and {δn} is a sequence in (-∞,+∞) such that(i)
∑n=1∞‍λn=∞;(ii)
limsupn→∞δn/rn≤0, or ∑n=1∞‍|δn|<∞. Then limn→∞an=0.Lemma 2.2 (see [5]).
LetH be a Hilbert space, K a closed convex subset of H, and T:K→K a nonexpansive mapping with nonempty fixed points set F(T). If {xn} is a sequence in K weakly converging to x and if {(I-T)xn} converges strongly to y, then (I-T)x=y.The following lemma is not hard to prove.Lemma 2.3.
LetH be a Hilbert space, K a closed convex subset of H, f:H→H a contraction with coefficient 0<h<1, and A a strongly positive linear bounded operator with coefficient δ>0. Then, for 0<γ<(δ/h),
(2.2)〈x-y,(A-γf)x-(A-γf)y〉≥(δ-γh)∥x-y∥2,x,y∈H.
That is, A-γf is strongly monotone with coefficient δ-γh.Recall the metric (nearest point) projectionPK from a real Hilbert space H to a closed convex subset K of H is defined as follows: given x∈H, PKx is the only point in K with the property
(2.3)∥x-PKx∥=miny∈K∥x-y∥.PK is characterized as follows.Lemma 2.4.
LetK be a closed convex subset of a real Hilbert space H. Given that x∈H and y∈K. Then y=PKx if and only if there holds the inequality
(2.4)〈x-y,y-z〉≥0,∀z∈K.Lemma 2.5.
Assume thatA is a strongly positive linear-bounded operator on a Hilbert space H with coefficient δ>0 and 0<ρ≤∥A∥-1. Then ∥I-ρA∥≤(1-ρδ).Proof.
Recall that a standard result in functional analysis is that ifV is linear bounded self-adjoint operator on H, then
(2.5)∥V∥=sup{|〈Vx,x〉|:x∈H,∥x∥=1}.
Now for x∈H with ∥x∥=1, we see that
(2.6)〈(I-ρA)x,x〉=1-ρ〈Ax,x〉≥1-ρ∥A∥≥0
(i.e., I-ρA is positive). It follows that
(2.7)∥I-ρA∥=sup{〈(I-ρA)x,x〉:x∈H,∥x∥=1}=sup{1-ρ〈Ax,x〉:x∈H,∥x∥=1}≤1-ρδ.The following lemma is also not hard to prove by induction.Lemma 2.6.
Assume that{an} is a sequence of nonnegative real numbers such that
(2.8)an+1≤(1-λn)an+(λn+μn)M,
where M is a nonnegative constant and {λn},{μn} are sequences in [0,+∞) such that(i)
∑n=0∞‍λn=∞;(ii)
∑n=0∞‍μn<∞. Then {an} is bounded.Notation.
We use→ for strong convergence and ⇀ for weak convergence.


## 3. A general Iteration Algorithm with Bounded Linear Operator

LetH be a real Hilbert space, A be a bounded linear operator on H, and T be a nonexpansive mapping on H. Assume that the fixed point set F(T)={x∈H:Tx=x} of T is nonempty. Since F(T) is closed convex, the nearest point projection from H onto F(T) is well defined.Throughout the rest of this paper, we always assume thatA is strongly positive, that is, there exists a constant δ>0 such that
(3.1)〈Ax,x〉≥δ∥x∥2,∀x∈H.
(Note: δ>0 is throughout reserved to be the constant such that (3.1) holds.)Recall also that a contraction onH is a self-mapping f of H such that
(3.2)∥f(x)-f(y)∥≤h∥x-y∥,∀x,y∈H,
where h∈(0,1) is a constant which is called contractive coefficient of f.For given contractionf with contractive coefficient 0<h<1, and t∈[0,1),s∈(0,1),s≥t such that 0≤t≤s<∥A∥-1 and 0<γ<δ/h, consider a mapping St,s on H defined by
(3.3)St,sx=(I-sA)Tx+tγf(x)+(s-t)x,x∈H.
Assume that
(3.4)s-ts→0,
it is not hard to see that St,s is a contraction for sufficiently small s, indeed, by Lemma 2.5 we have
(3.5)∥St,sx-St,sy∥≤tγ∥f(x)-f(y)∥+∥(I-sA)(Tx-Ty)∥+∥(s-t)(x-y)∥≤(tγh+1-sδ+s-t)∥x-y∥=(1+t(γh-1)-s(δ-1))∥x-y∥.
Hence, St,s has a unique fixed point, denoted by xt,s, which uniquely solves the fixed point equation:
(3.6)xt,s=(I-sA)Txt,s+tγf(xt,s)+(s-t)xt,s.
Note that xt,s indeed depends on f as well, but we will suppress this dependence of xt,s on f for simplicity of notation throughout the rest of this paper. We will also always use γ to mean a number in (0,δ/h).The next proposition summarizes the basic properties ofxt,s,(t≤s).Proposition 3.1.
Letxt,s be defined via (3.6).(i)
{xt,s} is bounded for t∈[0,∥A∥-1),s∈(0,∥A∥-1).(ii)
lims→0∥xt,s-Txt,s∥=0.(iii)
{xt,s} defines a continuous surface for (t,s)∈[0,∥A∥-1)×(0,∥A∥-1),t≤s into H.Proof.
Observe, fors∈(0,∥A∥-1), that ∥I-sA∥≤1-sδ by Lemma 2.5.
To show  (i) pickp∈F(T). We then have
(3.7)∥xt,s-p∥=∥(I-sA)(Txt,s-p)+t(γf(xt,s)-Ap)-sAp+tAp∥≤(1-sδ)∥xt,s-p∥+t∥γf(xt,s)-Ap∥+(s-t)∥Ap∥≤(1-sδ)∥xt,s-p∥+s[γh∥xt,s-p∥+∥γf(p)-Ap∥]+(s-t)∥Ap∥≤[1-s(δ-γh)]∥xt,s-p∥+s∥γf(p)-Ap∥+(s-t)∥Ap∥.
It follows that
(3.8)∥xt,s-p∥≤∥γf(p)-Ap∥δ-γh+s-ts∥Ap∥δ-γh<+∞.
Hence {xt,s} is bounded.
(ii) Since the boundedness of{xt,s} implies that of {f(xt,s)} and {ATxt,s}, and observe that
(3.9)∥xt,s-Txt,s∥=∥tf(xt,s)-sATxt,s+(s-t)xt,s∥,
we have
(3.10)lims→0∥xt,s-Txt,s∥=0.
To prove (iii) taket,t0∈[0,∥A∥-1),s,s0∈(0,∥A∥-1),s≥t,s0≥t0 and calculate
(3.11)∥xt,s-xt0,s0∥=∥(t-t0)γf(xt,s)+t0γ(f(xt,s)-f(xt0,s0))-(s-s0)ATxt,s+(I-s0A)(Txt,s-Txt0,s0)+(s-t)(xt,s-xt0,s0)+(s-s0+t0-t)xt0,s0(f(xt,s)-f(xt0,s0))∥≤|t-t0|γ∥f(xt,s)∥+t0γh∥xt,s-xt0,s0∥+|s-s0|∥ATxt,s∥+(1-s0δ)∥xt,s-xt0,s0∥+(s-t)∥xt,s-xt0,s0∥+[|s-s0|+|t-t0|]∥xt0,s0∥,
which implies that
(3.12)(s0δ-t0γh+t-s)∥xt,s-xt0,s0∥≤|t-t0|γ∥f(xt,s)∥+|s-s0|∥ATxt,s∥+[|s-s0|+|t-t0|]∥xt0,s0∥→0
as t→t0,s→s0. Note that
(3.13)limt→t0,s→s0(s0δ-t0γh+t-s)=s0(δ-1)-t0(γh-1)>0,
it is obvious that
(3.14)limt→t0,s→s0∥xt,s-xt0,s0∥=0.
This completes the proof of Proposition 3.1.Our first main result below shows thatxt,s converges strongly as s→0 to a fixed point of T which solves some variational inequality.Theorem 3.2.
One has thatxt,s converges strongly as s→0(t≤s) to a fixed point x~ of T which solves the variational inequality:
(3.15)〈(A-γf)x~,x~-z〉≤0,z∈F(T).
Equivalently, One has PF(T)(I-A+γf)x~=x~, where PF(T)(·) is the nearest point projection from H onto F(T).Proof.
We first shows the uniqueness of a solution of the variational inequality (3.15), which is indeed a consequence of the strong monotonicity of A-γf. Suppose x~∈F(T) and x^∈F(T) both are solutions to (3.15), then
(3.16)〈(A-γf)x~,x~-x^〉≤0,〈(A-γf)x^,x^-x~〉≤0.
Adding up (3.16) gets
(3.17)〈(A-γf)x~-(A-γf)x^,x~-x^〉≤0.
The strong monotonicity of A-γf implies that x~=x^ and the uniqueness is proved. Below we use x~∈F(T) to denote the unique solution of (3.15).
To prove thatxt,s converges strongly to x~, we write, for a given z∈F(T),
(3.18)xt,s-z=t(γf(xt,s)-stAz)+(I-sA)(Txt,s-z)+(s-t)xt,s
to derive that
(3.19)∥xt,s-z∥2=〈t(γf(xt,s)-stAz)+(I-sA)(Txt,s-z)+(s-t)xt,s,xt,s-z〉=t〈γf(xt,s)-Az,xt,s-z〉+〈(I-sA)(Txt,s-z),xt,s-z〉+(s-t)〈xt,s-Az,xt,s-z〉≤(1-sδ)∥xt,s-z∥2+t〈γf(xt,s)-Az,xt,s-z〉+(s-t)〈xt,s-Az,xt,s-z〉.
It follows that
(3.20)∥xt,s-z∥2≤tsδ〈γf(xt,s)-Az,xt,s-z〉+s-tsδ〈xt,s-Az,xt,s-z〉=tsδ{γ〈f(xt,s)-f(z),xt,s-z〉+〈γf(z)-Az,xt,s-z〉}+s-tsδ〈xt,s-Az,xt,s-z〉≤tsδ{γh∥xt,s-z∥2+〈γf(z)-Az,xt,s-z〉}+s-tsδ〈xt,s-Az,xt,s-z〉,
which leads to (3.21)∥xt,s-z∥2≤tsδ-tγh〈γf(z)-Az,xt,s-z〉+s-tsδ-tγh〈xt,s-Az,xt,s-z〉.
Observe that condition (3.4) implies
(3.22)s-tsδ-tγh→0,
as s→0. Since xt,s is bounded as s→0,s≥t, then there exists real sequences {sn},{tn} in [0,1] such that sn→0,sn≥tn and {xtn,sn} converges weakly to a point x*∈H. Using Proposition 3.1 and Lemma 2.2, we see that x*∈F(T), therefore by (3.21), we see xtn,sn→x*. We next prove that x* solves the variational inequality (3.15). Since
(3.23)xt,s=(I-sA)Txt,s+tγf(xt,s)+(s-t)xt,s,
we derive that
(3.24)s(A-γf)xt,s=sAxt,s-xt,s+tγf(xt,s)-sγf(xt,s)+(I-sA)Txt,s+(s-t)xt,s=(I-sA)(Txt,s-xt,s)+(s-t)(xt,s-γf(xt,s)),
so that
(3.25)(A-γf)xt,s=1s(I-sA)(Txt,s-xt,s)+s-ts(xt,s-γf(xt,s)).
It follows that, for z∈F(T),
(3.26)〈(A-γf)xt,s,xt,s-z〉=1s〈(I-sA)(Txt,s-xt,s),xt,s-z〉+s-ts〈xt,s-γf(xt,s),xt,s-z〉=-1s〈(I-T)xt,s-(I-T)z,xt,s-z〉+〈A(I-T)xt,s,xt,s-z〉+s-ts〈xt,s-γf(xt,s),xt,s-z〉≤〈A(I-T)xt,s,x,st-z〉+s-ts〈xt,s-γf(xt,s),xt,s-z〉,
since I-T is monotone (i.e., 〈x-y,(I-T)x-(I-T)y〉≥0 for x,y∈H). This is due to the nonexpansivity of T. Now replacing t,s in (3.26) with tn,sn and letting n→∞, we, noticing that (I-T)xtn,sn→(I-T)x*=0 for x*∈F(T), obtain
(3.27)〈(A-γf)x*,x*-z〉≤0.
That is, x*∈F(T) is a solution of (3.15), hence x*=x~ by uniqueness. In a summary, we have shown that each cluster point of xt,s equals x~. Therefore, xt,s→x~ as s→0.
The variational inequality (3.15) can be rewritten as
(3.28)〈(I-A+γf)x~-x~,x~-z〉≥0,z∈F(T).
This, by Lemma 2.4, is equivalent to the fixed point equation
(3.29)PF(T)(I-A+γf)x~=x~.
This complete the proof.Takingt=s in Theorem 3.2, we getCorollary 3.3 (see [7]).
One has thatxt=xt,t converges strongly as t→0 to a fixed point x~ of T which solves the variational inequality:
(3.30)〈(A-γf)x~,x~-z〉≤0,z∈F(T).
Equivalently, One has PF(T)(I-A+γf)x~=x~, where PF(T)(·) is the nearest point projection from H onto F(T).Next we study a general iteration method as follows. The initial guessx0 is selected in H arbitrarily, and the (n+1)th iterate xn+1 is recursively defined by
(3.31)xn+1=(I-αnA)Txn+βnγf(xn)+(αn-βn)xn,
where {αn}⊂(0,1),βn∈[0,1),βn≤αn are sequences satisfying following conditions:(C1)αn→0;(C2)∑n=0∞‍αn=∞;(C3) either ∑n=0∞‍|αn+1-αn|<∞ or limn→∞(αn+1/αn)=1;(C4)∑n=0∞‍(αn-βn)<∞.Below is the second main result of this paper.Theorem 3.4.
Let{xn} be general by Algorithm (3.31) with the sequences {αn},{βn} of parameters satisfying conditions (C1)–(C4). Then {xn} converges strongly to x~ that is obtained in Theorem 3.2.Proof.
Sinceαn→0 by condition (C1), we may assume, without loss of generality, that αn<∥A∥-1 for all n.
We now observe that{xn} is bounded. Indeed, pick any p∈F(T) to obtain
(3.32)∥xn+1-p∥=∥(I-αnA)(Txn-p)+βn(γf(xn)-Ap)+(αn-βn)(xn-Ap)∥≤∥I-αnA∥∥Txn-p∥+βn∥γf(xn)-Ap∥+(αn-βn)∥xn-Ap∥≤(1-αnδ)∥xn-p∥+βn[γ∥f(xn)-f(p)∥+∥γf(p)-Ap∥]+(αn-βn)∥xn-Ap∥≤(1-δαn)∥xn-p∥+αnγh∥xn-p∥-αnγh∥xn-p∥+βnγh∥xn-p∥+βn∥γf(p)-Ap∥+(αn-βn)∥xn-Ap∥=[1-(δ-γh)αn]∥xn-p∥-(αn-βn)γh∥xn-p∥+βn∥γf(p)-Ap∥+(αn-βn)(∥xn-p∥+∥p-Ap∥)≤[1-(δ-γh)αn]∥xn-p∥+(αn-βn)(1-γh)∥xn-p∥+βn∥γf(p)-Ap∥+(αn-βn)∥p-Ap∥≤[1-[(δ-γh)αn-(αn-βn)|1-γh|]]∥xn-p∥+βn∥γf(p)-Ap∥+(αn-βn)∥p-Ap∥≤[1-[(δ-γh)αn-(αn-βn)|1-γh|]]∥xn-p∥+(δ-γh)αnMδ-γh,
where M≥∥γf(p)-Ap∥+∥p-Ap∥ is a constant. By Lemma 2.6 we see that {xn} is bounded.
As a result, noticing(3.33)xn+1-Txn=αnATxn+βnγf(xn)+(αn-βn)xn
and αn→0, we obtain
(3.34)xn+1-Txn→0.
But the key is to prove that
(3.35)xn+1-xn→0.
To see this, we calculate
(3.36)∥xn+1-xn∥=∥(I-αnA)(Txn-Txn-1)-(αn-αn-1)ATxn-1+γ[αn(f(xn)-f(xn-1))+(αn-αn-1)f(xn-1)]+(αn-βn)(xn-xn-1)∥≤(1-(δ-γh)αn)∥xn-xn-1∥+|αn-αn-1|∥ATxn-1-f(xn-1)∥+(αn-βn)∥xn-xn-1∥=(1+αn-βn-(δ-γh)αn)∥xn-xn-1∥+|αn-αn-1|∥ATxn-1-f(xn-1)∥=[1-[(δ-γh-1)αn+βn]]∥xn-xn-1∥+|αn-αn-1|∥ATxn-1-f(xn-1)∥.
Since
(3.37)∑n=0∞‍[(δ-γh-1)αn+βn]=∑n=0∞‍[(δ-γh)αn-(αn-βn)]=∞
and condition (C3) holds, an application of Lemma 2.1 to (3.36) implies (3.35) which combined with (3.34), in turns, implies
(3.38)xn-Txn→0.
Next we show that
(3.39)limsupn→∞〈Txn-x~,γf(x~)-Ax~〉≤0,
where x~ is obtained in Theorem 3.2.
To see this, we take a subsequence{xnk} of {xn} such that
(3.40)limsupn→∞〈xn-x~,γf(x~)-Ax~〉=limk→∞〈xnk-x~,γf(x~)-Ax~〉.
We may also assume that xnk⇀z. Note that z∈F(T) in virtue of Lemma 2.2 and (3.38). It follows from the variational inequality (3.15) that
(3.41)limsupn→∞〈xn-x~,γf(x~)-Ax~〉=〈z-x~,γf(x~)-Ax~〉≤0.
So (3.39) holds, thanks to (3.38).
Finally, we provexn→x~. To this end, we calculate
(3.42)∥xn+1-x~∥2=∥(I-αnA)(Txn-x~)+αn(γf(xn)-Ax~)-αnγf(xn)+βnγf(xn)+(αn-βn)xn∥2=∥(I-αnA)(Txn-x~)+αn(γf(xn)-Ax~)+(αn-βn)(xn-γf(xn))∥2=∥(I-αnA)(Txn-x~)∥2+∥αn(γf(xn)-Ax~)∥2+∥(αn-βn)(xn-γf(xn))∥2+2〈(I-αnA)(Txn-x~),αn(γf(xn)-Ax~)〉+2〈(I-αnA)(Txn-x~),(αn-βn)(xn-γf(xn))〉+2〈αn(γf(xn)-Ax~),(αn-βn)(xn-γf(xn))〉≤(1-αnδ)2∥xn-x~∥2+αn2∥γf(xn)-Ax~∥2+(αn-βn)2∥xn-γf(xn)∥2+2αn〈Txn-x~,γf(xn)-Ax~〉-2αn2〈A(Txn-x~),γf(xn)-Ax~〉+2(αn-βn)〈Txn-x~,xn-γf(xn)〉-2αn(αn-βn)〈A(Txn-x~),xn-γf(xn)〉+2αn(αn-βn)〈γf(xn)-Ax~,xn-γf(xn)〉≤(1-αnδ)2∥xn-x~∥2+αn2∥γf(xn)-Ax~∥2+(αn-βn)2∥xn-γf(xn)∥2+2αn〈Txn-x~,γf(xn)-γf(x~)+γf(x~)-Ax~〉-2αn2〈A(Txn-x~),γf(xn)-Ax~〉+2(αn-βn)〈Txn-x~,xn-γf(xn)〉-2αn(αn-βn)〈A(Txn-x~),xn-γf(xn)〉+2αn(αn-βn)〈γf(xn)-Ax~,xn-γf(xn)〉≤(1-αnδ)2∥xn-x~∥2+αn2∥γf(xn)-Ax~∥2+(αn-βn)2∥xn-γf(xn)∥2+2αn〈Txn-x~,γf(xn)-γf(x~)〉+2αn〈Txn-x~,γf(x~)-Ax~〉-2αn2〈A(Txn-x~),γf(xn)-Ax~〉+2(αn-βn)〈Txn-x~,xn-γf(xn)〉-2αn(αn-βn)〈A(Txn-x~),xn-γf(xn)〉+2αn(αn-βn)〈γf(xn)-Ax~,xn-γf(xn)〉≤(1-αnδ)2∥xn-x~∥2+αn2∥γf(xn)-Ax~∥2+(αn-βn)2∥xn-γf(xn)∥2+2αnγh∥xn-x~∥2+2αn〈Txn-x~,γf(x~)-Ax~〉-2αn2〈A(Txn-x~),γf(xn)-Ax~〉+2(αn-βn)〈Txn-x~,xn-γf(xn)〉-2αn(αn-βn)〈A(Txn-x~),xn-γf(xn)〉+2αn(αn-βn)〈γf(xn)-Ax~,xn-γf(xn)〉≤[(1-αnδ)2+2αnγh]∥xn-x~∥2+αn2∥γf(xn)-Ax~∥2+(αn-βn)2∥xn-γf(xn)∥2+2αn〈Txn-x~,γf(x~)-Ax~〉-2αn2〈A(Txn-x~),γf(xn)-Ax~〉+2(αn-βn)〈Txn-x~,xn-γf(xn)〉-2αn(αn-βn)〈A(Txn-x~),xn-γf(xn)〉+2αn(αnγf(xn)-Ax~,xn-γf(xn)〉≤[1-2(δ-γh)αn]∥xn-x~∥2+αn2δ2∥xn-x~∥+αn2∥γf(xn)-Ax~∥2+(αn-βn)2∥xn-γf(xn)∥2+2αn〈Txn-x~,γf(x~)-Ax~〉-2αn2〈A(Txn-x~),γf(xn)-Ax~〉+2(αn-βn)〈Txn-x~,xn-γf(xn)〉-2αn(αn-βn)〈A(Txn-x~),xn-γf(xn)〉+2αn(αn-βn)〈γf(xn)-Ax~,xn-γf(xn)〉≤[1-2(δ-γh)αn]∥xn-x~∥2+αn2δ2∥xn-x~∥+αn2∥γf(xn)-Ax~∥2+(αn-βn)2∥xn-γf(xn)∥2+2αn〈Txn-x~,γf(x~)-Ax~〉-2αn2〈A(Txn-x~),γf(xn)-Ax~〉+2(αn-βn)〈Txn-x~,xn-γf(xn)〉-2αn(αn-βn)〈A(Txn-x~),xn-γf(xn)〉+2αn(αn-βn)〈γf(xn)-Ax~,xn-γf(xn)〉≤[1-2(δ-γh)αn]∥xn-x~∥2+2αn〈Txn-x~,γf(x~)-Ax~〉+αnMn,
where
(3.43)Mn=αn∥γf(xn)-Ax~∥2+(αn-βn)2αn∥xn-γf(xn)∥2-2αn〈A(Txn-x~),γf(xn)-Ax~〉+2αn-βnαn〈Txn-x~,xn-γf(xn)〉-2(αn-βn)〈A(Txn-x~),xn-γf(xn)〉+2(αn-βn)〈γf(xn)-Ax~,xn-γf(xn)〉.
That is,
(3.44)∥xn+1-x~∥2≤[1-2(δ-γh)αn]∥xn-x~∥2+αn[2〈Txn-x~,γf(x~)-Ax~〉+Mn].
Since {xn} is bounded, by the conditions of Theorem 3.4, we get limn→∞Mn=0 and ∑n=0∞‍(δ-γh)αn=∞, this together with (3.39) implies that
(3.45)limsupn→∞[2〈Txn-x~,γf(x~)-Ax~〉+Mn]≤0.
Now applying Lemma 2.1 to (3.44) concludes that xn→x~. This complete the proof of Theorem 3.4.If pickαn=βn, we obtain the result of Marino and Xu [2].


## 4.   Approximate Iteration Algorithm and Error Estimate

In this section, we use the following approximate iteration algorithm:(4.1)yn+1=(I-sA)Tyn+tγf(yn)+(s-t)yn,
for an arbitrary initial y0∈H, to calculate the fixed point of nonexpansive mapping and solution of variational inequality with bounded linear operator A, where A,T,γ,s,t and others δ,h as in the Section 3.Meanwhile, thex~∈F(T) is obtained in Theorem 3.2 which is unique solution of variational inequality (3.15) and {xn}→x~, as n→∞,xt,s→x~ as s→0, where {xn} and xt,s are respectively defined by (3.31) and (3.6).The following lemma will be useful for the establish of formula of convergence rate estimate.Lemma 4.1 (Banach's contractive mapping principle).
LetH be a Banach space and S be a contraction from H into self, that is,
(4.2)∥Sx-Sy∥≤θ∥x-y∥,∀x,y∈H,
where 0<θ<1 is a constant. Then the Picard iterative sequence xn+1=Sxn, for arbitrary initial x0∈H, converges strongly to a unique fixed point x* of S and
(4.3)∥xn-x*∥≤θn1-θ∥x0-Sx0∥.For aboveT,A,f,γ,s,t,δ, we define the following contractive mapping:
(4.4)St,sy=(I-sA)Ty+γf(y)+(s-t)y
from H into self. In fact, it is not hard to see that St,s is a contraction for sufficiently small s, indeed, by Lemma 2.5 we have, for any x,y∈H, that
(4.5)∥St,sx-St,sy∥≤tγ∥f(x)-f(y)∥+∥(I-sA)(Tx-Ty)∥+∥(s-t)(x-y)∥≤(tγh+1-sδ+s-t)∥x-y∥=(1+t(γh-1)-s(δ-1))∥x-y∥.
By using Lemma 4.1, then there exists unique fixed point xt,s∈H of St,s and the iterative sequence
(4.6)yn+1=St,syn=(I-sA)Tyn+γf(yn)+(s-t)yn,y0∈H,
converges strongly to this fixed point xt,s. Meanwhile, from (4.3) and (4.5) we obtain
(4.7)∥yn-xt,s∥≤(1+t(γh-1)-s(δ-1))ns(δ-1)-t(γh-1)∥y0-St,sy0∥.
On the other hand, from (3.21) we have
(4.8)∥xt,s-x~∥2≤tsδ-tγh〈γf(x~)-Ax~,xt,s-x~〉+s-tsδ-tγh(〈xt,s-x~,xt,s-x~〉+〈x~-Ax~,xt,s-x~〉),
which leads to
(4.9)(1-s-tsδ-tγh)∥xt,s-x~∥2≤tsδ-tγh〈γf(x~)-Ax~,xt,s-x~〉+s-tsδ-tγh〈x~-Ax~,xt,s-x~〉.
Therefore,
(4.10)(1-s-tsδ-tγh)∥xt,s-x~∥≤tsδ-tγh∥γf(x~)-Ax~∥+s-tsδ-tγh∥x~-Ax~∥,∥xt,s-x~∥≤tsδ-tγh+t-s∥γf(x~)-Ax~∥+s-tsδ-tγh+t-s∥x~-Ax~∥.
Letting D1=∥γf(x~)-Ax~∥,D2=∥x~-Ax~∥, it follows that
(4.11)∥xt,s-x~∥≤tsδ-tγh+t-sD1+s-tsδ-tγh+t-sD2.
From inequality (4.11) together with (4.7), and letting D3=∥y0-St,sy0∥, we get
(4.12)∥yn-x~∥≤tsδ-tγh+t-sD1+s-tsδ-tγh+t-sD2+(1+t(γh-1)-s(δ-1))ns(δ-1)-t(γh-1)D3.
Inequality (4.12) is, namely, the error estimate for approximate fixed point yn. Now, we give several special cases of inequality (4.12).Error Estimate 1
Consider(4.13)limsupn→∞∥yn-x~∥≤tsδ-tγh+t-sD1+s-tsδ-tγh+t-sD2.Error Estimate 2
Ift=s, then
(4.14)∥yn-x~∥≤1(δ-γh)D1+(1-s(δ-γh))ns(δ-γh)D3,
which can be used to estimate error for iterative scheme
(4.15)yn+1=(I-sA)Tyn+sγf(yn),y0∈H.Error Estimate 3
IfA=I, then
(4.16)∥yn-x~∥≤tsδ-tγh+t-sD1+(1+t(γh-1)-s(δ-1))ns(δ-1)-t(γh-1)D3,
which can be used to estimate error for iterative scheme
(4.17)yn+1=(1-s)Tyn+tγf(yn)+(s-t)xn,y0∈H.



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*Source: 605389-2012-09-06.xml*