Let H be a real Hilbert space, let S, T be two nonexpansive mappings such that F(S)∩F(T)≠∅, let f be a contractive mapping, and let A be a strongly positive linear bounded operator on H. In this paper, we suggest and consider the strong converegence analysis of a new two-step iterative algorithms for finding the approximate solution of two nonexpansive mappings as xn+1=βnxn+(1−βn)Syn, yn=αnγf(xn)+(I−αnA)Txn, n≥0 is a real number and {αn}, {βn} are two sequences in (0,1) satisfying the following
control conditions:
(C1) limn→∞αn=0, (C3) 0<liminfn→∞βn≤limsupn→∞βn<1, then ∥xn+1−xn∥→0. We also discuss several special cases of this iterative algorithm.

1. Introduction

Let H be a real Hilbert space. Recall that a mapping f:H→H is a contractive mapping on H if there exists a constant α∈(0,1) such that

∥f(x)-f(y)∥≤α∥x-y∥,x,y∈H.
We denote by Π the collection of all contractive mappings on H, that is,

∏={f:H→Hisacontractivemapping}.

Let T:H→H be a nonexpansive mapping, namely,

∥Tx-Ty∥≤∥x-y∥,x,y∈H.

Iterative algorithms for nonexpansive mappings have recently been applied to solve convex minimization problems (see [1–4] and the references therein).

A typical problem is to minimize a quadratic function over the closed convex set of the fixed points of a nonexpansive mapping T on a real Hilbert space H:

minx∈C12〈Ax,x〉-〈x,b〉,
where C is a closed convex set of the fixed points a nonexpansive mapping T on H, b is a given point in H and A is a linear, symmetric and positive operator.

In [5] (see also [6]), the author proved that the sequence {xn} defined by the iterative method below with the initial point x0∈H chosen arbitrarily

xn+1=(1-αnA)Txn+αnb,n≥0,
converges strongly to the unique solution of the minimization problem (1.4) provided the sequence {αn} satisfies certain control conditions.

On the other hand, Moudafi [3] introduced the viscosity approximation method for nonexpansive mappings (see also [7] for further developments in both Hilbert and Banach spaces). Let f be a contractive mapping on H. Starting with an arbitrary initial point x0∈H, define a sequence {xn} in H recursively by

xn+1=(1-αn)Txn+αnf(xn),n≥0,
where {αn} is a sequence in (0,1), which satisfies some suitable control conditions.

Recently, Marino and Xu [8] combined the iterative algorithm (1.5) with the viscosity approximation algorithm (1.6), considering the following general iterative algorithm:

xn+1=(I-αnA)Txn+αnγf(xn),n≥0,
where 0<γ<γ̅/α.

In this paper, we suggest a new iterative method for finding the pair of nonexpansive mappings. As an application and as special cases, we also obtain some new iterative algorithms which can be viewed as an improvement of the algorithm of Xu [7] and Marino and Xu [8]. Also we show that the convergence of the proposed algorithms can be proved under weaker conditions on the parameter {αn}. In this respect, our results can be considered as an improvement of the many known results.

2. Preliminaries

In the sequel, we will make use of the following for our main results:

Lemma 2.1 (see [<xref ref-type="bibr" rid="B4">4</xref>]).

Let {sn} be a sequence of nonnegative numbers satisfying the condition
sn+1≤(1-αn)sn+αnβn,n≥0,
where {αn}, {βn} are sequences of real numbers such that

{αn}⊂[0,1] and ∑n=0∞αn=∞,

limn→∞βn≤0 or ∑n=0∞αnβn is convergent.

Then limn→∞sn=0.Lemma 2.2 (see [<xref ref-type="bibr" rid="B9">9</xref>, <xref ref-type="bibr" rid="B10">10</xref>]).

Let {xn} and {yn} be bounded sequences in a Banach space X and {βn} be a sequence in [0,1] with
0<lim infn→∞βn≤lim supn→∞βn<1.
Suppose that xn+1=(1-βn)yn+βnxn for all n≥0 and lim supn→∞(∥yn+1-yn∥-∥xn+1-xn∥)≤0. Then limn→∞∥yn-xn∥=0.

Lemma 2.3 (see [<xref ref-type="bibr" rid="B2">2</xref>] (demiclosedness Principle)).

Assume that T is a nonexpansive self-mapping of a closed convex subset C of a Hilbert space H. If T has a fixed point, then I-T is demiclosed, that is, whenever {xn} is a sequence in C weakly converging to some x∈C and the sequence {(I-T)xn} strongly converges to some y, it follows that (I-T)x=y, where I is the identity operator of H.

Lemma 2.4 (see [<xref ref-type="bibr" rid="B8">8</xref>]).

Let {xt} be generated by the algorithm xt=tγf(xt)+(I-tA)Txt. Then {xt} converges strongly as t→0 to a fixed point x* of T which solves the variational inequality
〈(A-γf)x*,x*-x〉≤0,x∈F(T).

Lemma 2.5 (see [<xref ref-type="bibr" rid="B8">8</xref>]).

Assume A is a strong positive linear bounded operator on a Hilbert space H with coefficient γ¯>0 and 0<ρ≤∥A∥-1. Then ∥I-ρA∥≤1-ργ¯.

3. Main Results

Let H be a real Hilbert space, let A be a bounded linear operator on H, and let S, T be two nonexpansive mappings on H such that F(S)∩F(T)≠∅. Throughout the rest of this paper, we always assume that A is strongly positive.

Now, let f∈Π with the contraction coefficient 0<α<1 and let A be a strongly positive linear bounded operator with coefficient γ̅>0 satisfying 0<γ<γ̅/α. We consider the following modified iterative algorithm:

xn+1=βnxn+(1-βn)Syn,yn=αnγf(xn)+(I-αnA)Txn,n≥0,
where γ>0 is a real number and {αn}, {βn} are two sequences in (0,1).

First, we prove a useful result concerning iterative algorithm (3.1) as follows.

Lemma 3.1.

Let {xn} be a sequence in H generated by the algorithm (3.1) with the sequences {αn} and {βn} satisfying the following control conditions:

limn→∞αn=0,

0<lim infn→∞βn≤lim supn→∞βn<1.

Then ∥xn+1-xn∥→0.Proof.

From the control condition (C1), without loss of generality, we may assume that αn≤∥A∥-1. First observe that ∥I-αnA∥≤1-αnγ̅ by Lemma 2.5.

Now we show that {xn} is bounded. Indeed, for any p∈F(S)∩F(T),
∥yn-p∥=∥αn(γf(xn)-Ap)+(I-αnA)(Txn-p)∥≤αn∥γf(xn)-γf(p)∥+αn∥γf(p)-Ap∥+(1-αnγ̅)∥Txn-p∥≤αnγα∥xn-p∥+αn∥γf(p)-Ap∥+(1-αnγ̅)∥xn-p∥=[1-(γ̅-γα)αn]∥xn-p∥+αn∥γf(p)-Ap∥.
At the same time,
∥xn+1-p∥=∥βn(xn-p)+(1-βn)(Syn-p)∥≤βn∥xn-p∥+(1-βn)∥Syn-p∥≤βn∥xn-p∥+(1-βn)∥yn-p∥.
It follows from (3.2) and (3.3) that
∥xn+1-p∥≤βn∥xn-p∥+(1-βn)[1-(γ̅-γα)αn]∥xn-p∥+αn(1-βn)∥γf(p)-Ap∥=[1-(γ̅-γα)αn(1-βn)]∥xn-p∥+(γ̅-γα)αn(1-βn)∥γf(p)-Ap∥γ̅-γα,
which implies that
∥xn-p∥≤max{∥x0-p∥,∥γf(p)-Ap∥γ̅-γα},n≥0.
Hence {xn} is bounded and so are {ATxn} and {f(xn)}.

From (3.1), we observe that
∥yn+1-yn∥=∥αn+1γf(xn+1)+(I-αn+1A)Txn+1-αnγf(xn)-(I-αnA)Txn∥=∥αn+1γ(f(xn+1)-f(xn))+(αn+1-αn)γf(xn)+(I-αn+1A)(Txn+1-Txn)+(αn-αn+1)ATxn∥≤αn+1γ∥f(xn+1)-f(xn)∥+(1-αn+1γ̅)∥Txn+1-Txn∥+|αn+1-αn|(∥γf(xn)∥+∥ATxn∥)≤αn+1γα∥xn+1-xn∥+(1-αn+1γ̅)∥xn+1-xn∥+|αn+1-αn|(∥γf(xn)∥+∥ATxn∥)=[1-(γ̅-γα)αn+1]∥xn+1-xn∥+|αn+1-αn|(∥γf(xn)∥+∥ATxn∥).
It follows that
∥Syn+1-Syn∥-∥xn+1-xn∥≤∥yn+1-yn∥-∥xn+1-xn∥=(γ̅-γα)αn+1∥xn+1-xn∥+|αn+1-αn|(∥γf(xn)∥+∥ATxn∥),
which implies, from (C1) and the boundedness of {xn}, {f(xn)}, and {ATxn}, that
lim supn→∞(∥Syn+1-Syn∥-∥xn+1-xn∥)≤0.
Hence, by Lemma 2.2, we have
∥Syn-xn∥→0asn→∞.
Consequently, it follows from (3.1) that
limn→∞∥xn+1-xn∥=limn→∞(1-βn)∥Syn-xn∥=0.
This completes the proof.

Remark 3.2.

The conclusion ∥xn+1-xn∥→0 is important to prove the strong convergence of the iterative algorithms which have been extensively studied by many authors, see, for example, [3, 6, 7].

If we take S=I in (3.1), we have the following iterative algorithm:

xn+1=βnxn+(1-βn)yn,yn=αnγf(xn)+(I-αnA)Txn,n≥0.
Now we state and prove the strong convergence of iterative scheme (3.11).

Theorem 3.3.

Let {xn} be a sequence in H generated by the algorithm (3.11) with the sequences {αn} and {βn} satisfying the following control conditions:

limn→∞αn=0,

limn→∞αn=∞,

0<lim infn→∞βn≤lim supn→∞βn<1.

Then {xn} converges strongly to a fixed point x* of T which solves the variational inequality
〈(A-γf)x*,x*-x〉≤0,x∈F(T).Proof.

From Lemma 3.1, we have
∥xn+1-xn∥→0.

On the other hand, we have
∥xn-Txn∥≤∥xn+1-xn∥+∥xn+1-Txn∥=∥xn+1-xn∥+∥β(xn-Txn)+(1-βn)(yn-Txn)∥≤∥xn+1-xn∥+βn∥xn-Txn∥+(1-βn)∥yn-Txn∥≤∥xn+1-xn∥+βn∥xn-Txn∥+(1-βn)αn(∥γf(xn)∥+∥ATxn∥),
that is,
∥xn-Txn∥≤11-βn∥xn+1-xn∥+αn(∥γf(xn)∥+∥ATxn∥),
this together with (C1), (C3), and (3.13), we obtain
limn→∞∥xn-Txn∥=0.

Next, we show that, for any x*∈F(T),
lim supn→∞〈yn-x*,γf(x*)-Ax*〉≤0.

In fact, we take a subsequence {xnk} of {xn} such that
lim supn→∞〈xn-x*,γf(x*)-Ax*〉=limk→∞〈xnk-x*,γf(x*)-Ax*〉.
Since {xn} is bounded, we may assume that xnk†⇀z, where “⇀” denotes the weak convergence. Note that z∈F(T) by virtue of Lemma 2.3 and (3.16). It follows from the variational inequality (2.3) in Lemma 2.4 that
lim supn→∞〈xn-x*,γf(x*)-Ax*〉=〈z-x*,γf(x*)-Ax*〉≤0.
By Lemma 3.1 (noting S=I), we have
∥yn-xn∥→0.
Hence, we get
lim supn→∞〈yn-x*,γf(x*)-Ax*〉≤0.

Finally, we prove that {xn} converges to the point x*. In fact, from (3.2) we have
∥yn-x*∥≤∥xn-x*∥+αn∥γf(x*)-Ax*∥.
Therefore, from (3.16), we have
∥xn+1-x*∥2=∥βn(xn-x*)+(1-βn)(yn-x*)∥2≤βn∥xn-x*∥2+(1-βn)∥yn-x*∥2=βn∥xn-x*∥2+(1-βn)∥αn(γf(xn)-Ax*)+(I-αnA)(Txn-x*)∥2≤βn∥xn-x*∥2+(1-βn)[(1-αnγ̅)2∥xn-x*∥2+2αn〈γf(xn)-Ax*,yn-x*〉]=[1-2αnγ̅+(1-βn)αn2γ̅2]∥xn-x*∥2+2αn〈γf(xn)-γf(x*),yn-x*〉+2αn〈γf(x*)-Ax*,yn-x*〉≤[1-2αnγ̅+(1-βn)αn2γ̅2]∥xn-x*∥2+2αnγα∥xn-x*∥∥yn-x*∥+2αn〈γf(x*)-Ax*,yn-x*〉≤[1-2αn(γ̅-γα)]∥xn-x*∥2+(1-βn)αn2γ̅2∥xn-x*∥2+2αn2γα∥xn-x*∥∥γf(x*)-Ax*∥+2αn〈γf(x*)-Ax*,yn-x*〉.
Since {xn}, f(x*) and Ax* are all bounded, we can choose a constant M>0 such that
1γ̅-γα{(1-βn)γ̅22∥xn-x*∥2+γα∥xn-x*∥∥γf(x*)-Ax*∥}≤M,n≥0.
It follows from (3.23) that
∥xn+1-x*∥2≤[1-2(γ̅-αγ)αn]∥xn-x*∥2+2(γ̅-αγ)αnδn,
where
δn=αnM+1γ̅-γα〈γf(x*)-Ax*,yn-x*〉.
By (C1) and (3.17), we get
lim supn→∞βn≤0.
Now, applying Lemma 2.1 and (3.25), we conclude that xn→x*. This completes the proof.

Taking T=I in (3.1), we have the following iterative algorithm:

xn+1=βnxn+(1-βn)Syn,yn=αnγf(xn)+(I-αnA)xn,n≥0.

Now we state and prove the strong convergence of iterative scheme (3.28).

Theorem 3.4.

Let {xn} be a sequence in H generated by the algorithm (3.28) with the sequences {αn} and {βn} satisfying the following control conditions:

limn→∞αn=0,

limn→∞αn=∞,

0<lim infn→∞βn≤lim supn→∞βn<1.

Then {xn} converges strongly to a fixed point x* of S which solves the variational inequality
〈(A-γf)x*,x*-x〉≤0,x∈F(S).Proof.

From Lemma 3.1, we have
∥xn-Syn∥→0.
Thus, we have
∥xn-Sxn∥≤∥xn-Syn∥+∥Syn-Sxn∥≤∥xn-Syn∥+∥yn-xn∥≤∥xn-Syn∥+αn(∥γf(xn)∥+∥Axn∥)→0.
By the similar argument as (3.17), we also can prove that
lim supn→∞〈yn-x*,γf(x*)-Ax*〉≤0.
From (3.28), we obtain
∥xn+1-x*∥2=∥βn(xn-x*)+(1-βn)(Syn-x*)∥2≤βn∥xn-x*∥2+(1-βn)∥Syn-x*∥2≤βn∥xn-x*∥2+(1-βn)∥yn-x*∥2=βn∥xn-x*∥2+(1-βn)∥αn(γf(xn)-Ax*)+(I-αnA)(xn-x*)∥2≤βn∥xn-x*∥2+(1-βn){(1-αnγ̅)2∥xn-x*∥2+2〈γf(xn)-Ax*,yn-x*〉}.
The remainder of proof follows from the similar argument of Theorem 3.3. This completes the proof.

From the above results, we have the following corollaries.

Corollary 3.5.

Let {xn} be a sequence in H generated by the following algorithm
xn+1=βnxn+(1-βn)yn,yn=αnf(xn)+(1-αn)Txn,n≥0,
where the sequences {αn} and {βn} satisfy the following control conditions:

limn→∞αn=0,

limn→∞αn=∞,

0<lim infn→∞βn≤lim supn→∞βn<1.

Then {xn} converges strongly to a fixed point x* of T which solves the variational inequality
〈(I-f)x*,x*-x〉≤0,x∈F(T).Corollary 3.6.

Let {xn} be a sequence in H generated by the following algorithm
xn+1=βnxn+(1-βn)Syn,yn=αnf(xn)+(1-αn)xn,n≥0,
where the sequences {αn} and {βn} satisfy the following control conditions:

limn→∞αn=0,

limn→∞αn=∞,

0<lim infn→∞βn≤lim supn→∞βn<1.

Then {xn} converges strongly to a fixed point x* of S which solves the variational inequality
〈(I-f)x*,x*-x〉≤0,x∈F(S).Remark 3.7.

Theorems 3.3 and 3.4 provide the strong convergence results of the algorithms (3.11) and (3.28) by using the control conditions (C1) and (C2), which are weaker conditions than the previous known ones. In this respect, our results can be considered as an improvement of the many known results.

DeutschF.YamadaI.Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappingsGeobelK.KirkW. A.MoudafiA.Viscosity approximation methods for fixed-points problemsXuH.-K.Iterative algorithms for nonlinear operatorsXuH.-K.An iterative approach to quadratic optimizationYamadaI.ButnariuD.CensorY.ReichS.The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappingsXuH.-K.Viscosity approximation methods for nonexpansive mappingsMarinoG.XuH.-K.A general iterative method for nonexpansive mappings in Hilbert spacesSuzukiT.Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integralsSuzukiT.Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces