C is a bounded closed convex subset of a Hilbert space H, T and S:C→C are two asymptotically nonexpansive mappings such that ST=TS. We establish a strong convergence theorem for S and T in Hilbert space by hybrid method. The results generalize and unify many corresponding results.

1. Introduction

Let C be a bounded closed convex subset of a Hilbert
space H.
Recall that a mapping T:C→C is said to be asymptotically nonexpansive
mapping if∥Tnx−Tny∥≤tn∥x−y∥∀x,y∈C,where tn→1(n→∞). We may assume that tn≥1 for all n=1,2,3,…. Denote by F(T) the set of fixed points of T.
Throughout this paper T and S:C→C are two commutative asymptotically
nonexpansive mappings with asymptotical coefficients {tn} and {sn},
respectively. Suppose that F:=F(T)∩F(S)≠∅ ([1, Goebel and Kirk's theorem] makes it
possible). It is well known that F(T) and F(S) are convex and closed [1, 2], so is F. PK denotes the metric projection from H onto a closed convex subset K of H and ωw(xn) denotes the weak w-limit set of {xn}.
It is well known that a Hilbert space H satisfies Opial's condition [3], that is, if a
sequence {xn} converges weakly to an element y∈H and y≠z,
thenliminfn→∞∥xn−y∥<liminfn→∞∥xn−z∥.

Up to now, fixed points iteration processes for
nonexpansive and asymptotically nonexpansive mappings have been studied
extensively by many authors to solve nonlinear operator equations as well as
variational inequalities [4–6]. There are many strong convergence theorems
for nonexpansive and asymptotically nonexpansive mappings in Hilbert
space [7, 8].

Especially, Shimizu and Takahashi [7] studied the
following iteration process of nonexpansive mappings for arbitrary x0∈C:xn+1=αnx0+(1−αn)2(n+1)(n+2)∑k=0n∑i+j=kSiTjxn,where {αn}⊆[0,1],limn→∞αn=0,∑n=0∞αn=∞. And then they proved that {xn} converges strongly to PF(x0).
This result was extended to two commutative asymptotically nonexpansive
mappings by Shioji and Takahashi [9].

Recently, some attempts to the modified Mann iteration
method are made so that strong convergence is guaranteed. And for hybrid method
proposed by Haugazeau [10], Kim and Xu [8] introduced the following iteration
processes for asymptotically nonexpansive mapping T:x0∈Cchosenarbitrarily,yn=αnxn+(1−αn)Tnxn,Cn={v∈C:∥yn−v∥2≤∥xn−v∥2+θn},Qn={v∈C:〈xn−v,x0−xn〉≥0},xn+1=PCn⋂Qn(x0),where θn=(1−αn)(tn2−1)(diamC)2→0 as n→∞.
Then proved that {xn} converges strongly to PF(x0).
This result was generalized to two asymptotically nonexpansive mappings by
Plubtieng and Ungchittrakool [11].

On the basis of (1.3) and (1.4),
we propose a new iteration processes for two commutative asymptotically
nonexpansive mappings S and T:x0∈Cchosenarbitrarily,yn=αnxn+(1−αn)2(n+1)(n+2)∑k=0n∑i+j=kSiTjxn,Cn={v∈C:∥yn−v∥2≤∥xn−v∥2+θn},Qn={v∈C:〈xn−v,x0−xn〉≥0},xn+1=PCn⋂Qn(x0),where θn=(1−αn)(gn2−1)(diamC)2,gn=(2/(n+1)(n+2))∑k=0n∑i+j=ksitj,
for every n=1,2,…. The purpose of this paper is to prove {xn} converges strongly to PF(x0).

2. Auxiliary Lemmas

This section collects some lemmas which will be used
to prove the main results in the next section.

Lemma 2.1 (see [<xref ref-type="bibr" rid="B7">7</xref>]).

Letting Ln=(n+1)(n+2)/2,
there holds the identity in a Hilbert space H:∥yn−v∥2=1Ln∑k=0n∑i+j=k∥xi,j−v∥2−1Ln∑k=0n∑i+j=k∥xi,j−yn∥2for {xi,j}i,j=0∞⊆H, yn=(1/Ln)∑k=0n∑i+j=kxi,j∈H and v∈H.

Lemma 2.2.

Let C be a bounded closed convex subset of a Hilbert
space H, S and T two commutative asymptotically nonexpansive
mappings of C into itself with asymptotical coefficients {sn} and {tn},
respectively. For any x∈C,
put Fn(x)=(2/(n+1)(n+2))∑k=0n∑i+j=kSiTjx.
Thenliml→∞limsupn→∞supx∈C∥Fn(x)−SlFn(x)∥=0,liml→∞limsupn→∞supx∈C∥Fn(x)−TlFn(x)∥=0.

Proof.

Put xi,j=SiTjx, v=SlFn(x) and Ln=(n+1)(n+2)/2.
It follows from Lemma 2.1 that∥Fn(x)−SlFn(x)∥2=1Ln∑k=0n∑i+j=k∥SiTjx−SlFn(x)∥2−1Ln∑k=0n∑i+j=k∥SiTjx−Fn(x)∥2=1Ln∑k=0l−1∑i+j=k∥SiTjx−SlFn(x)∥2+1Ln∑k=ln∑i+j=k,i≤l−1∥SiTjx−SlFn(x)∥2+1Ln∑k=ln∑i+j=k,i≥l∥SiTjx−SlFn(x)∥2−1Ln∑k=0n∑i+j=k∥SiTjx−Fn(x)∥2≤1Ln∑k=0l−1∑i+j=k∥SiTjx−SlFn(x)∥2+1Ln∑k=ln∑i+j=k,i≤l−1∥SiTjx−SlFn(x)∥2+1Ln∑k=ln∑i+j=k,i≥lsl2∥Si−lTjx−Fn(x)∥2−1Ln∑k=0n∑i+j=k∥SiTjx−Fn(x)∥2=1Ln∑k=0l−1∑i+j=k∥SiTjx−SlFn(x)∥2+1Ln∑k=ln∑i+j=k,i≤l−1∥SiTjx−SlFn(x)∥2+1Ln∑k=0n−l∑i+j=ksl2∥SiTjx−Fn(x)∥2−1Ln∑k=0n∑i+j=k∥SiTjx−Fn(x)∥2≤1Ln∑k=0l−1∑i+j=k∥SiTjx−SlFn(x)∥2+1Ln∑k=ln∑i+j=k,i≤l−1∥SiTjx−SlFn(x)∥2+1Ln∑k=0n−l∑i+j=k(sl2−1)∥SiTjx−Fn(x)∥2.Choose p∈F,
then there exists a constant M>0 such that∥SiTjx−p∥≤sitj∥x−p∥≤M2,∥Fn(x)−p∥≤1Ln∑k=0n∑i+j=k∥SiTjx−p∥≤M2,∥SlFn(x)−p∥≤sl∥Fn(x)−p∥≤M2, for all nonnegative integer i,j,l, and n.
Hence, ∥SiTjx−SlFn(x)∥≤M,∥SiTjx−Fn(x)∥≤M for all nonnegative integer i,j,l, and n.
Sosupx∈C∥Fn(x)−SlFn(x)∥2≤(l+1)l(n+2)(n+1)M2+2(n+1−l)l(n+2)(n+1)M2+(sl2−1)(n+2−l)(n+1−l)(n+2)(n+1)M2→0(n→∞,l→∞).Similarly, we can
proveliml→∞limsupn→∞supx∈C∥Fn(x)−TlFn(x)∥=0.

Remark 2.3.

Lemma 2.2 extends
[7, Lemma 1].

Lemma 2.4.

Let S and T be two commutative asymptotically nonexpansive
mappings defined on a bounded closed convex subset C of a Hilbert space H with asymptotical coefficients {sn} and {tn},
respectively. Let Ln=((n+1)(n+2)/2).
If {xn} is a sequence in C such that {xn} converges weakly to some x∈C and {xn−(1/Ln)∑k=0n∑i+j=kSiTjxn} converges strongly to 0, then x∈F(S)∩F(T).

Proof.

We claim that {Slx} converges strongly to x as l→∞.
If not, there exist a positive number ε0 and a subsequence {lm} of {l} such that ∥Slmx−x∥≥ε0 for all m.
However, we have∥xn−Slmx∥≤∥xn−1Ln∑k=0n∑i+j=kSiTjxn∥+∥1Ln∑k=0n∑i+j=kSiTjxn−Slm(1Ln∑k=0n∑i+j=kSiTjxn)∥+∥Slm(1Ln∑k=0n∑i+j=kSiTjxn)−Slmx∥≤∥xn−1Ln∑k=0n∑i+j=kSiTjxn∥+∥1Ln∑k=0n∑i+j=kSiTjxn−Slm(1Ln∑k=0n∑i+j=kSiTjxn)∥+slm∥1Ln∑k=0n∑i+j=kSiTjxn−x∥≤∥xn−1Ln∑k=0n∑i+j=kSiTjxn∥+∥1Ln∑k=0n∑i+j=kSiTjxn−Slm(1Ln∑k=0n∑i+j=kSiTjxn)∥+slm∥1Ln∑k=0n∑i+j=kSiTjxn−xn∥+slm∥xn−x∥.By Opial's condition, for any y∈C with y≠x,
we haveliminfn→∞∥xn−x∥<liminfn→∞∥xn−y∥.Let r=liminfn→∞∥xn−x∥ and choose a positive number ρ such thatρ<r2+ε024−r.Then, there exists a subsequence {xnp} of {xn} such that limp→∞∥xnp−x∥=r and ∥xnp−x∥<r+(ρ/4) for all p.
By definition of {slm},
there exists a positive integer m0 such thatslm∥xnp−x∥<r+ρ4,for all m>m0.
Sincelimn→∞∥xn−1Ln∑k=0n∑i+j=kSiTjxn∥=0and {slm} is bounded, there exists a positive integer p0 such that∥xnp−1Lnp∑k=0np∑i+j=kSiTjxnp∥<ρ4,slm∥1Lnp∑k=0np∑i+j=kSiTjxnp−xnp∥<ρ4for all m and p>p0.
By {xnp}⊂C is bounded and Lemma 2.2, there exist m1>m0 and p1>0 such that∥1Lnp∑k=0np∑i+j=kSiTjxnp−Slm1(1Lnp∑k=0np∑i+j=kSiTjxnp)∥<ρ4for all p>p1.
By (2.7), (2.10), (2.12), and (2.13),
we have∥xnp−Slm1x∥<ρ4+ρ4+ρ4+r+ρ4=r+ρfor all p>max{p0,p1}.
However,∥xnp−Slm1x+x2∥2=12∥xnp−Slm1x∥2+12∥xnp−x∥2−14∥Slm1x−x∥2<(r+ρ)22+(r+ρ/4)22−ε024<(r+ρ)2−ε024<r2for all p>max{p0,p1}.
This contradicts (2.8).
So {Slx} converges strongly to x and then x∈F(S).
Similarly, we can get x∈F(T).
Hence, x is a common fixed point of S and T.

Lemma 2.5 (see [<xref ref-type="bibr" rid="B12">12</xref>]).

Let C be a bounded closed convex subset of a Hilbert
space H.
The set D:={v∈C:∥y−v∥2≤∥x−v∥2+〈z,v〉+b} is convex and closed for given x,y,z∈C and b∈ℝ.

3. Main Results

In this section, we prove our main theorem.

Theorem 3.1.

Let C be a bounded closed convex subset of a Hilbert H, T and S:C→C be two commutative asymptotically nonexpansive
mappings with asymptotical coefficients {tn} and {sn},
respectively. Suppose that 0≤αn≤a for all n,
where 0<a<1.
If F:=F(T)∩F(S)≠∅,
then the sequence generated by (1.5) converges strongly to PF(x0).

Proof.

Note that Cn is convex and closed for all n≥0 by Lemma 2.5. On the other hand, Qn is convex and closed. So is Cn∩Qn.

By definition of {tn} and {sn},
there exists M>0 such that ∥sitj−1∥≤M for all i,j≥0.
On the other hand, for arbitrary ε>0,
there exists N>0 such that ∥sitj−1∥<ε for all i,j>N.
Hence∥gn−1∥=∥2(n+1)(n+2)∑k=0n∑i+j=k(sitj−1)∥≤2(n+1)(n+2)∑k=0n∑i+j=k∥sitj−1∥≤2(n+1)(n+2)∑k=0n∑i+j=k,i≤N∥sitj−1∥+2(n+1)(n+2)∑k=0n∑i+j=k,j≤N∥sitj−1∥+2(n+1)(n+2)∑k=0n∑i+j=k,i≥N+1,j≥N+1∥sitj−1∥<2(N+1)M(n+2)+2(N+1)M(n+2)+ε.Thus limn→∞gn=1.
Obviously, limn→∞θn=0.

Next, we prove that F⊂Cn∩Qn.
Indeed, first of all∥yn−p∥2≤αn∥xn−p∥2+(1−αn)∥2(n+1)(n+2)∑k=0n∑i+j=kSiTjxn−p∥2≤αn∥xn−p∥2+(1−αn)gn2∥xn−p∥2=∥xn−p∥2+(1−αn)(gn2∥xn−p∥2−∥xn−p∥2)≤∥xn−p∥2+θnfor all p∈F.
So F⊂Cn.
It suffices to show that F⊂Qn for all n≥0. We prove this by induction. For n=0,
we have F⊂C=Q0. Assume that F⊂Qn. Since xn+1 is the projection of x0 onto Cn∩Qn,
we have〈xn+1−z,x0−xn+1〉≥0∀z∈Cn⋂QnAs F⊂Cn∩Qn, (3.3) holds for all z∈F,
in particular. This together with the definition of Qn+1 implies that F⊂Qn+1.
Hence, F⊂Cn∩Qn for all n≥0.

We will show that ∥xn+1−xn∥→0 as n→∞. By the definition of Qn,
we have that xn=PQn(x0).
It follows from xn+1∈Cn∩Qn⊂Qn that ∥xn−x0∥≤∥xn+1−x0∥.
This shows that the sequence {∥xn−x0∥} is increasing. Since C is bounded, we obtain that limn→∞∥xn−x0∥ exists. Notice again that from xn=PQn(x0) and xn+1∈Qn,
we have 〈xn+1−xn,xn−x0〉≥0.
Hence∥xn+1−xn∥2=∥(xn+1−x0)−(xn−x0)∥2=∥xn+1−x0∥2+∥xn−x0∥2−2〈xn+1−x0,xn−x0〉=∥xn+1−x0∥2−∥xn−x0∥2−2〈xn+1−x0−(xn−x0),xn−x0〉=∥xn+1−x0∥2−∥xn−x0∥2−2〈xn+1−xn,xn−x0〉≤∥xn+1−x0∥2−∥xn−x0∥2→0(n→∞).

Now we claim that ∥(2/(n+1)(n+2))∑k=0n∑i+j=kSiTjxn−xn∥→0 as n→∞. By the definition of yn,
we have∥2(n+1)(n+2)∑k=0n∑i+j=kSiTjxn−xn∥=11−αn∥yn−xn∥≤11−αn(∥yn−xn+1∥+∥xn+1−xn∥)≤11−a(∥yn−xn+1∥+∥xn+1−xn∥).Since xn+1∈Cn, ∥yn−xn+1∥2≤∥xn−xn+1∥2+θn→0 as n→∞.
So ∥yn−xn+1∥→0 as n→∞.
This implies that∥2(n+1)(n+2)∑k=0n∑i+j=kSiTjxn−xn∥→0(n→∞).

Since C is bounded closed convex, ωw(xn)≠∅.
It follows from (3.6) and Lemma 2.4 that ωw(xn)⊂F.
By the definition of Qn,
we have that ∥xn−x0∥≤∥PF(x0)−x0∥ for all n≥0. It follows from the weak lower semi-continuity
of the norm that ∥w−x0∥≤∥PF(x0)−x0∥ for all w∈ωw(xn).
Since ωw(xn)⊂F,
we have w=PF(x0) for all w∈ωw(xn).
Thus ωw(xn)={PF(x0)}.
Then, {xn} converges to PF(x0) weakly. By the fact∥xn−PF(x0)∥2=∥xn−x0+x0−PF(x0)∥2=∥xn−x0∥2+∥x0−PF(x0)∥2+2〈xn−x0,x0−PF(x0)〉≤2(∥PF(x0)−x0∥2+〈xn−x0,x0−PF(x0)〉)→0(n→∞),we have {xn} converges to PF(x0) strongly. This completes the proof.

The following corollary follows from Theorem 3.1.

Corollary 3.2.

Let C be a bounded closed convex subset of a Hilbert H, T and S:C→C be two commutative nonexpansive mappings.
Suppose that 0≤αn≤a for all n,
where 0<a<1.
If F:=F(T)∩F(S)≠∅,
then the sequence {xn} generated byx0∈C
chosen
arbitrarily
,yn=αnxn+(1−αn)2(n+1)(n+2)∑k=0n∑i+j=kSiTjxn,Cn={v∈C:∥yn−v∥2≤∥xn−v∥2},Qn={v∈C:〈xn−v,x0−xn〉≥0},xn+1=PCn∩Qn(x0),converges strongly to PF(x0).

Acknowledgments

This work is supported by National Natural Science Foundation of China (10771173).

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