Let E be a real reflexive Banach space with a uniformly Gâteaux differentiable norm. Let K be a nonempty bounded closed convex subset of E, and every nonempty closed convex bounded subset of K has the fixed point property for non-expansive self-mappings. Let f:K→K a contractive mapping and T:K→K be a uniformly continuous pseudocontractive mapping with F(T)≠∅. Let {λn}⊂(0,1/2) be a sequence satisfying the following conditions: (i) limn→∞λn=0; (ii) ∑n=0∞λn=∞. Define the sequence {xn} in K by x0∈K, xn+1=λnf(xn)+(1−2λn)xn+λnTxn, for all n≥0. Under some appropriate assumptions, we prove that the sequence {xn} converges strongly to a fixed point p∈F(T) which is the unique solution of the following variational inequality: 〈f(p)−p,j(z−p)〉≤0, for all z∈F(T).

1. Introduction

Let E be a real Banach space with dual E*. We denote by J the normalized duality mapping from E to 2E* defined by J(x)={f*∈E*:〈x,f*〉=‖x‖2=‖f*‖2},∀x∈E,
where 〈·,·〉 denotes the generalized duality pairing.

It is well known that, if E is smooth, then J is single-valued. In the sequel, we will denote the single-valued normalized duality mapping by j. We use D(T),R(T) to denote the domain and range of T, respectively.

An operator T:D(T)→R(T) is called pseudocontractive if there exists j(x-y)∈J(x-y) such that 〈Tx-Ty,j(x-y)〉≤‖x-y‖2,∀x,y∈D(T).

A point x∈K is a fixed point of T provided Tx=x. Denote by F(T) the set of fixed points of T, that is, F(T)={x∈K:Tx=x}.

Within the past 40 years or so, many authors have been devoted to the iterative construction of fixed points of pseudocontractive mappings (see [1–10]).

In 1974, Ishikawa [11] introduced an iterative scheme to approximate the fixed points of Lipschitzian pseudocontractive mappings and proved the following result.

Theorem 1.1 (see [<xref ref-type="bibr" rid="B7">11</xref>]).

If K is a compact convex subset of a Hilbert space H,T:K→K is a Lipschitzian pseudocontractive mapping. Define the sequence {xn} in K by
x0∈K,xn+1=(1-αn)xn+αnTyn,yn=(1-βn)xn+βnTxn,∀n≥0,
where {αn},{βn} are sequences of positive numbers satisfying the conditions

0≤αn≤βn<1,

limn→∞βn=0,

∑n=1∞αnβn=∞.

Then, the sequence {xn} converges strongly to a fixed point of T.

In connection with the iterative approximation of fixed points of pseudo-contractions, in 2001, Chidume and Mutangadura [12] provided an example of a Lipschitz pseudocontractive mapping with a unique fixed point for which the Mann iterative algorithm failed to converge. Chidume and Zegeye [13] introduced a new iterative scheme for approximating the fixed points of pseudocontractive mappings.

Theorem 1.2 (see [<xref ref-type="bibr" rid="B6">13</xref>]).

Let E be a real reflexive Banach space with a uniformly Gâteaux differentiable norm. Let K be a nonempty closed convex subset of E. Let T:K→K be a L-Lipschitzian pseudocontractive mapping such that F(T)≠∅. Suppose that every nonempty closed convex bounded subset of K has the fixed point property for nonexpansive self-mappings. Let {λn} and {θn} be two sequences in (0,1] satisfying the following conditions:

limn→∞θn=0,

λn(1+θn)≤1,∑n=0∞λnθn=∞,limn→∞(λn/θn)=0,

limn→∞((θn-1/θn-1)/λnθn)=0.

For given x1∈K arbitrarily, let the sequence {xn} be defined iteratively by
xn+1=(1-λn)xn+λnTxn-λnθn(xn-x1),∀n≥1.
Then, the sequence {xn} defined by (1.4) converges strongly to a fixed point of T.

Prototypes for the iteration parameters are, for example, λn=1/(n+1)a and θn=1/(n+1)b for 0<b<a and a+b<1. But we observe that the canonical choices of λn=1/n and θn=1/n are impossible. This bring us a question.

Question 1.

Under what conditions, limn→∞λn=0 and ∑n=0∞λn=∞ are sufficient to guarantee the strong convergence of the iterative scheme (1.4) to a fixed point of T?

In this paper, we explore an iterative scheme to approximate the fixed points of pseudocontractive mappings and prove that, under some appropriate assumptions, the proposed iterative scheme converges strongly to a fixed point of T, which solves some variational inequality. Our results improve and extend many results given in the literature.

2. Preliminaries

Let K be a nonempty closed convex subset of a real Banach space E. Recall that a mapping f:K→K is called contractive if there exists a constant α∈(0,1) such that ‖f(x)-f(y)‖≤α‖x-y‖,∀x,y∈K.

Let μ be a continuous linear functional on l∞ and s=(a0,a1,…)∈l∞. We write μn(an) instead of μ(s). We call μ a Banach limit if μ satisfies ∥μ∥=μ(1)=1 and μn(an+1)=μn(an) for all (a0,a1,…)∈l∞.

If μ is a Banach limit, then we have the following.

For all n≥1, an≤cn implies μn(an)≤μn(cn).

μn(an+r)=μn(an) for any fixed positive integer r.

liminfn→∞an≤μn(an)≤limsupn→∞an for all (a0,a1,…)∈l∞.

If s=(a0,a1,…)∈l∞ with an→a, then μ(s)=μn(an)=a for any Banach limit μ.

For more details on Banach limits, we refer readers to [14]. We need the following lemmas for proving our main results.

Lemma 2.1 (see [<xref ref-type="bibr" rid="B11">15</xref>]).

Let E be a Banach space. Suppose that K is a nonempty closed convex subset of E and T:K→E is a continuous pseudocontractive mapping satisfying the weakly inward condition: T(x)∈IK(x)¯(IK(x)¯ is the closure of IK(x)) for each x∈K, where IK(x)={x+c(u-x):u∈Eandc≥1}. Then, for each z∈K, there exists a unique continuous path t↦zt∈K for all t∈[0,1), satisfying the following equation
zt=tTzt+(1-t)z.
Furthermore, if E is a reflexive Banach space with a uniformly Gâteaux differentiable norm and every nonempty closed convex bounded subset of K has the fixed point property for nonexpansive self-mappings, then, as t→1, zt converges strongly to a fixed point of T.

Lemma 2.2 (see [<xref ref-type="bibr" rid="B3">16</xref>]).

(1) If E is smooth Banach space, then the duality mapping J is single valued and strong-weak* continuous.

(2) If E is a Banach space with a uniformly Gâteaux differentiable norm, then the duality mapping J:E→E* is single valued and norm to weak star uniformly continuous on bounded sets of E.

Lemma 2.3 (see [<xref ref-type="bibr" rid="B8">17</xref>]).

Let {an} be a sequence of nonnegative real numbers satisfying an+1≤(1-αn)an+αnβn for all n≥0, where {αn}⊂(0,1), and {βn} two sequences of real numbers such that ∑n=0∞αn=∞ and limsupn→∞βn≤0. Then {an} converges to zero as n→∞.

Lemma 2.4 (see [<xref ref-type="bibr" rid="B2">18</xref>]).

Let E be a real Banach space, and let J be the normalized duality mapping. Then, for any given x,y∈E,
‖x+y‖2≤‖x‖2+2〈y,j(x+y)〉,∀j(x+y)∈J(x+y).

Lemma 2.5 (see [<xref ref-type="bibr" rid="B14">14</xref>]).

Let a be a real number, and let (x0,x1,…,)∈l∞ such that μnxn≤a for all Banach limits. If limsupn→∞(xn+1-xn)≤0, then limsupn→∞xn≤a.

3. Main Results

Now, we are ready to give our main results in this paper.

Theorem 3.1.

Let E be a real reflexive Banach space with a uniformly Gâteaux differentiable norm. Let K be a nonempty bounded closed convex subset of E, and every nonempty closed convex bounded subset of K has the fixed point property for nonexpansive self-mappings. Let f:K→K a contractive mapping and T:K→K be a uniformly continuous pseudocontractive mapping with F(T)≠∅. Let {λn}⊂(0,1/2] be a sequence satisfying the conditions:

limn→∞λn=0,

∑n=0∞λn=∞.

Define the sequence {xn} in K by
x0∈K,xn+1=λnf(xn)+(1-2λn)xn+λnTxn,∀n≥0.
If limn→∞∥xn-Txn∥=0, then the sequence {xn} converges strongly to a fixed point p∈F(T), which is the unique solution of the following variational inequality:
〈f(p)-p,j(z-p)〉≤0,∀z∈F(T).Proof.

Take p∈F(T), and let S=I-T. Then, we have
〈Sx-Sy,j(x-y)〉≥0.
From (3.1), we obtain
xn=xn+1+λnxn-λnTxn+λnxn-λnf(xn)=xn+1+λnxn+λnSxn-λnf(xn)=xn+1+λn[xn+1+λnxn+λnSxn-λnf(xn)]+λnSxn-λnf(xn)=(1+λn)xn+1+λn2(xn+Sxn)-λn2f(xn)+λnSxn-λnf(xn)=(1+λn)xn+1+λnSxn+1+λn2(xn+Sxn)-λn2f(xn)+λn(Sxn-Sxn+1)-λnf(xn).
By (3.4), we have
xn-p=(1+λn)(xn+1-p)+λn(Sxn+1-Sp)+λn2(xn+Sxn)-λn2f(xn)+λn(Sxn-Sxn+1)+λn(p-f(xn)).
Combining (3.3) and (3.5), we have
〈xn-p-λn2(xn+Sxn)+λn2f(xn)-λn(Sxn-Sxn+1)+λn(f(xn)-p),j(xn+1-p)〉=(1+λn)‖xn+1-p‖2+λn〈Sxn+1-Sp,j(xn+1-p)〉≥(1+λn)‖xn+1-p‖2.

Next, we prove that limsupn→∞〈f(p)-p,j(xn-p)〉≤0. Indeed, taking z=f(p) in Lemma 2.1, we have
zt-xn=(1-t)(Tzt-xn)+t(f(p)-xn),
and, hence,
‖zt-xn‖2=(1-t)〈Tzt-xn,j(zt-xn)〉+t〈f(p)-xn,j(zt-xn)〉=(1-t)〈Tzt-Txn,j(zt-xn)〉+(1-t)〈Txn-xn,j(zt-xn)〉+t〈f(p)-zt,j(zt-xn)〉+t‖zt-xn‖2≤‖zt-xn‖2+(1-t)‖Txn-xn‖‖zt-xn‖+t〈f(p)-zt,j(zt-xn)〉.
Therefore, we have
〈zt-f(p),j(zt-xn)〉≤1-tt‖Txn-xn‖‖zt-xn‖≤M11-tt‖Txn-xn‖,
where M1>0 is some constant such that ∥zt-xn∥≤M1 for all t∈(0,1] and n≥1. Letting n→∞, we have
limsupn→∞〈zt-f(p),j(zt-xn)〉≤0.
From Lemma 2.1, we know zt→p as t→0. Since the duality mapping J:E→E* is norm to weak star uniformly continuous from Lemma 2.2, we have
limsupn→∞〈f(p)-p,j(xn-p)〉≤0.
From (3.6), we have
(1+λn)‖xn+1-p‖2≤〈xn-p-λn2(xn+Sxn)+λn2f(xn)-λn(Sxn-Sxn+1),j(xn+1-p)〉≤‖xn-p‖‖xn+1-p‖+M2λn2+M2λn‖Sxn+1-Sxn‖+λn‖f(xn)-f(p)‖‖xn+1-p‖+λn〈f(p)-p,j(xn+1-p)〉≤‖xn-p‖‖xn+1-p‖+M2λn2+M2λn‖Sxn+1-Sxn‖+λnα‖xn-p‖‖xn+1-p‖+λn〈f(p)-p,j(xn+1-p)〉≤1+λnα2(‖xn-p‖2+‖xn+1-p‖2)+M2λn2+M2λn‖Sxn+1-Sxn‖+λn〈f(p)-p,j(xn+1-p)〉,
where M2 is a constant such that
sup{‖xn+Sxn‖‖xn+1-p‖+‖f(xn)‖‖xn+1-p‖+‖xn+1-p‖,n≥0}≤M2.
It follows that
‖xn+1-p‖2≤1+λnα1+(2-α)λn‖xn-p‖2+M2λn2+M2λn‖Sxn+1-Sxn‖+λn1+(2-α)λn〈f(p)-p,j(xn+1-p)〉=[1-2(1-α)1+(2-α)λnλn]‖xn-p‖2+2(1-α)λn1+(2-α)λn×{1+(2-α)λn2(1-α)M2λn+1+(2-α)λn2(1-α)M2‖Sxn+1-Sxn‖+12(1-α)〈f(p)-p,j(xn+1-p)〉}=(1-αn)‖xn-p‖2+αnβn,
where
αn=2(1-α)1+(2-α)λnλn,βn=1+(2-α)λn2(1-α)M2λn+1+(2-α)λn2(1-α)M2‖Sxn+1-Sxn‖+12(1-α)〈f(p)-p,j(xn+1-p)〉.
Note that
‖xn+1-xn‖≤λn‖xn‖+λn‖Txn‖+λn‖xn-f(xn)‖⟶0(n⟶∞).
By the uniformly continuity of T, we have
‖Sxn+1-Sxn‖⟶0(n⟶∞).
Hence, it is clear that ∑n=0∞αn=∞ and limsupn→∞βn≤0.

Finally, applying Lemma 2.3 to (3.14), we can conclude that xn→p. This completes the proof.

From Theorem 3.1, we can prove the following corollary.

Corollary 3.2.

Let E be a real reflexive Banach space with a uniformly Gâteaux differentiable norm. Let K be a nonempty bounded closed convex subset of E, and every nonempty closed convex bounded subset of K has the fixed point property for nonexpansive self-mappings. Let T:K→K be a uniformly continuous pseudocontractive mapping with F(T)≠∅. Let {λn}⊂(0,1/2] be a sequence satisfying the conditions:

limn→∞λn=0,

∑n=0∞λn=∞.

Define the sequence {xn} in K by
u,x0∈K,xn+1=λnu+(1-2λn)xn+λnTxn,∀n≥0.
Then, the sequence {xn} converges strongly to a fixed point of T if and only if limn→∞∥xn-Txn∥=0.Theorem 3.3.

Let E be a uniformly smooth Banach space and K a nonempty bounded closed convex subset of E. Let f:K→K be a contractive mapping and T:K→K a uniformly continuous pseudocontractive mapping with F(T)≠∅. Let {λn}⊂(0,1/2] be a sequence satisfying the conditions:

limn→∞λn=0,

∑n=0∞λn=∞.

If limn→∞∥xn-Txn∥=0, then the sequence {xn} defined by (3.1) converges strongly to a fixed point p∈F(T), which is the unique solution of the following variational inequality:
〈f(p)-p,j(z-p)〉≤0,∀z∈F(T).Proof.

Since every uniformly smooth Banach space E is reflexive and whose norm is uniformly Gâteaux differentiable, at the same time, every closed convex and bounded subset of K has the fixed point property for nonexpansive mappings. Hence, from Theorem 3.1, we can obtain the result. This completes the proof.

From Theorem 3.3, we can prove the following corollary.

Corollary 3.4.

Let E be a uniformly smooth Banach space and K a nonempty bounded closed convex subset of E. Let T:K→K be a uniformly continuous pseudocontractive mapping with F(T)≠∅. Let {λn}⊂(0,1/2] be a sequence satisfying the conditions:

limn→∞λn=0,

∑n=0∞λn=∞.

Define the sequence {xn} in K by
u,x0∈K,xn+1=λnu+(1-2λn)xn+λnTxn,∀n≥0.
Then, the sequence {xn} converges strongly to a fixed point of T if and only if limn→∞∥xn-Txn∥=0.Theorem 3.5.

Let K be a nonempty bounded closed convex subset of a real reflexive Banach space E with a uniformly Gâteaux differentiable norm. Let f:K→K a contractive mapping and T:K→K be a uniformly continuous pseudocontractive mapping. Let {λn}⊂(0,1/2] be a sequence satisfying the conditions:

limn→∞λn=0,

∑n=0∞λn=∞.

If D∩F(T)≠∅, where D is defined as (3.22) below, then the sequence {xn} defined by (3.1) converges strongly to a fixed point p∈F(T), which is the unique solution of the following variational inequality:
〈f(p)-p,j(z-p)〉≤0,∀z∈F(T).Proof.

First, we note that the sequence {xn} is bounded. Now, if we define g(x)=μn∥xn-x∥2, then g(x) is convex and continuous. Also, we can easily prove that g(x)→∞ as ∥x∥→∞. Since E is reflexive, there exists y∈K such that g(y)=infx∈Kg(x). So the set
D={y∈K:g(y)=infx∈Kg(x)}≠∅.
Clearly, D is closed convex subset of K.

Now, we can take p∈D∩F(T) and t∈(0,1). By the convexity of K, we have that (1-t)p+tf(p)∈K. It follows that
g(p)≤g((1-t)p+tf(p)).
By Lemma 2.4, we have
‖xn-p-t(f(p)-p)‖2≤‖xn-p‖2-2t〈f(p)-p,j(xn-p-t(f(p)-p))〉.
Taking the Banach limit in (3.24), we have
μn‖xn-p-t(f(p)-p)‖2≤μn‖xn-p‖2-2tμn〈f(p)-p,j(xn-p-t(f(p)-p))〉.
This implies
2tμn〈f(p)-p,j(xn-p-t(f(p)-p))〉≤g(p)-g((1-t)p+tf(p)).
Therefore, it follows from (3.23) and (3.26) that
μn〈f(p)-p,j(xn-p-t(f(p)-p))〉≤0.
Since the normalized duality mapping j is single valued and norm-weak* uniformly continuous on bounded subset of E, we have
〈f(p)-p,j(xn-p)〉-〈f(p)-p,j(xn-p-t(f(p)-p))〉⟶0(t⟶0).
This implies that, for any ϵ>0, there exists δ>0 such that, for all t∈(0,δ) and n≥1,
〈f(p)-p,j(xn-p)〉-〈f(p)-p,j(xn-p-t(f(p)-p))〉<ϵ.
Taking the Banach limit and noting that (3.27), we have
μn〈f(p)-p,j(xn-p)〉≤μn〈f(p)-p,j(xn-p-t(f(p)-p))〉+ϵ≤ϵ.
By the arbitrariness of ϵ, we obtain
μn〈f(p)-p,j(xn-p)〉≤0.
At the same time, we note that
‖xn+1-xn‖≤λn(‖f(xn)‖+2‖xn‖+‖Txn‖)⟶0(n⟶∞).
Since {xn-p},{f(p)-p} are bounded and the duality mapping j is single valued and norm topology to weak star topology uniformly continuous on bounded sets in Banach space E with a uniformly Gâteaux differentiable norm, it follows that
limn→∞{〈f(p)-p,j(xn+1-p)〉-〈f(p)-p,j(xn-p)〉}=0.
From (3.31), (3.33), and Lemma 2.5, we conclude that
limsupn→∞〈f(p)-p,j(xn+1-p)〉≤0.

Finally, by the similar arguments as that the proof in Theorem 3.1, it is easy prove that the sequence {xn} converges to a fixed point of T. This completes the proof.

From Theorem 3.5, we can easily to prove the following result.

Corollary 3.6.

Let K be a nonempty bounded closed convex subset of a real reflexive Banach space E with a uniformly Gâteaux differentiable norm. Let f:K→K be a contractive mapping and T:K→K a uniformly continuous pseudocontractive mapping. Let {λn}⊂(0,1/2] be a sequence satisfying the conditions:

limn→∞λn=0,

∑n=0∞λn=∞.

Define the sequence {xn} in K by
u,x0∈K,xn+1=λnu+(1-2λn)xn+λnTxn,∀n≥0.
If D∩F(T)≠∅, where D is defined as (3.22), then the sequence {xn} defined by (3.35) converges strongly to a fixed point p∈F(T).Acknowledgment

This research was partially supported by Youth Foundation of Taizhou University (2011QN11).

ChangS. S.On the convergence problems of Ishikawa and Mann iterative processes with error for Φ-pseudo contractive type mappingsChidumeC. E.MooreC.The solution by iteration of nonlinear equations in uniformly smooth Banach spacesLiuQ. H.The convergence theorems of the sequence of Ishikawa iterates for hemicontractive mappingsMannW. R.Mean value methods in iterationOsilikeM. O.Iterative solution of nonlinear equations of the Φ-strongly accretive typeReichS.Iterative methods for accretive setsYaoY.MarinoG.LiouY. C.A hybrid method for monotone variational inequalities involving pseudocontractionsYaoY.LiouY. C.KangS. M.Iterative methods for k-strict pseudo-contractive mappings in Hilbert spacesYaoY.LiouY. C.YaoJ.-C.New relaxed hybrid-extragradient method for fixed point problems, a general system of variational inequality problems and generalized mixed equilibrium problemsYaoY.ChoY. J.LiouY. C.Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problemsIshikawaS.Fixed points and iteration of a nonexpansive mapping in a Banach spaceChidumeC. E.MutangaduraS. A.An example of the Mann iteration method for Lipschitz pseudocontractionsChidumeC. E.ZegeyeH.Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive mapsShiojiN.TakahashiW.Strong convergence of approximated sequences for nonexpansive mappings in Banach spacesMoralesC. H.JungJ. S.Convergence of paths for pseudocontractive mappings in Banach spacesChangS. S.ChoY. J.ZhouH.KimT. H.XuH. K.Strong convergence of modified Mann iterationsChangS. S.ChoY. J.LeeB. S.JungJ. S.KangS. M.Iterative approximations of fixed points and solutions for strongly accretive and strongly pseudo-contractive mappings in Banach spaces