Demiclosedness principle for total asymptotically pseudocontractive mappings in Hilbert spaces is established. The strong convergence to a fixed point of total asymptotically pseudocontraction in Hilbert spaces is obtained based on demiclosedness principle, metric projective operator, and hybrid iterative method. The main results presented in this paper extend and improve the corresponding results of Zhou (2009), Qin, Cho, and Kang (2011) and of many other authors.

1. Introduction

Throughout this article we assume that H is a real Hilbert space, whose inner product and norm are denoted by 〈·,·〉 and ∥·∥, respectively; C is a nonempty closed convex subset of H; ℕ and ℝ+ denote the natural number set and the set of nonnegative real numbers, respectively. Let T:C→C be a nonlinear mapping; F(T) denotes the set of fixed points of mapping T. We use “→” to stand for strong convergence and “⇀” for weak convergence.

Recently, the iterative approximation of fixed points for asymptotically pseudocontractive mappings, total asymptotically pseudocontractive mappings in Hilbert, or Banach spaces has been studied extensively by many authors, see for example [1–5].

The asymptotically pseudocontractions and total asymptotically pseudo-contractions are defined as follows.

Definition 1.1 (see [<xref ref-type="bibr" rid="B3">3</xref>]).

T:C→C is said to be asymptotically pseudocontraction if there exists a sequence {kn} of positive real numbers with kn≥1, limn→+∞kn=1, such that
(1.1)〈Tnx-Tny,x-y〉≤kn∥x-y∥2
holds; in [4], T:C→C is said to be total asymptotically pseudo-contraction if there exists sequences {μn},{νn} with μn,νn→0 as n→∞ and strictly increasing continuous functions ψ:ℝ+→ℝ+ with ψ(0)=0, such that
(1.2)〈Tnx-Tny,x-y〉≤∥x-y∥2+μnψ(∥x-y∥)+νn,n∈ℕ
holds; Zhou [3, page 3144] have proved the following Theorem.

Theorem Zhou

Let C be a bounded and closed convex subset of a real Hilbert space H. Let T:C→C be a uniformly L-Lipschitzian asymptotically pseudo-contraction with a fixed point. Suppose the control sequence αn is chosen so that αn∈[a,b], for some a,b∈(0,1/(1+L)). Let xn be a sequence generated in the following manner:
(1.3)x0∈C,chosenarbitrarily;n≥0,yn=(1-αn)xn+αnTnxn,Cn={z∈C:αn[1-(1+L)αn]∥xn-Tnxn∥2≤〈xn-z,yn-Tnyn〉+(kn-1)(diamC)2},Qn={z∈C:〈xn-z,x0-xn〉}≥0,xn+1=PCn∩Qnx0.
Then the iterative sequence {xn} converges strongly to PF(T)x0.

Qin et al. [5] introduced the class of total asymptotically pseudocontractive mappings in Hilbert spaces and established the following weak convergence theorem of fixed points.

Theorem Qin

Let C be a bounded and closed convex subset of a real Hilbert space H. Let T:C→C be a uniformly L-Lipschitzian total asymptotically pseudo-contractive mapping. Assume that F(T) is nonempty and there exist positive constants M and M* such that ψ(λ)≤M*λ2 for all λ≥M. Let xn be a sequence generated in the following manner:
(1.4)x1∈C,yn=(1-βn)xn+βnTnxn,xn+1=(1-αn)xn+αnTnyn,∀n≥1,
where αn and βn are sequences in (0,1). Assume that the following restrictions are satisfied:

Σn=1∞μn<∞ and Σn=1∞νn<∞.

a<αn<βn<b for some a>0 and some b∈(0,L-2[1+L2-1]).

Then the iterative sequence {xn} converges weakly to fixed point of T.

The purpose of this article is to prove the strong convergence for total asymptotically pseudo-contraction in Hilbert spaces. The results presented in the article improve and extend the corresponding results of Zhou [3], Qin et al. [5], and many other authors.

2. Preliminaries

A mapping T:C→C is said to be uniformly L-Lipschitzian if there exists some L>0 such that
(2.1)∥Tnx-Tny∥≤L∥x-y∥
holds for all x,y∈C and for all n∈ℕ. Let C be a nonempty closed convex subset of a real Hilbert space H. For every point x∈H there exists a unique nearest point in C, denoted by PCx, such that ∥x-PCx∥≤∥x-y∥ holds for all y∈C, where PC is called the metric projection of H onto C.

In order to prove the results of this article, we will need the following lemmas.

Lemma 2.1 (see [<xref ref-type="bibr" rid="B6">6</xref>]).

Let C be a nonempty closed convex subset of a real Hilbert space H. Given x∈H and z∈C, then z=PCx if and only if there holds the relation
(2.2)〈x-z,y-z〉≤0,∀y∈C.

Lemma 2.2 (see [<xref ref-type="bibr" rid="B6">6</xref>]).

Let C be a nonempty closed convex subset of a real Hilbert space H and PC:H→C the metric projection from H onto C. Then the following inequality holds:
(2.3)∥y-PCx∥2+∥x-PCx∥2≤∥x-y∥2,∀x∈H,y∈C.

3. Main ResultsTheorem 3.1 (demiclosedness principle).

Let C be a nonempty bounded and closed convex subset of a real Hilbert space H. Let T:C→C be a uniformly L-Lipschitzian and total asymptotically pseudo-contraction. Suppose there exists M*>0, such that ψ(ξn)≤M*ξn, then I-T is demiclosed at zero, where I is the identical mapping.

Proof.

Assume that {xn}⊂C, with xn⇀x and xn-Txn→0 as n→∞. We want to prove x∈C and x=Tx. Since C is a closed convex subset of H, so x∈C. In the following we prove x=Tx.

Now we choose α∈(0,1/(1+L)) and let yα,m=(1-α)x+αTmx for arbitrary fixed m≥1. Because T is uniformly L-Lipschitzian, we have
(3.1)∥xn-Tmxn∥≤∥xn-Txn∥+∥Txn-T2(t)xn∥+⋯+∥Tm-1xn-Tmxn∥≤mL∥xn-Txn∥→0,asn→∞.
Since T is total asymptotically pseudo-contraction, we have
(3.2)〈x-yα,m,(I-Tm)yα,m〉=〈x-xn,(I-Tm)yα,m〉+〈xn-yα,m,(I-Tm)yα,m〉=〈x-xn,(I-Tm)yα,m〉+〈xn-yα,m,(I-Tm)xn〉+〈xn-yα,m,(I-Tm)yα,m-(I-Tm)xn〉≤〈x-xn,(I-Tm)yα,m〉+〈xn-yα,m,(I-Tm)xn〉+μmψ(∥xn-yα,m∥)+νm.
By assumption xn⇀x, xn-Txn→0 and ∥xn-Tmxn∥→0 as n→∞, we have
(3.3)〈x-yα,m,(I-Tm)yα,m〉≤μmψ(∥xn-yα,m∥)+νm≤μmM*∥xn-yα,m∥+νm≤μmM*(diamC)+νm.
By the L-Lipschitz of T and the definition of yα,m, we have
(3.4)〈x-yα,m,(I-Tm)x-(I-Tm)yα,m〉≤(1+L)∥x-yα,m∥2≤(1+L)α2∥x-Tmx∥2.
Thus we have
(3.5)∥x-Tmx∥2=〈x-Tmx,x-Tmx〉=1α〈x-yα,m,x-Tmx〉=1α〈x-yα,m,x-Tmx-(yα,m-Tmyα,m)〉+1α〈x-yα,m,(yα,m-Tmyα,m)〉≤α(1+L)∥x-Tmx∥2+1α〈x-yα,m,yα,m-Tmyα,m〉≤α(1+L)∥x-Tmx∥2+1α(μmM*(diamC)+νm),
which implies that
(3.6)α[1-α(1+L)]∥x-Tmx∥2≤μmM*(diamC)+νm,∀m∈ℕ.
When m→∞,μm,νm→0, so we have ∥x-Tmx∥→0,m→∞, that is, Tmx→x,m→∞, so Tm+1x→Tx,m→∞. By the continuity of T, we have Tx=x.

Theorem 3.2.

Let C be a nonempty bounded and closed convex subset of a real Hilbert space H. Let T:C→C be a uniformly L-Lipschitzian and total asymptotically pseudo-contraction. Suppose there exists M*>0, such that ψ(ξn)≤M*ξn, then F(T) is a closed convex subset of C.

Proof.

Since T is uniformly L-Lipschitzian continuous, F(T) is closed. We need to show that F(T) is convex. We let p1,p2∈F(T), and p=tp1+(1-t)p2 for t∈(0,1). We take α∈(0,1/(1+L)) and let yα,n=(1-α)p+αTnp, n∈ℕ. Then for any z∈F(T), we have
(3.7)∥p-Tnp∥2=〈p-Tnp,p-Tnp〉=1α〈p-yα,n,p-Tnp〉=1α〈p-yα,n,p-Tnp-(yα,n-Tnyα,n)〉+1α〈p-yα,n,yα,n-Tnyα,n〉≤1+Lα∥p-yα,n∥2+1α〈p-z,yα,n-Tnyα,n〉+1α〈z-yα,n,yα,n-Tnyα,n〉≤1+Lα∥p-yα,n∥2+1α〈p-z,yα,n-Tnyα,n〉+1α(μnM*(diamC)+νn)≤α(1+L)∥p-Tnp∥2+1α〈p-z,yα,n-Tnyα,n〉+1α(μmM*(diamC)+νm).
This implies that
(3.8)α[1-α(1+L)]∥p-Tnp∥2≤〈p-z,yα,n-Tnyα,n〉+μmM*(diamC)+νm.
Now we take z=p1,p2, multiplying t and 1-t on both sides of above inequality, respectively, and adding up, and we can get
(3.9)α[1-α(1+L)]∥p-Tnp∥2≤μmM*(diamC)+νm.
By n→∞, we get Tnp→p. Since T is continuous, we have Tn+1p→Tp as n→∞, so that p=Tp.

Theorem 3.3.

Let C be a bounded and closed convex subset of a real Hilbert space H. Let T:C→C be a uniformly L-Lipschitzian and total asymptotically pseudo-contraction. Suppose there exists M*>0, such that ψ(ξn)≤M*ξn, F(T)≠∅, αn is a sequence in [a,b], where a,b∈(0,1/(1+L)). Let xn be a sequence generated by
(3.10)x1=x∈C,∀n∈ℕ,yn=(1-αn)xn+αnTnxn,Hn={z∈C:αn[1-(1+L)αn]∥xn-Tnxn∥2≤〈xn-z,yn-Tnyn〉+ζn},Wn={z∈C:〈xn-z,x-xn〉}≥0,xn+1=PHn∩Wnx,
where ζn=μmM*(diamC)+νm, then the iterative sequence {xn} converges strongly to PF(T)x in C.

Proof.

We divide the proof into seven steps.

(I) PF(T) is well defined for every x∈C.

By Theorem 3.2, we know F(T) is closed and convex subset of C. Moreover, by our assumption that F(T) is nonempty, therefore, PF(T)x is well defined for every x∈C.

(II) Hn and Wn are closed and convex for all n∈ℕ.

From the definitions of Wn and Hn, it is obvious that Hn and Wn are closed and convex for each n∈ℕ. We omit the details.

(III) We prove F(T)⊂Hn⋂Wn for each n∈ℕ.

We first show F(T)⊂Hn. Let z∈F(T), by (3.10), and the uniform L-Lipschitz continuity of T and the total asymptotical pseudo-contractiveness of T, we have
(3.11)∥xn-Tnxn∥2=〈xn-Tnxn,xn-Tnxn〉=1αn〈xn-yn,xn-Tnxn〉=1αn〈xn-yn,xn-Tnxn-(yn-Tnyn)〉+1αn〈xn-yn,(yn-Tnyn)〉≤1+Lαn∥xn-yn∥2+1αn〈xn-z,yn-Tnyn〉+1αn(μnM*(diamC)+νn)=(1+L)αn∥xn-Tnxn∥2+1αn〈xn-z,yn-Tnyn〉+1αn(μnM*(diamC)+νn).
This implies that
(3.12)αn[1-αn(1+L)]∥xn-Tnxn∥2≤〈xn-z,yn-Tnyn〉+(μnM*(diamC)+νn).
This shows that z∈Hn for all n∈ℕ. So F(T)⊂Hn for all n∈ℕ. Next we prove F(T)⊂Wn for all n∈ℕ. By induction, for n=1, we have F(T)⊂C=W1. Assume that F(T)⊂Wn. Since xn+1 is the projection of x onto Hn⋂Wn, by Lemma 2.1, we have
(3.13)〈xn+1-z,x-xn+1〉≥0,
for any z∈Hn⋂Wn, by the definition of Wn+1, this shows that z∈Wn+1. So F(T)⊂Hn⋂Wn, for all n∈ℕ.

(IV) We prove that limn→∞∥xn-x∥ exists.

From (3.10) and Lemma 2.1, we have xn=Pwnx, this with xn+1∈Wn+1 show ∥xn-x∥≤∥xn+1-x∥, for all n∈ℕ. As z∈F(T)⊂Wn, we also have ∥xn-x∥≤∥z-x∥, for all n∈ℕ. Consequently, limn→∞∥xn-x∥ exists and {xn} is bounded.

(V) We prove that ∥xn+1-xn∥→0 as n→∞.

By Lemma 2.2, we have
(3.14)∥xn+1-xn∥2≤∥xn+1-x∥2-∥xn-x∥2→0,
as n→∞.

(VI) Now we prove ∥xn-Txn∥→0 as n→∞.

It follows from ∥xn+1-xn∥→0 as n→∞, xn+1⊂Hn, {yn} is bounded, {Tnyn} is bounded, and αn∈(a,b) that
(3.15)αn[1-αn(1+L)]∥xn-Tnxn∥2≤〈xn-z,yn-Tnyn〉+(μnM*(diamC)+νn)≤∥xn-z∥∥yn-Tnyn∥+(μnM*(diamC)+νn)→0,n→∞.
So ∥xn-Tnxn∥→0 as n→∞. Additional
(3.16)∥xn-Txn∥≤∥xn+1-xn∥+∥Tn+1xn+1-xn+1∥+∥Tn+1xn-Tn+1xn+1∥+∥Txn-Tn+1xn∥≤(L+1)∥xn+1-xn∥+∥Tn+1xn+1-xn+1∥+L∥xn-Tnxn∥.
So ∥xn-Txn∥→0 as n→∞.

(VII) Finally, we prove xn→PF(T)x as n→∞.

Let xnk be a subsequence of xnk such that xnk⇀x^∈C, then by Theorem 3.1, we have x^∈F(T). We let ω∈PF(T)x. For any n∈ℕ, xn+1=PHn⋂Wnx and ω∈PF(T)x⊂Hn⋂Wn, so we get ∥xn+1-x∥≤∥ω-x∥.

On the other hand, from the weak lower semicontinuity of the norm, we have
(3.17)∥x^-x∥2=∥x^∥2-2〈x^,x〉+∥x∥2≤liminfn→∞(∥xnk∥2-2〈∥xnk∥2,x〉+∥x∥2)=liminfn→∞∥xnk-x∥2≤limsupn→∞∥xnk-x∥2≤∥ω-x∥2.
From the definition of PF(T)x, we obtain x^=ω and hence limsupn→∞∥xnk-x∥2=∥ω-x∥2. So we have limsupk→∞∥xnk∥=∥ω∥. Thus we obtain that xnk converges strongly to PF(T)x. Since xnk is an arbitrary weakly convergent sequence of xn, we can conclude that xn converges strongly to PF(T)x. This completes the proof of Theorem 3.3.

Remark 3.4.

Theorem 3.3 extends the main results of Zhou [3] and improves the main results of Qin et al. [5] and of many other authors.

Acknowledgments

The authors are grateful to the referees for their helpful and useful comments. This research is partially supported by the Fundamental Research Funds for the Central Universities (JBK120117), by the National Research Foundation of Yibin University (No. 2011B07) and by Scientific Research Fund Project of SiChuan Provincial Education Department (No. 12ZB345).

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