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# On the convergence of hybrid projection algorithms for total quasi-asymptotically pseudo-contractive mapping

*Journal of Inequalities and Applications*
**volume 2013**, Article number: 375 (2013)

## Abstract

In this paper, a hybrid projection algorithm for a total quasi-asymptotically pseudo-contractive mapping is introduced in a Hilbert space. A strong convergence theorem of the proposed algorithm to a fixed point of a total quasi-asymptotically pseudo-contractive mapping is proved. Our main result extends and improves many corresponding results.

**MSC:**47H05, 47H09.

## 1 Introduction

Throughout this paper, we always assume that *H* is a real Hilbert space, whose inner product and norm are denoted by \u3008\cdot ,\cdot \u3009 and \parallel \cdot \parallel. The symbol → is denoted by a strong convergence. Let *C* be a nonempty closed and convex subset of *H*, and let T:C\to C be a mapping. In this paper, we denote the fixed point set of *T* by \mathcal{F}(T), that is, \mathcal{F}(T):=\{x\in C:Tx=x\}.

Recall that *T* is said to be *asymptotically nonexpansive* if there exists a sequence \{{k}_{n}\}\subset [1,\mathrm{\infty}) with {k}_{n}\to 1 as n\to \mathrm{\infty} such that

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] as a generalization of the class of nonexpansive mappings.

*T* is said to be *asymptotically nonexpansive in the intermediate sense* if it is continuous and the following inequality holds:

Noticing that if we define

then {\rho}_{n}\to 0 as n\to \mathrm{\infty}. It follows that (1.2) is reduced to

The class of mappings, which are asymptotically nonexpansive in the intermediate sense, was introduced by Bruck *et al.* [2] (see also [3]). It is worth mentioning that the class of mappings which are asymptotically nonexpansive in the intermediate sense contains properly the class of asymptotically nonexpansive mappings.

Recall that *T* is said to be *asymptotically pseudocontractive* if there exists a sequence \{{k}_{n}\}\subset [1,\mathrm{\infty}) with {k}_{n}\to 1 as n\to \mathrm{\infty} such that

It is not hard to see that (1.5) is equivalent to

The class of an asymptotically pseudocontractive mapping was introduced by Schu [4] (see also [5]). In [6], Rhoades gave an example to show that the class of asymptotically pseudocontractive mappings contains properly the class of asymptotically nonexpansive mappings, see [6] for more details. Zhou [7] showed that every uniformly Lipschitz and asymptotically pseudocontractive mapping, which is also uniformly asymptotically regular, has a fixed point.

*T* is said to be an *asymptotically pseudocontractive mapping in the intermediate sense* if there exists a sequence \{{k}_{n}\}\subset [1,\mathrm{\infty}) with {k}_{n}\to 1 as n\to \mathrm{\infty} such that

Put

It follows that {\tau}_{n}\to 0 as n\to \mathrm{\infty}. Then, (1.8) is reduced to the following:

The class of asymptotically pseudocontractive mappings in the intermediate sense was introduced by Qin *et al.* [8].

Recall that *T* is said to be *total asymptotically pseudocontractive* if there exist sequences \{{k}_{n}\},\{{\nu}_{n}\}\subset [0,\mathrm{\infty}) with {k}_{n},{\nu}_{n}\to 0 as n\to \mathrm{\infty} such that

where \varphi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) is a continuous and strictly increasing function with \varphi (0)=0. The class of a total asymptotically pseudocontractive mapping was introduced by Qin [9].

It is easy to see that (1.10) is equivalent to the following: for all n\ge 1, x,y\in C,

If \varphi (\lambda )={\lambda}^{2}, then (1.10) is reduced to

Put

If \varphi (\lambda )={\lambda}^{2}, then the class of total asymptotically pseudocontractive mappings is reduced to the class of asymptotically pseudocontractive mappings in the intermediate sense.

In this paper, we introduce and study the following mapping.

**Definition 1.1** A mapping T:C\to C is said to be total quasi-asymptotically pseudo-contractive if \mathcal{F}(T)\ne \mathrm{\varnothing}, and there exist sequences \{{\mu}_{n}\}\subset [0,\mathrm{\infty}) and \{{\xi}_{n}\}\subset [0,\mathrm{\infty}) with {\mu}_{n}\to 0 and {\xi}_{n}\to 0 as n\to \mathrm{\infty} such that

where \varphi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) is a continuous and strictly increasing function with \varphi (0)=0.

It is easy to see that (1.14) is equivalent to the following:

**Remark 1** It is clear that every total asymptotically pseudo-contractive mapping with F(T)\ne \mathrm{\varnothing} is total quasi-asymptotically pseudo-contractive, but the converse maybe not true.

**Remark 2** If \varphi (\lambda )={\lambda}^{2}, the (1.14) is reduced to

**Remark 3** Put

If \varphi (\lambda )={\lambda}^{2}, then the class of total quasi-asymptotically pseudo-contractive mappings is reduced to the class of quasi-asymptotically pseudo-contractive mappings in the intermediate sense.

Recently, the iterative approximation of fixed points for asymptotically pseudo-contractive mappings, total asymptotically pseudo-contractive mappings in Hilbert, or Banach spaces has been studied extensively by many authors, see, for example, [7, 9–13]. In this paper, we shall consider and study a total quasi-asymptotically pseudo-contractive mapping as a generalization of (total) asymptotically pseudo-contractive mappings. Furthermore, we shall introduce an iterative algorithm for finding a fixed point of a total quasi-asymptotically pseudo-contractive mapping.

## 2 Preliminaries

A mapping T:C\to C is said to be uniformly *L*-Lipschitzian if there exists some L>0 such that

Let *C* be a nonempty closed convex subset of a real Hilbert space *H*. For every point x\in H, there exists a unique nearest point in *C*, denoted by {P}_{C}x, such that \parallel x-{P}_{C}x\parallel \le \parallel x-y\parallel holds for all y\in C, where {P}_{C} is said to be the metric projection of *H* onto *C*.

In order to prove our main results, we also need the following lemmas.

**Lemma 2.1** [14]

*Let* *C* *be a nonempty closed convex subset of a real Hilbert space* *H* *and let* {P}_{C} *be the metric projection from* *H* *onto* *C* (*i*.*e*., *for* x\in H, {P}_{C} *is the only point in* *C* *such that* \parallel x-{P}_{C}x\parallel =inf\{\parallel x-z\parallel :z\in C\}). *Given* x\in H *and* z\in C, *then* z={P}_{C}x *if and only if the relation*

*holds*.

**Lemma 2.2** *Let* *C* *be a nonempty bounded and closed convex subset of a real Hilbert space H*. *Let* T:C\to C *be a uniformly* *L*-*Lipschitzian and total quasi*-*asymptotically pseudo*-*contractive mapping with* \mathcal{F}(T)\ne \mathrm{\varnothing}. *Suppose there exist positive constants* *M* *and* {M}^{\ast} *such that* \varphi (\zeta )\le {M}^{\ast}{\zeta}^{2} *for all* \zeta >M. *Then* \mathcal{F}(T) *is a closed convex subset of* *C*.

*Proof* Since *ϕ* is an increasing function, it follows that \varphi (\zeta )\le \varphi (M) if \zeta \le M and \varphi (\zeta )\le {M}^{\ast}{\zeta}^{2} if \zeta \ge M. In either case, we can always obtain that

Since *T* is uniformly *L*-Lipschitzian continuous, \mathcal{F}(T) is closed. We need to show that \mathcal{F}(T) is convex. To this end, let {p}_{i}\in \mathcal{F}(T) (i=1,2), and write p=t{p}_{1}+(1-t){p}_{2} for t\in (0,1). We take \alpha \in (0,\frac{1}{1+L}), and define {y}_{\alpha ,n}=(1-\alpha )p+\alpha {T}^{n}p for each n\in \mathbb{N}. Then, for all z\in \mathcal{F}(T), we have from (2.3) that

This implies that

Now, we take z={p}_{i} (i=1,2) in (2.4), multiplying *t* and (1-t) on the both sides of the above inequality (2.4), respectively, and adding up, and we can get

Letting n\to \mathrm{\infty} in (2.5), we obtain {T}^{n}p\to p. Since *T* is continuous, we have {T}^{n+1}p\to Tp as n\to \mathrm{\infty}, therefore, p=Tp. This proves that \mathcal{F}(T) is a closed convex subset of *C*. □

## 3 Main results

In this section, we shall give our main results of this paper.

**Theorem 3.1** *Let* *C* *be a nonempty bounded and closed convex subset of a real Hilbert space* *H*. *Let* T:C\to C *be a uniformly* *L*-*Lipschitzian and total quasi*-*asymptotically pseudo*-*contractive mapping with* \mathcal{F}(T)\ne \mathrm{\varnothing}. *Suppose that there exist positive constants* *M* *and* {M}^{\ast} *such that* \varphi (\zeta )\le {M}^{\ast}{\zeta}^{2} *for all* \zeta >M. *Let* \{{x}_{n}\} *be a sequence generated by the following iterative scheme*:

*where* {\theta}_{n}={\mu}_{n}[\varphi (M)+{M}^{\ast}{(diamC)}^{2}]+{\xi}_{n}, \{{\alpha}_{n}\} *is a sequence in* [a,b] *with* a,b\in (0,\frac{1}{1+L}). *Then the sequence* \{{x}_{n}\} *converges strongly to a point* {P}_{\mathcal{F}(T)}{x}_{1}, *where* {P}_{\mathcal{F}(T)} *is the projection from* *C* *onto* \mathcal{F}(T).

*Proof* We split the proof into seven steps.

Step 1. Show that {P}_{\mathcal{F}(T)}{x}_{1} is well defined for every {x}_{1}\in C.

By Lemma 2.2, we know that \mathcal{F}(T) is a closed and convex subset of *C*. Therefore, in view of the assumption of \mathcal{F}(T)\ne \mathrm{\varnothing}, {P}_{\mathcal{F}(T)}{x}_{1} is well defined for every {x}_{1}\in C.

Step 2. Show that {C}_{n} and {Q}_{n} are closed and convex for all n\ge 1.

From the definitions of {C}_{n} and {Q}_{n}, it is obvious that {C}_{n} and {Q}_{n} are closed and convex for all n\ge 1. We omit the details.

Step 3. Show that \mathcal{F}(T)\subset {C}_{n}\cap {Q}_{n} for all n\ge 1.

To this end, we first prove that \mathcal{F}(T)\subset {C}_{n} for all n\ge 1. This can be proved by induction on *n*. It is obvious that \mathcal{F}(T)\subset {C}_{1}=C. Assume that \mathcal{F}(T)\subset {C}_{n} for some n\in \mathbb{N}. Then, using the uniform *L*-Lipschitzian continuity of *T*, the total quasi-asymptotic pseudo-contractiveness of *T* and (2.3), we have for any w\in \mathcal{F}(T) that

which implies that

which shows that w\in {C}_{n+1}. By the mathematical induction principle, \mathcal{F}(T)\subset {C}_{n} for all n\ge 1.

Next, we prove \mathcal{F}(T)\subset {Q}_{n} for all n\ge 1. By induction, for n=1, we have \mathcal{F}(T)\subset C={Q}_{1}. Assume that \mathcal{F}(T)\subset {Q}_{n} for some n\in \mathbb{N}. Since {x}_{n} is the projection of {x}_{1} onto {C}_{n}\cap {Q}_{n}, by Lemma 2.1, we have

Since \mathcal{F}(T)\subset {C}_{n}\cap {Q}_{n}, we easily see that

which implies that \mathcal{F}(T)\subset {Q}_{n+1}. This proves that \mathcal{F}(T)\subset {C}_{n}\cap {Q}_{n} for all n\ge 1.

Step 4. Show that {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-{x}_{1}\parallel exists.

In view of (3.1) and Lemma 2.1, we have {x}_{n}={P}_{{Q}_{n}}{x}_{1} and {x}_{n+1}\in {Q}_{n}, which implies

On the other hand, since \mathcal{F}(T)\subset {Q}_{n}, we also have

Therefore, {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-{x}_{1}\parallel exists and \{{x}_{n}\} is bounded.

Step 5. Show that \{{x}_{n}\} is a Cauchy sequence.

Noticing the construction of {C}_{n}, one has {C}_{m}\subset {C}_{n} and {x}_{m}={P}_{{C}_{m}}{x}_{1}\in {C}_{n} for any positive integer m>n. From (3.2), we have

It follows that

Letting n\to \mathrm{\infty} in (3.4), one has {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-{x}_{n+m}\parallel =0, \mathrm{\forall}m\ge n. Hence, \{{x}_{n}\} is a Cauchy sequence. Since *H* is a Hilbert space and *C* is closed and convex, one can assume that {x}_{n}\to q\in C as n\to \mathrm{\infty}.

Step 6. Show that {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-T{x}_{n}\parallel =0.

It follows from {x}_{n+1}\in {C}_{n} and (3.1) that

Since \{{y}_{n}\} is bounded, \{{T}^{n}{y}_{n}\} is bounded, {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n+1}-{x}_{n}\parallel =0 and {\alpha}_{n}\in (a,b), we have from (3.5) that

On the other hand, we notice that

From {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n+1}-{x}_{n}\parallel =0 and {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-{T}^{n}{x}_{n}\parallel =0, we have

It follows that T{x}_{n}\to q as n\to \mathrm{\infty}. Since T is continuous, one has that *q* is a fixed point of *T*; that is, q\in \mathcal{F}(T).

Step 7. Finally, we prove q={P}_{\mathcal{F}(T)}{x}_{1}.

By taking the limit in (3.3), we have

which implies that q={P}_{\mathcal{F}(T)}{x}_{1} by using Lemma 2.1. This completes the proof. □

Since every total asymptotically pseudo-contractive mapping with \mathcal{F}(T)\ne \mathrm{\varnothing} is total quasi-asymptotically pseudo-contractive, we immediately obtain the following corollary:

**Corollary 3.2** *Let* *C* *be a nonempty bounded and closed convex subset of a real Hilbert space* *H*. *Let* T:C\to C *be a uniformly* *L*-*Lipschitzian and total asymptotically pseudo*-*contractive mapping with* \mathcal{F}(T)\ne \mathrm{\varnothing}. *Suppose there exist positive constants* *M* *and* {M}^{\ast} *such that* \varphi (\zeta )\le {M}^{\ast}{\zeta}^{2} *for all* \zeta >M. *Let* \{{x}_{n}\} *be a sequence generated by the following iterative scheme*:

*where* {\theta}_{n}={\mu}_{n}[\varphi (M)+{M}^{\ast}{(diamC)}^{2}]+{\xi}_{n}, \{{\alpha}_{n}\} *is a sequence in* [a,b] *with* a,b\in (0,\frac{1}{1+L}). *Then the sequence* \{{x}_{n}\} *converges strongly to a point* {P}_{\mathcal{F}(T)}{x}_{1}, *where* {P}_{\mathcal{F}(T)} *is the projection from* *C* *onto* \mathcal{F}(T).

**Remark 3.3** Since the class of the total quasi-asymptotically pseudo-contractive mappings includes the class of asymptotically pseudocontractive mappings, the class of asymptotically pseudocontractive mappings in the intermediate sense, the class of the total asymptotically pseudo-contractive mappings, the class of quasi-asymptotically pseudo-contractive mappings in the intermediate sense as special cases, Theorem 3.1 improves the corresponding results in Zhou [7], Qin *et al.* [9], Chang [10] and Qin *et al.* [12].

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## Acknowledgements

The authors are grateful to the referees for their helpful and useful comments. The first author is supported by the Project of Shandong Province Higher Educational Science and Technology Program (grant No. J13LI51) and the Natural Science Foundation of Shandong Yingcai University (grant No. 12YCZDZR03). The second author is supported by the National Natural Science Foundation of China under grant (11071279) and the Project of Shandong Province Higher Educational Science and Technology Program (grant No. J13LI51).

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Wang, ZM., Su, Y. On the convergence of hybrid projection algorithms for total quasi-asymptotically pseudo-contractive mapping.
*J Inequal Appl* **2013**, 375 (2013). https://doi.org/10.1186/1029-242X-2013-375

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DOI: https://doi.org/10.1186/1029-242X-2013-375