## Abstract

Let *E* be a real *q*-uniformly smooth Banach space, which is also uniformly convex (for example, {L}_{p} or {\ell}_{p} spaces, 1<p<\mathrm{\infty}), and *C* be a nonempty bounded closed convex subset of *E*. Let T:C\to C be a *k*-strictly asymptotically pseudocontractive map with a nonempty fixed point set. A hybrid algorithm is constructed to approximate fixed points of such maps. Furthermore, strong convergence of the proposed algorithm is established.

## 1 Introduction

Let *E* be a real Banach space and {E}^{\ast} be the dual of *E*. We denote the value of {x}^{\ast}\in {E}^{\ast} at x\in E by \u3008x,{x}^{\ast}\u3009. The normalized duality mapping *J* from *E* to {2}^{{E}^{\ast}} is defined by

for all x\in E. It is known that a Banach space *E* is smooth if and only if the normalized duality mapping *J* is single valued. Some properties of the duality mapping have been given in [1, 2].

Let *C* be a nonempty subset of *E*. The mapping T:C\to C is called *nonexpansive* if

for all x,y\in C. Also, *T* is called *uniformly* *L*-*Lipschitz* if there exists a constant L>0 such that

for all x,y\in C and each n\ge 1. The mapping T:C\to C is called *k*-*strictly asymptotically pseudocontractive* if there exist a sequence \{{k}_{n}\} in [1,\mathrm{\infty}) with {lim}_{n\to \mathrm{\infty}}{k}_{n}=1 and a constant k\in [0,1), and for any x,y\in C, there exists j(x-y)\in J(x-y) such that

for each n\ge 1. If *I* denotes the identity operator, then (1.1) can be written in the form

The class of *k*-strictly asymptotically pseudocontractive mappings was first introduced in Hilbert spaces by Qihou [3]. In Hilbert spaces, *j* is the identity and it is shown [4] that (1.1) (and hence (1.2)) is equivalent to the inequality

which is the inequality considered by Qihou [3]. In the same paper, the author proved strong convergence of the modified Mann iteration processes for *k*-strictly asymptotically pseudocontractive mappings in Hilbert spaces. The modified Mann iteration scheme was introduced by Schu [5, 6] and has been used by several authors (see, for example, [7–12]). In [13] Osilike extended Qihou’s result from Hilbert spaces to much more general real *q*-uniformly smooth Banach spaces, 1<q<\mathrm{\infty}.

The classes of nonexpansive and asymptotically nonexpansive mappings are important classes of mappings because they have applications to solutions of differential equations which have been studied by several authors (see, *e.g.*, [14–16] and references contained therein). It would be of interest to study the class of *k*-strictly asymptotically pseudocontractive mappings in view of the fact that it is closely related to the above two classes.

On the other hand, using the metric projection, Matsushita and Takahashi [17] introduced the following iterative algorithm for nonexpansive mappings: {x}_{0}=x\in C and

where \overline{co}D denotes the convex closure of the set *D*, *J* is the normalized duality mapping, \{{t}_{n}\} is a sequence in (0,1) with {t}_{n}\to 0, and {P}_{{C}_{n}\cap {D}_{n}} is the metric projection from *E* onto {C}_{n}\cap {D}_{n}. Then, they proved that \{{x}_{n}\} generated by (1.3) converges strongly to a fixed point of the mapping *T*.

In this paper, motivated by these facts, we introduce the following iterative algorithm for finding fixed points of a *k*-strictly asymptotically pseudocontractive mapping *T* in a uniformly convex and *q*-uniformly smooth Banach space: {x}_{1}=x\in C, {C}_{0}={D}_{0}=C and

where \overline{co}D denotes the convex closure of the set *D*, *J* is the normalized duality mapping, \{{t}_{n}\} is a sequence in (0,1) with {t}_{n}\to 0, and {P}_{{C}_{n}\cap {D}_{n}} is the metric projection from *E* onto {C}_{n}\cap {D}_{n}.

The purpose of this paper is to establish a strong convergence theorem of the iterative algorithm (1.4) for *k*-strictly asymptotically pseudocontractive mappings in a uniformly convex and *q*-uniformly smooth Banach space.

## 2 Preliminaries

The *modulus of smoothness* of a Banach space *E* is the function {\rho}_{E}:[0,\mathrm{\infty})\to [0,\mathrm{\infty}) defined by

*E* is *uniformly smooth* if and only if {lim}_{t\to {0}^{+}}{\rho}_{E}(t)/t=0. Let q>1. The Banach space *E* is said to be *q*-*uniformly smooth* if there exists a constant c>0 such that {\rho}_{E}(t)\le c{t}^{q}. Hilbert spaces, {L}_{p} (or {\ell}_{p}) spaces, 1<p<\mathrm{\infty}, and the Sobolev spaces, {W}_{m}^{p}, 1<p<\mathrm{\infty}, are *q*-uniformly smooth.

When \{{x}_{n}\} is a sequence in *E*, we denote strong convergence of \{{x}_{n}\} to x\in E by {x}_{n}\to x and weak convergence by {x}_{n}\rightharpoonup x. The Banach space *E* is said to have the Kadec-Klee property if for every sequence \{{x}_{n}\} in *E*, {x}_{n}\rightharpoonup x and \parallel {x}_{n}\parallel \to \parallel x\parallel imply that {x}_{n}\to x. Every uniformly convex Banach space has the Kadec-Klee property [1].

Let *C* be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space *E*. Then for any x\in E, there exists a unique point {x}_{0}\in C such that

The mapping {P}_{C}:E\to C defined by {P}_{C}x={x}_{0} is called the *metric projection* from *E* onto *C*. Let x\in E and u\in C. Then it is known that u={P}_{C}x if and only if

for all y\in C ( see [1, 18]).

In the sequel, we need the following results.

**Proposition 2.1** (See [19])

*Let* *C* *be a bounded closed convex subset of a uniformly convex Banach space* *E*. *Then there exists a strictly increasing convex continuous function* \gamma :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) *with* \gamma (0)=0 *depending only on the diameter of* *C* *such that*

*holds for any nonexpansive mapping* T:C\to E, *any elements* {x}_{1},\dots ,{x}_{n} *in* *C*, *and any numbers* {\lambda}_{1},\dots ,{\lambda}_{n}\ge 0 *with* {\lambda}_{1}+\cdots +{\lambda}_{n}=1.

**Corollary 2.2** [[20], Corollary 1.2]

*Under the same suppositions as in Proposition * 2.1, *there exists a strictly increasing convex continuous function* \gamma :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) *with* \gamma (0)=0 *depending only on the diameter of* *C* *such that*

*holds for any nonexpansive mapping* T:C\to E, *any elements* {x}_{1},\dots ,{x}_{n} *in* *C*, *and any numbers* {\lambda}_{1},\dots ,{\lambda}_{n}\ge 0 *with* {\lambda}_{1}+\cdots +{\lambda}_{n}=1. (*Note that* *γ* *does not depend on* *T*.)

In order to utilize Corollary 2.2 for *k*-strictly asymptotically pseudocontractive mappings, we need the following lemmas.

**Lemma 2.3** [4]

*Let* *E* *be a real Banach space*, *C* *be a nonempty subset of* *E*, *and* T:C\to C *be a* *k*-*strictly asymptotically pseudocontractive mapping*. *Then* *T* *is uniformly* *L*-*Lipschitzian*.

**Lemma 2.4** [[21], Lemma 3.1]

*Let* *E* *be a real* *q*-*uniformly smooth Banach space and* *C* *be a nonempty convex subset of* *E*. *Let* T:C\to C *be a* *k*-*strictly asymptotically pseudocontractive map*, *and let* \{{\alpha}_{n}\} *be a real sequence in* [0,1]. *Define* {S}_{n}:C\to C *by* {S}_{n}x:=(1-{\alpha}_{n})x+{\alpha}_{n}{T}^{n}x *for all* x\in C. *Then for all* x,y\in C, *we have*

*where* *L* *is the uniformly Lipschitzian constant of* *T* *and* {c}_{q}>0 *is the constant which appeared in* [[21], *Theorem * 2.1].

Let \beta =min\{1,{[\frac{q}{2}(1-k){(1+L)}^{-(q-2)}/{c}_{q}]}^{1/(q-1)}\} and choose \alpha \in (0,\beta ). Set {\alpha}_{n}=\alpha for all n\ge 1 in Lemma 2.4 and observe that {\parallel {S}_{n}x-{S}_{n}y\parallel}^{q}\le (1+\frac{q}{2}\alpha ({k}_{n}-1)){\parallel x-y\parallel}^{q}. Thus,

for all x,y\in C and each n\ge 1.

**Theorem 2.5** [[21], Theorem 3.1]

*Let* *E* *be a real* *q*-*uniformly smooth Banach space which is also uniformly convex*. *Let* *C* *be a nonempty closed convex subset of* *E* *and* T:C\to C *be a k*-*strictly asymptotically pseudocontractive mapping with a nonempty fixed point set*. *Then* (I-T) *is demiclosed at zero*, *i*.*e*., *if* {x}_{n}\rightharpoonup x *and* {x}_{n}-T{x}_{n}\to 0, *then* x\in F(T), *where* F(T) *is the set of all fixed points of* *T*.

## 3 Strong convergence theorem

In this section, we study the iterative algorithm (1.4) for finding fixed points of *k*-strictly asymptotically pseudocontractive mappings in a uniformly convex and *q*-uniformly smooth Banach space. We first prove that the sequence \{{x}_{n}\} generated by (1.4) is well defined. Then, we prove that \{{x}_{n}\} converges strongly to {P}_{F(T)}x, where {P}_{F(T)} is the metric projection from *E* onto F(T).

**Lemma 3.1** *Let* *C* *be a nonempty closed convex subset of a reflexive*, *strictly convex*, *and smooth Banach space* *E*, *and let* T:C\to C *be a mapping*. *If* F(T)\ne \mathrm{\varnothing}, *then the sequence* \{{x}_{n}\} *generated by* (1.4) *is well defined*.

*Proof* It is easy to check that {C}_{n}\cap {D}_{n} is closed and convex and F(T)\subset {C}_{n} for each n\in \mathbb{N}. Moreover, {D}_{1}=C and so F(T)\subset {C}_{1}\cap {D}_{1}. Suppose F(T)\subset {C}_{k}\cap {D}_{k} for k\in \mathbb{N}. Then there exists a unique element {x}_{k+1}\in {C}_{k}\cap {D}_{k} such that {x}_{k+1}={P}_{{C}_{k}\cap {D}_{k}}x. If u\in F(T), then it follows from (2.1) that

which implies u\in {D}_{k+1}. Therefore, F(T)\subset {C}_{k+1}\cap {D}_{k+1}. By the mathematical induction, we obtain that F(T)\subset {C}_{n}\cap {D}_{n} for all n\in \mathbb{N}. Therefore, \{{x}_{n}\} is well defined. □

In order to prove our main result, the following lemma is needed.

**Lemma 3.2** *Let* *C* *be a nonempty bounded closed convex subset of a real* *q*-*uniformly smooth and uniformly convex Banach space* *E*. *Let* T:C\to C *be a* *k*-*strictly asymptotically pseudocontractive mapping with* \{{k}_{n}\} *such that* F(T)\ne \mathrm{\varnothing}. *Let* \{{x}_{n}\} *be the sequence generated by* (1.4), *then for any* j\in \mathbb{N},

*Proof* Fix j\in \mathbb{N} and put m=n-j. Since {x}_{n}={P}_{{C}_{n-1}\cap {D}_{n-1}}x, we have {x}_{n}\in {C}_{n-1}\subseteq \cdots \subseteq {C}_{m}. Since {t}_{m}>0, there exist {y}_{1},\dots ,{y}_{N}\in C and {\lambda}_{1},\dots ,{\lambda}_{N}\ge 0 with {\lambda}_{1}+\cdots +{\lambda}_{N}=1 such that

and \parallel {y}_{i}-{T}^{m}{y}_{i}\parallel \le {t}_{m}\parallel {x}_{m}-{T}^{m}{x}_{m}\parallel for all i\in \{1,\dots ,N\}. It follows from Lemma 2.3 that *T* is uniformly *L*-Lipschitzian. Put M={sup}_{x\in C}\parallel x\parallel, u={P}_{F(T)}x and {r}_{0}={sup}_{n\ge 1}(1+L)\parallel {x}_{n}-u\parallel. Thus,

for all i\in \{1,\dots ,N\}. Define {H}_{m}:C\to E by

for all x\in C, where {a}_{m}={(1+\frac{q}{2}\alpha ({k}_{m}-1))}^{1/q} and {S}_{m} is as in (2.2). It follows from (2.2) that {H}_{m} is nonexpansive. Using (3.2) and the fact that \parallel {y}_{i}-{S}_{m}{y}_{i}\parallel =\alpha \parallel {y}_{i}-{T}^{m}{y}_{i}\parallel, we have

for all i\in \{1,\dots ,N\}. It follows from Corollary 2.2, (3.1), and (3.3) that

Since and , it follows from the last inequality that {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-{H}_{m}{x}_{n}\parallel =0. Thus, {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-{S}_{m}{x}_{n}\parallel =0 and so {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-{T}^{m}{x}_{n}\parallel =0. This completes the proof. □

**Theorem 3.3** *Let* *C* *be a nonempty bounded closed convex subset of a real* *q*-*uniformly smooth and uniformly convex Banach space* *E*. *Let* T:C\to C *be a k*-*strictly asymptotically pseudocontractive mapping with* \{{k}_{n}\} *such that* F(T)\ne \mathrm{\varnothing}. *Let* \{{x}_{n}\} *be the sequence generated by* (1.4). *Then* \{{x}_{n}\} *converges strongly to the element* {P}_{F(T)}x *of* F(T), *where* {P}_{F(T)} *is the metric projection from* *E* *onto* F(T).

*Proof* Put u={P}_{F(T)}x. Since F(T)\subset {C}_{n}\cap {D}_{n} and {x}_{n+1}={P}_{{C}_{n}\cap {D}_{n}}x, we have that

for all n\in \mathbb{N}. By Lemma 3.2, we have

Since \{{x}_{n}\} is bounded, there exists \{{x}_{{n}_{i}}\}\subset \{{x}_{n}\} such that {x}_{{n}_{i}}\rightharpoonup v. It follows from Theorem 2.5 (demiclosedness of *T*) that v\in F(T). From the weakly lower semicontinuity of norm and (3.4), we obtain

This together with the uniqueness of {P}_{F(T)}x implies u=v, and hence {x}_{{n}_{i}}\rightharpoonup u. Therefore, we obtain {x}_{n}\rightharpoonup u. Furthermore, we have that

Since *E* is uniformly convex, using the Kadec-Klee property, we have that x-{x}_{n}\to x-u. It follows that {x}_{n}\to u. This completes the proof. □

## References

- 1.
Agarwal RP, Regan DO, Sahu DR: Convexity, smoothness and duality mappings. In

*Fixed Point Theory for Lipschitzian-type Mappings with Applications*. Springer, New York; 2009:49–115. - 2.
Cioranescu I:

*Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems*. Kluwer Academic, Dordrecht; 1990. - 3.
Qihou L: Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings.

*Nonlinear Anal.*1996, 26(11):1835–1842. 10.1016/0362-546X(94)00351-H - 4.
Osilike MO, Aniagbosor SC, Akuchu BG: Fixed points of asymptotically demicontractive mappings in arbitrary Banach spaces.

*Panam. Math. J.*2002, 12(2):77–88. - 5.
Schu J: Iterative construction of fixed point of asymptotically nonexpansive mappings.

*J. Math. Anal. Appl.*1991, 158: 407–413. 10.1016/0022-247X(91)90245-U - 6.
Schu J: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings.

*Bull. Aust. Math. Soc.*1991, 43: 153–159. 10.1017/S0004972700028884 - 7.
Ofoedu EU: Strong convergence theorem for uniformly

*L*-Lipschitzian asymptotically pseudocontractive mapping in real Banach space.*J. Math. Anal. Appl.*2006, 321(2):722–728. 10.1016/j.jmaa.2005.08.076 - 8.
Rafiq A: On iterations for families of asymptotically pseudocontractive mappings.

*Appl. Math. Lett.*2011, 24: 33–38. 10.1016/j.aml.2010.08.005 - 9.
Osilike MO, Igbokwe DI: Convergence theorems for asymptotically pseudocontractive maps.

*Bull. Korean Math. Soc.*2002, 39(3):389–399. - 10.
Tang Y, Liu L: Note on some results for asymptotically pseudocontractive mappings and asymptotically nonexpansive mappings.

*Fixed Point Theory Appl.*2006., 2006: Article ID 24978 - 11.
Tan KK, Xu HK: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process.

*J. Math. Anal. Appl.*1993, 178: 301–308. 10.1006/jmaa.1993.1309 - 12.
Dehghan H: Approximating fixed points of asymptotically nonexpansive mappings in Banach spaces by metric projections.

*Appl. Math. Lett.*2011, 24: 1584–1587. 10.1016/j.aml.2011.03.051 - 13.
Osilike MO: Iterative approximations of fixed points of asymptotically demicontractive mappings.

*Indian J. Pure Appl. Math.*1998, 29(12):1291–1300. - 14.
Chidume CE, Zegeye H: Strong convergence theorems for common fixed points of uniformly

*L*-Lipschitzian pseudocontractive semi-groups.*Appl. Anal.*2007, 86(3):353–366. 10.1080/00036810601156730 - 15.
Shioji N, Takahashi W: Strong convergence theorems for asymptotically nonexpansive semi-groups in Hilbert spaces.

*Nonlinear Anal.*1998, 34: 87–99. 10.1016/S0362-546X(97)00682-2 - 16.
Suzuki T: On strong convergence to common fixed points of nonexpansive semi-groups in Banach spaces.

*Proc. Am. Math. Soc.*2003, 131: 2133–2136. 10.1090/S0002-9939-02-06844-2 - 17.
Matsushita S, Takahashi W: Approximating fixed points of nonexpansive mappings in a Banach space by metric projections.

*Appl. Math. Comput.*2008, 196: 422–425. 10.1016/j.amc.2007.06.006 - 18.
Alber YI: Metric and generalized projection operators in Banach spaces: properties and applications. Lecture Notes in Pure and Applied Mathematics. In

*Theory and Applications of Nonlinear Operators of Accretive and Monotone Type*. Dekker, New York; 1996:15–50. - 19.
Bruck RE: On the convex approximation property and the asymptotic behaviour of nonlinear contractions in Banach spaces.

*Isr. J. Math.*1981, 38: 304–314. 10.1007/BF02762776 - 20.
Kruppel M: On an inequality for nonexpansive mappings in uniformly convex Banach spaces.

*Rostock. Math. Kolloqu.*1997, 51: 25–32. - 21.
Osilike MO, Udomene A, Igbokwe DI, Akuchu BG: Demiclosedness principle and convergence theorems for

*k*-strictly asymptotically pseudocontractive maps.*J. Math. Anal. Appl.*2007, 326(2):1334–1345. 10.1016/j.jmaa.2005.12.052

## Acknowledgements

The authors are grateful to the reviewers for their useful comments.

## Author information

### Affiliations

### Corresponding author

## Additional information

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

## Rights and permissions

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## About this article

### Cite this article

Dehghan, H., Shahzad, N. Strong convergence of a CQ method for *k*-strictly asymptotically pseudocontractive mappings.
*Fixed Point Theory Appl* **2012, **208 (2012). https://doi.org/10.1186/1687-1812-2012-208

Received:

Accepted:

Published:

DOI: https://doi.org/10.1186/1687-1812-2012-208

### Keywords

- strong convergence
- CQ method
*k*-strictly asymptotically pseudocontractive mapping