# Strong convergence by the shrinking effect of two half-spaces and applications

- Muhammad Aqeel Ahmad Khan
^{1, 2}Email author and - Hafiz Fukhar-ud-din
^{2, 3}

**2013**:30

https://doi.org/10.1186/1687-1812-2013-30

© Khan and Fukhar-ud-din; licensee Springer 2013

**Received: **28 June 2012

**Accepted: **22 January 2013

**Published: **11 February 2013

## Abstract

This paper provides a new hybrid-type shrinking projection method for strong convergence results in a Hilbert space. The paper continues - by utilizing the proposed hybrid algorithm - with a strong convergence towards an approximate common element of the set of solutions of a finite family of generalized equilibrium problems and the set of common fixed points of two finite families of *k*-strict pseudo-contractions in a Hilbert space. Comparatively, our results improve and extend various results announced in the current literature.

**MSC:**47H05, 47H09, 49H05.

## Keywords

*δ*-inverse strongly monotone map

## 1 Introduction

*H*be a real Hilbert space equipped with the inner product $\u3008\cdot ,\cdot \u3009$ and the induced norm $\parallel \cdot \parallel $ and let

*C*be a nonempty closed convex subset of

*H*. A map $T:C\to C$ is said to be (i) Lipschitzian if $\parallel Tx-Ty\parallel \le L\parallel x-y\parallel $ for some $L>0$ and for all $x,y\in C$; (ii) nonexpansive if $\parallel Tx-Ty\parallel \le \parallel x-y\parallel $ for all $x,y\in C$; (iii)

*k*-strict pseudo-contraction if there exists a constant $k\in [0,1)$ such that

Denote by $F(T)$ the set of all fixed points of *T*.

*T*is nonexpansive if and only if

*T*is a 0-strict pseudo-contraction. Moreover,

*T*is said to be pseudo-contraction if $k=1$ and a strong pseudo-contraction if there exists a positive constant $\lambda \in (0,1)$ such that $T-\lambda I$ is a pseudo-contraction. Therefore, the class of

*k*-strict pseudo-contractions falls into the one between the classes of nonexpansive maps and pseudo-contractions. We further remark that the class of strong pseudo-contractions is independent of the class of

*k*-strict pseudo-contractions. If

*T*is a

*k*-strict pseudo-contraction, then

*T*satisfies the Lipschitz condition

Recent developments in fixed point theory reflect that an iterative construction of fixed points of nonlinear maps is an active area of research. Note that the class of *k*-strict pseudo-contractions is prominent among the classes of nonlinear maps in the current literature. Although strict pseudo-contractions have more powerful applications than nonexpansive maps for solving inverse problems, iterative algorithms for strict pseudo-contractions are far less developed than those for nonexpansive maps.

In fixed point theory, various iterative schemas for computing approximate fixed points of nonlinear maps have been proposed and analyzed. Such iterative schemas can be compared w.r.t. their efficiency and convergence properties. In many situations, strong convergence of an iterative algorithm of a nonlinear map is much more desirable than weak convergence. Obviously, a trivial choice for approximation is the classical Picard algorithm (*i.e.*, ${x}_{n+1}={T}^{n}({x}_{0})$). On the other hand, if we take $X=\mathbb{R}$, $C:=[0,1]$, $T(x):=1-x$ and ${x}_{0}:=0$, then the Picard algorithm alternates between 0 and 1 and does not converge to the fixed point $\frac{1}{2}$. The classical Mann algorithm, which prevails the Picard algorithm, exhibits weak convergence even in the setting of a Hilbert space. Moreover, Chidume and Mutangadura [2] constructed an example for a Lipschitz pseudo-contraction with a unique fixed point for which the Mann algorithm fails to converge. These facts indicate that the iterative schemas should be modified for the desirable strong convergence properties.

They showed that algorithm (1.2) converges strongly to ${P}_{F(T)}x$ under some appropriate conditions. Moreover, algorithm (1.2) is also known as the CQ-method for the Mann algorithm because, at each step, the Mann algorithm is used to construct the sets ${C}_{k}$ and ${Q}_{k}$ which are in turn used to construct the next iterate of ${x}_{k+1}$ and hence the name.

*et al.*[6] introduced another type of the hybrid method which guarantees strong convergence by the shrinking effect of a sequence of closed convex sets $\{{C}_{k+1}\}$. More precisely, their algorithm reads as follows:

Very recently, Dong *et al.* [7] proposed a shrinking projection method similar to (1.3) for nonexpansive maps in a Hilbert space setting. They, in fact, established a strong convergence result by the shrinking effect of one of the half-spaces as defined in (1.2), namely ${Q}_{k}$.

Note that if ${A}_{i}\equiv 0$ and ${f}_{i}(x,y)=f(x,y)$ for all $i\ge 1$, then the problem (1.4) reduces to the classical equilibrium problem $EP(f)$. That is, to find $x\in C$ such that $f(x,y)\ge 0$. Moreover, if ${f}_{i}(x,y)\equiv 0$ and ${A}_{i}=A$ for all $i\ge 1$, then the problem (1.4) reduces to the classical variational inequality problem $VI(C,A)$. That is, to find $x\in C$ such that $\u3008Ax,y-x\u3009\ge 0$.

Equilibrium problems provide a unified approach to address a variety of problems arising in various disciplines of science. The problem (1.4) is very general in the sense that it includes - as special cases - optimization problem, minimax problem, variational inequality problem, Nash equilibrium problem in noncooperative games and others; see, for instance, [8–10]. Combettes and Hirstoaga [9] introduced an iterative method to find an approximate solution of $EP(f)$. Since then, numerous algorithms have been analyzed to find a common element of the set of solutions of $EP(f)$ or $GEP(f)$ and the set of fixed points of a nonlinear map; see, for example, [10–16] and the references therein.

In 2009, Ceng *et al.* [17] introduced an implicit-type algorithm for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a strict pseudo-contraction in a real Hilbert space. Recently, Kim *et al.* [18] and Kangtunyakarn [19] approximated a common element of the set of solutions of two generalized equilibrium problems and the set of fixed points of a strict pseudo-contraction using the shrinking projection algorithm as defined in (1.3). Quite recently, Cholamjiak and Suantai [20] established a strong convergence result regarding a system of generalized equilibrium problems and a countable family of strict pseudo-contractions in a real Hilbert space.

Inspired and motivated by the work of Nakajo and Takahashi [5], Takahashi *et al.* [6], Dong *et al.* [7], Ceng *et al.* [17] and Cholamjiak and Suantai [20], we propose a hybrid method based on the shrinking effect of the two half-spaces, namely ${C}_{n}$ and ${Q}_{n}$, of the underlying Hilbert space *H*. The proposed algorithm approximates a common element of the set of solutions of a finite family of generalized equilibrium problems and the set of common fixed points of two finite families of *k*-strict pseudo-contractions. Our results refine and improve various results announced in the current literature.

## 2 Preliminaries

*H*be a real Hilbert space, a map ${P}_{C}:H\to C$ defined by

is known as a metric projection or a nearest point projection of *H* onto *C*. Moreover, ${P}_{C}$ is characterized as nonexpansive in Hilbert spaces. We know that for $x\in H$ and $z\in C$, $z={P}_{C}x$ is equivalent to $\u3008x-z,z-u\u3009\ge 0$ for all $u\in C$.

*δ*-inverse-strongly monotone map, if there exists $\delta >0$ such that

*A*is a

*δ*-inverse-strongly monotone map of

*C*onto

*H*, then

*A*is $(\frac{1}{\delta})$-Lipschitz continuous. Moreover, for $x,y\in C$ and $r>0$, we have

If $r\le 2\delta $, then $I-rA$ is a nonexpansive map from *C* onto *H*.

The following crucial results for a *k*-strict pseudo-contraction can be found in [[21], Proposition 2.1].

**Lemma 2.1** *Let* *C* *be a nonempty closed convex subset of a real Hilbert space* *H*. *If* $T:C\to H$ *is a* *k*-*strict pseudo*-*contraction*, *then the fixed point set* $F(T)$ *is closed and convex so that the projection* ${P}_{F(T)}$ *is well defined*.

**Lemma 2.2** *Let* *C* *be a nonempty closed convex subset of a real Hilbert space* *H* *and* $T:C\to C$ *be a* *k*-*strict pseudo*-*contraction*. *Then* $(I-T)$ *is demiclosed*, *that is*, *if* $\{{x}_{n}\}$ *is a sequence in* *C* *with* ${x}_{n}\rightharpoonup x$ *and* ${x}_{n}-T{x}_{n}\to 0$, *then* $x\in F(T)$.

The following lemma is well known in the context of a real Hilbert space.

**Lemma 2.3**

*Let*

*H*

*be a real Hilbert space*,

*then the following identity holds*:

*For solving the equilibrium problem*,

*let us assume that the bifunction*

*f*

*satisfies the following conditions*(

*cf*. [8]

*and*[9]):

- (A1)
$f(x,x)=0$

*for all*$x\in C$; - (A2)
*f**is monotone*,*i*.*e*., $f(x,y)+f(y,x)\le 0$*for all*$x,y\in C$; - (A3)
*for all*$x,y,z\in C$, ${lim\hspace{0.17em}sup}_{t\downarrow 0}f(tz+(1-t)x,y)\le f(x,y)$; - (A4)
$f(x,\cdot )$

*is convex and lower semicontinuous for all*$x\in C$.

A central result in the theory of equilibrium problems - for an approximate solution of $EP(f)$ - is the following result due to [8].

**Lemma 2.4**

*Let*

*C*

*be a closed convex subset of a real Hilbert space*

*H*,

*and let*$f:C\times C\to \mathbb{R}$

*be a bifunction satisfying*(A1)-(A4).

*For*$r>0$

*and*$x\in H$,

*there exists*$z\in C$

*such that*

*Moreover*,

*define a map*${V}_{r}:H\to C$

*by*

*for all*$x\in H$.

*Then the following hold*:

- (1)
$EP(f)$

*is closed and convex*; - (2)
${V}_{r}$

*is single*-*valued*; - (3)${V}_{r}$
*is a firmly nonexpansive*-*type map*,*i*.*e*.,${\parallel {V}_{r}x-{V}_{r}y\parallel}^{2}\le \u3008{V}_{r}x-{V}_{r}y,x-y\u3009\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}\phantom{\rule{0.5em}{0ex}}x,y\in H;$ - (4)
$F({V}_{r})=EP(f)$.

## 3 Main result

Let *H* be a real Hilbert space and *C* be its nonempty closed convex subset. Let ${T}_{i}(modN),{S}_{i}(modN):C\to C$ be two finite families of *k*-strict pseudo-contractions such that $k=max\{{k}_{i}:1\le i\le N\}$. Let ${f}_{i}(modN):C\times C\to \mathbb{R}$ be a finite family of bifunctions and ${A}_{i}(modN):C\to H$ be a finite family of *δ*-inverse-strongly monotone maps such that $\delta =max\{{\delta}_{i}:1\le i\le N\}$.

Throughout this section, we assume that the mod*N* function takes values in the indexing set $I=\{1,2,3,\dots ,N\}$ and the set of common fixed points of two finite families of *k*-strict pseudo-contractions ${\{{T}_{i}\}}_{i=1}^{N}$ and ${\{{S}_{i}\}}_{i=1}^{N}$ is nonempty, that is, $F=({\bigcap}_{i=1}^{N}F({T}_{i}))\cap ({\bigcap}_{i=1}^{N}F({S}_{i}))\ne \mathrm{\varnothing}$.

**Algorithm**

It is worth mentioning that at each iteration, algorithm (3.1) computes a projection onto intersection of two half-spaces. Since these half-spaces are closed and convex, so one can follow Dykstra’s algorithm [22] for the computation of such a projection.

The main result of this section is as follows.

**Theorem 3.1**

*Let*

*C*

*be a nonempty closed convex subset of a real Hilbert space*

*H*

*and let*${T}_{i}(modN),{S}_{i}(modN):C\to C$

*be two finite families of*

*k*-

*strict pseudo*-

*contractions*.

*Let*${f}_{i}(modN):C\times C\to \mathbb{R}$

*be a finite family of bifunctions satisfying*(A1)-(A4)

*and let*${A}_{i}(modN):C\to H$

*be a finite family of*

*δ*-

*inverse*-

*strongly monotone maps*.

*Let*$\{{r}_{n,i}\}\subset (0,\mathrm{\infty})$

*and*$\{{\alpha}_{n,i}\}$, $\{{\beta}_{n,i}\}$

*be two control sequences such that*

- (C1)
$0\le k<a\le {\alpha}_{n,i},{\beta}_{n,i}\le b<1$;

- (C2)
$0<r<{r}_{n,i}<s<2\delta $

*for all*$i\ge 1$.

*Assume that* $\mathcal{F}:=[{\bigcap}_{i=1}^{N}F({T}_{i})]\cap [{\bigcap}_{i=1}^{N}F({S}_{i})]\cap [{\bigcap}_{i=1}^{N}GEP({f}_{i},{A}_{i})]\ne \mathrm{\varnothing}$, *then the sequence* $\{{x}_{n}\}$ *generated by* (3.1) *converges strongly to* $x={P}_{\mathcal{F}}{x}_{1}$, *where* ${P}_{\mathcal{F}}$ *is the metric projection of* *H* *onto* ℱ.

*Proof*It follows from Lemma 2.4 that ${u}_{n,i}$ can be written as ${u}_{n,i}={V}_{{r}_{n,i}}({x}_{n}-{r}_{n,i}{A}_{i}{x}_{n})$ for all $n\ge 1$. Moreover, it follows from the nonexpansiveness of $I-{r}_{n,i}{A}_{i}$ that

where $p={\bigcap}_{i=1}^{N}{T}_{i}p={\bigcap}_{i=1}^{N}{S}_{i}p={\bigcap}_{i=1}^{N}{V}_{{r}_{n,i}}(p-{r}_{n,i}{A}_{i}p)$.

Hence, ${C}_{k+1}$ is convex, and consequently ${C}_{n+1}$ is convex for each $n\ge 1$.

*ω*-limit of $\{{x}_{n}\}$. Since $\{{x}_{n}\}$ is bounded, therefore $\omega ({x}_{n})\ne \mathrm{\varnothing}$. Let $q\in \omega ({x}_{n})$, there exists a subsequence $\{{x}_{{n}_{j}}\}$ of $\{{x}_{n}\}$ such that ${x}_{{n}_{j}}\rightharpoonup q$. It follows from the estimate (3.17) that ${u}_{{n}_{j},i}\rightharpoonup q$. We first show that $q\in GEP({f}_{1},{A}_{1})$, where ${f}_{1}={f}_{{n}_{j}}$ for some $j\ge 1$. From ${u}_{{n}_{j},i}={V}_{{r}_{{n}_{j},i}}(I-{r}_{{n}_{j},i}{A}_{i}){x}_{n}$, for all $n\ge 1$, we have

*A*that $\u3008{y}_{t}-{u}_{{n}_{j},i},{A}_{1}{y}_{t}-{A}_{1}{u}_{{n}_{j},i}\u3009\ge 0$. Hence, (A4) implies that

Therefore, we conclude that ${lim}_{j\to \mathrm{\infty}}{x}_{{n}_{j}}=q={P}_{\mathcal{F}}{x}_{1}$. From the arbitrariness of $\{{x}_{{n}_{j}}\}$, we get that ${lim}_{n\to \mathrm{\infty}}{x}_{n}={P}_{\mathcal{F}}{x}_{1}$. This completes the proof. □

In particular, if ${T}_{i}$ and ${S}_{i}$ - in algorithm (3.1) - are two finite families of nonexpansive maps, then the following result holds.

**Corollary 3.2**

*Let*

*C*

*be a nonempty closed convex subset of a real Hilbert space*

*H*

*and let*${T}_{i}(modN),{S}_{i}(modN):C\to C$

*be two finite families of nonexpansive maps*.

*Let*${f}_{i}(modN):C\times C\to \mathbb{R}$

*be a finite family of bifunctions satisfying*(A1)-(A4)

*and let*${A}_{i}(modN):C\to H$

*be a finite family of*

*δ*-

*inverse*-

*strongly monotone maps*.

*Let*$\{{r}_{n,i}\}\subset (0,\mathrm{\infty})$

*and*$\{{\alpha}_{n,i}\}$, $\{{\beta}_{n,i}\}$

*be two control sequences such that*

- (C1)
$0\le k<a\le {\alpha}_{n,i},{\beta}_{n,i}\le b<1$;

- (C2)
$0<r<{r}_{n,i}<s<2\delta $

*for all*$i\ge 1$.

*Assume that* $\mathcal{F}:=[{\bigcap}_{i=1}^{N}F({T}_{i})]\cap [{\bigcap}_{i=1}^{N}F({S}_{i})]\cap [{\bigcap}_{i=1}^{N}GEP({f}_{i},{A}_{i})]\ne \mathrm{\varnothing}$, *then the sequence* $\{{x}_{n}\}$ *generated by* (3.1) *converges strongly to* $x={P}_{\mathcal{F}}{x}_{1}$, *where* ${P}_{\mathcal{F}}$ *is the metric projection of* *H* *onto* ℱ.

In order to address variational inequality problems coupled with the fixed point problems, we prove the following result with a slight modification of algorithm (3.1).

**Theorem 3.3**

*Let*

*C*

*be a nonempty closed convex subset of a real Hilbert space*

*H*

*and let*${T}_{i}(modN),{S}_{i}(modN):C\to C$

*be two finite families of*

*k*-

*strict pseudo*-

*contractions*.

*Let*${f}_{i}(modN):C\times C\to \mathbb{R}$

*be a finite family of bifunctions satisfying*(A1)-(A4)

*and let*${A}_{i}(modN):C\to H$

*be a finite family of*

*δ*-

*inverse*-

*strongly monotone maps*.

*Let*$\{{r}_{n,i}\}\subset (0,\mathrm{\infty})$

*and*$\{{\alpha}_{n,i}\}$, $\{{\beta}_{n,i}\}$

*be two control sequences such that*

- (C1)
$0\le k<a\le {\alpha}_{n,i},{\beta}_{n,i}\le b<1$;

- (C2)
$0<r<{r}_{n,i}<s<2\delta $

*for all*$i\ge 1$.

*Assume that*$\mathcal{F}:=[{\bigcap}_{i=1}^{N}F({T}_{i})]\cap [{\bigcap}_{i=1}^{N}F({S}_{i})]\cap [{\bigcap}_{i=1}^{N}VI(C,{A}_{i})]\ne \mathrm{\varnothing}$,

*then the sequence*$\{{x}_{n}\}$

*generated by*

*converges strongly to* $x={P}_{\mathcal{F}}{x}_{1}$, *where* ${P}_{\mathcal{F}}$ *is the metric projection of* *H* *onto* ℱ.

*Proof*Set ${f}_{i}(x,y)\equiv 0$ for each $i\ge 1$, then

This implies that ${h}_{n,i}={u}_{n,i}:={P}_{C}({x}_{n}-{r}_{n,i}{A}_{i}{x}_{n})$. The desired result then follows from Theorem 3.1 immediately. □

As an application of Theorem 3.1 - by substituting ${A}_{i}\equiv 0$ for all $i\ge 1$ in algorithm (3.1) - we have the following result for a finite family of equilibrium problems.

**Theorem 3.4** *Let* *C* *be a nonempty closed convex subset of a real Hilbert space* *H* *and let* ${T}_{i}(modN),{S}_{i}(modN):C\to C$ *be two finite families of* *k*-*strict pseudo*-*contractions*. *Let* ${f}_{i}(modN):C\times C\to \mathbb{R}$ *be a finite family of bifunctions satisfying* (A1)-(A4). *Let* $\{{\alpha}_{n,i}\}$, $\{{\beta}_{n,i}\}$ *be two control sequences such that*

(C1) $0\le k<a\le {\alpha}_{n,i},{\beta}_{n,i}\le b<1$.

*Assume that*$\mathcal{F}:=[{\bigcap}_{i=1}^{N}F({T}_{i})]\cap [{\bigcap}_{i=1}^{N}F({S}_{i})]\cap [{\bigcap}_{i=1}^{N}EP({f}_{i})]\ne \mathrm{\varnothing}$,

*then the sequence*$\{{x}_{n}\}$

*generated by*

*converges strongly to* $x={P}_{\mathcal{F}}{x}_{1}$, *where* ${P}_{\mathcal{F}}$ *is the metric projection of* *H* *onto* ℱ.

**Remark 3.5**Additionally - in Theorem 3.4 - if we set ${\beta}_{n,i}\equiv 0$, then Theorem 3.4 sets analogue [[17], Theorem 3.3] in the following aspects:

- (i)
from a single

*k*-strict pseudo-contraction to a finite family of maps; - (ii)
from an equilibrium problem to a finite family of generalized equilibrium problems.

## Declarations

### Acknowledgements

We wish to thanks the referees for careful reading and helpful comments which led the manuscript to the present form. The author M.A.A. Khan gratefully acknowledges the support from German Science Foundation (DFG Project KO 1737/5-1) and Higher Education Commission of Pakistan. The author H. Fukhar-ud-din is grateful to King Fahd University of Petroleum & Minerals for support during this research.

## Authors’ Affiliations

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