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Strong convergence by the shrinking effect of two halfspaces and applications
Fixed Point Theory and Applications volume 2013, Article number: 30 (2013)
Abstract
This paper provides a new hybridtype 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 kstrict pseudocontractions in a Hilbert space. Comparatively, our results improve and extend various results announced in the current literature.
MSC:47H05, 47H09, 49H05.
1 Introduction
Let 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 TxTy\parallel \le L\parallel xy\parallel $ for some $L>0$ and for all $x,y\in C$; (ii) nonexpansive if $\parallel TxTy\parallel \le \parallel xy\parallel $ for all $x,y\in C$; (iii) kstrict pseudocontraction if there exists a constant $k\in [0,1)$ such that
Denote by $F(T)$ the set of all fixed points of T.
In 1967, Browder and Petryshyn [1] introduced the class of strict pseudocontractions as an important generalization of the class of nonexpansive maps. Clearly, T is nonexpansive if and only if T is a 0strict pseudocontraction. Moreover, T is said to be pseudocontraction if $k=1$ and a strong pseudocontraction if there exists a positive constant $\lambda \in (0,1)$ such that $T\lambda I$ is a pseudocontraction. Therefore, the class of kstrict pseudocontractions falls into the one between the classes of nonexpansive maps and pseudocontractions. We further remark that the class of strong pseudocontractions is independent of the class of kstrict pseudocontractions. If T is a kstrict pseudocontraction, 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 kstrict pseudocontractions is prominent among the classes of nonlinear maps in the current literature. Although strict pseudocontractions have more powerful applications than nonexpansive maps for solving inverse problems, iterative algorithms for strict pseudocontractions 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 $X=\mathbb{R}$, $C:=[0,1]$, $T(x):=1x$ 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 pseudocontraction 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.
The hybrid projection algorithm in mathematical programming was introduced by Haugazeau [3] in 1968. Later, many researchers studied the hybrid projection method and its various modifications for strong convergence results. In 2000, Solodov and Svaiter [4] established a strong convergence result for finding zeros of maximal monotone operators. They proposed and analyzed the following algorithm:
In fact, algorithm (1.1) enforces strong convergence by combining the classical proximal point algorithm with simple projection steps onto intersection of two halfspaces ${H}_{k}$ and ${W}_{k}$ containing the solution set. Inspired by the seminal work of Solodov and Svaiter [4], Nakajo and Takahashi [5] proposed the following hybrid method for nonexpansive maps in Hilbert spaces:
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 CQmethod 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.
In 2008, Takahashi 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 halfspaces as defined in (1.2), namely ${Q}_{k}$.
For $i=1,2,3,\dots ,N$ , let ${A}_{i}:C\to H$ be a finite family of nonlinear maps and ${f}_{i}:C\times C\to \mathbb{R}$ (the set of reals) be a finite family of bifunctions. A generalized equilibrium problem is to find the set
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 $\u3008Ax,yx\u3009\ge 0$.
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 implicittype algorithm for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a strict pseudocontraction 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 pseudocontraction 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 pseudocontractions 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 halfspaces, 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 kstrict pseudocontractions. Our results refine and improve various results announced in the current literature.
2 Preliminaries
Throughout the paper, we write ${x}_{n}\to x$ (resp. ${x}_{n}\rightharpoonup x$) to indicate strong convergence (resp. weak convergence). Let 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 $\u3008xz,zu\u3009\ge 0$ for all $u\in C$.
Let $A:C\to H$ be a δinversestrongly monotone map, if there exists $\delta >0$ such that
If A is a δinversestrongly 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 $IrA$ is a nonexpansive map from C onto H.
The following crucial results for a kstrict pseudocontraction 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 kstrict pseudocontraction, 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 kstrict pseudocontraction. Then $(IT)$ 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 $x,y,z\in C$, ${lim\hspace{0.17em}sup}_{t\downarrow 0}f(tz+(1t)x,y)\le f(x,y)$;

(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 singlevalued;

(3)
${V}_{r}$ is a firmly nonexpansivetype map, i.e.,
$${\parallel {V}_{r}x{V}_{r}y\parallel}^{2}\le \u3008{V}_{r}x{V}_{r}y,xy\u3009\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}\phantom{\rule{0.5em}{0ex}}x,y\in H;$$ 
(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 kstrict pseudocontractions 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 δinversestrongly monotone maps such that $\delta =max\{{\delta}_{i}:1\le i\le N\}$.
Throughout this section, we assume that the modN 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 kstrict pseudocontractions ${\{{T}_{i}\}}_{i=1}^{N}$ 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
Our hybrid algorithm reads as follows:
It is worth mentioning that at each iteration, algorithm (3.1) computes a projection onto intersection of two halfspaces. Since these halfspaces 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 kstrict pseudocontractions. 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 δinversestrongly 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 $p={\bigcap}_{i=1}^{N}{T}_{i}p={\bigcap}_{i=1}^{N}{S}_{i}p={\bigcap}_{i=1}^{N}{V}_{{r}_{n,i}}(p{r}_{n,i}{A}_{i}p)$.
We proceed to show that algorithm (3.1) is well defined. It is obvious from the definitions of respective sets that ${C}_{n+1}$ is closed and ${Q}_{n+1}$ is closed and convex. Now, we show that ${C}_{n+1}$ is convex. Since ${C}_{1}=C$ is convex, we assume that ${C}_{k}$ is convex for some $k\ge 2$. For any $z\in {C}_{k}$, the inequality $\parallel {y}_{n,i}z\parallel \le \parallel {x}_{n}z\parallel $ is equivalent to
Hence, ${C}_{k+1}$ is convex, and consequently ${C}_{n+1}$ is convex for each $n\ge 1$.
Next, we show by induction that $\mathcal{F}\subset {C}_{n+1}\cap {Q}_{n+1}$ for all $n\ge 1$. Obviously, $\mathcal{F}\subset {C}_{1}\cap {Q}_{1}=C$. Let $p\in \mathcal{F}$. From (2.1), (3.1) and Lemma 2.3, we have the following estimate:
Since ${\alpha}_{n,i}k>0$ (by (C1)), therefore the above estimate (3.2) yields
Moreover,
Reasoning as above and utilizing (3.3), the estimate (3.4) implies that
This implies that $p\in {C}_{n+1}$ for all $n\ge 1$. It suffices to show that $p\in {Q}_{n+1}$ for all $n\ge 1$. We prove this by induction. Note that $\mathcal{F}\subset {Q}_{1}=C$ is obvious. Assume that $\mathcal{F}\subset {Q}_{k}$ also $\mathcal{F}\subset {C}_{k}\cap {Q}_{k}$ for some $k\ge 2$. This implies ${x}_{k}$ is a projection of ${x}_{1}$ onto ${C}_{k}\cap {Q}_{k}$, and consequently we have
Since $\mathcal{F}\subset {C}_{k}\cap {Q}_{k}$, we have
Hence, $p\in {Q}_{k+1}$, and consequently $\mathcal{F}\subset {C}_{n+1}\cap {Q}_{n+1}$ for all $n\ge 1$. Since ℱ is now closed and convex, so it follows from Lemma 2.1 that ${P}_{\mathcal{F}}$ is well defined. Note that ${x}_{n+1}={P}_{{C}_{n+1}\cap {Q}_{n+1}}{x}_{1}$, therefore $\parallel {x}_{n+1}{x}_{1}\parallel \le \parallel p{x}_{1}\parallel $ for all $p\in \mathcal{F}\subset {C}_{n+1}\cap {Q}_{n+1}$. In particular, we have $\parallel {x}_{n+1}{x}_{1}\parallel \le \parallel {P}_{\mathcal{F}}{x}_{1}{x}_{1}\parallel $. This implies that $\{{x}_{n}\}$ is bounded. On the other hand, ${x}_{n}={P}_{{C}_{n}\cap {Q}_{n}}{x}_{1}$ and ${x}_{n+1}\in {C}_{n+1}\cap {Q}_{n+1}\subset {Q}_{n+1}$, we have
That is, the sequence $\{\parallel {x}_{n}{x}_{1}\parallel \}$ is nondecreasing. This implies ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}{x}_{1}\parallel $ exists. Note that
Taking lim sup on both sides of the above estimate, we have ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\parallel {x}_{n+1}{x}_{n}\parallel}^{2}=0$. That is,
Since ${x}_{n+1}\in {C}_{n+1}$, we have $\parallel {y}_{n,i}{x}_{n+1}\parallel \le \parallel {x}_{n}{x}_{n+1}\parallel $. This implies
Moreover, it follows from (3.6), (3.7) and the following inequality:
that
Note that
Since ${\parallel {y}_{n,i}p\parallel}^{2}\le {\parallel {z}_{n,i}p\parallel}^{2}\le {\parallel {u}_{n,i}p\parallel}^{2}$, therefore utilizing (3.9), we get
Rearranging the terms in the above estimate and utilizing (C2), we have
Hence, (3.8) implies that
Since $k<a\le {\beta}_{n,i}\le b<1$, then the following variant of the estimate (3.4) implies that
Again, utilizing (3.8), we have
Observe that $\parallel {y}_{n,i}{z}_{n,i}\parallel =(1{\beta}_{n,i})\parallel {S}_{i}{z}_{n,i}{z}_{n,i}\parallel $. It follows from the fact that $k<a\le {\beta}_{n,i}\le b<1$ and (3.11) that
Moreover, $\parallel {x}_{n}{z}_{n,i}\parallel \le \parallel {x}_{n}{y}_{n,i}\parallel +\parallel {y}_{n,i}{z}_{n,i}\parallel $. Letting $n\to \mathrm{\infty}$ on both sides and utilizing (3.8) and (3.12), we have
Reasoning as above, that is, $k<a\le {\alpha}_{n,i}\le b<1$, we consider the following variant of the estimate (3.2):
Hence, we conclude from the above estimate and (3.13) that
On the other hand, $\parallel {z}_{n,i}{u}_{n,i}\parallel =(1{\alpha}_{n,i})\parallel {T}_{i}{u}_{n,i}{u}_{n,i}\parallel $. Making use of the fact that $k<a\le {\alpha}_{n,i}\le b<1$ and (3.14), we get
Moreover, we conclude from the estimates (3.12) and (3.15) that
As a direct consequence of the estimates (3.8) and (3.16), we have
Next, we show that $\omega ({x}_{n})\subset \mathcal{F}$, where $\omega ({x}_{n})$ is the set of all weak ωlimit 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
From (A2), we have
Let ${y}_{t}=ty+(1t)q$ for $0<t<1$ and $y\in C$. Since $q\in C$, this implies that ${y}_{t}\in C$. It follows from the estimate (3.18) that
Since ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{{n}_{j}}{u}_{{n}_{j},i}\parallel =0$, therefore ${lim}_{n\to \mathrm{\infty}}\parallel {A}_{1}{x}_{{n}_{j}}{A}_{1}{u}_{{n}_{j},i}\parallel =0$. Moreover, it follows from the monotonicity of 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
Using (3.20), (A1) and (A4), the following estimate:
implies that
Letting $t\to 0$, we have ${f}_{1}(q,y)+\u3008yq,{A}_{1}q\u3009\ge 0$ for all $y\in C$. Thus, $q\in GEP({f}_{1},{A}_{1})$. In a similar fashion, we have some $k\ge 1$ such that ${f}_{2}={f}_{{n}_{k}}$ and $q\in GEP({f}_{2},{A}_{2})$. Therefore, $q\in {\bigcap}_{i=1}^{N}GEP({f}_{i},{A}_{i})$. Since ${u}_{{n}_{j},i}\rightharpoonup q$, so it follows from (3.14) and Lemma 2.2 that $q\in {\bigcap}_{i=1}^{N}F({T}_{i})$. Reasoning as above, we can show that $q\in {\bigcap}_{i=1}^{N}F({S}_{i})$. Hence, $q\in \mathcal{F}$. Let $x={P}_{\mathcal{F}}{x}_{1}$, which implies that $x={P}_{\mathcal{F}}{x}_{1}\in {C}_{n+1}$. Since ${x}_{n+1}={P}_{{C}_{n+1}\cap {Q}_{n+1}}{x}_{1}\in {C}_{n+1}$, we have
On the other hand, we have
That is,
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 δinversestrongly 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 kstrict pseudocontractions. 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 δinversestrongly 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
is equivalent to
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 kstrict pseudocontractions. 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 kstrict pseudocontraction to a finite family of maps;

(ii)
from an equilibrium problem to a finite family of generalized equilibrium problems.
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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 1737/51) and Higher Education Commission of Pakistan. The author H. Fukharuddin is grateful to King Fahd University of Petroleum & Minerals for support during this research.
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Khan, M.A.A., Fukharuddin, H. Strong convergence by the shrinking effect of two halfspaces and applications. Fixed Point Theory Appl 2013, 30 (2013). https://doi.org/10.1186/16871812201330
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Keywords
 nonexpansive map
 strict pseudocontraction
 shrinking projection method
 CQmethod
 equilibrium problem
 δinverse strongly monotone map