# A hybrid projection method for solving a common solution of a system of equilibrium problems and fixed point problems for asymptotically strict pseudocontractions in the intermediate sense in Hilbert spaces

- Chatchawan Watchararuangwit
^{1}, - Pongrus Phuangphoo
^{1, 2}and - Poom Kumam
^{1}Email author

**2012**:252

**DOI: **10.1186/1029-242X-2012-252

© Watchararuangwit et al.; licensee Springer 2012

**Received: **15 June 2012

**Accepted: **15 October 2012

**Published: **30 October 2012

## Abstract

In this paper, we introduce a new iterative algorithm which is constructed by using the hybrid projection method for finding a common solution of a system of equilibrium problems of bifunctions satisfying certain conditions and a common solution of fixed point problems of a family of uniformly Lipschitz continuous and asymptotically ${\lambda}_{i}$-strict pseudocontractive mappings in the intermediate sense. We prove the strong convergence theorem for a new iterative algorithm under some mild conditions in Hilbert spaces. Finally, we also give a numerical example which supports our results.

**MSC:**47H05, 47H09, 47H10.

### Keywords

asymptotically strict pseudocontraction in the intermediate sense hybrid projection method system of equilibrium problems fixed point problems## 1 Introduction

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

*H*with the inner product $\u3008\cdot ,\cdot \u3009$ and the norm $\parallel \cdot \parallel $. Let ${\{{F}_{m}\}}_{m\in \mathrm{\Gamma}}$ be a family of bifunctions from $C\times C$ into ℝ, where ℝ is the set of real numbers and Γ is an arbitrary index set. The

*system of equilibrium problems*is to find $x\in C$ such that

*equilibrium problem*of finding $x\in C$ such that

The set of solutions of (1.3) is denoted by $EP(F)$.

Recall the following definitions.

*monotone*if

*A*is called

*α*-

*inverse*-

*strongly monotone*[1, 2], if there exists a positive real number

*α*such that

Clearly, if *A* is *α*-inverse-strongly monotone, then *A* is monotone.

*A*is called

*β*-

*strongly monotone*if there exists a positive real number

*β*such that

*A*is called

*L*-

*Lipschitz continuous*if there exists a positive real number

*L*such that

It is easy to see that if *A* is an *α*-inverse-strongly monotone mapping from *C* into *H*, then *A* is $\frac{1}{\alpha}$-Lipschitz continuous.

In 2009, Qin *et al.* [3] introduced the following algorithm for a finite family of asymptotically ${\lambda}_{i}$-strictly pseudocontractions.

*the explicit iterative sequence*of a finite family of asymptotically ${\lambda}_{i}$-strictly pseudocontractions $\{{S}_{1},{S}_{2},\dots ,{S}_{N}\}$. Since for each $n\ge 1$, it can be written as $n=(h-1)N+i$, where $i=i(n)\in \{1,2,3,\dots ,N\}$, $h=h(n)\ge 1$ is a positive integer and $h(n)\to \mathrm{\infty}$, as $n\to \mathrm{\infty}$, we can rewrite the above table in the following compact form:

Next, Sahu *et al.* [4] introduced new iterative schemes for asymptotically strictly pseudocontractive mappings in the intermediate sense. To be more precise, they proved the following theorem.

**Theorem (SXY)**

*Let*

*C*

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

*H*

*and*$T:C\to C$

*be a uniformly continuous asymptotically*

*κ*-

*strictly pseudocontractive mapping in the intermediate sense with a sequence*$\{{\gamma}_{n}\}$

*such that*$F(T)$

*is nonempty and bounded*.

*Let*$\{{\alpha}_{n}\}$

*be a sequence in*$[0,1]$

*such that*$0<\delta \le {\alpha}_{n}\le 1-\kappa $

*for all*$n\in \mathbb{N}$.

*Let*$\{{x}_{n}\}\subset C$

*be a sequence generated by the following*(

*CQ*)

*algorithm*:

*where* ${\theta}_{n}={c}_{n}+{\gamma}_{n}{\mathrm{\Delta}}_{n}$ *and* ${\mathrm{\Delta}}_{n}=sup\{\parallel {x}_{n}-z\parallel :z\in F(T)\}<\mathrm{\infty}$. *Then* $\{{x}_{n}\}$ *converges strongly to* ${P}_{F(T)}(u)$, *where* ${P}_{F(T)}$ *is a metric projection from* *H* *into* $F(T)$.

In 2010, Hu and Cai [5] considered the asymptotically strictly pseudocontractive mappings in the intermediate sense concerning the equilibrium problem. They obtained the following result in a real Hilbert space. Next, Ceng *et al.* [6] introduced the viscosity approximation method for a modified Mann iteration process for asymptotically strict pseudocontractive mappings in the intermediate sense and they proved the strong convergence of a general CQ-algorithm and extended the concept of asymptotically strictly pseudocontractive mappings in the intermediate sense to the Banach space setting called nearly asymptotically strictly pseudocontractive mappings in the intermediate sense. Finally, they established a weak convergence theorem for a fixed point of nearly asymptotically strictly pseudocontractive mappings in the intermediate sense which are not necessarily Lipschitz continuous mappings.

**Theorem (HC)** *Let* *C* *be a nonempty closed and convex subset of a real Hilbert space* *H* *and* $N\ge 1$ *be an integer*, $\varphi :C\to C$ *be a bifunction satisfying* (A1)-(A4), *and* $A:C\to H$ *be an* *α*-*inverse*-*strongly monotone mapping*. *Let for each* $1\le i\le N$, ${T}_{i}:C\to C$ *be a uniformly continuous* ${k}_{i}$-*strictly asymptotically pseudocontractive mapping in the intermediate sense for some* $0\le {k}_{i}<1$ *with sequences* $\{{\gamma}_{n,i}\}\subset [0,\mathrm{\infty})$ *such that* ${lim}_{n\to \mathrm{\infty}}{\gamma}_{n,i}=0$ *and* $\{{c}_{n,i}\}\subset [0,\mathrm{\infty})$ *such that* ${lim}_{n\to \mathrm{\infty}}{c}_{n,i}=0$. *Let* $k=max\{{k}_{i}:1\le i\le N\}$, ${\gamma}_{n}=max\{{\gamma}_{n,i}:1\le i\le N\}$, *and* ${c}_{n}=max\{{c}_{n,i}:1\le i\le N\}$. *Assume that* $\u03dc={\bigcap}_{i=1}^{N}F({T}_{i})\cap EP(\varphi )$ *is nonempty and bounded*. *Let* $\{{\alpha}_{n}\}$ *and* $\{{\beta}_{n}\}$ *be sequences in* $[0,1]$ *such that* $0<a\le {\alpha}_{n}\le 1$, $0<\delta \le {\beta}_{n}\le 1-k$ *for all* $n\in \mathbb{N}$, *and* $0<b\le {r}_{n}\le c<2\alpha $.

*Let*$\{{x}_{n}\}$

*and*$\{{u}_{n}\}$

*be sequences generated by the following algorithm*:

*where* ${\theta}_{n}={c}_{h(n)}+{\gamma}_{h(n)}{\rho}_{n}^{2}\to 0$, *as* $n\to \mathrm{\infty}$, *and* ${\rho}_{n}=sup\{\parallel {x}_{n}-v\parallel :v\in \u03dc\}<\mathrm{\infty}$. *Then* $\{{x}_{n}\}$ *converges strongly to* ${P}_{\u03dc}({x}_{0})$.

In 2011, Duan and Zhao [7] introduced new iterative schemes for finding a common solution set of a system of equilibrium problems and a solution of a fixed point set of asymptotically strict pseudocontractions in the intermediate sense and they proved these schemes converge strongly.

In 2012, Shui Ge [8] introduced a new hybrid algorithm with variable coefficients for a fixed point problem of a uniformly Lipschitz continuous mapping and asymptotically pseudocontractive mapping in the intermediate sense on unbounded domains and he proved strong convergence in a real Hilbert space.

**Theorem (Ge)**

*Let*

*C*

*be a nonempty*,

*closed*,

*and convex subset of a real Hilbert space*

*H*, $T:C\to C$

*be a uniformly*

*L*-

*Lipschitz continuous mapping and asymptotically pseudocontractive mapping in the intermediate sense with sequences*$\{{k}_{n}\}\subset [1,\mathrm{\infty})$

*and*$\{{v}_{n}\}\subset [0,\mathrm{\infty})$.

*Let*${q}_{n}=2{k}_{n}-1$

*for each*$n\in \mathbb{N}$.

*Let*$\{{x}_{n}\}$

*be the sequence generated by the following hybrid algorithm with variable coefficients*:

*where*

*Assume that the positive real number* ${r}_{0}$ *is chosen so that* ${B}_{{r}_{0}}({x}_{1})\cap Fix(T)\ne \mathrm{\varnothing}$ *and that* $\{{\alpha}_{n}\}$ *and* $\{{\beta}_{n}\}$ *are sequences in* $(0,1)$ *such that* $a\le {\alpha}_{n}\le {\beta}_{n}\le b$ *for some* $a>0$ *and for some* $b\in (0,\frac{1}{2+L})$.

*Then* $\{{x}_{n}\}$ *converges strongly to a fixed point of* *T*.

In this paper, motivated and inspired by the previously mentioned above results, we introduce a new iterative algorithm by the hybrid projection method for finding a common solution of a system of equilibrium problems of bifunctions satisfying certain conditions and a common solution of fixed point problems of a family of uniformly Lipschitz continuous and asymptotically ${\lambda}_{i}$-strict pseudocontractive mappings in the intermediate sense in a real Hilbert space. Then, we prove a strong convergence theorem of the iterative algorithm generated by this conditions. Finally, we also give a numerical example which supports our results. The results obtained in this paper extend and improve several recent results in this area.

## 2 Preliminaries

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

*C*be a closed and convex subset of

*H*. For any point $x\in H$, there exists a unique nearest point in

*C*, denoted by ${P}_{C}(x)$, such that

*H*onto

*C*defined by the following:

*H*onto

*C*. It is also known that ${P}_{C}$ satisfies

- (1)
→ for strong convergence and ⇀ for weak convergence.

- (2)
${\omega}_{w}({x}_{n})=\{x:\mathrm{\exists}{x}_{{n}_{j}}\rightharpoonup x\}$ denotes the weak

*w*-limit set of $\{{x}_{n}\}$. - (3)
A nonlinear mapping

*S*: $C\to C$ is a self-mapping in*C*. We denote the set of fixed points of*S*by $F(S)$ (*i.e.*, $F(S)=\{x\in C:Sx=x\}$). Recall the following definitions.

**Definition 2.1**Let

*S*be a mapping from

*C*to

*C*. Then

- (1)
*S*is said to be*nonexpansive*if$\parallel Sx-Sy\parallel \le \parallel x-y\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in C.$(2.1) - (2)
*S*is said to be*uniformly Lipschitz continuous*if there exists a constant $L>0$ such that$\parallel {S}^{n}x-{S}^{n}y\parallel \le L\parallel x-y\parallel ,\phantom{\rule{1em}{0ex}}\text{for all integers}n\ge 1,\mathrm{\forall}x,y\in C.$(2.2) - (3)
*S*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$\parallel {S}^{n}x-{S}^{n}y\parallel \le {k}_{n}\parallel x-y\parallel ,\phantom{\rule{1em}{0ex}}\text{for all integers}n\ge 1,\mathrm{\forall}x,y\in C.$(2.3)

*C*is a nonempty, bounded, closed, and convex subset of a real Hilbert space

*H*, then every asymptotically nonexpansive self-mapping has a fixed point. Further, the set $F(S)$ of fixed points of

*S*is closed and convex.

- (4)

*et al.*(see [10, 11]) as a generalization of the class of asymptotically nonexpansive mappings. It is known that if

*C*is a nonempty, bounded, closed, and convex subset of a real Hilbert space

*H*, then every asymptotically nonexpansive self-mapping in the intermediate sense has a fixed point (see [12]).

- (5)
*S*is said to be*contractive*if there exists a coefficient $k\in (0,1)$ such that$\parallel Sx-Sy\parallel \le k\parallel x-y\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in C.$(2.5) - (6)
*S*is said to be a*λ*-*strict pseudocontraction*if there exists a coefficient $\lambda \in [0,1)$ such that${\parallel Sx-Sy\parallel}^{2}\le {\parallel x-y\parallel}^{2}+\lambda {\parallel (I-S)x-(I-S)y\parallel}^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in C.$(2.6)

*S*is a nonexpansive mapping, then

*S*is a strict pseudocontraction with $\lambda =0$. We also remark that if $\lambda =1$, then

*S*is called a pseudocontractive mapping.

- (7)
*S*is said to be an*asymptotically**λ*-*strict pseudocontraction*with the sequence $\{{d}_{n}\}$ (see also [13]) if there exists a sequence $\{{d}_{n}\}\subset [0,\mathrm{\infty})$ with ${d}_{n}\to 0$ as $n\to \mathrm{\infty}$ and a constant $\lambda \in [0,1)$ such that(2.7)

*S*is an asymptotically nonexpansive mapping, then

*S*is an asymptotically strict pseudocontraction with $\lambda =0$. We also remark that if $\lambda =1$, then

*S*is said to be an asymptotically pseudocontractive mapping which was introduced by Schu [15] in 1991.

- (8)

The class of asymptotically strict pseudocontractions in the intermediate sense was introduced by Sahu, Xu, and Yao [4] as a generalization of a class of asymptotically strict pseudocontractions.

For solving the equilibrium problem, let us give the following assumptions for the bifunction *F* and the set *C*:

(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 each $x,y,z\in C$, ${lim\hspace{0.17em}sup}_{t\to 0}F(tz+(1-t)x,y)\le F(x,y)$;

(A4) for each $x\in C$, $y\mapsto F(x,y)$ is convex and lower semicontinuous.

**Lemma 2.2** ([16])

*Let*

*C*

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

*H*.

*For any*$x,y,z\in H$

*and given also a real number*$a\in \mathbb{R}$,

*the set*

*is closed and convex*.

**Lemma 2.3** ([17])

*Let*

*C*

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

*H*.

*Let*$F:C\times C\to \mathbb{R}$

*satisfy*(A1)-(A4),

*and let*$r>0$

*and*$x\in H$.

*Then there exists*$z\in C$

*such that*

**Lemma 2.4** ([18])

*Assume that*$F:C\times C\to \mathbb{R}$

*satisfies*(A1)-(A4).

*For*$r>0$

*and*$x\in H$,

*define a mapping*${T}_{r}:H\to C$

*as follows*:

*Then the following hold*:

- (1)
${T}_{r}$

*is single*-*valued*; - (2)${T}_{r}$
*is firmly nonexpansive*,*i*.*e*.,*for any*$x,y\in H$,${\parallel {T}_{r}x-{T}_{r}y\parallel}^{2}\le \u3008{T}_{r}x-{T}_{r}y,x-y\u3009;$(2.10) - (3)
$F({T}_{r})=EP(F)$;

*and* - (4)
$EP(F)$

*is closed and convex*.

*Let*

*H*

*be a real Hilbert space*.

*Then the following identities hold*:

- (i)
${\parallel x-y\parallel}^{2}={\parallel x\parallel}^{2}-{\parallel y\parallel}^{2}-2\u3008x-y,y\u3009$, $\mathrm{\forall}x,y\in H$.

- (ii)
${\parallel tx+(1-t)y\parallel}^{2}=t{\parallel x\parallel}^{2}+(1-t){\parallel y\parallel}^{2}-t(1-t){\parallel x-y\parallel}^{2}$, $\mathrm{\forall}t\in [0,1]$, $\mathrm{\forall}x,y\in H$.

- (iii)
${\parallel x+y\parallel}^{2}={\parallel x\parallel}^{2}+2\u3008y,x+y\u3009$.

**Lemma 2.6** ([4])

*Let* *C* *be a nonempty closed and convex subset of a real Hilbert space* *H*, *and* $S:C\to C$ *be a uniformly* *L*-*Lipschitz continuous and asymptotically* *λ*-*strict pseudocontraction in the intermediate sense*. *Then* $F(S)$ *is closed and convex*.

**Lemma 2.7** ([4])

*Let* *C* *be a nonempty closed and convex subset of a real Hilbert space* *H* *and* $S:C\to C$ *be a uniformly* *L*-*Lipschitz continuous and asymptotically* *λ*-*strict pseudocontraction in the intermediate sense*. *Then the mapping* $I-S$ *is demiclosed at zero*, *that is*, *if the sequence* $\{{x}_{n}\}$ *in* *C* *is such that* ${x}_{n}\rightharpoonup \overline{x}$ *and* ${x}_{n}-S{x}_{n}\to 0$, *then* $\overline{x}\in F(S)$.

**Lemma 2.8** ([20])

*Let*

*C*

*be a nonempty closed and convex subset of a real Hilbert space H*.

*Let*$\{{x}_{n}\}$

*be a sequence in*

*H*

*and*$u\in H$,

*and let*$q={P}_{C}u$.

*Suppose that*$\{{x}_{n}\}$

*is such that*${\omega}_{n}({x}_{n})\subset C$

*and satisfies the condition*

*Then* ${x}_{n}\to q$.

**Lemma 2.9** ([4])

*Let*

*C*

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

*H*.

*Let*$S:C\to C$

*be an asymptotically*

*λ*-

*strict pseudocontractive mapping in the intermediate sense with the sequence*${\gamma}_{n}$.

*Then*

*for all* $x,y\in C$ *and* $n\in \mathbb{N}$.

## 3 Main results

In this section, we prove a strong convergence theorem which solves the problem of finding a common solution of a system of equilibrium problems and a common solution of fixed point problems in Hilbert spaces.

**Theorem 3.1** *Let* *C* *be a nonempty closed and convex subset of a real Hilbert space* *H*. *Let* $M\ge 1$ *be a positive integer*. *Let* ${\{{F}_{m}\}}_{m=1}^{M}:C\times C\to \mathbb{R}$ *be a bifunction satisfying* (A1)-(A4). *Let* ${\{{S}_{i}\}}_{i=1}^{N}:C\to C$ *be a uniformly Lipschitz continuous and asymptotically* ${\lambda}_{i}$-*strict pseudocontractive mapping in the intermediate sense for some* $0\le {\lambda}_{i}<1$ *with the sequences* $\{{c}_{n,i}\}\subset [0,\mathrm{\infty})$ *such that* ${lim}_{n\to \mathrm{\infty}}{c}_{n,i}=0$ *and* $\{{d}_{n,i}\}\subset [0,\mathrm{\infty})$ *such that* ${lim}_{n\to \mathrm{\infty}}{d}_{n,i}=0$. *Let* $\lambda =max\{{\lambda}_{i}:1\le i\le N\}$, ${c}_{n}=max\{{c}_{n,i}:1\le i\le N\}$ *and* ${d}_{n}=max\{{d}_{n,i}:1\le i\le N\}$. *Assume that* $\mathrm{\Omega}:=({\bigcap}_{m=1}^{M}SEP({F}_{m}))\cap ({\bigcap}_{i=1}^{N}F({S}_{i}))$ *is nonempty and bounded*. *Let* $\{{\alpha}_{n}\}$, $\{{\beta}_{n}\}$ *be sequences in* $[0,1]$ *such that* $0<a\le {\alpha}_{n}\le 1$, $0<b\le {\beta}_{n}\le 1-\lambda $, $a,b\in \mathbb{R}$, $\mathrm{\forall}n\in \mathbb{N}$ *and* $\{{r}_{m,n}\}$ *be a sequence in* $(0,\mathrm{\infty})$ *such that* ${lim}_{n\to \mathrm{\infty}}{r}_{m,n}>0$.

*Let*$\{{x}_{n}\}$

*be a sequence generated by the following algorithm*:

*where* ${\theta}_{n}={c}_{h(n)}+{d}_{h(n)}{\rho}_{n}^{2}\to 0$, *as* $n\to \mathrm{\infty}$ *and* ${\rho}_{n}=sup\{\parallel {x}_{n}-w\parallel :w\in \mathrm{\Omega}\}<\mathrm{\infty}$ *and* $n=(h(n)-1)N+i(n)$, *where* $i(n)\in \{1,2,3,\dots ,N\}$. *Then* $\{{x}_{n}\}$ *converges strongly to some point* ${p}^{\ast}$, *where* ${p}^{\ast}={P}_{\mathrm{\Omega}}({x}_{1})$.

*Proof* The proof is split into seven steps.

Step 1. We will show that ${P}_{\mathrm{\Omega}}$ is well defined.

From Lemma 2.4, we get ${\bigcap}_{m=1}^{M}SEP({F}_{m})$ is closed and convex. From the assumption of ${\{{S}_{i}\}}_{i=1}^{N}$ and Lemma 2.6, it follows that ${\bigcap}_{i=1}^{N}F({S}_{i})$ is closed and convex.

Therefore, $\mathrm{\Omega}:=({\bigcap}_{m=1}^{M}SEP({F}_{m}))\cap ({\bigcap}_{i=1}^{N}F({S}_{i}))$ is closed and convex. Hence, ${P}_{\mathrm{\Omega}}$ is well defined.

Step 2. We will show that ${C}_{n}$ is closed and convex for each $n\ge 1$.

By the assumption of ${C}_{n+1}$, it is easy to see that ${C}_{n}$ is closed for each $n\ge 1$. We only show that ${C}_{n}$ is convex for each $n\ge 1$.

*k*. For each $w\in {C}_{k}$, we see that

In view of the convexity of ${C}_{k}$, we see that $\overline{w}\in {C}_{k}$. This implies that $\overline{w}\in {C}_{k+1}$. Therefore, ${C}_{k+1}$ is convex. Hence, ${C}_{n}$ is closed and convex for each $n\ge 1$.

Step 3. We will show that $\mathrm{\Omega}\subset {C}_{n}$ for each $n\ge 1$.

Put ${\mathrm{\Theta}}_{n}^{m}:={T}_{{r}_{m,n}}^{{F}_{m}}{T}_{{r}_{m-1,n}}^{{F}_{m-1}}\cdots {T}_{{r}_{2,n}}^{{F}_{2}}{T}_{{r}_{1,n}}^{{F}_{1}}{x}_{n}$ for every $m\in \{1,2,3,\dots ,M\}$ and ${\mathrm{\Theta}}_{n}^{0}=I$ for all $n\in \mathbb{N}$. Therefore, ${u}_{n}={\mathrm{\Theta}}_{n}^{M}{x}_{n}$. It is obvious that $\mathrm{\Omega}\subset {C}_{1}=H$. Suppose that $\mathrm{\Omega}\subset {C}_{k}$ for some $k\ge 1$.

*k*. Taking $p\in \mathrm{\Omega}$ and for each $m\in \{1,2,3,\dots ,M\}$, we see that ${T}_{{r}_{m,n}}^{{F}_{m}}$ is nonexpansive and ${T}_{{r}_{m,n}}^{{F}_{m}}p=p$. We note that

Therefore, $p\in {C}_{k+1}$, and so $\mathrm{\Omega}\subset {C}_{n}$ for each $n\ge 1$. Hence, $\{{x}_{n}\}$ is well defined.

Step 4. We will show that $\{{x}_{n}\}$ is bounded.

*H*, there exists a unique $q\in \mathrm{\Omega}$ such that $q={P}_{\mathrm{\Omega}}{x}_{1}$. By the assumption, we have ${x}_{n}={P}_{{C}_{n}}{x}_{1}$ for any $q\in \mathrm{\Omega}\subset {C}_{n}$. Then

This implies that $\{{x}_{n}\}$ is bounded. Therefore, $\{{u}_{n}\}$, $\{{z}_{n}\}$, and $\{{y}_{n}\}$ are also bounded.

Step 5. We will show that $\parallel {u}_{n}-{S}_{i}{u}_{n}\parallel \to 0$ and $\parallel {x}_{n}-{S}_{i}{x}_{n}\parallel \to 0$ as $n\to \mathrm{\infty}$, $\mathrm{\forall}i\in \{1,2,3,\dots ,N\}$.

It is obvious that the relations $h(n)=h(n-N)+1$ and $i(n)=i(n-N)$ hold.

Step 6. We will show that ${p}^{\ast}\in \mathrm{\Omega}:=({\bigcap}_{i=1}^{N}F({S}_{i}))\cap ({\bigcap}_{m=1}^{M}SEP({F}_{m}))$.

(6.1) We will show that ${p}^{\ast}\in {\bigcap}_{i=1}^{N}F({S}_{i})$.

We take ${p}^{\ast}\in {\omega}_{w}({x}_{n})$ and assume that ${x}_{{n}_{j}}\rightharpoonup {p}^{\ast}$ for some subsequence $\{{x}_{{n}_{j}}\}$ of $\{{x}_{n}\}$.

(6.2) We will show that ${p}^{\ast}\in {\bigcap}_{m=1}^{M}SEP({F}_{m})$.

*t*, for each $m\in \{1,2,3,\dots ,M\}$, we get

Therefore, ${p}^{\ast}\in {\bigcap}_{m=1}^{M}SEP({F}_{m})$, and so ${p}^{\ast}\in \mathrm{\Omega}$.

Step 7. We will show that $\{{x}_{n}\}$ converges strongly to ${P}_{\mathrm{\Omega}}{x}_{1}$.

Since Ω is a nonempty closed and convex subset of *H*, there exists a unique ${p}^{\ast}\in \mathrm{\Omega}$ such that ${p}^{\ast}={P}_{\mathrm{\Omega}}({x}_{1})$. It follows from Lemma 2.8 that ${x}_{n}\to {p}^{\ast}$, where ${p}^{\ast}={P}_{\mathrm{\Omega}}({x}_{1})$. This completes proof. □

## 4 Deduced theorems

If we take $M=1$ in Theorem 3.1, then we obtain the following result.

**Theorem 4.1** *Let* *C* *be a nonempty closed and convex subset of a real Hilbert space* *H*. *Let* $M\ge 1$ *be a positive integer*. *Let* $F:C\times C\to \mathbb{R}$ *be a bifunction satisfying* (A1)-(A4). *Let* ${\{{S}_{i}\}}_{i=1}^{N}:C\to C$ *be a uniformly Lipschitz continuous and asymptotically* ${\lambda}_{i}$-*strict pseudocontractive mapping in the intermediate sense for some* $0\le {\lambda}_{i}<1$ *with the sequences* $\{{c}_{n,i}\}\subset [0,\mathrm{\infty})$ *such that* ${lim}_{n\to \mathrm{\infty}}{c}_{n,i}=0$ *and* $\{{d}_{n,i}\}\subset [0,\mathrm{\infty})$ *such that* ${lim}_{n\to \mathrm{\infty}}{d}_{n,i}=0$. *Let* $\lambda =max\{{\lambda}_{i}:1\le i\le N\}$, ${c}_{n}=max\{{c}_{n,i}:1\le i\le N\}$ *and* ${d}_{n}=max\{{d}_{n,i}:1\le i\le N\}$. *Assume that* $\mathrm{\Omega}:=EP(F)\cap ({\bigcap}_{i=1}^{N}F({S}_{i}))$ *is nonempty and bounded*. *Let* $\{{\alpha}_{n}\}$, $\{{\beta}_{n}\}$ *be sequences in* $[0,1]$ *such that* $0<a\le {\alpha}_{n}\le 1$, $0<b\le {\beta}_{n}\le 1-\lambda $, $a,b\in \mathbb{R}$, $\mathrm{\forall}n\in \mathbb{N}$, $\{{r}_{m,n}\}$ *be a sequence in* $(0,\mathrm{\infty})$ *such that* ${lim}_{n\to \mathrm{\infty}}{r}_{m,n}>0$.

*Let*$\{{x}_{n}\}$

*be a sequence generated by the following algorithm*:

*where* ${\theta}_{n}={c}_{h(n)}+{d}_{h(n)}{\rho}_{n}^{2}\to 0$, *as* $n\to \mathrm{\infty}$ *and* ${\rho}_{n}=sup\{\parallel {x}_{n}-w\parallel :w\in \mathrm{\Omega}\}<\mathrm{\infty}$ *and* $n=(h(n)-1)N+i(n)$, *where* $i(n)\in \{1,2,3,\dots ,N\}$. *Then* $\{{x}_{n}\}$ *converges strongly to some point* ${p}^{\ast}$, *where* ${p}^{\ast}={P}_{\mathrm{\Omega}}({x}_{1})$.

**Remark 4.2** Theorem 4.1 improves and extends the theorem of Tada and Takahashi [21] and the corollary of Duan and Zhao [7].

If we set ${F}_{m}\equiv 0$ and ${r}_{m,n}=1$ for all $m\in \{1,2,3,\dots ,N\}$ in Theorem 3.1, then we obtain the following result.

**Theorem 4.3** *Let* *C* *be a nonempty closed and convex subset of a real Hilbert space* *H*. *Let* $M\ge 1$ *be a positive integer*. *Let* ${\{{S}_{i}\}}_{i=1}^{N}:C\to C$ *be a uniformly Lipschitz continuous and asymptotically* ${\lambda}_{i}$-*strict pseudocontractive mapping in the intermediate sense for some* $0\le {\lambda}_{i}<1$ *with the sequences* $\{{c}_{n,i}\}\subset [0,\mathrm{\infty})$ *such that* ${lim}_{n\to \mathrm{\infty}}{c}_{n,i}=0$ *and* $\{{d}_{n,i}\}\subset [0,\mathrm{\infty})$ *such that* ${lim}_{n\to \mathrm{\infty}}{d}_{n,i}=0$. *Let* $\lambda =max\{{\lambda}_{i}:1\le i\le N\}$, ${c}_{n}=max\{{c}_{n,i}:1\le i\le N\}$ *and* ${d}_{n}=max\{{d}_{n,i}:1\le i\le N\}$. *Assume that* $\mathrm{\Omega}:={\bigcap}_{i=1}^{N}F({S}_{i})$ *is nonempty and bounded*. *Let* $\{{\alpha}_{n}\}$, $\{{\beta}_{n}\}$ *be sequences in* $[0,1]$ *such that* $0<a\le {\alpha}_{n}\le 1$, $0<b\le {\beta}_{n}\le 1-\lambda $, $a,b\in \mathbb{R}$, $\mathrm{\forall}n\in \mathbb{N}$, $\{{r}_{m,n}\}$ *be a sequence in* $(0,\mathrm{\infty})$ *such that* ${lim}_{n\to \mathrm{\infty}}{r}_{m,n}>0$.

*Let*$\{{x}_{n}\}$

*be a sequence generated by the following algorithm*:

*where* ${\theta}_{n}={c}_{h(n)}+{d}_{h(n)}{\rho}_{n}^{2}\to 0$, *as* $n\to \mathrm{\infty}$ *and* ${\rho}_{n}=sup\{\parallel {x}_{n}-w\parallel :w\in \mathrm{\Omega}\}<\mathrm{\infty}$ *and* $n=(h(n)-1)N+i(n)$, *where* $i(n)\in \{1,2,3,\dots ,N\}$. *Then* $\{{x}_{n}\}$ *converges strongly to some point* ${p}^{\ast}$, *where* ${p}^{\ast}={P}_{\mathrm{\Omega}}({x}_{1})$.

**Remark 4.4** Theorem 4.1 improves and extends the theorem of Sahu, Xu, and Yao [4], the theorem of Qin, Cho, Kang, and Shang [3] and the corollary of Duan and Zhao [7].

## 5 Numerical examples

In this section, in order to demonstrate the effectiveness, realization and convergence of algorithm of Theorem 3.1, we consider the following simple example that was presented in reference [4].

**Example 5.1**Let $x\in \mathbb{R}$ and $C=[0,1]$. For each $x\in C$, we define

where $0<k<1$.

It is easy to see that $S:C\to C$ is discontinuous at $x=\frac{1}{2}$ and *S* is not Lipschitz continuous.

Set ${C}_{1}=[0,\frac{1}{2}]$ and ${C}_{2}=(\frac{1}{2},1]$.

for all $x,y\in C$ and $n\in \mathbb{N}$ and for some $K>0$.

Therefore, *S* is an asymptotically *k*-strict pseudocontractive mapping in the intermediate sense.

In Theorem 3.1, we set $N=1$, ${F}_{m}\equiv 0$, ${\beta}_{n}=1-k$, ${\alpha}_{n}=\frac{n+1}{2n}$. We apply it to find the fixed point of *S* of Example 5.1.

*S*. The convergence of each iteration is also shown in Figure 1 for comparison.

**The numerical results for an initial guess**
${\mathit{x}}_{\mathbf{1}}\mathbf{=}\mathbf{0.2}\mathbf{,}\mathbf{0.5}\mathbf{,}\mathbf{0.8}$

n (iterative number) | Initial guess | ||
---|---|---|---|

${\mathit{x}}_{\mathbf{1}}\mathbf{=}\mathbf{0.2}$ | ${\mathit{x}}_{\mathbf{1}}\mathbf{=}\mathbf{0.5}$ | ${\mathit{x}}_{\mathbf{1}}\mathbf{=}\mathbf{0.8}$ | |

10 | 0.1467 | 0.2049 | 0.2105 |

20 | 0.0163 | 0.0205 | 0.0209 |

30 | 0.0016 | 0.0019 | 0.0020 |

40 | 1.5149 × 10 | 1.8819 × 10 | 1.9196 × 10 |

50 | 1.4889 × 10 | 1.8494 × 10 | 1.8864 × 10 |

## Declarations

### Acknowledgements

This research was supported by the Faculty of Science, KMUTT Research Fund 2553-2554.

## Authors’ Affiliations

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