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

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.

1 Introduction

Let C be a closed and convex subset of a real Hilbert space H with the inner product $〈\cdot ,\cdot 〉$ 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

$$F_m(x,y)\ge 0,m\in \mathrm{\Gamma },\mathrm{\forall }y\in C.$$

(1.1)

The set of solutions of (1.1) is denoted by $SEP(F_m)$, where $m\in \mathrm{\Gamma }$, that is,

$$SEP(F_m)=\{x\in C:F_m(x,y)\ge 0,\mathrm{\forall }y\in C\}.$$

(1.2)

If Γ is a singleton, then the problem (1.1) is reduced to the equilibrium problem of finding $x\in C$ such that

$$F(x,y)\ge 0,\mathrm{\forall }y\in C.$$

(1.3)

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

Recall the following definitions.

A mapping $A:C\to H$ is called monotone if

$$〈Ax-Ay,x-y〉\ge 0,\mathrm{\forall }x,y\in C.$$

(1.4)

A mapping A is called α-inverse-strongly monotone [1, 2], if there exists a positive real number α such that

$$〈Ax-Ay,x-y〉\ge \alpha {\parallel Ax-Ay\parallel }^2,\mathrm{\forall }x,y\in C.$$

(1.5)

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

A mapping A is called β-strongly monotone if there exists a positive real number β such that

$$〈Ax-Ay,x-y〉\ge \beta {\parallel x-y\parallel }^2,\mathrm{\forall }x,y\in C.$$

(1.6)

A mapping A is called L-Lipschitz continuous if there exists a positive real number L such that

$$\parallel Ax-Ay\parallel \le L\parallel x-y\parallel ,\mathrm{\forall }x,y\in C.$$

(1.7)

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.

Let $x_0\in C$ and ${\{{\alpha }_n\}}_{n=0}^{\mathrm{\infty }}$ be a sequence in $(0,1)$. The sequence $\{x_n\}$ is as follows:

$$\{\begin{array}{c}{x}_1={\alpha }_0{x}_0+(1-{\alpha }_0)S_1{x}_0,\hfill \\ x_2={\alpha }_1{x}_1+(1-{\alpha }_1)S_2{x}_1,\hfill \\ x_3={\alpha }_2{x}_2+(1-{\alpha }_2)S_3{x}_2,\hfill \\ ⋮\hfill \\ x_N={\alpha }_{N-1}{x}_{N-1}+(1-{\alpha }_{N-1})S_N{x}_{N-1},\hfill \\ x_{N+1}={\alpha }_N{x}_N+(1-{\alpha }_N)S_1^2{x}_N,\hfill \\ x_{N+2}={\alpha }_{N+1}{x}_{N+1}+(1-{\alpha }_{N+1})S_2^2{x}_{N+1},\hfill \\ ⋮\hfill \\ x_{2N}={\alpha }_{2N-1}{x}_{2N-1}+(1-{\alpha }_{2N-1})S_N^2{x}_{2N-1},\hfill \\ x_{2N+1}={\alpha }_{2N}{x}_{2N}+(1-{\alpha }_{2N})S_1^3{x}_{2N},\hfill \\ ⋮.\hfill \end{array}$$

(1.8)

It is called 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:

$$x_n={\alpha }_{n-1}{x}_{n-1}+(1-{\alpha }_{n-1})S_{i(n)}^{h(n)}{x}_{x-1},\mathrm{\forall }n\ge 1.$$

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:

$$\{\begin{array}{c}u=x_1\in C\mathit{\text{chosen arbitrarily}},\hfill \\ y_n=(1-{\alpha }_n)x_n+{\alpha }_n{T}^n{x}_n,\hfill \\ C_n=\{z\in C:{\parallel y_n-z\parallel }^2\le {\parallel x_n-z\parallel }^2+{\theta }_n\},\hfill \\ Q_n=\{z\in C:〈x_n-z,u-x_n〉\ge 0\},\hfill \\ x_{n+1}=P_{C_n\cap Q_n}(u),\mathrm{\forall }n\in \mathbb{N},\hfill \end{array}$$

(1.9)

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 $Ϝ={\bigcap }_{i=1}^{N}F(T_i)\cap EP(\varphi )$ 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:

(1.10)

where ${\theta }_n=c_{h(n)}+{\gamma }_{h(n)}{\rho }_n^2\to 0$, as $n\to \mathrm{\infty }$, and ${\rho }_n=sup\{\parallel x_n-v\parallel :v\in Ϝ\}<\mathrm{\infty }$. Then $\{x_n\}$ converges strongly to $P_{Ϝ}(x_0)$.

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 $q_n=2k_n-1$ for each $n\in \mathbb{N}$. Let $\{x_n\}$ be the sequence generated by the following hybrid algorithm with variable coefficients:

$$\{\begin{array}{c}{x}_1\in C\mathit{\text{chosen arbitrarily}},\hfill \\ C_1=C,\hfill \\ z_n=(1-{\hat{\beta }}_n)x_n+{\hat{\beta }}_n{T}^n{x}_n,\hfill \\ y_n=(1-{\hat{\alpha }}_n)x_n+{\hat{\alpha }}_n{T}^n{z}_n,\hfill \\ C_n=\{u\in C_n:{\parallel y_n-u\parallel }^2\le {\parallel x_n-u\parallel }^2\hfill \\ \phantom{C_n=}-{\hat{\alpha }}_n{\hat{\beta }}_n(1-{\hat{\beta }}_n-{\hat{\beta }}_n^2{L}^2-q_n{\hat{\beta }}_n){\parallel x_n-T^n{x}_n\parallel }^2+{\alpha }_n{\theta }_n\},\hfill \\ x_{n+1}=P_{C_{n+1}}(x_1),\mathrm{\forall }n\in \mathbb{N},\hfill \end{array}$$

(1.11)

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

Let H be a real Hilbert space with the inner product $〈\cdot ,\cdot 〉$ 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

$$\parallel x-P_{C}x\parallel \le \parallel x-y\parallel ,\mathrm{\forall }y\in C.$$

$P_C$ is called the metric projection of H onto C defined by the following:

$$P_C(x)=arg\underset{y\in C}{min}\parallel x-y\parallel .$$

We know that $P_C$ is a nonexpansive mapping H onto C. It is also known that $P_C$ satisfies

$${\parallel P_{C}x-P_{C}y\parallel }^2\le 〈P_{C}x-P_{C}y,x-y〉,\mathrm{\forall }x,y\in H,$$

and

$$〈x-P_{C}x,z-P_{C}x〉\ge 0,\mathrm{\forall }z\in C.$$

We will adopt the following notations:

• (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

  $$\parallel Sx-Sy\parallel \le \parallel x-y\parallel ,\mathrm{\forall }x,y\in C.$$

  (2.1)

• (2) S is said to be uniformly Lipschitz continuous if there exists a constant $L>0$ such that

  $$\parallel S^{n}x-S^{n}y\parallel \le L\parallel x-y\parallel ,\text{for all integers }n\ge 1,\mathrm{\forall }x,y\in C.$$

  (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

  $$\parallel S^{n}x-S^{n}y\parallel \le k_n\parallel x-y\parallel ,\text{for all integers }n\ge 1,\mathrm{\forall }x,y\in C.$$

  (2.3)

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk (see [9]) in 1972. 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 has a fixed point. Further, the set $F(S)$ of fixed points of S is closed and convex.

• (4) S is said to be asymptotically nonexpansive in the intermediate sense [10, 11] if it is continuous and the following inequality holds:

  $$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\underset{x,y\in C}{sup}(\parallel S^{n}x-S^{n}y\parallel -\parallel x-y\parallel )\le 0.$$

  (2.4)

Putting ${\xi }_n=max\{0,{sup}_{x,y\in C}(\parallel S^{n}x-S^{n}y\parallel -\parallel x-y\parallel )\}$, we see that ${\xi }_n\to 0$ as $n\to \mathrm{\infty }$. Then (2.4) is reduced to

$$\parallel S^{n}x-S^{n}y\parallel \le \parallel x-y\parallel +{\xi }_n,\mathrm{\forall }x,y\in C.$$

The class of asymptotically nonexpansive mappings in the intermediate sense was introduced by Kirk and Bruck 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

  $$\parallel Sx-Sy\parallel \le k\parallel x-y\parallel ,\mathrm{\forall }x,y\in C.$$

  (2.5)

• (6) S is said to be a λ-strict pseudocontraction if there exists a coefficient $\lambda \in [0,1)$ such that

  $${\parallel Sx-Sy\parallel }^2\le {\parallel x-y\parallel }^2+\lambda {\parallel (I-S)x-(I-S)y\parallel }^2,\mathrm{\forall }x,y\in C.$$

  (2.6)

The class of strict pseudocontractions was introduced by Brower and Petryshyn (see [1]) in 1967. Clearly, if 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)

The class of asymptotically strict pseudocontractions was introduced by Qihou [14] in 1996. Clearly, if 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) S is said to be an asymptotically λ-strict pseudocontraction in the intermediate sense with the sequence $\{d_n\}$ [4, 5] 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

  

  (2.8)

Putting $c_n=max\{0,{sup}_{x,y\in C}({\parallel S^{n}x-S^{n}y\parallel }^2-(1+d_n){\parallel x-y\parallel }^2-\lambda {\parallel (x-s^{n}x)-(y-s^{n}y)\parallel }^2)\}$, we see that $c_n\to 0$ as $n\to \mathrm{\infty }$. Then (2.8) is reduced to

$${\parallel S^{n}x-S^{n}y\parallel }^2\le (1+d_n){\parallel x-y\parallel }^2+\lambda {\parallel (x-s^{n}x)-(y-s^{n}y)\parallel }^2+c_n,\mathrm{\forall }x,y\in C,\mathrm{\forall }n\in \mathbb{N}.$$

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

$$\{v\in C:{\parallel y-v\parallel }^2\le {\parallel x-v\parallel }^2+〈z,v〉+a\}$$

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

$$F(z,y)+\frac{1}{r}〈y-z,z-x〉\ge 0,\mathrm{\forall }y\in C.$$

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:

$$T_r(x)=\{z\in C:F(z,y)+\frac{1}{r}〈y-z,z-x〉\ge 0,\mathrm{\forall }y\in C\}.$$

(2.9)

Then the following hold:

• (1) $T_r$ is single-valued;

• (2) $T_r$ is firmly nonexpansive, i.e., for any $x,y\in H$,

  $${\parallel T_{r}x-T_{r}y\parallel }^2\le 〈T_{r}x-T_{r}y,x-y〉;$$

  (2.10)

• (3) $F(T_r)=EP(F)$; and

• (4) $EP(F)$ is closed and convex.

Lemma 2.5 ([7, 19])

Let H be a real Hilbert space. Then the following identities hold:

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

• (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) ${\parallel x+y\parallel }^2={\parallel x\parallel }^2+2〈y,x+y〉$.

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

$$\parallel x_n-u\parallel \le \parallel u-q\parallel ,\mathrm{\forall }n\in \mathbb{N}.$$

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

$$\parallel S^{n}x-S^{n}y\parallel \le \frac{1}{1-\lambda }(\lambda \parallel x-y\parallel +\sqrt{(1+(1-\lambda ){\gamma }_n){\parallel x-y\parallel }^2+(1-\lambda )c_n}),$$

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:

$$\{\begin{array}{c}{x}_1\in C\mathit{\text{chosen arbitrarily}},\hfill \\ C_1=H,\hfill \\ u_n=T_{r_{M,n}}^{F_M}{T}_{r_{M-1,n}}^{F_{M-1}}\cdots T_{r_{2,n}}^{F_2}{T}_{r_{1,n}}^{F_1}{x}_n,\hfill \\ z_n=(1-{\beta }_n)u_n+{\beta }_n{S}_{i(n)}^{h(n)}{u}_n,\hfill \\ y_n=(1-{\alpha }_n)u_n+{\alpha }_n{z}_n,\hfill \\ C_{n+1}=\{w\in C_n:{\parallel y_n-w\parallel }^2\le {\parallel x_n-w\parallel }^2+{\theta }_n\},\hfill \\ x_{n+1}=P_{C_{n+1}}(x_1),\mathrm{\forall }n\in \mathbb{N},\hfill \end{array}$$

(3.1)

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 $n\ge 1$.

Note that $C_1=H$ is convex. Suppose that $C_k$ is convex for some $k\ge 1$. Next, we show that $C_{k+1}$ is convex for the same k. For each $w\in C_k$, we see that

$${\parallel y_k-w\parallel }^2\le {\parallel x_k-w\parallel }^2+{\theta }_k$$

is equivalent to

$$2〈x_k-y_k,w〉\le {\parallel x_k\parallel }^2-{\parallel y_k\parallel }^2+{\theta }_k.$$

(3.2)

Taking $w_1$ and $w_2$ in $C_{k+1}$ and putting $\overline{w}=t{w}_1+(1-t)w_2$, it follows that $w_1,w_2\in C_k$, and so

$$2〈x_k-y_k,w_1〉\le {\parallel x_k\parallel }^2-{\parallel y_k\parallel }^2+{\theta }_k$$

(3.3)

and

$$2〈x_k-y_k,w_2〉\le {\parallel x_k\parallel }^2-{\parallel y_k\parallel }^2+{\theta }_k.$$

(3.4)

Combining (3.3) with (3.4), we obtain that

$$2〈x_k-y_k,\overline{w}〉\le {\parallel x_k\parallel }^2-{\parallel y_k\parallel }^2+{\theta }_k.$$

That is,

$${\parallel y_k-\overline{w}\parallel }^2\le {\parallel x_k-\overline{w}\parallel }^2+{\theta }_k.$$

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 $k\ge 1$.

Next, we show that $\mathrm{\Omega }\subset C_{k+1}$ for the same 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

$$\parallel u_n-p\parallel =\parallel {\mathrm{\Theta }}_n^M{x}_n-{\mathrm{\Theta }}_n^{M}p\parallel \le \parallel x_n-p\parallel ,\mathrm{\forall }n\in \mathbb{N}.$$

(3.5)

We observe that

$$\begin{array}{rcl}{\parallel z_n-p\parallel }^2& =& {\parallel (1-{\beta }_n)(u_n-p)+{\beta }_n(S_{i(n)}^{h(n)}(u_n-p))\parallel }^2\\ =& (1-{\beta }_n){\parallel u_n-p\parallel }^2+{\beta }_n{\parallel S_{i(n)}^{h(n)}{u}_n-p\parallel }^2-{\beta }_n(1-{\beta }_n){\parallel S_{i(n)}^{h(n)}{u}_n-u_n\parallel }^2\\ \le & (1-{\beta }_n){\parallel u_n-p\parallel }^2+{\beta }_n[(1+d_{h(n)}){\parallel u_n-p\parallel }^2+\lambda {\parallel S_{i(n)}^{h(n)}{u}_n-u_n\parallel }^2+c_{h(n)}]\\ -{\beta }_n(1-{\beta }_n){\parallel S_{i(n)}^{h(n)}{u}_n-u_n\parallel }^2\\ \le & (1+d_{h(n)}){\parallel u_n-p\parallel }^2-{\beta }_n(1-{\beta }_n-\lambda ){\parallel S_{i(n)}^{h(n)}{u}_n-u_n\parallel }^2+{\beta }_n{c}_{h(n)}\\ \le & (1+d_{h(n)}){\parallel u_n-p\parallel }^2+{\beta }_n{c}_{h(n)}.\end{array}$$

(3.6)

By virtue of convexity of ${\parallel \cdot \parallel }^2$, one has

$$\begin{array}{rcl}{\parallel y_n-p\parallel }^2& =& {\parallel (1-{\alpha }_n)(u_n-p)+{\alpha }_n(z_n-p)\parallel }^2\\ \le & (1-{\alpha }_n){\parallel u_n-p\parallel }^2+{\alpha }_n{\parallel z_n-p\parallel }^2.\end{array}$$

(3.7)

Substituting (3.5) and (3.6) into (3.7), we obtain

$$\begin{array}{rcl}{\parallel y_n-p\parallel }^2& \le & (1-{\alpha }_n){\parallel u_n-p\parallel }^2+{\alpha }_n{\parallel z_n-p\parallel }^2\\ \le & (1-{\alpha }_n){\parallel u_n-p\parallel }^2+{\alpha }_n[(1+d_{h(n)}){\parallel u_n-p\parallel }^2+{\beta }_n{c}_{h(n)}]\\ \le & {\parallel u_n-p\parallel }^2+d_{h(n)}{\parallel u_n-p\parallel }^2+{\beta }_n{c}_{h(n)}\\ \le & {\parallel u_n-p\parallel }^2+d_{h(n)}{\parallel x_n-p\parallel }^2+c_{h(n)}\\ =& {\parallel u_n-p\parallel }^2+{\theta }_n\end{array}$$

(3.8)

$$\le {\parallel x_n-p\parallel }^2+{\theta }_n.$$

(3.9)

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.

Since Ω is a nonempty closed and convex subset of 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

$$\parallel x_n-x_1\parallel \le \parallel q-x_1\parallel =\parallel P_{\mathrm{\Omega }}{x}_1-x_1\parallel .$$

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 $n\to \mathrm{\infty }$, $\mathrm{\forall }i\in \{1,2,3,\dots ,N\}$.

Since $x_n=P_{C_n}{x}_1$ and $x_n=P_{C_n}{x}_1\in C_{n+1}\subset C_n$, we have

$$\begin{array}{rcl}0& \le & 〈x_1-x_n,x_n-x_{n+1}〉\\ =& 〈x_1-x_n,x_n-x_1+x_1-x_{n+1}〉\\ \le & -{\parallel x_1-x_n\parallel }^2+\parallel x_1-x_n\parallel \parallel x_1-x_{n+1}\parallel .\end{array}$$

(3.10)

Therefore, ${\parallel x_1-x_n\parallel }^2\le \parallel x_1-x_n\parallel \parallel x_1-x_{n+1}\parallel$, and so

$$\parallel x_n-x_1\parallel =\parallel x_1-x_n\parallel \le \parallel x_1-x_{n+1}\parallel .$$

(3.11)

Thus, the sequence $\{\parallel x_n-x_1\parallel \}$ is nondecreasing. Since $\{x_n\}$ is bounded, ${lim}_{n\to \mathrm{\infty }}\parallel x_n-x_1\parallel$ exists. On the other hand, from (3.10), we have

$$\begin{array}{rcl}{\parallel x_n-x_{n+1}\parallel }^2& =& {\parallel x_n-x_1+x_1-x_{n+1}\parallel }^2\\ =& {\parallel x_n-x_1\parallel }^2+2〈x_n-x_1,x_1-x_{n+1}〉+{\parallel x_1-x_{n+1}\parallel }^2\\ =& {\parallel x_n-x_1\parallel }^2+2〈x_n-x_1,x_1-x_n+x_n-x_{n+1}〉+{\parallel x_1-x_{n+1}\parallel }^2\\ =& {\parallel x_n-x_1\parallel }^2-2{\parallel x_n-x_1\parallel }^2+2〈x_n-x_1,x_n-x_{n+1}〉+{\parallel x_1-x_{n+1}\parallel }^2\\ \le & {\parallel x_1-x_{n+1}\parallel }^2-{\parallel x_n-x_1\parallel }^2.\end{array}$$

(3.12)

The fact that ${lim}_{n\to \mathrm{\infty }}\parallel x_n-x_1\parallel$ exists implies that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-x_{n+1}\parallel =0.$$

(3.13)

It is easy to see that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-x_{n+i}\parallel =0,\mathrm{\forall }i=1,2,3,\dots ,N.$$

Since $x_{n+1}=P_{C_{n+1}}{x}_1\in C_{n+1}$, we have

$${\parallel y_n-x_{n+1}\parallel }^2\le {\parallel x_n-x_{n+1}\parallel }^2+{\theta }_n.$$

It follows that

$$\begin{array}{rcl}{\parallel y_n-x_n\parallel }^2& =& {\parallel y_n-x_{n+1}+x_{n+1}-x_n\parallel }^2\\ =& {\parallel y_n-x_{n+1}\parallel }^2+{\parallel x_{n+1}-x_n\parallel }^2+2〈y_n-x_{n+1},x_{n+1}-x_n〉\\ \le & {\parallel x_n-x_{n+1}\parallel }^2+{\theta }_n+{\parallel x_{n+1}-x_n\parallel }^2+2〈y_n-x_{n+1},x_{n+1}-x_n〉\\ \le & 2{\parallel x_{n+1}-x_n\parallel }^2+2\parallel y_n-x_{n+1}\parallel \parallel x_{n+1}-x_n\parallel +{\theta }_n.\end{array}$$

(3.14)

Since ${\theta }_n\to 0$ as $n\to \mathrm{\infty }$ and from (3.13), we obtain

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-y_n\parallel =0.$$

(3.15)

For each $p\in \mathrm{\Omega }$, it follows from the firmly nonexpansive $T_{r_{m,n}}^{F_m}$ that for each $m\in \{1,2,3,\dots ,M\}$, we have

$$\begin{array}{rcl}{\parallel {\mathrm{\Theta }}_n^m{x}_n-p\parallel }^2& =& {\parallel T_{r_{m,n}}^{F_m}{\mathrm{\Theta }}_n^{m-1}{x}_n-T_{r_{m,n}}^{F_m}p\parallel }^2\\ \le & 〈{\mathrm{\Theta }}_n^m{x}_n-p,{\mathrm{\Theta }}_n^{m-1}{x}_n-p〉\\ =& \frac{1}{2}({\parallel {\mathrm{\Theta }}_n^m{x}_n-p\parallel }^2+{\parallel {\mathrm{\Theta }}_n^{m-1}{x}_n-p\parallel }^2-{\parallel {\mathrm{\Theta }}_n^m{x}_n-{\mathrm{\Theta }}_n^{m-1}{x}_n\parallel }^2),\\ \text{for all }1\le m\le M.\end{array}$$

Thus, we get

$${\parallel {\mathrm{\Theta }}_n^m{x}_n-p\parallel }^2\le {\parallel {\mathrm{\Theta }}_n^{m-1}{x}_n-p\parallel }^2-{\parallel {\mathrm{\Theta }}_n^m{x}_n-{\mathrm{\Theta }}_n^{m-1}{x}_n\parallel }^2,\text{for all }1\le m\le M.$$

(3.16)

This implies that for each $m\in \{1,2,3,\dots ,M\}$,

$$\begin{array}{rcl}{\parallel {\mathrm{\Theta }}_n^m{x}_n-p\parallel }^2& \le & {\parallel {\mathrm{\Theta }}_n^0{x}_n-p\parallel }^2-{\parallel {\mathrm{\Theta }}_n^m{x}_n-{\mathrm{\Theta }}_n^{m-1}{x}_n\parallel }^2-{\parallel {\mathrm{\Theta }}_n^{m-1}{x}_n-{\mathrm{\Theta }}_n^{m-2}{x}_n\parallel }^2\\ -\cdots -{\parallel {\mathrm{\Theta }}_n^2{x}_n-{\mathrm{\Theta }}_n^1{x}_n\parallel }^2-{\parallel {\mathrm{\Theta }}_n^1{x}_n-{\mathrm{\Theta }}_n^0{x}_n\parallel }^2\\ \le & {\parallel x_n-p\parallel }^2-{\parallel {\mathrm{\Theta }}_n^m{x}_n-{\mathrm{\Theta }}_n^{m-1}{x}_n\parallel }^2.\end{array}$$

Therefore, by the convexity of ${\parallel \cdot \parallel }^2$ and (3.8) and the nonexpansivity of $T_{r_{m,n}}^{F_m}$, we get

$$\begin{array}{rcl}{\parallel y_n-p\parallel }^2& \le & {\parallel u_n-p\parallel }^2+{\theta }_n\\ =& {\parallel {\mathrm{\Theta }}_n^M{x}_n-{\mathrm{\Theta }}_n^{M}p\parallel }^2+{\theta }_n\\ \le & {\parallel {\mathrm{\Theta }}_n^m{x}_n-p\parallel }^2+{\theta }_n\\ \le & {\parallel x_n-p\parallel }^2-{\parallel {\mathrm{\Theta }}_n^m{x}_n-{\mathrm{\Theta }}_n^{m-1}{x}_n\parallel }^2+{\theta }_n.\end{array}$$

It follows that

$$\begin{array}{rcl}{\parallel {\mathrm{\Theta }}_n^m{x}_n-{\mathrm{\Theta }}_n^{m-1}{x}_n\parallel }^2& \le & {\parallel x_n-p\parallel }^2-{\parallel y_n-p\parallel }^2+{\theta }_n\\ \le & \parallel x_n-y_n\parallel (\parallel x_n-p\parallel +\parallel y_n-p\parallel )+{\theta }_n.\end{array}$$

(3.17)

From (3.15) and (3.17), we obtain

$$\underset{n\to \mathrm{\infty }}{lim}\parallel {\mathrm{\Theta }}_n^m{x}_n-{\mathrm{\Theta }}_n^{m-1}{x}_n\parallel =0,\mathrm{\forall }m\in \{1,2,3,\dots ,M\}.$$

(3.18)

Then we have

$$\begin{array}{rcl}\parallel u_n-x_n\parallel & \le & \parallel u_n-{\mathrm{\Theta }}_n^{M-1}{x}_n\parallel +\parallel {\mathrm{\Theta }}_n^{M-1}{x}_n-{\mathrm{\Theta }}_n^{M-2}{x}_n\parallel \\ +\cdots +\parallel {\mathrm{\Theta }}_n^1{x}_n-{\mathrm{\Theta }}_n^0{x}_n\parallel \to 0,\text{as }n\to \mathrm{\infty }.\end{array}$$

Therefore,

$$\underset{n\to \mathrm{\infty }}{lim}\parallel u_n-x_n\parallel =0.$$

(3.19)

From (3.13) and (3.19), we get

$$\parallel u_{n+1}-u_n\parallel \le \parallel u_{n+1}-x_{n+1}\parallel +\parallel x_{n+1}-x_n\parallel +\parallel x_n-u_n\parallel \to 0,\text{as }n\to \mathrm{\infty }.$$

(3.20)

It follows that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel u_{n+i}-u_n\parallel =0,\mathrm{\forall }i\in \{1,2,3,\dots ,N\}.$$

(3.21)

Since for any positive integer $n\ge N$, we can write $n=(h(n)-1)N+i(n)$, where $i(n)\in \{1,2,3,\dots ,N\}$, note that

$$\begin{array}{rcl}\parallel u_n-S_n{u}_n\parallel & \le & \parallel u_n-S_{i(n)}^{h(n)}{u}_n\parallel +\parallel S_{i(n)}^{h(n)}{u}_n-S_n{u}_n\parallel \\ =& \parallel u_n-S_{i(n)}^{h(n)}{u}_n\parallel +\parallel S_{i(n)}^{h(n)}{u}_n-S_{i(n)}{u}_n\parallel .\end{array}$$

(3.22)

From the conditions $0<a\le {\alpha }_n\le 1$ and $0<b\le {\beta }_n\le 1-\lambda$, we get

$$\begin{array}{rcl}\parallel u_n-S_{i(n)}^{h(n)}{u}_n\parallel & =& \frac{1}{{\beta }_n}\parallel z_n-u_n\parallel \\ =& \frac{1}{{\alpha }_n{\beta }_n}\parallel y_n-u_n\parallel \\ \le & \frac{1}{ab}(\parallel y_n-x_n\parallel +\parallel x_n-u_n\parallel ).\end{array}$$

From (3.15) and (3.19), we obtain

$$\underset{n\to \mathrm{\infty }}{lim}\parallel u_n-S_{i(n)}^{h(n)}{u}_n\parallel =0.$$

(3.23)

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

Therefore, we compute

$$\begin{array}{rcl}\parallel S_{i(n)}^{h(n)-1}{u}_n-u_n\parallel & \le & \parallel S_{i(n)}^{h(n)-1}{u}_n-S_{i(n-N)}^{h(n)-1}{u}_{n-N+1}\parallel +\parallel S_{i(n-N)}^{h(n)-1}{u}_{n-N+1}-S_{i(n-N)}^{h(n-N)}{u}_{n-N}\parallel \\ +\parallel S_{i(n-N)}^{h(n-N)}{u}_{n-N}-u_{n-N}\parallel +\parallel u_{n-N}-u_{n-N+1}\parallel +\parallel u_{n-N+1}-u_n\parallel \\ =& \parallel S_{i(n)}^{h(n)-1}{u}_n-S_{i(n)}^{h(n)-1}{u}_{n-N+1}\parallel +\parallel S_{i(n-N)}^{h(n-N)}{u}_{n-N+1}-S_{i(n-N)}^{h(n-N)}{u}_{n-N}\parallel \\ +\parallel S_{i(n-N)}^{h(n-N)}{u}_{n-N}-u_{n-N}\parallel +\parallel u_{n-N}-u_{n-N+1}\parallel +\parallel u_{n-N+1}-u_n\parallel .\end{array}$$

Applying Lemma 2.9 and (3.21), we get

$$\underset{n\to \mathrm{\infty }}{lim}\parallel u_n-S_{i(n)}^{h(n)-1}{u}_n\parallel =0.$$

(3.24)

From (3.22) and (3.24), it follows that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel u_n-S_n{u}_n\parallel =0.$$

(3.25)

Since

$$\parallel u_n-S_{n+i}{u}_n\parallel \le \parallel u_n-u_{n+i}\parallel +\parallel u_{n+i}-S_{n+i}{u}_{n+i}\parallel +\parallel S_{n+i}{u}_{n+i}-S_{n+i}{u}_n\parallel \to 0,\text{as }n\to \mathrm{\infty }$$

for any $i\in \{1,2,3,\dots ,N\}$, which gives that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel u_n-S_i{u}_n\parallel =0,\mathrm{\forall }i\in \{1,2,3,\dots ,N\}.$$

(3.26)

Moreover, for each $i\in \{1,2,3,\dots ,N\}$, we obtain

$$\parallel x_n-S_i{x}_n\parallel \le \parallel x_n-u_n\parallel +\parallel u_n-S_i{u}_n\parallel +\parallel S_i{u}_n-S_i{x}_n\parallel \to 0,\text{as }n\to \mathrm{\infty }.$$

This implies that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-S_i{x}_n\parallel =0,\mathrm{\forall }i\in \{1,2,3,\dots ,N\}.$$

(3.27)

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 $\{x_n\}$.

Note that $S_i$ is uniformly Lipschitz continuous and (3.27), we obtain

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-S_i^k{x}_n\parallel =0,\mathrm{\forall }k\in \mathbb{N}.$$

(3.28)

It follows from Lemma 2.7 that

$$p^{\ast }\in \bigcap _{i=1}^{N}F(S_i).$$

(3.29)

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

By Lemma 2.3, for each $m\in \{1,2,3,\dots ,M\}$, we have

$$F_m({\mathrm{\Theta }}_n^m{x}_n,y)+\frac{1}{r_n}〈y-{\mathrm{\Theta }}_n^m{x}_n,{\mathrm{\Theta }}_n^m{x}_n-{\mathrm{\Theta }}_n^{m-1}{x}_n〉\ge 0,\mathrm{\forall }y\in C.$$

From (A2), we get

$$\frac{1}{r_n}〈y-{\mathrm{\Theta }}_n^m{x}_n,{\mathrm{\Theta }}_n^m{x}_n-{\mathrm{\Theta }}_n^{m-1}{x}_n〉\ge F_m(y,{\mathrm{\Theta }}_n^m{x}_n),\mathrm{\forall }y\in C.$$

Taking $n=n_j$, we get

$$〈y-{\mathrm{\Theta }}_{n_j}^m{x}_{n_j},\frac{{\mathrm{\Theta }}_{n_j}^m{x}_{n_j}-{\mathrm{\Theta }}_{n_j}^{m-1}{x}_{n_j}}{r_{n_j}}〉\ge F_m(y,{\mathrm{\Theta }}_{n_j}^m{x}_{n_j}),\mathrm{\forall }y\in C.$$

From (3.18), we obtain that ${\mathrm{\Theta }}_{n_j}^m{x}_{n_j}\rightharpoonup p^{\ast }$ as $j\to \mathrm{\infty }$ for each $m\in \{1,2,3,\dots ,M\}$ (especially $u_{n_j}={\mathrm{\Theta }}_{n_j}^M{x}_{n_j}$). Considering this together with (3.18) and (A4), we have for each $m\in \{1,2,3,\dots ,M\}$ that

$$0\ge F_m(y,p^{\ast }),\mathrm{\forall }y\in C.$$

For any $0<t\le 1$ and $y\in C$, we let $y_t=ty+(1-t)p^{\ast }$. Since $y\in C$ and $p^{\ast }\in C$, we obtain that $y_t\in C$, and so $F_m(y_t,p^{\ast })\le 0$. It follows that

$$0=F_m(y_t,y_t)\le t{F}_m(y_t,y)+(1-t)F_m(y_t,p^{\ast })\le t{F}_m(y_t,y).$$

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

$$F_m(y_t,y)\ge 0,\mathrm{\forall }y\in C.$$

Letting $t\to 0$, from (A3), we get

$$F_m(p^{\ast },y)\ge 0,\mathrm{\forall }y\in C.$$

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 $P_{\mathrm{\Omega }}{x}_1$.

Set $p^{\ast }=P_{\mathrm{\Omega }}(x_1)$, then

$$\parallel x_{n+1}-x_1\parallel \le \parallel p^{\ast }-x_1\parallel ,\mathrm{\forall }n\in \mathbb{N}.$$

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:

$$\{\begin{array}{c}{x}_1\in C\mathit{\text{chosen arbitrarily}},\hfill \\ C_1=H,\hfill \\ u_n=T_{r_n}^F{x}_n,\hfill \\ z_n=(1-{\beta }_n)u_n+{\beta }_n{S}_{i(n)}^{h(n)}{u}_n,\hfill \\ y_n=(1-{\alpha }_n)u_n+{\alpha }_n{z}_n,\hfill \\ C_{n+1}=\{w\in C_n:{\parallel y_n-w\parallel }^2\le {\parallel x_n-w\parallel }^2+{\theta }_n\},\hfill \\ x_{n+1}=P_{C_{n+1}}(x_1),\mathrm{\forall }n\in \mathbb{N},\hfill \end{array}$$

(4.1)

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:

$$\{\begin{array}{c}{x}_1\in C\mathit{\text{chosen arbitrarily}},\hfill \\ C_1=H,\hfill \\ z_n=(1-{\beta }_n)x_n+{\beta }_n{S}_{i(n)}^{h(n)}{x}_n,\hfill \\ y_n=(1-{\alpha }_n)x_n+{\alpha }_n{z}_n,\hfill \\ C_{n+1}=\{w\in C_n:{\parallel y_n-w\parallel }^2\le {\parallel x_n-w\parallel }^2+{\theta }_n\},\hfill \\ x_{n+1}=P_{C_{n+1}}(x_1),\mathrm{\forall }n\in \mathbb{N},\hfill \end{array}$$

(4.2)

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

$$Sx=\{\begin{array}{cc}kx,\hfill & \text{if }x\in [0,\frac{1}{2}];\hfill \\ 0,\hfill & \text{if }x\in (\frac{1}{2},1],\hfill \end{array}$$

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 each $x,y\in C_1$, we have

$$|S^{n}x-S^{n}y|=k^n|x-y|\le |x-y|,\mathrm{\forall }x,y\in C_1\text{ and }\mathrm{\forall }n\in \mathbb{N}.$$

For each $x,y\in C_2$, we have

$$|S^{n}x-S^{n}y|=0\le |x-y|,\mathrm{\forall }x,y\in C_2\text{ and }\mathrm{\forall }n\in \mathbb{N}.$$

For each $x\in C_1$ and $y\in C_2$, we have

$$\begin{array}{rcl}|S^{n}x-S^{n}y|& =& |k^{n}x-0|\\ \le & |k^n(x-y)+k^{n}y|\\ \le & k^n|x-y|+k^n|y|\\ \le & |x-y|+k^n,\mathrm{\forall }n\in \mathbb{N}.\end{array}$$

It follows that

$$\begin{array}{rcl}{|S^{n}x-S^{n}y|}^2& \le & {(|x-y|+k^n)}^2\\ \le & {|x-y|}^2+k{|x-S^{n}x-(y-S^{n}y)|}^2+k^{n}K,\end{array}$$

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.

Under the above assumption in Theorem 3.1 is simplified as follows:

$$\{\begin{array}{c}{x}_1\in H\text{ chosen arbitrarily},\hfill \\ C_1=H,\hfill \\ z_n=k{x}_n+(1-k)S^n{x}_n,\hfill \\ y_n=(\frac{n-1}{2n})x_n+(\frac{n+1}{2n})z_n,\hfill \\ C_{n+1}=\{w\in C_n:{\parallel y_n-w\parallel }^2\le {\parallel x_n-w\parallel }^2+{\theta }_n\},\hfill \\ x_{n+1}=P_{C_{n+1}}{x}_1,\mathrm{\forall }n\in \mathbb{N}.\hfill \end{array}$$

(5.1)

In fact, in one-dimensional case, $C_{n+1}$ is a closed interval. If we set $[a_{n+1},b_{n+1}]:=C_{n+1}$, then the projection point $x_{n+1}$ of $x_1\in C$ onto $C_{n+1}$ can be expressed as

$$x_{n+1}=P_{\mathrm{\Omega }}(x_1)\{\begin{array}{cc}{x}_1,\hfill & \text{if }{x}_1\in [a_{n+1},b_{n+1}];\hfill \\ b_{n+1},\hfill & \text{if }{x}_1>b_{n+1};\hfill \\ a_{n+1},\hfill & \text{if }{x}_1<a_{n+1}.\hfill \end{array}$$

The numerical results for an initial guess $x_1=0.2,0.5,0.8$ are shown in Table 1. From the table, we see that the iterations converge to 0 which is the unique fixed point of S. The convergence of each iteration is also shown in Figure 1 for comparison.

Figure 1

The convergence comparison of different initial values ${\mathit{x}}_{\mathbf{1}}$ .

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