The Modified Inertial Iterative Algorithm for Solving Split Variational Inclusion Problem for Multi-Valued Quasi Nonexpansive Mappings with Some Applications

Abstract

Based on the very recent work by Shehu and Agbebaku in Comput. Appl. Math. 2017, we introduce an extension of their iterative algorithm by combining it with inertial extrapolation for solving split inclusion problems and fixed point problems. Under suitable conditions, we prove that the proposed algorithm converges strongly to common elements of the solution set of the split inclusion problems and fixed point problems.

1. Introduction

The split monotone variational inclusion problem (SMVIP) was introduced by Moudafi [1]. This problem is as follows:

$$\mathrm{Find}\mathrm{a}\mathrm{point}{x}^{*}\in H_1\mathrm{such}\mathrm{that}0\in \widehat{f}\left(x^{*}\right)+B_1\left(x^{*}\right)$$

(1)

and such that

$$y^{*}=A{x}^{*}\in H_2\mathrm{solves}0\in \widehat{g}\left(y^{*}\right)+B_2\left(y^{*}\right),$$

(2)

where 0 is the zero vector, $H_1$ and $H_2$ are real Hilbert spaces, $\widehat{f}$ and $\widehat{g}$ are given single-valued operators defined on $H_1$ and $H_2$, respectively, $B_1$ and $B_2$ are multi-valued maximal monotone mappings defined on $H_1$ and $H_2$, respectively, and A is a bounded linear operator defined on $H_1$ to $H_2$.

It is well known (see [1]) that

$$0\in \widehat{f}\left(x^{*}\right)+B_1\left(x^{*}\right)⟺x^{*}=J_{\lambda }^{B_1}(x^{*}-\lambda \widehat{f}\left(x^{*}\right)),$$

and that

$$0\in \widehat{g}\left(y^{*}\right)+B_2\left(y^{*}\right)⟺y^{*}=J_{\lambda }^{B_2}(y^{*}-\lambda \widehat{g}\left(y^{*}\right)),y^{*}=A{x}^{*},$$

where $J_{\lambda }^{B_1}:={(I+\lambda B_1)}^{-1}$ and $J_{\lambda }^{B_2}:={(I+\lambda B_2)}^{-1}$ are the resolvent operators of $B_1$ and $B_2$, respectively, with $\lambda >0$. Note that $J_{\lambda }^{B_1}$ and $J_{\lambda }^{B_2}$ are nonexpansive and firmly nonexpansive.

Recently, Shehu and Agbebaku [2] proposed an algorithm involving a step-size selected and proved strong convergence theorem for split inclusion problem and fixed point problem for multi-valued quasi-nonexpansive mappings. In [1], Moudafi pointed out that the problem (SMVIP) [3,4,5] includes, as special cases, the split variational inequality problem [6], the split zero problem, the split common fixed point problem [7,8,9] and the split feasibility problem [10,11], which have already been studied and used in image processing and recovery [12], sensor networks in computerized tomography and data compression for models of inverse problems [13].

If $\widehat{f}\equiv 0$ and $\widehat{g}\equiv 0$ in the problem (SMVIP), then the problem reduces to the split variational inclusion problem (SVIP) as follows:

$$\mathrm{Find}\mathrm{a}\mathrm{point}{x}^{*}\in H_1\mathrm{such}\mathrm{that}0\in B_1\left(x^{*}\right)$$

(3)

and such that

$$y^{*}=A{x}^{*}\in H_2\mathrm{solves}0\in B_2\left(y^{*}\right).$$

(4)

Note that the problem (SVIP) is equivalent to the following problem:

$$\mathrm{Find}\mathrm{a}\mathrm{point}{x}^{*}\in H_1\mathrm{such}\mathrm{that}{x}^{*}=J_{\lambda }^{B_1}\left(x^{*}\right)\mathrm{and}{y}^{*}=J_{\lambda }^{B_2}\left(y^{*}\right),y^{*}=A{x}^{*}$$

for some $\lambda >0$.

We denote the solution set of the problem (SVIP) by $\Omega$, i.e.,

$$\Omega =\{x^{*}\in H_1:0\in B_1\left(x^{*}\right)\mathrm{and}0\in B_2\left(y^{*}\right),y^{*}=A{x}^{*}\}.$$

Many works have been developed to solve the split variational inclusion problem (SVIP). In 2002, Byrne et al. [7] introduced the iterative method $\left\{x_n\right\}$ as follows: For any $x_0\in H_1$,

$$x_{n+1}=J_{\lambda }^{B_1}(x_n+\gamma A^{*}(J_{\lambda }^{B_2}-I)A{x}_n)$$

(5)

for each $n\ge 0$, where $A^{*}$ is the adjoint of the bounded linear operator A, $\gamma \in (0,2/L)$, $L=\parallel A^{*}A\parallel$ and $\lambda >0$. They have shown the weak and strong convergence of the above iterative method for solving the problem (SVIP).

Later, inspired by the above iterative algorithm, many authors have extended the algorithm $\left\{x_n\right\}$ generated by (5). In particular, Kazmi and Rizvi [4] proposed an algorithm $\left\{x_n\right\}$ for approximating a solution of the problem (SVIP) as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}_n=J_{\lambda }^{B_1}(x_n+{\gamma }_n{A}^{*}(J_{\lambda }^{B_2}-I)A{x}_n),\hfill \\ x_{n+1}={\alpha }_n{f}_n\left(x_n\right)+(1-{\alpha }_n)S{u}_n\hfill \end{array}\right.\end{array}$$

(6)

for each $n\ge 0$, where $\left\{{\alpha }_n\right\}$ is a sequence in $(0,1)$, $\lambda >0$, $\gamma \in (0,1/L)$, L is the spectral radius of the operator $A^{*}A$, $f:H_1\to H_1$ is a contraction and $S:H_1\to H_1$ is a nonexpansive mapping. In 2015, Sitthithakerngkiet et al. [5] proposed an algorithm $\left\{x_n\right\}$ for solving the problem (SVIP) and the fixed point problem (FPP) of a countable family of nonexpansive mappings as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{y}_n=J_{\lambda }^{B_1}(x_n+{\gamma }_n{A}^{*}(J_{\lambda }^{B_2}-I)A{x}_n),\hfill \\ x_{n+1}={\alpha }_{n}f\left(x_n\right)+(1-{\alpha }_{n}D)S_n{y}_n\hfill \end{array}\right.\end{array}$$

(7)

for each $n\ge 0$, where $\left\{{\alpha }_n\right\}$ is a sequence in $(0,1)$, $\lambda >0$, $\gamma \in (0,1/L)$, L is the spectral radius of the operator $A^{*}A$, $f:H_1\to H_1$ is a contraction, $D:H_1\to H_2$ is strongly positive bounded linear operator and, for each $n\ge 1$, $S_n:H_1\to H_1$ is a nonexpansive mapping.

In both their works, they obtained some strong convergence results by using their proposed iterative methods (for some more results on algorithms, see [14,15]).

Recall that a point $x^{*}\in H_1$ is called a fixed point of a given multi-valued mapping $S:H_1\to 2^{H_1}$ if

$$x^{*}\in S{x}^{*}$$

(8)

and the fixed point problem (FPP) for a multi-valued mapping $S:H_1\to 2^{H_1}$ is as follows:

$$\mathrm{Find}\mathrm{a}\mathrm{point}{x}^{*}\in H_1\mathrm{such}\mathrm{that}{x}^{*}\in S{x}^{*}.$$

The set of fixed points of the multi-valued mapping S is denoted by $F\left(S\right)$.

As applications, the fixed point theory for multi-valued mappings was applied to various fields, especially mathematical economics and game theory (see [16,17,18]).

Recently, motivated by the results of Byrne et al. [7], Kazmi and Rizvi [4] and Sitthithakerngkiet [5], Shehu and Agbebaku [2] introduced the split fixed point inclusion problem (SFPIP) from the problems (SVIP) and (FPP) for a multi-valued quasi-nonexpansive mapping $S:H_1\to 2^{H_1}$ as follows:

$$\mathrm{Find}\mathrm{a}\mathrm{point}{x}^{*}\in H_1\mathrm{such}\mathrm{that}0\in B_1\left(x^{*}\right),x^{*}\in S{x}^{*}$$

(9)

and such that

$$y^{*}=A{x}^{*}\in H_2\mathrm{solves}0\in B_2\left(y^{*}\right),$$

(10)

where $H_1$ and $H_2$ are real Hilbert spaces, $B_1$ and $B_2$ are multi-valued maximal monotone mappings defined on $H_1$ and $H_2$, respectively, and A is a bounded linear operator defined on $H_1$ to $H_2$.

Note that the problem (SFPIP) is equivalent to the following problem: for some $\lambda >0$,

$$\mathrm{Find}\mathrm{a}\mathrm{point}{x}^{*}\in H_1\mathrm{such}\mathrm{that}{x}^{*}=J_{\lambda }^{B_1}\left(x^{*}\right),x^{*}\in S{x}^{*}\mathrm{and}A{x}^{*}=J_{\lambda }^{B_2}\left(A{x}^{*}\right).$$

The solution set of the problem (SFPIP) is denoted by $F\left(S\right)\bigcap \Omega$, i.e.,

$$F\left(S\right)\bigcap \Omega =\{x^{*}\in H_1:0\in B_1\left(x^{*}\right),x^{*}\in S{x}^{*}\mathrm{and}0\in B_2\left(A{x}^{*}\right)\}.$$

Notice that, if S is the identity operator, then the problem (SFPIP) reduces to the problem (SVIP). Moreover, if $J_{\lambda }^{B_1}=J_{\lambda }^{B_2}=A=I$, then the problem (SFPIP) reduces to the problem (FPP) for a multi-valued quasi-nonexpansive mapping.

Furthermore, Shehu and Agbebaku [2] introduced an algorithm $\left\{x_n\right\}$ for solving the problem (SFPIP) for a multi-valued quai-nonexpasive mapping S as follows: For any $x_1\in H_1$,

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}_n=J_{\lambda }^{B_1}(x_n+{\gamma }_n{A}^{*}(J_{\lambda }^{B_2}-1)A{x}_n),\hfill \\ x_{n+1}={\alpha }_n{f}_n\left(x_n\right)+{\beta }_n{x}_n+{\delta }_n(\sigma w_n+(1-\sigma )u_n),w_n\in S{x}_n,\hfill \end{array}\right.\end{array}$$

(11)

for each $n\ge 1$, where $\left\{{\alpha }_n\right\}$, $\left\{{\beta }_n\right\}$ and $\left\{{\delta }_n\right\}$ are the real sequences in $(0,1)$ such that

$${\alpha }_n+{\beta }_n+{\delta }_n=1,\sigma \in (0,1),{\gamma }_n:={\tau }_n\frac{\parallel (J_{\lambda }^{B_2}-I)A{x}_n{\parallel }^2}{\parallel A^{*}(J_{\lambda }^{B_2}-I){\parallel }^2},$$

where $0<a\le {\tau }_n\le b<1$, and $\left\{f_n\left(x\right)\right\}$ is the uniform convergence sequence for any x in a bounded subset D of $H_1$, and proved that the sequences $\left\{u_n\right\}$ and $\left\{x_n\right\}$ generated by (11) both converge strongly to $p\in F\left(S\right)\cap \Omega$, where $p=P_{F\left(S\right)\cap \Omega }f\left(p\right)$.

In optimization theory, the second-order dynamical system, which is called the heavy ball method, is used to accelerate the convergence rate of algorithms. This method is a two-step iterative method for minimizing a smooth convex function which was firstly introduced by Polyak [19].

The following is a modified heavy ball method for the improvement of the convergence rate, which was introduced by Nesterov [20]:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{y}_n=x_n+{\theta }_n(x_n-x_{n-1}),\hfill \\ x_{n+1}=y_n-{\lambda }_n\nabla f\left(y_n\right)\hfill \end{array}\right.\end{array}$$

for each $n\ge 1$, where ${\lambda }_n>0$, ${\theta }_n\in [0,1)$ is an extrapolation factor. Here, the term ${\theta }_n(x_n-x_{n-1})$ is the inertia (for more recent results on the inertial algorithms, see [21,22]).

The following method is called the inertial proximal point algorithm, which was introduced by Alvarez and Attouch [23]. This method combined the proximal point algorithm [24] with the inertial extrapolation [25,26]:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{y}_n=x_n+{\theta }_n(x_n-x_{n-1}),\hfill \\ x_{n+1}={(I+{\lambda }_n\widehat{T})}^{-1}\left(y_n\right)\hfill \end{array}\right.\end{array}$$

(12)

for each $n\ge 1$, where I is identity operator and $\widehat{T}$ is a maximal monotone operator. It was proven that, if a positive sequence ${\lambda }_n$ is non-decreasing, ${\theta }_n\in [0,1)$ and the following summability condition holds:

$$\begin{array}{c}\hfill \sum _{n=1}^{\infty }{\theta }_n{\parallel x_n-x_{n-1}\parallel }^2<\infty ,\end{array}$$

(13)

then $\left\{x_n\right\}$ generated by (12) converges to a zero point of T.

In fact, recently, some authors have pointed out some problems in this summability condition (13) given in [27], that is, to satisfy this summability condition (13) of the sequence $\left\{x_n\right\}$, one needs to calculate $\left\{{\theta }_n\right\}$ at each step. Recently, Bot et al. [28] improved this condition, that is, they got rid of the summability condition (13) and replaced the other conditions.

In this paper, inspired by the results of Shehu and Agbebaku [2], Nesterov [20] and Alvarez and Attouch [23], we proposed a new algorithm by combining the iterative algorithm (11) with the inertial extrapolation for solving the problem (SFPIP) and prove some strong convergence theorems of the proposed algorithm to show the existence of a solution of the problem (SFPIP). Furthermore, as applications, we consider our proposed algorithm for solving the variational inequality problem and give some applications in game theory.

2. Preliminaries

In this section, we recall some definitions and results which will be used in the proof of the main results.

Let $H_1$ and $H_2$ be two real Hilbert spaces with the inner product $⟨·,·⟩$ and the norm $\parallel ·\parallel$. Let C be a nonempty closed and convex subset of $H_1$ and D be a nonempty bounded subset of $H_1$. Let $A:H_1\to H_2$ be a bounded linear operator and $A^{*}:H_2\to H_1$ be the adjoint of A.

Let $\left\{x_n\right\}$ be a sequence in H, we denote the strong and weak convergence of a sequence $\left\{x_n\right\}$ by $x_n\to x$ and $x_n\rightharpoonup x$, respectively.

Recall that a mapping $T:C\to C$ is said to be:

(1)

Lipschitz if there exists a positive constant $\alpha$ such that, for all $x,y\in C$,

$$\parallel Tx-Ty\parallel \le \alpha \parallel x-y\parallel .$$

If $\alpha \in (0,1)$ and $\alpha =1$, then the mapping T is contractive and nonexpansive, respectively.

(2)

firmly nonexpansive if

$${\parallel Tx-Ty\parallel }^2\le ⟨Tx-Ty,x-y⟩$$

for all $x,y\in C$.

A mapping $P_C$ is said to be the metric projection of $H_1$ onto C if, for all point $x\in H_1$, there exists a unique nearest point in C, denoted by $P_{C}x$, such that

$$\parallel x-P_{C}x\parallel \le \parallel x-y\parallel$$

for all $y\in C$.

It is well known that $P_C$ is nonexpansive mapping and satisfies

$$⟨x-y,P_{C}x-P_{C}y⟩\le {\parallel P_{C}x-P_{C}y\parallel }^2$$

for all $x,y\in H_1$. Moreover, $P_{C}x$ is characterized by the fact $P_{C}x\in C$ and

$$⟨x-P_{C}x,y-P_{C}x⟩\le 0$$

for all $y\in C$ and $x\in H_1$ (see [6,22]).

A multi-valued mapping $B_1:H_1\to 2^{H_1}$ is said to be monotone if, for all $x,y\in H_1$, $u\in B_1\left(x\right)$ and $v\in B_1\left(y\right)$,

$$⟨x-y,u-v⟩\ge 0.$$

A monotone mapping $B_1:H_1\to 2^{H_1}$ is said to be maximal if the graph $G\left(B_1\right)$ of $B_1$ is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping $B_1$ is maximal if and only if, for all $(x,u)\in H_1\times H_1$,

$$⟨x-y,u-v⟩\ge 0$$

for all $(y,v)\in G\left(B_1\right)$ implies that $u\in B_1\left(x\right)$.

Let $B_1:H_1\to 2^{H_1}$ be a multi-valued maximal monotone mapping. Then the resolvent mapping$J_{\lambda }^{B_1}:H_1\to H_1$ associated with $B_1$ is defined by

$$J_{\lambda }^{B_1}\left(x\right):={(I+\lambda B_1)}^{-1}\left(x\right)$$

for all $x\in H_1$ and for some $\lambda >0$, where I is the identity operator on $H_1$. It is well known that, for any $\lambda >0$, the resolvent operator $J_{\lambda }^{B_1}$ is single-valued firmly nonexpansive (see [2,5,6,14]).

Definition 1.

Suppose that $\left\{f_n\left(x\right)\right\}$ is a sequence of functions defined on a bounded set D. Then $f_n\left(x\right)$ converges uniformly to the function $f\left(x\right)$ on D if, for all $x\in D$,

$$f_n\left(x\right)\to f\left(x\right)\mathrm{as}n\to \infty .$$

Let $f_n:D\to H_1$ be a uniformly convergent sequence of contraction mappings on D, i.e., there exists ${\mu }_n\in (0,1)$ such that

$$f_n\left(x\right)-f_n\left(y\right)\parallel \le {\mu }_n\parallel x-y\parallel$$

for all $x,y\in D$.

Let $CB\left(H_1\right)$ denote the family of nonempty closed and bounded subsets of $H_1$. The Hausdorff metric on $CB\left(H_1\right)$ is defined by

$$\widehat{H}(x,y)=max\left\{\underset{x\in A}{sup}\underset{y\in B}{inf}\parallel x-y\parallel ,\underset{y\in B}{sup}\underset{x\in A}{inf}\parallel x-y\parallel \right\}$$

for all $A,B\in CB\left(H_1\right)$ (see [18]).

Definition 2.

[2] Let $S:H_1\to CB\left(H_1\right)$ be a multi-valued mapping. Assume that $p\in H_1$ is a fixed point of S, that is, $p\in Sp$. The mapping S is said to be:

(1) 

nonexpansive if, for all $x,y\in H_1$,

$$\widehat{H}(Sx,Sy)\le \parallel x-y\parallel .$$

(2) 

quasi-nonexpansive if $F\left(S\right)\ne \varnothing$ and, for all $x\in H_1$ and $p\in F\left(S\right)$,

$$\widehat{H}(Sx,Sp)\le \parallel x-p\parallel$$

Definition 3.

[2] A single-valued mapping $S:H\to H$ is said to be demiclosed at the origin if, for any sequence $\left\{x_n\right\}\subset H$ with $x_n\rightharpoonup x$ and $S{x}_n\to 0$, we have $Sx=0$.

Definition 4.

[2] A multi-valued mapping $S:H_1\to CB\left(H_1\right)$ is said to be demiclosed at the origin if, for any sequence $\left\{x_n\right\}\subset H$ with $x_n\rightharpoonup x$ and $d(x_n,S{x}_n)\to 0$, we have $x\in Sx$.

Lemma 1.

[29,30] Let H be a Hilbert space. Then, for any $x,y,z\in X$ and $\alpha ,\beta ,\gamma \in [0,1]$ with $\alpha +\beta +\gamma =1$, we have

$$\begin{array}{c}\hfill {\parallel \alpha x+\beta y+\gamma z\parallel }^2={\alpha \parallel x\parallel }^2+{\beta \parallel y\parallel }^2+{\gamma \parallel z\parallel }^2{-\alpha \beta \parallel x-y\parallel }^2{-\alpha \gamma \parallel x-z\parallel }^2-\beta \gamma {\parallel y-z\parallel }^2.\end{array}$$

Lemma 2.

[2,31] Let H be a real Hilbert space. Then the following results hold:

(1)

${\parallel x-y\parallel }^2={\parallel x\parallel }^2-2⟨x,y⟩+{\parallel y\parallel }^2$.

(2)

${\parallel x+y\parallel }^2={\parallel x\parallel }^2+2⟨x,y⟩+{\parallel y\parallel }^2$.

(3)

${\parallel x+y\parallel }^2\le {\parallel x\parallel }^2+2⟨y,x+y⟩$ for all $x,y\in H$.

Lemma 3.

[2,32,33] Let $\left\{a_n\right\},\left\{c_n\right\}\subset {\mathbb{R}}_{+}$, $\left\{{\sigma }_n\right\}\subset (0,1)$ and $\left\{b_n\right\}\subset \mathbb{R}$ be sequences such that

$$a_{n+1}\le (1-{\sigma }_n)a_n+b_n+c_n\mathrm{for}\mathrm{all}n\ge 0.$$

Assume ${\sum }_{n=0}^{\infty }\left|c_n\right|<\infty$. Then the following results hold:

(1)

If $b_n\le \beta {\sigma }_n$ for some $\beta \ge 0$, then $\left\{a_n\right\}$ is a bounded sequence.

(2)

If we have

$$\sum _{n=0}^{\infty }{\sigma }_n=\infty \mathrm{and}\underset{n\to \infty }{lim\; sup}\frac{b_n}{{\sigma }_n}\le 0,$$

then ${lim}_{n\to \infty }{a}_n=0$.

Lemma 4.

[32,33] Let $\left\{s_n\right\}$ be a sequence of non-negative real numbers such that

$$s_{n+1}\le (1-{\lambda }_n)s_n+{\lambda }_n{t}_n+r_n$$

for each $n\ge 1$, where

(a)

$\left\{{\lambda }_n\right\}\subset [0,1]$ and ${\sum }_{n=1}^{\infty }{\lambda }_n=\infty$;

(b)

$limsup{t}_n\le 0$;

(c)

$r_n\ge 0$ and ${\sum }_{n=1}^{\infty }{r}_n<\infty$.

Then $s_n\to 0$ as $n\to \infty$.

3. The Main Results

In this section, we prove some strong convergence theorems of the proposed algorithm for solving the problem (SFPIP).

Theorem 1.

Let $H_1$, $H_2$ be two real Hilbert spaces, $A:H_1\to H_2$ be bounded operator with adjoint operator $A^{*}$ and $B_1:H_1\to 2^{H_1}$, $B_2:H_2\to 2^{H_2}$ be maximal monotone mappings. Let $S:H_1\to CB\left(H_1\right)$ be a multi-valued quasi-nonexpansive mapping and S be demiclosed at the origin. Let $\left\{f_n\right\}$ be a sequence of ${\mu }_n$-contractions $f_n:H_1\to H_1$ with $0<{\mu }_{*}\le {\mu }_n\le {\mu }^{*}<1$ and $\left\{f_n\left(x\right)\right\}$ be uniformly convergent for any x in a bounded subset D of $H_1$. Suppose that $F\left(S\right)\cap \Omega \ne \varnothing$. For any $x_0,x_1\in H_1$, let the sequences $\left\{y_n\right\}$, $\left\{u_n\right\}$, $\left\{z_n\right\}$ and $\left\{x_n\right\}$ be generated by

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{y}_n=x_n+{\theta }_n(x_n-x_{n-1}),\hfill \\ u_n=J_{\lambda }^{B_1}(y_n+{\gamma }_n{A}^{*}(J_{\lambda }^{B_2}-I)A{y}_n),\hfill \\ z_n=\xi v_n+(1-\xi )u_n,v_n\in S{x}_n,\hfill \\ x_{n+1}={\alpha }_n{f}_n\left(x_n\right)+{\beta }_n{x}_n+{\delta }_n{z}_n\hfill \end{array}\right.\end{array}$$

(14)

for each $n\ge 1$, where $\xi \in (0,1)$, ${\gamma }_n:={\tau }_n\frac{\parallel (J_{\lambda }^{B_2}-I)A{y}_n{\parallel }^2}{\parallel A^{*}(J_{\lambda }^{B_2}-I)A{y}_n{\parallel }^2}$ with $0<{\tau }_{*}\le {\tau }_n\le {\tau }^{*}<1$, $\left\{{\theta }_n\right\}\subset [0,\overline{\omega })$ for some $\overline{\omega }>0$ and $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\},\left\{{\delta }_n\right\}\in (0,1)$ with ${\alpha }_n+{\beta }_n+{\delta }_n=1$ satisfying the following conditions:

(C1)

${lim}_{n\to \infty }{\alpha }_n=0$;

(C2)

${\sum }_{n=1}^{\infty }{\alpha }_n=\infty$;

(C3)

$0<{ϵ}_1\le {\beta }_n$ and $0<{ϵ}_2\le {\delta }_n$;

(C4)

${lim}_{n\to \infty }\frac{{\theta }_n}{{\alpha }_n}\parallel x_n-x_{n-1}\parallel =0$.

Then $\left\{x_n\right\}$ generated by $\left(14\right)$ converges strongly to $p\in F\left(S\right)\cap \Omega$, where $p=P_{F\left(S\right)\cap \Omega }f\left(p\right)$.

Proof. 

First, we show that $\left\{x_n\right\}$ is bounded. Let $p=P_{F\left(S\right)\cap \Omega }f\left(p\right)$. Then $p\in F\left(S\right)\cap \Omega$ and so $J_{\lambda }^{B_1}p=p$ and $J_{\lambda }^{B_2}Ap=Ap$. By the triangle inequality, we get

$$\begin{array}{c}\parallel y_n-p\parallel =\parallel x_n+{\theta }_n(x_n-x_{n-1})-p\parallel \hfill \\ \hfill \le \parallel x_n-p\parallel +{\theta }_n\parallel x_n-x_{n-1}\parallel .\end{array}$$

(15)

By the Cauchy-Schwarz inequality and Lemma 2 (1) and (2), we get

$$\begin{array}{c}\parallel y_n{-p\parallel }^2=\parallel x_n+{\theta }_n(x_n-x_{n-1}){-p\parallel }^2\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}=\parallel x_n{-p\parallel }^2+{\theta }_n^2{\parallel x_n-x_{n-1}\parallel }^2+2{\theta }_n⟨x_n-p,x_n-x_{n-1}⟩\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\le \parallel x_n{-p\parallel }^2+{\theta }_n^2\parallel x_n-x_{n-1}{\parallel }^2+2{\theta }_n\parallel x_n-x_{n-1}\parallel \parallel x_n-p\parallel .\hfill \end{array}$$

(16)

By using (15) and the fact that S is quasi-nonexpansive S, we get

$$\begin{array}{c}\parallel z_n-p\parallel =\parallel \xi v_n+(1-\xi )u_n-p\parallel \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}=\parallel \xi (v_n-p)+(1-\xi )(u_n-p)\parallel \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\le \xi \parallel v_n-p\parallel +(1-\xi )\parallel u_n-p\parallel \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\le \xi d(v_n,Sp)+(1-\xi )\parallel y_n-p\parallel \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\le \xi \widehat{H}(S{x}_n,Sp)+(1-\xi )[\parallel x_n-p\parallel +{\theta }_n\parallel x_n-x_{n-1}\parallel ]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\le \xi \parallel x_n-p\parallel +(1-\xi )\parallel x_n-p\parallel +(1-\xi ){\theta }_n\parallel x_n-x_{n-1}\parallel \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\le \parallel x_n-p\parallel +{\theta }_n\parallel x_n-x_{n-1}\parallel ,\hfill \end{array}$$

(17)

which implies that

$$\begin{array}{cc}\parallel z_n{-p\parallel }^2\hfill & \le (\parallel x_n-p\parallel +{\theta }_n\parallel x_n-x_{n-1}{\parallel )}^2\hfill \\ & \hfill =\parallel x_n{-p\parallel }^2+2{\theta }_n\parallel x_n-x_{n-1}\parallel \parallel x_n-p\parallel +{\theta }_n^2{\parallel x_n-x_{n-1}\parallel }^2.\end{array}$$

(18)

Since $J_{\lambda }^{B_1}$ is nonexpansive, by Lemma 2 (2), we get

$$\begin{array}{c}\parallel u_n{-p\parallel }^2=\parallel J_{\lambda }^{B_I}(y_n+{\gamma }_n{A}^{*}(J_{\lambda }^{B_2}-I)A{y}_n){-p\parallel }^2\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}=\parallel J_{\lambda }^{B_1}(y_n+{\gamma }_n{A}^{*}(J_{\lambda }^{B_2}-I)A{y}_n)-J_{\lambda }^{B_1}{p\parallel }^2\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\le \parallel y_n+{\gamma }_n{A}^{*}(J_{\lambda }^{B_2}-I)A{y}_n{-p\parallel }^2\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}=\parallel y_n{-p\parallel }^2+{\gamma }_n^2{\parallel A^{*}(J_{\lambda }^{B_2}-I)A{y}_n\parallel }^2+2{\gamma }_n⟨y_n-p,A^{*}(J_{\lambda }^{B_2}-I)A{y}_n⟩.\hfill \end{array}$$

(19)

Again, by Lemma 2 (2), we get

$$\begin{array}{c}⟨y_n-p,A^{*}(J_{\lambda }^{B_2}-I)A{y}_n⟩\hfill \\ =⟨A(y_n-p),(J_{\lambda }^{B_2}-I)A{y}_n⟩\hfill \\ =⟨J_{\lambda }^{B_2}A{y}_n-Ap-(J_{\lambda }^{B_2}-I)A{y}_n,(J_{\lambda }^{B_2}-I)A{y}_n⟩\hfill \\ =⟨J_{\lambda }^{B_2}A{y}_n-Ap,(J_{\lambda }^{B_2}-I)A{y}_n⟩-⟨(J_{\lambda }^{B_2}-I)A{y}_n,(J_{\lambda }^{B_2}-I)A{y}_n⟩\hfill \\ =⟨J_{\lambda }^{B_2}A{y}_n-Ap,(J_{\lambda }^{B_2}-I)A{y}_n⟩-{\parallel (J_{\lambda }^{B_2}-I)A{y}_n\parallel }^2\hfill \\ =\frac{1}{2}(\parallel J_{\lambda }^{B_2}A{y}_n{-Ap\parallel }^2+{\parallel (J_{\lambda }^{B_2}-I)A{y}_n\parallel }^2\hfill \\ -\parallel J_{\lambda }^{B_2}A{y}_n-Ap-(J_{\lambda }^{B_2}-I)A{y}_n{\parallel }^2)-{\parallel (J_{\lambda }^{B_2}-I)A{y}_n\parallel }^2\hfill \\ =\frac{1}{2}\left(\parallel J_{\lambda }^{B_2}A{y}_n{-Ap\parallel }^2+\parallel (J_{\lambda }^{B_2}-I)A{y}_n{\parallel }^2-{\parallel J_{\lambda }^{B_2}A{y}_n-Ap-J_{\lambda }^{B_2}A{y}_n+A{y}_n\parallel }^2\right)\hfill \\ -\parallel (J_{\lambda }^{B_2}-I)A{y}_n{\parallel }^2\hfill \\ =\frac{1}{2}\left(\parallel J_{\lambda }^{B_2}A{y}_n{-Ap\parallel }^2+\parallel (J_{\lambda }^{B_2}-I)A{y}_n{\parallel }^2-{\parallel A{y}_n-Ap\parallel }^2\right)-{\parallel (J_{\lambda }^{B_2}-I)A{y}_n\parallel }^2\hfill \\ =\frac{1}{2}\left(\parallel J_{\lambda }^{B_2}A{y}_n{-Ap\parallel }^2-\parallel A{y}_n{-Ap\parallel }^2-{\parallel (J_{\lambda }^{B_2}-I)A{y}_n\parallel }^2\right)\hfill \\ \le \frac{1}{2}\left(\parallel A{y}_n{-Ap\parallel }^2-\parallel A{y}_n{-Ap\parallel }^2-{\parallel (J_{\lambda }^{B_2}-I)A{y}_n\parallel }^2\right)\hfill \\ =-\frac{1}{2}{\parallel (J_{\lambda }^{B_2}-I)A{y}_n\parallel }^2.\hfill \end{array}$$

(20)

Using (20) into (19), we get

$$\begin{array}{c}\parallel u_n{-p\parallel }^2\le \parallel y_n{-p\parallel }^2+{\gamma }_n^2\parallel A^{*}(J_{\lambda }^{B_2}-I)A{y}_n{\parallel }^2-{\gamma }_n{\parallel (J_{\lambda }^{B_2}-I)A{y}_n\parallel }^2\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}=\parallel y_n{-p\parallel }^2-{\gamma }_n\left(\parallel (J_{\lambda }^{B_2}-I)A{y}_n{\parallel }^2-{\gamma }_n{\parallel A^{*}(J_{\lambda }^{B_2}-I)A{y}_n\parallel }^2\right).\hfill \end{array}$$

(21)

By the definition of ${\gamma }_n$, (21) can then be written as follows:

$$\begin{array}{c}\hfill \parallel u_n{-p\parallel }^2\le \parallel y_n{-p\parallel }^2-{\gamma }_n(1-{\tau }_n)\parallel (J_{\lambda }^{B_2}-I)A{y}_n{\parallel }^2\le {\parallel y_n-p\parallel }^2.\end{array}$$

Thus we have

$$\parallel u_n-p\parallel \le \parallel y_n-p\parallel .$$

(22)

Using the condition (C3) and (17), we get

$$\begin{array}{c}\parallel x_{n+1}-p\parallel =\parallel {\alpha }_n{f}_n\left(x_n\right)+{\beta }_n{x}_n+{\delta }_n{z}_n-p\parallel \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}=\parallel {\alpha }_n(f_n\left(x_n\right)-f_n\left(p\right))+{\alpha }_n(f_n\left(p\right)-p)+{\beta }_n(x_n-p)+{\delta }_n(z_n-p)\parallel \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\le {\alpha }_n\parallel f_n\left(x_n\right)-f_n\left(p\right)\parallel +{\alpha }_n\parallel f_n\left(p\right)-p\parallel +{\beta }_n\parallel x_n-p\parallel +{\delta }_n\parallel z_n-p\parallel \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\le {\alpha }_n{\mu }_n\parallel x_n-p\parallel +{\alpha }_n\parallel f_n\left(p\right)-p\parallel +{\beta }_n\parallel x_n-p\parallel +{\delta }_n(\parallel x_n-p\parallel \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}+(1-\xi ){\theta }_n\parallel x_n-x_{n-1}\parallel )\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\le ({\alpha }_n{\mu }^{*}+({\beta }_n+{\delta }_n))\parallel x_n-p\parallel +(1-\xi ){\delta }_n{\theta }_n\parallel x_n-x_{n-1}\parallel +{\alpha }_n\parallel f_n\left(p\right)-p\parallel \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}=(1-{\alpha }_n(1-{\mu }^{*})\parallel x_n-p\parallel +(1-\xi ){\delta }_n{\alpha }_n\frac{{\theta }_n}{{\alpha }_n}\parallel x_n-x_{n-1}\parallel +{\alpha }_n\parallel f_n\left(p\right)-p\parallel .\hfill \end{array}$$

Since $\left\{f_n\right\}$ is the uniform convergence on D, there exists a constant $M>0$ such that

$$\parallel f_n\left(p\right)-p\parallel \le M$$

for each $n\ge 1$. So we can choose ${\displaystyle \beta :=\frac{M}{1-{\mu }^{*}}}$ and set

$$a_n:=\parallel x_n-p\parallel ,b_n:={\alpha }_n\parallel f_n\left(p\right)-p\parallel ,$$

$${\displaystyle c_n:=(1-\xi ){\delta }_n{\alpha }_n\frac{{\theta }_n}{{\alpha }_n}\parallel x_n-x_{n-1}\parallel ,{\sigma }_n:={\alpha }_n(1-{\mu }^{*}).}$$

By Lemma 3 (1) and our assumptions, it follows that $\left\{x_n\right\}$ is bounded. Moreover, $\left\{u_n\right\}$ and $\left\{y_n\right\}$ are also bounded.

Now, by Lemma 2, we get

$$\begin{array}{c}\parallel x_{n+1}{-p\parallel }^2\hfill \\ =\parallel {\alpha }_n(f_n\left(x_n\right)-f_n\left(p\right))+{\alpha }_n(f_n\left(p\right)-p)+{\beta }_n(x_n-p)+{\delta }_n(z_n-p){\parallel }^2\hfill \\ \le \parallel {\alpha }_n(f_n\left(x_n\right)-f_n\left(p\right))+{\beta }_n(x_n-p)+{\delta }_n(z_n-p){\parallel }^2+2{\alpha }_n⟨f_n\left(p\right)-p,x_{n+1}-p⟩\hfill \\ =\parallel {\beta }_n(x_n-p)+{\delta }_n(z_n-p){\parallel }^2+{\alpha }_n^2{\parallel f_n\left(x_n\right)-f_n\left(p\right)\parallel }^2\hfill \\ +2{\alpha }_n⟨f_n\left(x_n\right)-f_n\left(p\right),{\beta }_n(x_n-p)+{\delta }_n(z_n-p)⟩+2{\alpha }_n⟨f_n\left(p\right)-p,x_{n+1}-p⟩\hfill \\ \le {\beta }_n^2\parallel x_n{-p\parallel }^2+{\delta }_n^2\parallel z_n{-p\parallel }^2+2{\beta }_n{\delta }_n⟨x_n-p,z_n-p⟩+{\alpha }_n^2{\mu }_n^2{\parallel x_n-p\parallel }^2\hfill \\ +2{\alpha }_n⟨f_n\left(p\right)-p,x_{n+1}-p⟩+2{\alpha }_n\parallel f_n\left(x_n\right)-f_n\left(p\right)\parallel \parallel {\beta }_n(x_n-p)+{\delta }_n(z_n-p)\parallel \hfill \\ \le {\beta }_n^2\parallel x_n{-p\parallel }^2+{\delta }_n^2{\parallel z_n-p\parallel }^2+{\beta }_n{\delta }_n\left(\parallel x_n{-p\parallel }^2+\parallel z_n{-p\parallel }^2-{\parallel x_n-z_n\parallel }^2\right)\hfill \\ +{\alpha }_n^2{\mu }^{*2}\parallel x_n{-p\parallel }^2+2{\alpha }_n{\mu }_n\parallel x_n-p\parallel \left({\beta }_n\parallel x_n-p\parallel +{\delta }_n\parallel z_n-p\parallel \right)\hfill \\ +2{\alpha }_n⟨f_n\left(p\right)-p,x_{n+1}-p⟩\hfill \\ \le {\beta }_n({\beta }_n+{\delta }_n)\parallel x_n{-p\parallel }^2+{\delta }_n({\beta }_n+{\delta }_n)\parallel z_n{-p\parallel }^2-{\beta }_n{\delta }_n\parallel x_n-z_n{\parallel }^2+{\alpha }_n^2{\mu }^{*2}{\parallel x_n-p\parallel }^2\hfill \\ +2{\mu }^{*}{\alpha }_n({\beta }_n+{\delta }_n)\parallel x_n{-p\parallel }^2+2{\mu }^{*}{\alpha }_n(1-\xi ){\delta }_n{\theta }_n\parallel x_n-x_{n-1}\parallel \parallel x_n-p\parallel \hfill \\ +2{\alpha }_n⟨f_n\left(p\right)-p,x_{n+1}-p⟩\hfill \\ \le {\beta }_n({\beta }_n+{\delta }_n)\parallel x_n{-p\parallel }^2+{\delta }_n({\beta }_n+{\delta }_n)(\parallel x_n{-p\parallel }^2+{\theta }_n^2{\parallel x_n-x_{n-1}\parallel }^2\hfill \\ +2{\theta }_n\parallel x_n-x_{n-1}\parallel \parallel x_n-p\parallel )-{\beta }_n{\delta }_n\parallel x_n-z_n{\parallel }^2+{\alpha }_n^2{\mu }^{*2}{\parallel x_n-p\parallel }^2\hfill \\ +2{\mu }^{*}{\alpha }_n({\beta }_n+{\delta }_n)\parallel x_n{-p\parallel }^2+2{\mu }^{*}{\alpha }_n(1-\xi ){\delta }_n{\theta }_n\parallel x_n-x_{n-1}\parallel \parallel x_n-p\parallel \hfill \\ +2{\alpha }_n⟨f_n\left(p\right)-p,x_{n+1}-p⟩\hfill \\ =\left({(1-{\alpha }_n)}^2+{\alpha }_n^2{\mu }^{*2}+2{\mu }^{*}{\alpha }_n(1-{\alpha }_n)\right)\parallel x_n{-p\parallel }^2-{\beta }_n{\delta }_n{\parallel x_n-z_n\parallel }^2\hfill \\ +2\left(1-{\alpha }_n(1-{\mu }^{*}(1-\xi ))\right){\delta }_n{\theta }_n\parallel x_n-x_{n-1}\parallel \parallel x_n-p\parallel +(1-{\alpha }_n){\delta }_n{\theta }_n^2{\parallel x_n-x_{n-1}\parallel }^2\hfill \\ +2{\alpha }_n⟨f_n\left(p\right)-p,x_{n+1}-p⟩.\hfill \end{array}$$

(23)

Now, we consider two steps for the proof as follows:

Case 1. Suppose that there exists $n_0\in \mathbb{N}$ such that $\{\parallel x_n-p{\parallel \}}_{n=n_0}^{\infty }$ is non-increasing and then $\{\parallel x_n-p\parallel \}$ converges. By Lemma 1, we get

$$\begin{array}{c}\parallel x_{n+1}{-p\parallel }^2=\parallel {\alpha }_n{f}_n\left(x_n\right)+{\beta }_n{x}_n+{\delta }_n{z}_n{-p\parallel }^2\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}={\alpha }_n\parallel f_n\left(x_n\right){-p\parallel }^2+{\beta }_n\parallel x_n{-p\parallel }^2+{\delta }_n\parallel z_n{-p\parallel }^2-{\alpha }_n{\beta }_n{\parallel f_n\left(x_n\right)-x_n\parallel }^2\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}-{\alpha }_n{\gamma }_n\parallel f_n\left(x_n\right)-z_n{\parallel }^2-{\beta }_n{\gamma }_n{\parallel x_n-z_n\parallel }^2\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\le {\alpha }_n\parallel f_n\left(x_n\right){-p\parallel }^2+{\beta }_n\parallel x_n{-p\parallel }^2+{\delta }_n{\parallel z_n-p\parallel }^2\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\le {\alpha }_n\parallel f_n\left(x_n\right){-p\parallel }^2+{\beta }_n{\parallel x_n-p\parallel }^2+{\delta }_n\left(\xi \parallel x_n{-p\parallel }^2+(1-\xi ){\parallel u_n-p\parallel }^2\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\le {\alpha }_n\parallel f_n\left(x_n\right){-p\parallel }^2+({\beta }_n+\xi {\delta }_n)\parallel x_n{-p\parallel }^2+(1-\xi ){\delta }_n{\parallel u_n-p\parallel }^2,\hfill \end{array}$$

which implies that

$$\begin{array}{c}\hfill -\parallel u_n{-p\parallel }^2\le \frac{1}{(1-\xi ){\delta }_n}\left({\alpha }_n\parallel f_n\left(x_n\right){-p\parallel }^2+({\beta }_n+\xi {\delta }_n)\parallel x_n{-p\parallel }^2-{\parallel x_{n+1}-p\parallel }^2\right).\end{array}$$

(24)

Applying (16) and (24) to (21), we get

$$\begin{array}{c}{\gamma }_n(\parallel (J_{\lambda }^{B_2}-I)A{y}_n{\parallel }^2-{\gamma }_n\parallel A^{*}(J_{\lambda }^{B_2}-I)A{y}_n{\parallel }^2)\hfill \\ \le \parallel y_n{-p\parallel }^2-{\parallel u_n-p\parallel }^2\hfill \\ \le \parallel x_n{-p\parallel }^2+2{\theta }_n\parallel x_{n-1}-p\parallel \parallel x_n-p\parallel +{\theta }_n^2{\parallel x_n-x_{n-1}\parallel }^2\hfill \\ +\frac{1}{(1-\xi ){\delta }_n}({\alpha }_n\parallel f_n\left(x_n\right){-p\parallel }^2+({\beta }_n+\xi {\delta }_n)\parallel x_n{-p\parallel }^2-\parallel x_{n+1}-p{\parallel }^2)\hfill \\ =\frac{{\beta }_n+{\delta }_n}{(1-\xi ){\delta }_n}\parallel x_n{-p\parallel }^2+\frac{{\alpha }_n}{(1-\xi ){\delta }_n}\parallel f_n\left(x_n\right){-p\parallel }^2-\frac{1}{(1-\xi ){\delta }_n}{\parallel x_{n+1}-p\parallel }^2\hfill \\ +{\theta }_n\parallel x_n-x_{n-1}\parallel \left(2\parallel x_n-p\parallel +{\theta }_n\parallel x_n-x_{n-1}\parallel \right)\hfill \\ \le \frac{1}{(1-\xi ){ϵ}_2}(\parallel x_n{-p\parallel }^2-\parallel x_{n+1}{-p\parallel }^2)+\frac{{\alpha }_n}{(1-\xi ){ϵ}_2}(\parallel f_n\left(x_n\right){-p\parallel }^2-{\parallel x_n-p\parallel }^2\hfill \\ +\frac{{\theta }_n}{{\alpha }_n}\parallel x_n-x_{n-1}\parallel \left(2\parallel x_n-p\parallel +{\alpha }_n\frac{{\theta }_n}{{\alpha }_n}\parallel x_n-x_{n-1}\parallel \right)).\hfill \end{array}$$

Since $\{\parallel x_n-p\parallel \}$ is convergent, we have $\parallel x_n-p\parallel -\parallel x_{n+1}-p\parallel \to 0$ as $n\to \infty$. By the conditions (C2) and (C4), we get

$${\gamma }_n(\parallel (J_{\lambda }^{B_2}-I)A{y}_n{\parallel }^2-{\gamma }_n\parallel A^{*}(J_{\lambda }^{B_2}-I)A{y}_n{\parallel }^2)\to 0\mathrm{as}n\to \infty .$$

From the definition of ${\gamma }_n$, we get

$$\frac{{\tau }_n(1-{\tau }_n){\parallel (J_{\lambda }^{B_2}-I)A{y}_n\parallel }^4}{\parallel A^{*}(J_{\lambda }^{B_2}-I)A{y}_n{\parallel }^2}\to 0\mathrm{as}n\to \infty$$

or

$$\frac{\parallel (J_{\lambda }^{B_2}-I)A{y}_n{\parallel }^2}{\parallel A^{*}(J_{\lambda }^{B_2}-I)A{y}_n\parallel }\to 0\mathrm{as}n\to \infty .$$

Since

$$\begin{array}{c}\hfill \parallel A^{*}(J_{\lambda }^{B_2}-I)A{y}_n\parallel \le \parallel A^{*}\parallel \parallel (J_{\lambda }^{B_2}-I)A{y}_n\parallel =\parallel A\parallel \parallel (J_{\lambda }^{B_2}-I)A{y}_n\parallel ,\end{array}$$

it is easy to see that

$$\begin{array}{cc}\hfill \parallel (J_{\lambda }^{B_2}-I)A{y}_n\parallel & \le \parallel A\parallel \frac{\parallel (J_{\lambda }^{B_2}-I)A{y}_n{\parallel }^2}{\parallel A^{*}(J_{\lambda }^{B_2}-I)A{y}_n\parallel }.\hfill \end{array}$$

Consequently, we get

$$\begin{array}{c}\hfill \parallel (J_{\lambda }^{B_2}-I)A{y}_n\parallel \to 0\mathrm{as}n\to \infty \end{array}$$

(25)

and also

$$\parallel A^{*}(J_{\lambda }^{B_2}-I)A{y}_n\parallel \to 0\mathrm{as}n\to \infty .$$

(26)

Similarly, from (23) and our assumptions, we get

$$\begin{array}{c}\parallel x_n-z_n{\parallel }^2\hfill \\ =\frac{1}{{\beta }_n{\delta }_n}\{\parallel x_n{-p\parallel }^2-\parallel x_{n+1}{-p\parallel }^2+(1-{\alpha }_n){\delta }_n{\theta }_n^2{\parallel x_n-x_{n-1}\parallel }^2\hfill \\ +2\left(1-{\alpha }_n(1-{\mu }^{*}(1-\xi ))\right){\delta }_n{\theta }_n\parallel x_n-x_{n-1}\parallel \parallel x_n-p\parallel \hfill \\ +{\alpha }_n\left[\left({\alpha }_n(1+{\mu }^{*2})-2(1-{\mu }^{*}(1-{\alpha }_n))\right){\parallel x_n-p\parallel }^2+2⟨f_n\left(p\right)-p,x_{n+1}-p⟩\right]\}\hfill \\ \le \frac{1}{{ϵ}_1{ϵ}_2}\{\parallel x_n{-p\parallel }^2-\parallel x_{n+1}{-p\parallel }^2+\frac{{\theta }_n}{{\alpha }_n}\parallel x_n-x_{n-1}\parallel [{\delta }_n(1-{\alpha }_n){\alpha }_n^2\frac{{\theta }_n}{{\alpha }_n}\parallel x_n-x_{n-1}\parallel \hfill \\ +2{\delta }_n\left(1-{\alpha }_n(1-{\mu }^{*}(1-\xi ))\right){\theta }_n\parallel x_n-p\parallel ]+{\alpha }_n[2⟨f_n\left(p\right)-p,x_{n+1}-p⟩\hfill \\ +\left({\alpha }_n(1+{\mu }^{*2})-2(1-{\mu }^{*}(1-{\alpha }_n))\right){\parallel x_n-p\parallel }^2\left]\right\}\to 0\mathrm{as}n\to \infty .\hfill \end{array}$$

Therefore, we have

$$\begin{array}{c}\hfill \parallel x_n-z_n\parallel \to 0\mathrm{as}n\to \infty .\end{array}$$

(27)

By the condition (C2) and (27), we get

$$\begin{array}{c}\parallel x_{n+1}-x_n\parallel =\parallel {\alpha }_n{f}_n\left(x_n\right)+{\beta }_n{x}_n+{\delta }_n{z}_n-x_n\parallel \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\le {\alpha }_n\parallel f_n\left(x_n\right)-x_n\parallel +{\delta }_n\parallel x_n-z_n\parallel \to 0\mathrm{as}n\to \infty .\hfill \end{array}$$

Thus we have

$$\begin{array}{c}\hfill \parallel x_{n+1}-z_n\parallel \le \parallel x_{n+1}-x_n\parallel +\parallel x_n-z_n\parallel \to 0\mathrm{as}n\to \infty .\end{array}$$

Since $J_{\lambda }^{B_1}$ is firmly nonexpansive, we have

$$\begin{array}{c}\parallel u_n{-p\parallel }^2\hfill \\ =\parallel J_{\lambda }^{B_1}(y_n+{\gamma }_n{A}^{*}(J_{\lambda }^{B_2}-I)A{y}_n)-J_{\lambda }^{B_1}{p\parallel }^2\hfill \\ \le ⟨u_n-p,y_n+{\gamma }_n{A}^{*}(J_{\lambda }^{B_2}-I)A{y}_n-p⟩\hfill \\ =\frac{1}{2}\left(\parallel u_n{-p\parallel }^2+\parallel y_n+{\gamma }_n{A}^{*}(J_{\lambda }^{B_2}-I)A{y}_n{-p\parallel }^2-{\parallel u_n-y_n-{\gamma }_n{A}^{*}(J_{\lambda }^{B_2}-I)A{y}_n\parallel }^2\right)\hfill \\ =\frac{1}{2}(\parallel u_n{-p\parallel }^2+\parallel y_n{-p\parallel }^2+{\gamma }_n^2{\parallel A^{*}(J_{\lambda }^{B_2}-I)A{y}_n\parallel }^2+2⟨y_n-p,{\gamma }_n{A}^{*}(J_{\lambda }^{B_2}-I)A{y}_n⟩\hfill \\ -\parallel u_n-y_n{\parallel }^2-{\gamma }_n^2{\parallel A^{*}(J_{\lambda }^{B_2}-I)A{y}_n\parallel }^2+2⟨u_n-y_n,{\gamma }_n{A}^{*}(J_{\lambda }^{B_2}-I)A{y}_n⟩)\hfill \\ \le \frac{1}{2}\left(\parallel y_n{-p\parallel }^2+\parallel y_n{-p\parallel }^2-{\parallel u_n-y_n\parallel }^2+2⟨u_n-p,{\gamma }_n{A}^{*}(J_{\lambda }^{B_2}-I)A{y}_n⟩\right)\hfill \\ \le \frac{1}{2}\left(2\parallel y_n{-p\parallel }^2-\parallel u_n-y_n{\parallel }^2+2{\gamma }_n\parallel u_n-p\parallel \parallel A^{*}(J_{\lambda }^{B_2}-I)A{y}_n\parallel \right)\hfill \\ \le \parallel y_n{-p\parallel }^2-\frac{1}{2}\parallel u_n-y_n{\parallel }^2+{\gamma }_n\parallel u_n-p\parallel \parallel A^{*}(J_{\lambda }^{B_2}-I)A{y}_n\parallel \hfill \end{array}$$

or

$$\parallel u_n-y_n{\parallel }^2\le 2\left(\parallel y_n{-p\parallel }^2-\parallel u_n{-p\parallel }^2+{\gamma }_n\parallel u_n-p\parallel \parallel A^{*}(J_{\lambda }^{B_2}-1)A{y}_n\parallel \right).$$

(28)

From (28), (16), (24) and (26) and our assumptions, it follows that

$$\begin{array}{c}\parallel u_n-y_n{\parallel }^2\le 2[\parallel x_n{-p\parallel }^2+2{\theta }_n\parallel x_n-x_{n-1}\parallel \parallel x_n-p\parallel +{\theta }_n^2{\parallel x_n-x_{n-1}\parallel }^2\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}+\frac{1}{(1-\xi ){\delta }_n}\left({\alpha }_n\parallel f_n\left(x_n\right){-p\parallel }^2+({\beta }_n+\xi {\delta }_n)\parallel x_n{-p\parallel }^2-{\parallel x_{n+1}-p\parallel }^2\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}+{\gamma }_n\parallel u_n-p\parallel \parallel A^{*}(J_{\lambda }^{B_2}-1)A{y}_n\parallel ]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}=2[\frac{1}{(1-\xi ){ϵ}_2}\left(\parallel x_n{-p\parallel }^2-{\parallel x_{n+1}-p\parallel }^2\right)+{\gamma }_n\parallel u_n-p\parallel \parallel A^{*}(J_{\lambda }^{B_2}-1)A{y}_n\parallel \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}+\frac{{\alpha }_n}{(1-\xi ){ϵ}_2}(\parallel f_n\left(x_n\right){-p\parallel }^2-{\parallel x_n-p\parallel }^2\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}+\frac{{\theta }_n}{{\alpha }_n}\parallel x_n-x_{n-1}\parallel \left(2\parallel x_n-p\parallel +{\alpha }_n\frac{{\theta }_n}{{\alpha }_n}\parallel x_n-x_{n-1}\parallel \right)\left)\right]\to 0\mathrm{as}n\to \infty ,\hfill \end{array}$$

that is, we have

$$\parallel u_n-y_n\parallel \to 0\mathrm{as}n\to \infty .$$

(29)

From $y_n:=x_n+{\theta }_n(x_n-x_{n-1})$, we get

$$\begin{array}{c}\hfill \parallel y_n-x_n\parallel =\parallel x_n+{\theta }_n(x_n-x_{n-1})-x_n\parallel ={\alpha }_n\frac{{\theta }_n}{{\alpha }_n}\parallel x_n-x_{n-1}\parallel ,\end{array}$$

which, with the condition (C4), implies that

$$\parallel y_n-x_n\parallel \to 0\mathrm{as}n\to \infty .$$

(30)

In addition, using (27), (29) and (30), we obtain

$$\begin{array}{c}\parallel z_n-u_n\parallel \le \parallel u_n-y_n\parallel +\parallel y_n-z_n\parallel \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\le \parallel u_n-y_n\parallel +\parallel y_n-x_n\parallel +\parallel x_n-z_n\parallel \to 0\mathrm{as}n\to \infty .\hfill \end{array}$$

From $z_n:=\xi v_n+(1-\xi )u_n$, we get

$$\parallel v_n-u_n\parallel =\frac{1}{\xi }\parallel z_n-u_n\parallel \to 0\mathrm{as}n\to \infty .$$

(31)

Thus, by (29)–(31), we also get

$$\begin{array}{c}\parallel x_n-v_n\parallel \le \parallel x_n-u_n\parallel +\parallel u_n-v_n\parallel \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\le \parallel x_n-y_n\parallel +\parallel y_n-u_n\parallel +\parallel u_n-v_n\parallel \to 0\mathrm{as}n\to \infty .\hfill \end{array}$$

Therefore, we have

$$d(x_n,S{x}_n)\le \parallel x_n-v_n\parallel \to 0\mathrm{as}n\to \infty .$$

(32)

Since $\left\{x_n\right\}$ is bounded, there exists a subsequence $\left\{x_{n_k}\right\}$ of $\left\{x_n\right\}$ such that $x_{n_k}\rightharpoonup x^{*}\in H_1$ and, consequently, $\left\{u_{n_k}\right\}$ and $\left\{y_{n_k}\right\}$ converge weakly to the point $x^{*}$.

From (32), Lemma 4 and the demiclosedness principle for a multi-valued mapping S at the origin, we get $x^{*}\in S{x}^{*}$, which implies that

$$x^{*}\in F\left(S\right).$$

Next, we show that $x^{*}\in \Omega$. Let $(v,z)\in G\left(B_1\right)$, that is, $z\in B_1\left(v\right)$. On the other hand, $u_{n_k}=J_{\lambda }^{B_1}(y_{n_k}+{\gamma }_{n_k}{A}^{*}(J_{\lambda }^{B_2}-I)A{y}_{n_k})$ can be written as

$$y_{n_k}+{\gamma }_{n_k}{A}^{*}(J_{\lambda }^{B_1}-I)A{y}_{n_k}\in u_{n_k}+\lambda B_1\left(u_{n_k}\right),$$

or, equivalently,

$$\frac{(y_{n_k}-u_{n_K})+{\gamma }_{n_k}{A}^{*}(J_{\lambda }^{B_1}-I)A{y}_{n_k}}{\lambda }\in B_1\left(u_{n_k}\right).$$

Since $B_1$ is maximal monotone, we get

$$⟨v-u_{n_k},z-\frac{(y_{n_k}-u_{n_k})+{\gamma }_{n_k}{A}^{*}(J_{\lambda }^{B_2}-I)A{y}_{n_k}}{\lambda }⟩\ge 0.$$

Therefore, we have

$$\begin{array}{c}⟨v-u_{n_k},z⟩\ge ⟨v-u_{n_k},\frac{(y_{n_k}-u_{n_k})+{\gamma }_{n_k}{A}^{*}(J_{\lambda }^{B_2}-I)A{y}_{n_k}}{\lambda }⟩\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=⟨v-u_{n_k},\frac{y_{n_k}-u_{n_k}}{\lambda }⟩+⟨v-u_{n_k},\frac{{\gamma }_{n_k}{A}^{*}(J_{\lambda }^{B_2}-I)A{y}_{n_k}}{\lambda }⟩.\hfill \end{array}$$

(33)

Since $u_{n_k}\rightharpoonup x^{*}$, we have

$${\displaystyle \underset{k\to \infty }{lim}⟨v-u_{n_k},z⟩=⟨v-x^{*},z⟩.}$$

By (26) and (29), it follows that (33) becomes $⟨v-x^{*},z⟩\ge 0$, which implies that

$$0\in B_1\left(x^{*}\right).$$

Moreover, from (29), we know that $\left\{A{y}_{n_k}\right\}$ converges weakly to $A{x}^{*}$ and, by (25), the fact that $J_{\lambda }^{B_2}$ is nonexpansive and the demiclosedness principle for a multi-valued mapping, we have

$$0\in B_2\left(A{x}^{*}\right),$$

which implies that $x^{*}\in \Omega$. Thus $x^{*}\in F\left(S\right)\cap \Omega$. Since $\left\{f_n\left(x\right)\right\}$ is uniformly convergent on D, we get

$$\begin{array}{c}\underset{n\to \infty }{lim\; sup}⟨f_n\left(p\right)-p,x_{n+1}-p⟩=\underset{j\to \infty }{lim\; sup}⟨f_{n_j}\left(p\right)-p,x_{n_j+1}-p⟩\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=⟨f\left(p\right)-p,x^{*}-p⟩\le 0.\hfill \end{array}$$

From (23), we get

$$\begin{array}{c}\parallel x_{n+1}{-p\parallel }^2\hspace{1em}\le \left(1-2{\alpha }_n(1-{\mu }^{*}(1-{\alpha }_n))+{\alpha }_n^2(1+{\mu }^{*2})\right)\parallel x_n{-p\parallel }^2-{\beta }_n{\delta }_n{\parallel x_n-z_n\parallel }^2\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}+2\left(1-{\alpha }_n(1-{\mu }^{*}(1-\xi ))\right){\delta }_n{\theta }_n\parallel x_n-x_{n-1}\parallel \parallel x_n-p\parallel \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}+(1-{\alpha }_n){\delta }_n{\theta }_n^2{\parallel x_n-x_{n-1}\parallel }^2+2{\alpha }_n⟨f_n\left(p\right)-p,x_{n+1}-p⟩\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\le \left(1-2{\alpha }_n(1-{\mu }^{*})\right){\parallel x_n-p\parallel }^2+2{\alpha }_n(1-{\mu }^{*})\frac{⟨f_n\left(p\right)-p,x_{n+1}-p⟩}{1-{\mu }^{*}}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}+{\alpha }_n[{\delta }_n\frac{{\theta }_n}{{\alpha }_n}\parallel x_n-x_{n-1}\parallel (2\left(1-{\alpha }_n(1-{\mu }^{*}(1-\xi ))\right)\parallel x_n-p\parallel \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}+\left((1-{\alpha }_n){\alpha }_n\frac{{\theta }_n}{{\alpha }_n}\parallel x_n-x_{n-1}\parallel \right)+{\alpha }_n(1+{\mu }^{*2}){\parallel x_n-p\parallel }^2].\hfill \end{array}$$

By Lemma 4, we obtain

$$\underset{n\to \infty }{lim}{x}_n=p.$$

Case 2. Suppose that $\{\parallel x_n-p{\parallel \}}_{n=n_0}^{\infty }$ is not a monotonically decreasing sequence for some $n_0$ large enough. Set ${\mathsf{\Gamma }}_n={\parallel x_n-p\parallel }^2$ and let $\tau :\mathbb{B}\to \mathbb{N}$ be a mapping defined by

$$\tau \left(n\right):=max\{k\in \mathbb{N}:k\le n,{\mathsf{\Gamma }}_k\le {\mathsf{\Gamma }}_{k+1}\}$$

for all $n\ge n_0$. Obviously, $\tau$ is a non-decreasing sequence. Thus we have

$$0\le {\mathsf{\Gamma }}_{\tau \left(n\right)}\le {\mathsf{\Gamma }}_{\tau \left(n\right)+1}$$

for all $n\ge n_0$. That is, $\parallel x_{\tau \left(n\right)}-p\parallel \le \parallel x_{\tau \left(n\right)+1}-p\parallel$ for all $n\ge n_0$. Thus ${lim}_{n\to \infty }\parallel x_{\tau \left(n\right)}-p\parallel$ exists. As in Case 1, we can show that

$$\underset{n\to \infty }{lim}\parallel (J_{\lambda }^{B_2}-I)A{y}_{\tau \left(n\right)}\parallel =0,\underset{n\to \infty }{lim}\parallel A^{*}(J_{\lambda }^{B_2}-I)A{y}_{\tau \left(n\right)}\parallel =0,$$

(34)

$$\underset{n\to \infty }{lim}\parallel x_{\tau \left(n\right)+1}-x_{\tau \left(n\right)}\parallel =0,\underset{n\to \infty }{lim}\parallel u_{\tau \left(n\right)}-x_{\tau \left(n\right)}\parallel =0,$$

(35)

$$\underset{n\to \infty }{lim}\parallel v_{\tau \left(n\right)}-u_{\tau \left(n\right)}\parallel =0,\underset{n\to \infty }{lim}\parallel x_{\tau \left(n\right)}-v_{\tau \left(n\right)}\parallel =0.$$

(36)

Therefore, we have

$$d(x_{\tau \left(n\right)},S{x}_{\tau \left(n\right)})\le \parallel x_{\tau \left(n\right)}-v_{\tau \left(n\right)}\parallel \to 0\mathrm{as}n\to \infty .$$

(37)

Since $\left\{x_{\tau \left(n\right)}\right\}$ is bounded, there exists a subsequence $\left\{u_{\tau \left(n\right)}\right\}$ of $\left\{x_{\tau \left(n\right)}\right\}$ that converges weakly to a point $x^{*}\in H_1$. From $\parallel u_{\tau \left(n\right)}-x_{\tau \left(n\right)}\parallel \to 0$, it follows that $u_{\tau \left(n\right)}\rightharpoonup x^{*}\in H_1$.

Moreover, as in Case 1, we show that $x^{*}\in F\left(S\right)\cap \Omega$. Furthermore, since $\left\{f_n\left(x\right)\right\}$ is uniformly convergent on $D\subset H_1$, we obtain that

$$\underset{n\to \infty }{lim\; sup}⟨f_{\tau \left(n\right)}\left(p\right)-p,x_{\tau \left(n\right)+1}-p⟩\le 0.$$

From (23), we get

$$\begin{array}{c}\parallel x_{\tau \left(n\right)+1}{-p\parallel }^2\le \left(1-2{\alpha }_{\tau \left(n\right)}(1-{\mu }^{*}(1-{\alpha }_{\tau \left(n\right)}))+{\alpha }_{\tau \left(n\right)}^2(1+{\mu }^{*2})\right){\parallel x_{\tau \left(n\right)}-p\parallel }^2\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}-{\beta }_{\tau \left(n\right)}{\delta }_{\tau \left(n\right)}{\parallel x_{\tau \left(n\right)}-z_{\tau \left(n\right)}\parallel }^2+2{\alpha }_{\tau \left(n\right)}⟨f_{\tau \left(n\right)}\left(p\right)-p,x_{\tau \left(n\right)+1}-p⟩\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}+2\left(1-{\alpha }_{\tau \left(n\right)}(1-{\mu }^{*}(1-\xi ))\right){\delta }_{\tau \left(n\right)}{\theta }_{\tau \left(n\right)}\parallel x_{\tau \left(n\right)}-x_{\tau \left(n\right)-1}\parallel \parallel x_{\tau \left(n\right)}-p\parallel \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}+(1-{\alpha }_{\tau \left(n\right)}){\delta }_{\tau \left(n\right)}{\theta }_{\tau \left(n\right)}^2{\parallel x_{\tau \left(n\right)}-x_{\tau \left(n\right)-1}\parallel }^2\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\le \left(1-2{\alpha }_{\tau \left(n\right)}(1-{\mu }^{*})\right)\parallel x_{\tau \left(n\right)}{-p\parallel }^2+{\alpha }_{\tau \left(n\right)}^2(1+{\mu }^{*2}){\parallel x_{\tau \left(n\right)}-p\parallel }^2\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}+{\delta }_{\tau \left(n\right)}{\theta }_n\parallel x_{\tau \left(n\right)}-x_{\tau \left(n\right)-1}\parallel (2(1-{\alpha }_{\tau \left(n\right)}(1-{\mu }^{*}))\parallel x_{\tau \left(n\right)}-p\parallel \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}+(1-{\alpha }_{\tau \left(n\right)}){\theta }_{\tau \left(n\right)}\parallel x_{\tau \left(n\right)}-x_{\tau \left(n\right)-1}\parallel )+2{\alpha }_{\tau \left(n\right)}⟨f_{\tau \left(n\right)}\left(p\right)-p,x_{\tau \left(n\right)+1}-p⟩,\hfill \end{array}$$

which implies that

$$\begin{array}{c}2{\alpha }_{\tau \left(n\right)}(1-{\mu }^{*}){\parallel x_{\tau \left(n\right)}-p\parallel }^2\le \parallel x_{\tau \left(n\right)}{-p\parallel }^2-\parallel x_{\tau \left(n\right)+1}{-p\parallel }^2+{\alpha }_{\tau \left(n\right)}^2(1+{\mu }^{*2}){\parallel x_{\tau \left(n\right)}-p\parallel }^2\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}+{\delta }_{\tau \left(n\right)}{\theta }_n\parallel x_{\tau \left(n\right)}-x_{\tau \left(n\right)-1}\parallel (2(1-{\alpha }_{\tau \left(n\right)}(1-{\mu }^{*}))\parallel x_{\tau \left(n\right)}-p\parallel \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}+(1-{\alpha }_{\tau \left(n\right)}){\theta }_{\tau \left(n\right)}\parallel x_{\tau \left(n\right)}-x_{\tau \left(n\right)-1}\parallel )\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}+2{\alpha }_{\tau \left(n\right)}⟨f_{\tau \left(n\right)}\left(p\right)-p,x_{\tau \left(n\right)+1}-p⟩,\hfill \end{array}$$

or

$$\begin{array}{c}2(1-{\mu }^{*}){\parallel x_{\tau \left(n\right)}-p\parallel }^2\le {\alpha }_{\tau \left(n\right)}(1+{\mu }^{*2}){\parallel x_{\tau \left(n\right)}-p\parallel }^2+2⟨f_{\tau \left(n\right)}\left(p\right)-p,x_{\tau \left(n\right)+1}-p⟩\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}+{\delta }_{\tau \left(n\right)}\frac{{\theta }_{\tau \left(n\right)}}{{\alpha }_{\tau \left(n\right)}}\parallel x_{\tau \left(n\right)}-x_{\tau \left(n\right)-1}\parallel (2(1-{\alpha }_{\tau \left(n\right)}(1-{\mu }^{*}))\parallel x_{\tau \left(n\right)}-p\parallel \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}+(1-{\alpha }_{\tau \left(n\right)}){\alpha }_{\tau \left(n\right)}\frac{{\theta }_{\tau \left(n\right)}}{{\alpha }_{\tau \left(n\right)}}\parallel x_{\tau \left(n\right)}-x_{\tau \left(n\right)-1}\parallel ).\hfill \end{array}$$

Thus we have

$$\underset{n\to \infty }{lim\; sup}\parallel x_{\tau \left(n\right)}-p\parallel \le 0$$

and so

$$\underset{n\to \infty }{lim}\parallel x_{\tau \left(n\right)}-p\parallel =0.$$

(38)

By (35) and (38), we get

$$\parallel x_{\tau \left(n\right)+1}-p\parallel \le \parallel x_{\tau \left(n\right)+1}-x_{\tau \left(n\right)}\parallel +\parallel x_{\tau \left(n\right)}-p\parallel \to 0,n\to \infty .$$

Furthermore, for all $n\ge n_0$, it is easy to see that ${\mathsf{\Gamma }}_{\tau \left(n\right)}\le {\mathsf{\Gamma }}_{\tau \left(n\right)+1}$ if $n\ne \tau \left(n\right)$ (that is, $\tau \left(n\right)<n$) because of ${\mathsf{\Gamma }}_j\ge {\mathsf{\Gamma }}_{j+1}$ for $\tau \left(n\right)+1\le j\le n$. Consequently, it follows that, for all $n\ge n_0$,

$$0\le {\mathsf{\Gamma }}_n\le max\{{\mathsf{\Gamma }}_{\tau \left(n\right)},{\mathsf{\Gamma }}_{\tau \left(n\right)+1}\}={\mathsf{\Gamma }}_{\tau \left(n\right)+1}.$$

Therefore, $lim{\mathsf{\Gamma }}_n=0$, that is, $\left\{x_n\right\}$ converges strongly to the point $x^{*}$. This completes the proof. ☐

Remark 1.

[22] The condition (C4) is easily implemented in numerical results because the value of $\parallel x_n-x_{n-1}\parallel$ is known before choosing ${\theta }_n$. Indeed, we can choose the parameter ${\theta }_n$ such as

$$\begin{array}{c}\hfill {\theta }_n=\left\{\begin{array}{cc}min\left\{\overline{\omega },\frac{{\omega }_n}{\parallel x_n-x_{n-1}\parallel }\right\},\hfill & \mathit{if}\parallel x_n-x_{n-1}\parallel \ne 0,\hfill \\ \overline{\omega },\hfill & \mathit{otherwise},\hfill \end{array}\right.\end{array}$$

where $\left\{{\omega }_n\right\}$ is a positive sequence such that ${\omega }_n=o\left({\alpha }_n\right)$. Moreover, in the condition (C4), we can take ${\displaystyle {\alpha }_n=\frac{1}{n+1}}$, ${\displaystyle \overline{\omega }=\frac{4}{5}}$ and

$$\begin{array}{c}\hfill {\theta }_n=\left\{\begin{array}{cc}min\left\{\overline{\omega },\frac{{\alpha }_n^2}{\parallel x_n-x_{n-1}\parallel }\right\},\hfill & \mathit{if}\parallel x_n-x_{n-1}\parallel \ne 0,\hfill \\ \overline{\omega },\hfill & \mathit{otherwise},\hfill \end{array}\right.\end{array}$$

or

$$\begin{array}{c}\hfill {\theta }_n=\left\{\begin{array}{cc}min\left\{\frac{4}{5},\frac{1}{{(n+1)}^2\parallel x_n-x_{n-1}\parallel }\right\},\hfill & \mathit{if}\parallel x_n-x_{n-1}\parallel \ne 0,\hfill \\ \frac{4}{5},\hfill & \mathit{otherwise}.\hfill \end{array}\right.\end{array}$$

If the multi-valued quasi-nonexpansive mapping S in Theorem 1 is a single-valued quasi-nonexpansive mapping, then we obtain the following:

Corollary 1.

Let $H_1$ and $H_2$ be two real Hilbert spaces. Suppose that $A:H_1\to H_2$ is a bounded linear operator with adjoint operator $A^{*}$. Let $\left\{f_n\right\}$ be a sequence of ${\mu }_n$-contractions $f_n:H_1\to H_1$ with $0<{\mu }_{*}\le {\mu }_n\le {\mu }^{*}<1$ and $\left\{f_n\left(x\right)\right\}$ be uniformly convergent for any x in a bounded subset D of $H_1$. Suppose that $S:H_1\to H_1$ is a single-valued quasi-nonexpansive mapping, $I-S$ is demiclosed at the origin and $F\left(S\right)\cap \Omega \ne \varnothing$. For any $x_0,x_1\in H_1$, let the sequences $\left\{y_n\right\}$, $\left\{u_n\right\}$, $\left\{z_n\right\}$ and $\left\{x_n\right\}$ be generated by

$$\begin{array}{c}\left\{\begin{array}{c}{y}_n=x_n+{\theta }_n(x_n-x_{n-1}),\hfill \\ u_n=J_{\lambda }^{B_1}(y_n+{\gamma }_n{A}^{*}(J_{\lambda }^{B_2}-I)A{y}_n),\hfill \\ z_n=\xi S{x}_n+(1-\xi )u_n,\hfill \\ x_{n+1}={\alpha }_n{f}_n\left(x_n\right)+{\beta }_n{x}_n+{\delta }_n{z}_n\hfill \end{array}\right.\hfill \end{array}$$

(39)

for each $n\ge 1$, where $\xi \in (0,1)$, ${\gamma }_n:={\tau }_n\frac{\parallel (J_{\lambda }^{B_2}-I)A{y}_n{\parallel }^2}{\parallel A^{*}(J_{\lambda }^{B_2}-I)A{y}_n{\parallel }^2}$ with $0<{\tau }_{*}\le {\tau }_n\le {\tau }^{*}<1$, $\left\{{\theta }_n\right\}\subset [0,\overline{\omega })$ for some $\overline{\omega }>0$ and $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\},\left\{{\delta }_n\right\}\in (0,1)$ with ${\alpha }_n+{\beta }_n+{\delta }_n=1$ satisfying the following conditions:

(C1) 

${lim}_{n\to \infty }{\alpha }_n=0$;

(C2) 

${\sum }_{n=1}^{\infty }{\alpha }_n=\infty$;

(C3) 

$0<{ϵ}_1\le {\beta }_n$ and $0<{ϵ}_2\le {\delta }_n$;

(C4) 

${lim}_{n\to \infty }\frac{{\theta }_n}{{\alpha }_n}\parallel x_n-x_{n-1}\parallel =0$.

Then the sequence $\left\{x_n\right\}$ generated by $\left(39\right)$ converges strongly to a point $p\in F\left(S\right)\cap \Omega$, where $p=P_{F\left(S\right)\cap \Omega }f\left(p\right)$.

Remark 2.

If ${\theta }_n=0$, then the iterative scheme (14) in Theorem 1 reduces to the iterative (11).

4. Applications

In this section, we give some applications of the problem (SFPIP) in the variational inequality problem and game theory. First, we introduce variational inequality problem in [34] and game theory (see [35]).

4.1. The Variational Inequality Problem

Let C be a nonempty closed and convex subset of a real Hilbert space $H_1$. Suppose that an operator $F:H_1\to H_1$ is monotone.

Now, we consider the following variational inequality problem (VIP):

$$\mathrm{Find}\mathrm{a}\mathrm{point}{x}^{*}\in C\mathrm{such}\mathrm{that}⟨F{x}^{*},y-x^{*}⟩\ge 0\mathrm{for}\mathrm{all}y\in C.$$

(40)

The solution set of the problem (VIP) is denoted by $\mathsf{\Gamma }$.

Moreover, it is well-known that $x^{*}$ is a solution of the problem (VIP) if and only if $x^{*}$ is a solution of the problem (FPP) [34], that is, for any $\gamma >0$,

$$x^{*}=P_C(x^{*}-\gamma F{x}^{*}).$$

The following lemma is extracted from [2,36]. This lemma is used for finding a solution of the split inclusion problem and the variational inequality problem:

Lemma 5.

Let $H_1$ be a real Hilbert space, $F:H_1\to H_1$ be a monotone and L-Lipschitz operator on a nonempty closed and convex subset C of $H_1$. For any $\gamma >0$, let $T=P_C(I-\gamma F\left(P_C(I-\gamma F)\right))$. Then, for any $y\in \mathsf{\Gamma }$ and $L\gamma <1$, we have

$\parallel Tx-Ty\parallel \le \parallel x-y\parallel$,

$I-T$ is demiclosed at the origin and $F\left(T\right)=\mathsf{\Gamma }$.

Now, we apply our Theorem 1, by combining with Lemma 5, to find a solution of the problem (VIP), that is, a point in the set $\mathsf{\Gamma }$.

Let $B_1:H_1\to 2^{H_1}$ and $B_2:H_2\to 2^{H_2}$ be maximal monotone mappings defined on $H_1$ and $H_2$, respectively, and $A:H_1\to H_2$ be a bounded linear operator with its adjoint $A^{*}$.

Now, we consider the split fixed point variational inclusion problem (SFPVIP) as follows:

$$\mathrm{Find}\mathrm{a}\mathrm{point}{x}^{*}\in H_1\mathrm{such}\mathrm{that}0\in B_1\left(x^{*}\right),x^{*}\in \mathsf{\Gamma }$$

(41)

and

$$y^{*}=A{x}^{*}\in H_2\mathrm{such}\mathrm{that}0\in B_2\left(y^{*}\right).$$

(42)

Theorem 2.

Let $H_1$ and $H_2$ be two real Hilbert spaces, $A:H_1\to H_2$ be a bounded linear operator with its adjoint $A^{*}$. Let $\left\{f_n\right\}$ be a sequence of ${\mu }_n$-contractions $f_n:H_1\to H_1$ with $0<{\mu }_{*}\le {\mu }_n\le {\mu }^{*}<1$ and $\left\{f_n\left(x\right)\right\}$ be uniformly convergent for any x in a bounded subset D of $H_1$. For any $\lambda >0$, let $T=P_C(I-\gamma F\left(P_C(I-\gamma F)\right))$ with $L\gamma <1$, where $F:H_1\to H_1$ is a L-Lipschitz and monotone operator on $C\subset H_1$ and $F\left(T\right)\cap \Omega \ne \varnothing$. For any $x_0,x_1\in H_1$, let the sequences $\left\{y_n\right\}$, $\left\{u_n\right\}$, $\left\{z_n\right\}$ and $\left\{x_n\right\}$ be generated by

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{y}_n=x_n+{\theta }_n(x_n-x_{n-1}),\hfill \\ u_n=J_{\lambda }^{B_1}(y_n+{\gamma }_n{A}^{*}(J_{\lambda }^{B_2}-I)A{y}_n),\hfill \\ z_n=\xi T{x}_n+(1-\xi )u_n,\hfill \\ x_{n+1}={\alpha }_n{f}_n\left(x_n\right)+{\beta }_n{x}_n+{\delta }_n{z}_n\hfill \end{array}\right.\end{array}$$

(43)

for each $n\ge 1$, where $\xi \in (0,1)$, ${\gamma }_n:={\tau }_n\frac{\parallel (J_{\lambda }^{B_2}-I)A{y}_n{\parallel }^2}{\parallel A^{*}(J_{\lambda }^{B_2}-I)A{y}_n{\parallel }^2}$ with $0<{\tau }_{*}\le {\tau }_n\le {\tau }^{*}<1$, $\left\{{\theta }_n\right\}\subset [0,\overline{\omega })$ for some $\overline{\omega }>0$ and $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\},\left\{{\delta }_n\right\}\in (0,1)$ with ${\alpha }_n+{\beta }_n+{\delta }_n=1$ satisfying the following conditions:

(C1) 

${lim}_{n\to \infty }{\alpha }_n=0$;

(C2) 

${\sum }_{n=1}^{\infty }{\alpha }_n=\infty$;

(C3) 

$0<{ϵ}_1\le {\beta }_n$, $0<{ϵ}_2\le {\delta }_n$;

(C4) 

${lim}_{n\to \infty }\frac{{\theta }_n}{{\alpha }_n}\parallel x_n-x_{n-1}\parallel =0$.

Then the sequence $\left\{x_n\right\}$ generated by $\left(43\right)$ converges strongly to a point $p\in F\left(T\right)\cap \Omega =\mathsf{\Gamma }\cap \Omega$, where $p=P_{\mathsf{\Gamma }\cap \Omega }f\left(p\right)$.

Proof. 

Since $I-T$ is demiclosed at the origin and $F\left(T\right)=\mathsf{\Gamma }$, by using Lemma (5) and Corollary (1), the sequence $\left\{x_n\right\}$ converges strongly to a point $p\in F\left(T\right)\cap \Omega$, that is, the sequence $\left\{x_n\right\}$ converges strongly to a point $p\in \mathsf{\Gamma }$. ☐

4.2. Game Theory

Now, we consider a game of N players in strategic form

$$G=(p_i,S_i),$$

where $i=1,\cdots ,N$, $p_i:S=S_1\times S_2\times \cdots \times S_N\to \mathbb{R}$ is the pay-off function (continuous) of the ith player and $S_i\in {\mathbb{R}}^{M_i}$ is the set of strategy of the ith player such that $M_i=\left|S_i\right|$.

Let $S_i$ be nonempty compact and convex set, $s_i\in S_i$ be the strategy of the ith player and $s=(s_1,s_2,\cdots ,s_N)$ be the collective strategy of all players. For any $s\in S$ and $z_i\in S_i$ of the ith player for each i, the symbols $S_{-i}$, $s_{-i}$ and $(z_i,s_{-i})$ are defined by

• $S_{-i}:=(S_1\times \cdots \times S_{i-1}\times S_{i+1}\times \cdots \times S_N)$ is the set of strategies of the remaining players when $s_i$ was chosen by ith player,
• $s_{-i}:=(s_1,\cdots ,s_{i-1},s_{i+1},\cdots ,s_N)$ is the strategies of the remaining players when ith player has $s_i$ and
• $(z_i,s_{-i}):=(s_1,\cdots ,s_{i-1},z_i,s_{i+1},\cdots ,s_N)$ is the strategies of the situation that $z_i$ was chosen by ith player when the rest of the remaining players have chosen $s_{-i}$.

Moreover, $\overline{s_i}$ is a special strategy of the ith player, supporting the player to maximize his pay-off, which equivalent to the following:

$${\displaystyle p_i(\overline{s_i},s_{-i})=\underset{z_i\in S_i}{max}{p}_i(z_i,s_{-i}).}$$

Definition 5.

[37,38] Given a game of N players in strategic form, the collective strategies $s^{*}\in S$ is said to be a Nash equilibrium point if

$${\displaystyle p_i\left(s^{*}\right)=\underset{z_i\in S_i}{max}{p}_i(z_i,s_i^{*})}$$

for all $i=1,\cdots ,N$ and $s_i^{*}\in S_{-i}$.

If no player can change his strategy to bring advantages, then the collective strategies $s^{*}=(s_i^{*},s_{-i}^{*})$ is a Nash equilibrium point. Furthermore, a Nash equilibrium point is the collective strategies of all players, i.e., $s_i^{*}$ (for each $i\ge 1$) is the best response of ith player. There is a multi-valued mapping $T_i:S_{-i}\to 2^{S_i}$ such that

$$\begin{array}{c}{T}_i\left(s_{-i}\right)=argmax{p}_i(z_i,s_{-i})\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}=\{s_i\in S_i:p_i(s_i,s_{-i})=\underset{z_i\in S_i}{max}{p}_i(z_i,s_{-i})\}\hfill \end{array}$$

for all $s_{-i}\in S_{-i}$. Therefore, we can define the mapping $T:S\to 2^S$ by

$$T:=T_1\times T_2\times \cdots \times T_N$$

such that the Nash equilibrium point is the collective strategies $s^{*}$, where $s^{*}\in F\left(T\right)$. Note that $s^{*}\in F\left(T\right)$ is equivalent to $s_i^{*}\in T\left(s_{-i}^{*}\right)$.

Let $H_1$ and $H_2$ be two real Hilbert spaces, $B_1:H_1\to 2^{H_1}$ and $B_2:H_2\to 2^{H_2}$ be multi-valued mappings. Suppose S is nonempty compact and convex subset of $H_1={\mathbb{R}}^{M_N}$, $H_2=\mathbb{R}$ and the rest of the players have made their best responses $s_{-i}^{*}$. For each $s\in S$, define a mapping $A:S\to H_2$ by

$$As=p_i\left(s\right)-p_i(z_i,s_{-i}^{*})$$

, where $p_i$ is linear, bounded and convex. Indeed, A is also linear, bounded and convex.

The Nash equilibrium problem (NEP) is the following:

$$\mathrm{Find}\mathrm{a}\mathrm{point}{s}^{*}\in S\mathrm{such}\mathrm{that}A{s}^{*}>0,0\in H_2.$$

(44)

However, the solution to the problem (NEP) may not be single-valued. Then the problem (NEP) reduces to finding the fixed point problem (FPP) of a multi-valued mapping, i.e.,

$$\mathrm{Find}\mathrm{a}\mathrm{point}{s}^{*}\in S\mathrm{such}\mathrm{that}{s}^{*}\in T{s}^{*},$$

(45)

where T is multi-valued pay-off function.

Now, we apply our Theorem 1 to find a solution to the problem (FPP).

Let $B_1:H_1\to 2^{H_1}$ and $B_2:H_2\to 2^{H_2}$ be maximal monotone mappings defined on $H_1$ and $H_2$, respectively, and $A:H_1\to H_2$ be a bounded linear operator with its adjoint $A^{*}$.

Now, we consider the following problem:

$$\mathrm{Find}\mathrm{a}\mathrm{point}{s}^{*}\in H_1\mathrm{such}\mathrm{that}0\in B_1\left(s^{*}\right),s^{*}\in T{s}^{*}$$

(46)

and

$$y^{*}=A{s}^{*}\in H_2\mathrm{such}\mathrm{that}0\in B_2\left(y^{*}\right).$$

(47)

Theorem 3.

Assume that $B_1$ and $B_2$ are maximal monotone mappings defined on Hilbert spaces $H_1$ and $H_2$, respectively. Let $T:S\to CB\left(S\right)$ be a multi-valued quasi-nonexpansive mapping such that T is demiclosed at the origin. Let $\left\{f_n\right\}$ be a sequence of ${\mu }_n$-contractions $f_n:H_1\to H_1$ with $0<{\mu }_{*}\le {\mu }_n\le {\mu }^{*}<1$ and $\left\{f_n\left(x\right)\right\}$ be uniformly convergent for any x in a bounded subset D of $H_1$. Suppose that the problem(NEP)has a nonempty solution and $F\left(T\right)\cap \Omega \ne \varnothing$. For arbitrarily chosen $x_0,x_1\in H_1$, let the sequences $\left\{y_n\right\}$, $\left\{u_n\right\}$, $\left\{z_n\right\}$ and $\left\{x_n\right\}$ be generated by

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{y}_n=x_n+{\theta }_n(x_n-x_{n-1}),\hfill \\ u_n=J_{\lambda }^{B_1}(y_n+{\gamma }_n{A}^{*}(J_{\lambda }^{B_2}-I)A{y}_n),\hfill \\ z_n=\xi v_n+(1-\xi )u_n,v_n\in T{x}_n,\hfill \\ x_{n+1}={\alpha }_n{f}_n\left(x_n\right)+{\beta }_n{x}_n+{\delta }_n{z}_n\hfill \end{array}\right.\end{array}$$

(48)

for each $n\ge 1$, where $\xi \in (0,1)$, ${\gamma }_n:={\tau }_n\frac{\parallel (J_{\lambda }^{B_2}-I)A{y}_n{\parallel }^2}{\parallel A^{*}(J_{\lambda }^{B_2}-I)A{y}_n{\parallel }^2}$ with $0<{\tau }_{*}\le {\tau }_n\le {\tau }^{*}<1$, $\left\{{\theta }_n\right\}\subset [0,\overline{\omega })$ for some $\overline{\omega }>0$ and $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\},\left\{{\delta }_n\right\}\in (0,1)$ with ${\alpha }_n+{\beta }_n+{\delta }_n=1$ satisfying the following conditions:

(C1) 

${lim}_{n\to \infty }{\alpha }_n=0$;

(C2) 

${\sum }_{n=1}^{\infty }{\alpha }_n=\infty$;

(C3) 

$0<{ϵ}_1\le {\beta }_n$ and $0<{ϵ}_2\le {\delta }_n$;

(C4) 

${lim}_{n\to \infty }\frac{{\theta }_n}{{\alpha }_n}\parallel x_n-x_{n-1}\parallel =0$.

Then the sequence $\left\{x_n\right\}$ generated by Equation $\left(48\right)$ converges strongly to Nash equilibrium point.

Proof. 

By Theorem 1, the sequence $\left\{x_n\right\}$ converges strongly to a point $p\in F\left(T\right)\cap \Omega$, then the sequence $\left\{x_n\right\}$ converges strongly to a Nash equilibrium point. ☐