A new technique for convergence theorem of fixed point problem of quasi-nonexpansive mapping

Abstract

For the purpose of this paper, we use the method different from the relaxed extragradient method for finding a common element of the set of fixed points of a quasi-nonexpansive mapping, the set of solutions of equilibrium problems, and the set of solutions of a modified system of variational inequalities without demiclosed condition of W and $W_{\omega}:= (1-\omega )I+\omega W$, where W is a quasi-nonexpansive mapping and $\omega\in (0,\frac{1}{2} )$ in the framework of Hilbert space. By using our main result, we obtain a strong convergence theorem involving a finite family of nonspreading mappings and another corollary. Moreover, we give a numerical example to encourage our main theorem.

1 Introduction

Let C be a nonempty closed convex subset of a real Hilbert space H. Recall that the mapping $W: C \rightarrow C$ is called quasi-nonexpansive if

$$ \Vert Wp-q\Vert \leq \Vert p-q\Vert , $$

for all $p\in C$ and $q\in F(W)$. We denote by $F(W)$ the set of fixed points of W. Fixed point problems have been widely studied and developed in the literature.

Let Ψ be a bifunction of $C\times C$ into $\mathbb{R}$, where $\mathbb{R}$ is the set of real numbers. The equilibrium problem for $\Psi:C\times C \rightarrow\mathbb{R}$ is to find $p \in C$ such that

$$ \Psi(p,\zeta)\geq0, \quad \forall\zeta\in C. $$

(1.1)

We denote the set of solutions of (1.1) by $\operatorname{EP} (\Psi )$. Equilibrium problems were introduced by Blum and Oettli [1] in 1994 and included many well-known problems such as the variational inequality problem, the optimization problem, and the nonexpansive mapping and fixed point problem.

A mapping $D_{1}: C \rightarrow H$ is called $d_{1}$-inverse strongly monotone if there exists a positive real number $d_{1} > 0$ such that

$$ \langle D_{1}p-D_{1}\zeta, p-\zeta \rangle\geq d_{1}\Vert D_{1}p-D_{1}\zeta \Vert ^{2}, $$

for all $p,\zeta\in C$.

Let $B:C \rightarrow H$. The variational inequality is to find a point $\phi\in C$ such that

$$ \langle B\phi, \psi-\phi \rangle\geq0, $$

(1.2)

for all $\psi\in C$. The set of solutions of (1.2) is denoted by $\operatorname{VIP}(C,B)$. The variational inequalities were initially studied and introduced by Lions and Stampacchia [2].

The concept of quasi-nonexpansive mapping was investigated by Diaz and Metcalf [3]. In 2007, Su et al. [4] introduced strong convergence theorems for quasi-nonexpansive mappings, the monotone hybrid iteration method used to approximate the fixed point of quasi-nonexpansive mappings. In 2011, Tian and Jin [5] introduced an iterative method of a quasi-nonexpansive mapping in the framework of Hilbert space. They proved the strong convergence theorem of iterative scheme $\{p_{n}\}$ generated by (1.3) as follows.

Theorem 1.1

Let H be a real Hilbert space, let F be a κ-Lipschitzian and η-strongly monotone operator on H with $\kappa>0$, $\eta>0$ and let W be a quasi-nonexpansive mapping on H, and f is a L-Lipschitzian mapping with coefficient $L>0$ for all $p,\zeta\in H$. Assume the set $F(W)$ of fixed points of W is nonempty closed and convex. Let $0< \mu< \frac{2\eta}{\kappa^{2}}$, $0< \gamma< \mu ( \eta-\frac{\mu\kappa^{2}}{2} )/L = \tau/L$ and start with an arbitrary chosen $p_{0}\in H$, let the sequence $\{p_{n}\}$ be generated by

$$ p_{n+1}=\alpha_{n}\gamma f(p_{n})+(I- \alpha_{n} \mu F)W_{\omega}p_{n}, $$

(1.3)

where the sequence $\{\alpha_{n} \}\subset (0,1 )$ satisfies $\lim_{n \rightarrow\infty} \alpha_{n} =0$, and $\sum_{n=1}^{\infty} \alpha_{n} = \infty$. Also $\omega\in (0,\frac {1}{2} )$, $W_{\omega}:= (1-\omega)I+\omega W$ with two conditions on W:

• 1. $\Vert Wp-q \Vert \leq \Vert p-q \Vert$ for any $p\in H$, and $q \in F(W)$; this means that W is a quasi-nonexpansive mapping;

• 2. W is demiclosed on H; that is, if $\{\zeta_{k}\}\subset H$, $\zeta_{k}\rightharpoonup\xi$, and $(I-W)\zeta_{k} \rightarrow0$, then $\xi\in F(W)$.

Then $\{p_{n} \}$ converges strongly to the $p^{*} \in F(W)$ which is the unique solution of the VIP:

$$ \bigl\langle (\mu F-\gamma f)p^{*},p-p^{*} \bigr\rangle \leq0,\quad \forall p\in F(W). $$

Many strong convergence theorems of quasi-nonexpansive mapping W were proved by assuming the following conditions:

• 1. $W_{\omega}:= (1-\omega)I+\omega W$ for all $\omega\in (0, \frac{1}{2} )$,

• 2. W is demiclosed on H.

In 2012, Dong et al. [6] proved strong convergence theorem by using a relaxed extragradient method as follows.

Theorem 1.2

Let C be a nonempty closed convex subset of a real Hilbert space H. Let the mappings $D_{1},D_{2}:C\rightarrow H$ be $d_{1}$-inverse strongly monotone and $d_{2}$-inverse strongly monotone, respectively. Let Ψ be a bifunction from $C\times C\rightarrow\mathbb{R}$ satisfying (J1)-(J4) and let $\{W_{n}\}^{\infty}_{n=1}:C\rightarrow C$ be a countable family of nonexpansive mappings such that $\Omega:=\bigcap_{n=1}^{\infty}F(W_{n}) \cap \operatorname{EP}(\Psi) \cap F(G) \neq\emptyset$. Let $f:C \rightarrow C$ be a contraction with coefficient $\rho\in(0,1/2)$. Set $\beta_{0}=1$. For given $p_{1} \in C$ arbitrarily, let the sequences $\{ p_{n} \}$, $\{ \zeta_{n} \}$, $\{ \xi_{n} \}$, and $\{ \phi_{n} \}$ be generated by

$$ \textstyle\begin{cases} \Psi ( \phi_{n},\zeta ) + \frac{1}{g_{n}} \langle\zeta -\phi_{n}, \phi_{n} - p_{n} \rangle\geqslant0,\quad \forall \zeta\in C, \\ \xi_{n}=P_{C}(\phi_{n}-\lambda_{B} D_{2} \phi_{n}), \\ \zeta_{n}=\alpha_{n}f(p_{n})+(1-\alpha_{n})P_{C}(\xi_{n}-\lambda_{A} D_{1} \xi_{n}), \\ p_{n+1} = \beta_{n} p_{n} + \sigma_{n}\sum_{i=1}^{\infty}(\beta _{i-1}-\beta_{i})W_{i}\zeta_{n} \\ \hphantom{p_{n+1} ={}}{}+(1-\beta_{n})(1-\sigma_{n})P_{C}(\xi_{n}-\lambda_{A} D_{1} \xi_{n}),\quad \forall n\in\mathbb{N}, \end{cases} $$

(1.4)

where $\lambda_{A} \in(0,2d_{1})$, $\lambda_{B} \in(0,2d_{2})$, and the sequences $\{ \alpha_{n} \} \subset[0,1]$, $\{ \beta_{n} \} \subset[0,1]$, $\{ \sigma_{n} \}\subset[0,1]$, and $\{ g_{n} \}\subset(r,\infty)$, $r>0$, are such that

• (i) $\{\beta_{n}\}$ is strictly decreasing,

• (ii) $0 < \liminf_{n\rightarrow\infty} \beta_{n} < \limsup_{n\rightarrow\infty} \beta_{n} < 1$,

• (iii) $\lim_{n \rightarrow\infty} \alpha_{n} =0$ and $\sum_{n=1}^{\infty} \alpha_{n} = \infty$,

• (iv) $\sigma_{n}>1/2(1-\rho)$, $\sum_{n=1}^{\infty} \vert \sigma_{n} - \sigma_{n-1} \vert < \infty$,

• (v) $\sum_{n=1}^{\infty} \vert g_{n} - g_{n-1} \vert < \infty$.

Then the sequence $\{ p_{n} \}$ generated by (1.4) converges strongly to $p^{*}=P_{\Omega} \cdot f(p^{*})$, and $(p^{*}, \zeta ^{*} )$ is a solution of the general system of variational inequalities (1.5) where $\zeta^{*}=P_{C}(p^{*}-\lambda_{B} D_{2} p^{*})$.

Many authors used the extragradient method to prove fixed point theorem of nonlinear mappings.

Let $D_{1},D_{2}: C \rightarrow H$ be two mappings. In 2008, Ceng et al. [7] introduced a relaxed extragradient method for finding solutions of problem $( p^{*},\xi^{*} ) \in C \times C$ such that

$$ \textstyle\begin{cases} \langle\lambda_{A}D_{1}\xi^{*} + p^{*} - \xi^{*} , p-p^{*} \rangle\geq0, \quad \forall p\in C, \\ \langle\lambda_{B}D_{2}p^{*} + \xi^{*} - p^{*} , p-\xi^{*} \rangle\geq0, \quad \forall p\in C, \end{cases} $$

(1.5)

which is called a system of variational inequalities where $\lambda _{A}, \lambda_{B}>0$.

In 2013, Kangtunyakarn [8] modified (1.5) for finding $( p^{*},\xi^{*} ) \in C \times C$ such that

$$ \textstyle\begin{cases} \langle p^{*} - (I-\lambda_{A}D_{1} ) (ap^{*}+ (1-a )\xi^{*} ), p-p^{*} \rangle\geq0,\quad \forall p\in C, \\ \langle\xi^{*} - (I-\lambda_{B}D_{2} )p^{*}, p-\xi^{*} \rangle\geq0, \quad \forall p\in C, \end{cases} $$

(1.6)

which is called a modification of system of variational inequalities, for every $\lambda_{A}, \lambda_{B}>0$ and $a\in [ 0,1 ]$. If $a=0$, (1.6) reduces to (1.5). He introduced the relation between solutions of (1.6) and fixed point of the mapping G as follows.

Lemma 1.3

Let C be a nonempty closed convex subset of a real Hilbert space H and let $D_{1},D_{2}: C \rightarrow H$ be mappings. For every $\lambda _{A}, \lambda_{B} > 0$ and $a\in [ 0,1 ]$, the following statements are equivalent:

• 1. $( p^{*},\xi^{*} ) \in C \times C$ is a solution of problem (1.6),

• 2. $p^{*}$ is a fixed point of the mapping $G: C \rightarrow C$, i.e., $p^{*} \in F(G)$, defined by

  $$ G(p) = P_{C}(I-\lambda_{A}D_{1}) \bigl(ap+(1-a)P_{C}(I-\lambda_{B}D_{2})p\bigr), $$

where $\xi^{*}=P_{C}(I-\lambda_{B}D_{2})p^{*}$.

After we investigated Theorem 1.1, Theorem 1.2 and researchers in the same direction, we have the questions as follows:

• (1) Can we prove strong convergence theorem without demiclosed condition and $W_{\omega}:= (1-\omega)I+\omega W$, where W is a quasi-nonexpansive mapping and $\omega\in (0,\frac{1}{2} )$ in the framework of Hilbert space?

• (2) Can we prove strong convergence theorem without relaxed extragradient method?

In this paper, we give the answer for the mentioned questions and introduce the method of iterative scheme $\{p_{n}\}$ for finding a common element of the set of fixed points of a quasi-nonexpansive mapping, the set of solutions of equilibrium problems and the set of solutions of a modified system of variational inequalities. Applying our main result, we prove strong convergence theorem involving a finite family of nonspreading mappings and another corollary. Moreover, We also give a numerical example to support our main theorem.

2 Preliminaries

Let H be a real Hilbert space with inner product $\langle\cdot , \cdot \rangle$ and norm $\Vert \cdot \Vert$. In this paper, we use the symbol of weak and strong convergence by ‘⇀’ and ‘→’, respectively. For every $p \in H$, there exists a unique nearest point $P_{C}p$ in C such that $\Vert p- P_{C}p\Vert \leq \Vert p-\zeta \Vert$ for all $\zeta\in C$. $P_{C}$ is called the metric projection of H onto C.

Remark 2.1

It is well known that metric projection $P_{C}$ has the following properties:

• 1. $P_{C}$ is firmly nonexpansive, i.e.,

  $$ \Vert P_{C}p - P_{C}\zeta \Vert ^{2} \leq \langle P_{C}p - P_{C}\zeta, p - \zeta \rangle, \quad \forall p,\zeta\in H. $$

• 2. For each $p \in H$,

  $$ \xi=P_{C} (p ) \quad \Leftrightarrow\quad \langle p-\xi,\xi-\zeta \rangle\geq0, \quad \forall\zeta\in C. $$

Recall that H satisfies Opial’s condition [9], i.e., for any sequence $\{p_{n} \}$ with $p_{n} \rightharpoonup p$, the inequality

$$ \lim_{n \rightarrow\infty} \inf \Vert p_{n}-p\Vert < \lim _{n \rightarrow\infty} \inf \Vert p_{n}-\zeta \Vert $$

holds for every $\zeta\in H$ with $\zeta\neq p$.

Lemma 2.2

Let H be a real Hilbert space. Then we have the following well-known results:

• 1. $\Vert p \pm\zeta \Vert ^{2} = \Vert p\Vert ^{2} \pm2 \langle p, \zeta \rangle+ \Vert \zeta \Vert ^{2}$,

• 2. $\Vert p + \zeta \Vert ^{2} \leq \Vert p\Vert ^{2} + 2 \langle\zeta, p+\zeta \rangle$,

for all $p,\zeta\in H$.

Lemma 2.3

([10])

Let $( E , \langle\cdot,\cdot \rangle )$ be an inner product space. Then, for all $p, \zeta, \xi\in E$ and $\alpha_{1}, \alpha_{2}, \alpha_{3} \in[0,1]$ with $\alpha_{1} + \alpha_{2} + \alpha_{3} = 1$, we have

$$\begin{aligned} \Vert \alpha_{1} p + \alpha_{2} \zeta+ \alpha_{3} \xi \Vert ^{2} =& \alpha_{1} \Vert p \Vert ^{2} + \alpha_{2} \Vert \zeta \Vert ^{2} + \alpha_{3} \Vert \xi \Vert ^{2} - \alpha_{1} \alpha _{2} \Vert p-\zeta \Vert ^{2} \\ &{} - \alpha_{1} \alpha_{3} \Vert p-\xi \Vert ^{2} - \alpha_{2} \alpha _{3} \Vert \zeta-\xi \Vert ^{2}. \end{aligned}$$

For solving the equilibrium problem, we assume that the bifunction $\Psi :C\times C\rightarrow\mathbb{R}$ satisfies the following conditions:

• (J1) $\Psi(p,p)=0$ for all $p\in C$;

• (J2) Ψ is monotone, i.e., $\Psi(p,\zeta)+\Psi(\zeta,p)\leq 0$ for all $p,\zeta\in C$;

• (J3) for each $p,\zeta,\xi\in C$,

  $$\lim_{t \downarrow0} \Psi \bigl( t\xi+(1-t)p,\zeta \bigr) \leq\Psi (p, \zeta); $$

• (J4) for each $p\in C$, $\zeta\mapsto\Psi(p,\zeta)$ is convex and lower semicontinuous.

Lemma 2.4

([1])

Let C be a nonempty closed convex subset of H and let Ψ be a bifunction of $C \times C$ into $\mathbb{R}$ satisfying (J1)-(J4). Let $r>0$ and $p \in H$. Then there exists $\xi\in C$ such that

$$ \Psi(\xi,\zeta) + \frac{1}{r} \langle\zeta-\xi, \xi-p \rangle\geq0, \quad \forall\zeta\in C. $$

Lemma 2.5

([11])

Assume that $\Psi:C\times C\rightarrow\mathbb{R}$ satisfies (J1)-(J4). For $r >0$, define a mapping $W_{r}:H\rightarrow C$ as follows:

$$ W_{r}(p)= \biggl\{ \xi\in C : \Psi(\xi,\zeta)+\frac{1}{r} \langle\zeta-\xi ,\xi-p\rangle\geq0,\forall\zeta\in C \biggr\} , $$

for all $p\in H$. Then the following hold:

• (1) $W_{r}$ is single-valued;

• (2) $W_{r}$ is firmly nonexpansive, i.e., for any $p,\zeta\in H$,

  $$ \bigl\Vert W_{r}(p)-W_{r}(\zeta) \bigr\Vert ^{2} \leq \bigl\langle W_{r}(p)-W_{r}(\zeta),p- \zeta \bigr\rangle ; $$

• (3) $F (W_{r} ) = \operatorname{EP}(\Psi)$;

• (4) $\operatorname{EP}(\Psi)$ is closed and convex.

Lemma 2.6

([12])

Let $\{h_{n} \}$ be a sequence of nonnegative real numbers satisfying

$$h_{n+1} \leq (1- \alpha_{n} )h_{n} + \delta_{n},\quad \forall n \geq0, $$

where $\{\alpha_{n} \}$ is a sequence in $(0,1 )$ and $\{\delta_{n} \}$ is a sequence such that

• (1) $\sum_{n=1}^{\infty} \alpha_{n} = \infty$,

• (2) $\limsup_{n \rightarrow\infty} \frac{\delta_{n}}{\alpha _{n}} \leq0$ or $\sum_{n=1}^{\infty} \vert \delta_{n} \vert < \infty$.

Then $\lim_{n \rightarrow\infty} h_{n} = 0$.

Lemma 2.7

([13])

Let H be a real Hilbert space, let C be a nonempty closed convex subset of H and let $D_{1}$ be a mapping of C into H. Let $u \in C$. Then for $\lambda> 0$,

$$ u = P_{C} (I-\lambda D_{1} )u\quad \Leftrightarrow\quad u \in \operatorname{VIP} (C,D_{1} ), $$

where $P_{C}$ is the metric projection of H onto C.

Lemma 2.8

([14])

Let C be a nonempty closed convex subset of a real Hilbert space H and let $W: C\rightarrow C$ be a quasi-nonexpansive mapping with $F (W )\neq\emptyset$. Then $\operatorname{VIP} (C,I-W ) = F (W )$.

Remark 2.9

From Lemmas 2.7 and 2.8, we have

$$ F(W)=\operatorname{VIP} (C,I-W ) = F \bigl( P_{C} \bigl(I-\lambda (I-W ) \bigr) \bigr), $$

for all $\lambda> 0$.

3 Main result

Theorem 3.1

Let C be a nonempty closed convex subset of a real Hilbert space H, let $\Psi_{1},\Psi_{2}:C\times C\rightarrow\mathbb{R}$ be bifunctions satisfying (J1)-(J4) and let $W:C\rightarrow C$ be a quasi-nonexpansive mapping. Let $D_{1},D_{2}:C\rightarrow H$ be $d_{1},d_{2}$-inverse strongly monotone mappings, respectively. Define the mapping $G:C\rightarrow C$ by $G(p) = P_{C}(I-\lambda_{A}D_{1}) (ap+(1-a)P_{C}(I-\lambda _{B}D_{2})p )$ for all $p \in C$ and $a \in [0,1 ]$. Assume $\mathcal{F}= \operatorname{EP}(\Psi_{1}) \cap \operatorname{EP}(\Psi_{2}) \cap F(G) \cap F(W) \neq \emptyset$. Suppose that $p_{1},u\in C$ and let $\{ p_{n} \}$, $\{ \phi _{n} \}$, and $\{ \psi_{n} \}$ be sequences generated by

$$ \textstyle\begin{cases} \Psi_{1} ( \phi_{n},\zeta ) + \frac{1}{g_{n}} \langle \zeta-\phi_{n}, \phi_{n} - p_{n} \rangle\geqslant0,\quad \forall\zeta\in C, \\ \Psi_{2} ( \psi_{n},\zeta ) + \frac{1}{h_{n}} \langle \zeta-\psi_{n}, \psi_{n} - p_{n} \rangle\geqslant0,\quad \forall\zeta\in C, \\ p_{n+1} = \alpha_{n}u + \beta_{n}p_{n} + \gamma_{n}P_{C} (I - \lambda_{n} (I-W) ) \phi_{n} + \delta_{n} G(\psi_{n}),\quad \forall n\in\mathbb{N} , \end{cases} $$

(3.1)

where the sequences $\lambda_{A} \in(0,2d_{1})$, $\lambda_{B} \in (0,2d_{2})$ and $\{ \alpha_{n} \}, \{ \beta_{n} \}, \{ \gamma_{n} \}, \{ \delta_{n} \} \subseteq[0,1]$ with $\alpha_{n}+\beta_{n}+\gamma _{n}+\delta_{n}=1$ for all $n \in\mathbb{N}$. Suppose the following conditions hold:

• (i) $\lim_{n \rightarrow\infty} \alpha_{n} =0$ and $\sum_{n=1}^{\infty} \alpha_{n} = \infty$,

• (ii) $0 < c \leq\beta_{n}, \gamma_{n}, \delta_{n} \leq d < 1$ for some $c, d > 0$ and for all $n \geq1$,

• (iii) $0 < e \leq g_{n},h_{n} \leq f$ for some $e, f > 0$ and for all $n \geq1$,

• (iv) $\sum_{n=1}^{\infty} \lambda_{n} < \infty$ and $0 < \lambda_{n} < 1$,

• (v) $\sum_{n=1}^{\infty} \vert \alpha_{n+1} - \alpha_{n} \vert < \infty$, $\sum_{n=1}^{\infty} \vert \beta_{n+1} - \beta_{n} \vert < \infty$, $\sum_{n=1}^{\infty} \vert \gamma_{n+1} - \gamma_{n} \vert < \infty$, $\sum_{n=1}^{\infty} \vert \lambda_{n+1} - \lambda_{n} \vert < \infty$, $\sum_{n=1}^{\infty} \vert g_{n+1} - g_{n} \vert < \infty$, $\sum_{n=1}^{\infty} \vert h_{n+1} - h_{n} \vert < \infty$.

Then $\{ p_{n} \}$, $\{ \phi_{n} \}$, and $\{ \psi_{n} \}$ converge strongly to $p_{0}=P_{\mathcal{F}}u$ and $(p_{0},\xi_{0} )$ is a solution of (1.6) where $\xi_{0}=P_{C}(I-\lambda_{B}D_{2})p_{0}$.

Proof

First, we show that G is a nonexpansive mapping. Let $p,\zeta\in C$. Since $D_{1}$, $D_{2}$ are $d_{1},d_{2}$-inverse strongly monotone, $\lambda_{A} \in(0,2d_{1})$, and $\lambda_{B} \in (0,2d_{2})$, we have

$$\begin{aligned}& \bigl\Vert (I-\lambda_{A}D_{1})p-(I-\lambda_{A}D_{1}) \zeta\bigr\Vert ^{2} \\& \quad =\Vert p-\zeta \Vert ^{2}-2 \lambda_{A} \langle p-\zeta,D_{1}p-D_{1}\zeta \rangle+\lambda_{A}^{2}\Vert D_{1} p-D_{1}\zeta \Vert ^{2} \\& \quad \leq \Vert p-\zeta \Vert ^{2} - 2d_{1} \lambda_{A} \Vert D_{1} p-D_{1} \zeta \Vert ^{2} + \lambda_{A}^{2} \Vert D_{1} p-D_{1} \zeta \Vert ^{2} \\& \quad =\Vert p-\zeta \Vert ^{2} + \lambda_{A}( \lambda_{A}-2d_{1})\Vert D_{1} p-D_{1} \zeta \Vert ^{2} \\& \quad \leq \Vert p-\zeta \Vert ^{2}. \end{aligned}$$

Then $I-\lambda_{A}D_{1}$ is a nonexpansive mapping. Similarly $I-\lambda _{B}D_{2}$ is a nonexpansive mapping. Then G is a nonexpansive mapping.

Next, we show $\{ p_{n} \}$ is bounded. Let $\xi\in\mathcal{F}$, then $\phi_{n}=W_{g_{n}}p_{n}$ and $\psi_{n}=W_{h_{n}}p_{n}$. It is clear that $\Vert \phi_{n}-\xi \Vert \leq \Vert p_{n}-\xi \Vert$ and $\Vert \psi_{n}-\xi \Vert \leq \Vert p_{n}-\xi \Vert$. By Remark 2.9, we have

$$ \xi\in F \bigl( P_{C}\bigl(I-\lambda_{n}(I-W) \bigr) \bigr). $$

(3.2)

Observe that

$$\begin{aligned} \Vert W\phi_{n}-\xi \Vert ^{2} &=\bigl\Vert ( \phi_{n}-\xi)-(I-W)\phi _{n} \bigr\Vert ^{2} \\ &=\Vert \phi_{n}-\xi \Vert ^{2} - 2 \bigl\langle \phi_{n}-\xi, (I-W)\phi_{n} \bigr\rangle +\bigl\Vert (I-W) \phi_{n}\bigr\Vert ^{2} \\ &\leq \Vert \phi_{n}-\xi \Vert ^{2}. \end{aligned}$$

It implies that

$$ \bigl\Vert (I-W)\phi_{n}\bigr\Vert ^{2} \leq2 \bigl\langle \phi_{n}-\xi,(I-W)\phi _{n} \bigr\rangle . $$

(3.3)

From (3.2) and (3.3), we have

$$\begin{aligned} \bigl\Vert P_{C} \bigl(I-\lambda_{n}(I-W) \bigr)\phi_{n}-\xi\bigr\Vert ^{2} =& \bigl\Vert P_{C} \bigl(I-\lambda_{n}(I-W) \bigr)\phi_{n}-P_{C} \bigl(I-\lambda _{n}(I-W) \bigr)\xi\bigr\Vert ^{2} \\ \leq& \bigl\Vert (\phi_{n}-\xi)-\lambda_{n} \bigl((I-W) \phi_{n}-(I-W)\xi \bigr) \bigr\Vert ^{2} \\ =& \Vert \phi_{n}-\xi \Vert ^{2} -2 \lambda_{n}\bigl\langle \phi_{n}-\xi ,(I-W) \phi_{n} \bigr\rangle \\ &{}+ \lambda_{n}^{2}\bigl\Vert (I-W)\phi_{n} \bigr\Vert ^{2} \\ \leq& \Vert \phi_{n}-\xi \Vert ^{2} + \lambda_{n} ( \lambda _{n}-1 )\bigl\Vert (I-W) \phi_{n}\bigr\Vert ^{2} \\ \leq& \Vert \phi_{n}-\xi \Vert ^{2}. \end{aligned}$$

(3.4)

From the definition of $p_{n}$ and (3.4), we have

$$\begin{aligned} \Vert p_{n+1}-\xi \Vert =&\bigl\Vert \alpha_{n}(u- \xi)+\beta _{n}(p_{n}-\xi)+\gamma_{n} \bigl(P_{C}\bigl(I-\lambda_{n}(I-W)\bigr) \phi_{n}-\xi \bigr) \\ &{}+ \delta_{n} \bigl(G(\psi_{n})-\xi \bigr)\bigr\Vert \\ \leq& \alpha_{n}\Vert u-\xi \Vert + \beta_{n}\Vert p_{n}-\xi \Vert + \gamma_{n}\bigl\Vert P_{C} \bigl(I-\lambda_{n}(I-W) \bigr)\phi _{n}-\xi\bigr\Vert \\ &{}+ \delta_{n}\bigl\Vert G(\psi_{n})-\xi\bigr\Vert \\ \leq& \alpha_{n}\Vert u-\xi \Vert + \beta_{n}\Vert p_{n}-\xi \Vert + \gamma_{n}\Vert \phi_{n}- \xi \Vert + \delta_{n}\Vert \psi_{n}-\xi \Vert \\ \leq& \alpha_{n}\Vert u-\xi \Vert + \beta_{n}\Vert p_{n}-\xi \Vert + \gamma_{n}\Vert p_{n}-\xi \Vert + \delta_{n}\Vert p_{n}-\xi \Vert \\ =& \alpha_{n}\Vert u-\xi \Vert + (1 - \alpha_{n}) \Vert p_{n}-\xi \Vert . \end{aligned}$$

By induction, we can conclude that

$$ \Vert p_{n}-\xi \Vert \leq \max\bigl\{ \Vert u-\xi \Vert , \Vert p_{1}-\xi \Vert \bigr\} , $$

for all $n \geq1$. This implies that the sequence $\{p_{n}\}$ is bounded and so are $\{ \phi_{n} \}$, $\{ \psi_{n} \}$, $\{ (I-W) \phi_{n} \}$, and $\{ P_{C} (I - \lambda_{n} (I-W) ) \phi_{n} \}$.

Then we show that $\lim_{n \rightarrow\infty} \Vert p_{n+1} - p_{n} \Vert = 0$.

From the definition of $p_{n}$ and nonexpansiveness of G, we have

$$\begin{aligned} \Vert p_{n+1}-p_{n} \Vert =& \bigl\Vert (\alpha_{n}-\alpha_{n-1})u + \beta_{n}(p_{n} - p_{n-1}) + (\beta_{n} - \beta_{n-1})p_{n-1} \\ &{}+ \gamma_{n} \bigl( P_{C}\bigl(I-\lambda_{n}(I-W) \bigr)\phi_{n} - P_{C}\bigl(I-\lambda _{n-1}(I-W) \bigr)\phi_{n-1} \bigr) \\ &{}+ ( \gamma_{n} - \gamma_{n-1} )P_{C}\bigl(I- \lambda_{n-1}(I-W)\bigr)\phi _{n-1} \\ &{}+ \delta_{n} \bigl( G (\psi_{n} ) - G (\psi _{n-1} ) \bigr) + ( \delta_{n} - \delta_{n-1} )G ( \psi _{n-1} ) \bigr\Vert \\ \leq& |\alpha_{n}-\alpha_{n-1}| \Vert u\Vert + \beta_{n} \Vert p_{n} - p_{n-1}\Vert + | \beta_{n} - \beta_{n-1}|\Vert p_{n-1}\Vert \\ &{}+ \gamma_{n}\bigl\Vert P_{C}\bigl(I- \lambda_{n}(I-W)\bigr)\phi_{n} - P_{C}\bigl(I- \lambda _{n-1}(I-W)\bigr)\phi_{n-1}\bigr\Vert \\ &{}+ | \gamma_{n} - \gamma_{n-1} | \bigl\Vert P_{C}\bigl(I-\lambda_{n-1}(I-W)\bigr)\phi _{n-1} \bigr\Vert \\ &{}+ \delta_{n} \bigl\Vert G(\psi_{n}) - G( \psi_{n-1})\bigr\Vert + |\delta_{n}-\delta _{n-1}| \bigl\Vert G(\psi_{n-1})\bigr\Vert \\ \leq& |\alpha_{n}-\alpha_{n-1}| \Vert u\Vert + \beta_{n} \Vert p_{n} - p_{n-1}\Vert + | \beta_{n} - \beta_{n-1}|\Vert p_{n-1}\Vert \\ &{}+ \gamma_{n}\bigl\Vert ( \phi_{n}- \phi_{n-1} ) - \bigl(\lambda _{n}(I-W)\phi_{n}- \lambda_{n}(I-W)\phi_{n-1} \bigr) \\ &{}- \bigl( \lambda_{n}(I-W)\phi_{n-1}- \lambda_{n-1}(I-W)\phi_{n-1} \bigr)\bigr\Vert \\ &{}+ | \gamma_{n} - \gamma_{n-1} | \bigl\Vert P_{C}\bigl(I-\lambda_{n-1}(I-W)\bigr)\phi _{n-1} \bigr\Vert + \delta_{n} \Vert \psi_{n} - \psi_{n-1}\Vert \\ &{}+ |\delta_{n}-\delta_{n-1}| \bigl\Vert G( \psi_{n-1})\bigr\Vert \\ \leq& |\alpha_{n}-\alpha_{n-1}| \Vert u\Vert + \beta_{n} \Vert p_{n} - p_{n-1}\Vert + | \beta_{n} - \beta_{n-1}|\Vert p_{n-1}\Vert \\ &{}+ \gamma_{n} \Vert \phi_{n}-\phi_{n-1} \Vert + \lambda_{n}\bigl\Vert (I-W)\phi_{n}-(I-W) \phi_{n-1}\bigr\Vert \\ &{}+ |\lambda_{n} - \lambda_{n-1}| \bigl\Vert (I-W) \phi_{n-1} \bigr\Vert \\ &{}+ | \gamma_{n} - \gamma_{n-1} | \bigl\Vert P_{C}\bigl(I-\lambda_{n-1}(I-W)\bigr)\phi _{n-1} \bigr\Vert + \delta_{n} \Vert \psi_{n} - \psi_{n-1}\Vert \\ &{}+ |\delta_{n}-\delta_{n-1}| \bigl\Vert G( \psi_{n-1})\bigr\Vert . \end{aligned}$$

(3.5)

On the other hand, from $\phi_{n} = W_{g_{n}}p_{n}$ and $\phi_{n+1} = W_{g_{n+1}}p_{n+1}$, we have

$$ \Psi_{1} ( \phi_{n},\zeta ) + \frac{1}{g_{n}} \langle \zeta-\phi_{n}, \phi_{n}-p_{n} \rangle\geq0,\quad \forall \zeta\in C $$

(3.6)

and

$$ \Psi_{1} ( \phi_{n+1},\zeta ) + \frac{1}{g_{n+1}} \langle\zeta-\phi_{n+1}, \phi_{n+1}-p_{n+1} \rangle\geq0,\quad \forall\zeta\in C. $$

(3.7)

Putting $\zeta=\phi_{n+1}$ in (3.6) and $\zeta=\phi_{n}$ in (3.7), we have

$$ \Psi_{1} ( \phi_{n},\phi_{n+1} ) + \frac{1}{g_{n}} \langle\phi_{n+1}-\phi_{n}, \phi_{n}-p_{n} \rangle\geq0 $$

and

$$ \Psi_{1} ( \phi_{n+1},\phi_{n} ) + \frac{1}{g_{n+1}} \langle\phi_{n}-\phi_{n+1}, \phi_{n+1}-p_{n+1} \rangle\geq0. $$

From (J2), we have

$$ \biggl\langle \phi_{n+1}-\phi_{n}, \frac{\phi_{n}-p_{n}}{g_{n}} - \frac {\phi_{n+1}-p_{n+1}}{g_{n+1}} \biggr\rangle \geq0. $$

So

$$ \biggl\langle \phi_{n+1}-\phi_{n}, \phi_{n}- \phi_{n+1}+\phi_{n+1}-p_{n} - \frac{g_{n}}{g_{n+1}} ( \phi_{n+1}-p_{n+1} ) \biggr\rangle \geq0. $$

Then

$$\begin{aligned} \Vert \phi_{n+1}-\phi_{n} \Vert ^{2} &\leq \biggl\langle \phi _{n+1}-\phi_{n},p_{n+1}-p_{n}+ \phi_{n+1}-p_{n+1} - \frac {g_{n}}{g_{n+1}} ( \phi_{n+1}-p_{n+1} ) \biggr\rangle \\ &= \biggl\langle \phi_{n+1}-\phi_{n},p_{n+1}-p_{n}+ \biggl(1 - \frac {g_{n}}{g_{n+1}} \biggr) ( \phi_{n+1}-p_{n+1} ) \biggr\rangle \\ &\leq \Vert \phi_{n+1}-\phi_{n} \Vert \biggl( \Vert p_{n+1}-p_{n}\Vert + \biggl\vert 1 - \frac{g_{n}}{g_{n+1}} \biggr\vert \Vert \phi_{n+1}-p_{n+1} \Vert \biggr), \end{aligned}$$

and hence

$$\begin{aligned} \Vert \phi_{n+1}-\phi_{n} \Vert &\leq \Vert p_{n+1}-p_{n}\Vert + \frac{1}{g_{n+1}} \vert g_{n+1} - g_{n} \vert \Vert \phi _{n+1}-p_{n+1} \Vert \\ &\leq \Vert p_{n+1}-p_{n}\Vert + \frac{1}{e} \vert g_{n+1} - g_{n} \vert \Vert \phi_{n+1}-p_{n+1} \Vert . \end{aligned}$$

(3.8)

We use $\psi_{n} = W_{h_{n}}p_{n}$ and $\psi_{n+1} = W_{h_{n+1}}p_{n+1}$. By using the same method as (3.8), we have

$$ \Vert \psi_{n+1}-\psi_{n} \Vert \leq \Vert p_{n+1}-p_{n}\Vert + \frac{1}{e} \vert h_{n+1} - h_{n} \vert \Vert \psi_{n+1}-p_{n+1} \Vert . $$

(3.9)

From (3.5), (3.8), and (3.9), we have

$$\begin{aligned} \Vert p_{n+1}-p_{n} \Vert \leq& |\alpha_{n}- \alpha_{n-1}| \|u\| + \beta_{n} \|p_{n} - p_{n-1}\| + |\beta_{n} - \beta_{n-1}| \|p_{n-1}\| \\ &{}+ \gamma_{n} \biggl( \Vert p_{n+1}-p_{n} \Vert + \frac{1}{e} \vert g_{n+1} - g_{n} \vert \Vert \phi_{n+1}-p_{n+1} \Vert \biggr) \\ &{}+ \lambda_{n}\bigl\Vert (I-W)\phi_{n}-(I-W) \phi_{n-1}\bigr\Vert + |\lambda _{n} - \lambda_{n-1}| \bigl\Vert (I-W)\phi_{n-1} \bigr\Vert \\ &{}+ | \gamma_{n} - \gamma_{n-1} | \bigl\Vert P_{C}\bigl(I-\lambda_{n-1}(I-W)\bigr)\phi _{n-1} \bigr\Vert \\ &{}+ \delta_{n} \biggl( \Vert p_{n+1}-p_{n} \Vert + \frac{1}{e} \vert h_{n+1} - h_{n} \vert \Vert \psi_{n+1}-p_{n+1} \Vert \biggr) \\ &{}+ |\delta_{n}-\delta_{n-1}| \bigl\Vert G( \psi_{n-1})\bigr\Vert \\ \leq& |\alpha_{n}-\alpha_{n-1}| \|u\| + \beta_{n} \|p_{n} - p_{n-1}\| + | \beta_{n} - \beta_{n-1}|\|p_{n-1}\| \\ &{}+ \gamma_{n} \Vert p_{n+1}-p_{n}\Vert + \frac{1}{e} \vert g_{n+1} - g_{n} \vert \Vert \phi_{n+1}-p_{n+1} \Vert \\ &{}+ \lambda_{n}\bigl\Vert (I-W)\phi_{n}-(I-W) \phi_{n-1}\bigr\Vert + |\lambda _{n} - \lambda_{n-1}| \bigl\Vert (I-W)\phi_{n-1} \bigr\Vert \\ &{}+ | \gamma_{n} - \gamma_{n-1} | \bigl\Vert P_{C}\bigl(I-\lambda_{n-1}(I-W)\bigr)\phi _{n-1} \bigr\Vert + \delta_{n} \Vert p_{n+1}-p_{n} \Vert \\ &{}+ \frac{1}{e} \vert h_{n+1} - h_{n} \vert \Vert \psi _{n+1}-p_{n+1} \Vert + |\delta_{n}- \delta_{n-1}| \bigl\Vert G(\psi_{n-1})\bigr\Vert \\ \leq& ( 1-\alpha_{n} ) \|p_{n} - p_{n-1}\| + | \alpha _{n}-\alpha_{n-1}|M + |\beta_{n} - \beta_{n-1}|M \\ &{}+ | \gamma_{n} - \gamma_{n-1} |M + | \delta_{n}-\delta_{n-1}|M + |\lambda_{n} - \lambda_{n-1}|M + \lambda_{n}M \\ &{}+ \frac{1}{e} \vert g_{n+1} - g_{n} \vert M + \frac{1}{e} \vert h_{n+1} - h_{n} \vert M, \end{aligned}$$

where

$$\begin{aligned} M :=&\max_{n \in \mathbb{N}} \bigl\{ \|u\|, \|p_{n}\| , \bigl\Vert P_{C}\bigl(I-\lambda_{n}(I-W)\bigr) \phi_{n}\bigr\Vert , \bigl\Vert G(\psi_{n})\bigr\Vert , \bigl\Vert (I-W)\phi_{n}\bigr\Vert , \\ & \bigl\Vert (I-W)\phi_{n+1}-(I-W)\phi_{n}\bigr\Vert , \|\phi_{n}-p_{n}\| , \|\psi_{n}-p_{n} \| \bigr\} . \end{aligned}$$

From the conditions (i), (iv), (v), and Lemma 2.6, we have

$$ \lim_{n \rightarrow\infty} \Vert p_{n+1} - p_{n} \Vert = 0. $$

(3.10)

Since $W_{g_{n}}$ is a firmly nonexpansive mapping, we obtain

$$\begin{aligned} \Vert \phi_{n} - \xi \Vert ^{2} &= \Vert W_{g_{n}}p_{n} - W_{g_{n}}\xi \Vert ^{2} \\ &\leq \langle W_{g_{n}}p_{n} - W_{g_{n}}\xi, p_{n}-\xi \rangle \\ &\leq \langle\phi_{n} - \xi, p_{n} - \xi \rangle \\ &= \frac{1}{2} \bigl( \Vert \phi_{n} - \xi \Vert ^{2} + \Vert p_{n}-\xi \Vert ^{2}-\Vert \phi_{n}-p_{n} \Vert ^{2} \bigr). \end{aligned}$$

It implies that

$$ \Vert \phi_{n}-\xi \Vert ^{2} \leq \Vert p_{n}-\xi \Vert ^{2}-\Vert \phi_{n}-p_{n} \Vert ^{2}. $$

(3.11)

By using the same method as (3.11), we have

$$ \Vert \psi_{n}-\xi \Vert ^{2} \leq \Vert p_{n}-\xi \Vert ^{2}-\Vert \psi_{n}-p_{n} \Vert ^{2}. $$

(3.12)

From the definition of $p_{n}$, (3.4), (3.11), and (3.12), we have

$$\begin{aligned} \Vert p_{n+1}-\xi \Vert ^{2} =& \bigl\Vert \alpha_{n}(u-\xi)+\beta _{n}(p_{n}-\xi)+ \gamma_{n} \bigl(P_{C}\bigl(I-\lambda_{n}(I-W) \bigr)\phi_{n}-\xi \bigr) \\ &{}+ \bigl.\bigl.\delta_{n} \bigl(G(\psi_{n})-\xi \bigr)\bigr\Vert \bigr.^{2} \\ \leq& \alpha_{n}\Vert u-\xi \Vert ^{2} + \beta_{n}\Vert p_{n}-\xi \Vert ^{2} + \gamma_{n}\bigl\Vert P_{C} \bigl(I-\lambda _{n}(I-W) \bigr)\phi_{n}-\xi\bigr\Vert ^{2} \\ &{}+ \delta_{n}\bigl\Vert G(\psi_{n})-\xi\bigr\Vert ^{2} - \beta_{n}\gamma _{n}\bigl\Vert P_{C}\bigl(I-\lambda_{n}(I-W)\bigr)\phi_{n}-p_{n} \bigr\Vert ^{2} \\ &{}- \beta_{n}\delta_{n}\bigl\Vert G( \psi_{n})-p_{n} \bigr\Vert ^{2} \\ \leq& \alpha_{n}\Vert u-\xi \Vert ^{2} + \beta_{n}\Vert p_{n}-\xi \Vert ^{2} + \gamma_{n}\Vert \phi_{n}-\xi \Vert ^{2} + \delta _{n}\Vert \psi_{n}-\xi \Vert ^{2} \\ &{}- \beta_{n}\gamma_{n}\bigl\Vert P_{C} \bigl(I-\lambda_{n}(I-W)\bigr)\phi_{n}-p_{n} \bigr\Vert ^{2} - \beta_{n}\delta_{n}\bigl\Vert G(\psi_{n})-p_{n} \bigr\Vert ^{2} \\ \leq& \alpha_{n}\Vert u-\xi \Vert ^{2} + \beta_{n}\Vert p_{n}-\xi \Vert ^{2} + \gamma_{n} \bigl( \Vert p_{n}-\xi \Vert ^{2}- \Vert \phi_{n}-p_{n} \Vert ^{2} \bigr) \\ &{}+ \delta_{n} \bigl( \Vert p_{n}-\xi \Vert ^{2}-\Vert \psi _{n}-p_{n} \Vert ^{2} \bigr) - \beta_{n}\delta_{n}\bigl\Vert G( \psi _{n})-p_{n} \bigr\Vert ^{2} \\ &{}- \beta_{n}\gamma_{n}\bigl\Vert P_{C} \bigl(I-\lambda_{n}(I-W)\bigr)\phi_{n}-p_{n} \bigr\Vert ^{2} \\ =& \alpha_{n}\Vert u-\xi \Vert ^{2} + ( 1- \alpha_{n} )\Vert p_{n} -\xi \Vert ^{2} - \gamma_{n}\Vert \phi_{n}-p_{n} \Vert ^{2} \\ &{}- \delta_{n}\Vert \psi_{n}-p_{n} \Vert ^{2} - \beta_{n}\gamma _{n}\bigl\Vert P_{C}\bigl(I-\lambda_{n}(I-W)\bigr)\phi_{n}-p_{n} \bigr\Vert ^{2} \\ &{}- \beta_{n}\delta_{n}\bigl\Vert G( \psi_{n})-p_{n} \bigr\Vert ^{2} \\ \leq& \alpha_{n}\Vert u-\xi \Vert ^{2} + \Vert p_{n} -\xi \Vert ^{2} - \gamma_{n}\Vert \phi_{n}-p_{n} \Vert ^{2} - \delta_{n} \Vert \psi_{n}-p_{n} \Vert ^{2} \\ &{}- \beta_{n}\gamma_{n}\bigl\Vert P_{C} \bigl(I-\lambda_{n}(I-W)\bigr)\phi_{n}-p_{n} \bigr\Vert ^{2} - \beta_{n}\delta_{n}\bigl\Vert G(\psi_{n})-p_{n} \bigr\Vert ^{2}, \end{aligned}$$

which implies that

$$\begin{aligned} \gamma_{n}\Vert \phi_{n}-p_{n} \Vert ^{2} \leq& \alpha_{n}\Vert u-\xi \Vert ^{2} + \Vert p_{n} -\xi \Vert ^{2} - \Vert p_{n+1}-\xi \Vert ^{2} \\ \leq& \alpha_{n}\Vert u-\xi \Vert ^{2} + \Vert p_{n}-p_{n+1}\Vert \bigl(\Vert p_{n}-\xi \Vert + \Vert p_{n+1}-\xi \Vert \bigr). \end{aligned}$$

From the conditions (i), (ii), and (3.10), we have

$$ \lim_{n \rightarrow\infty} \Vert \phi_{n}-p_{n} \Vert = 0. $$

(3.13)

By using the same method as (3.13), we can imply that

$$ \lim_{n \rightarrow\infty} \Vert \psi_{n}-p_{n} \Vert = \lim_{n \rightarrow\infty} \bigl\Vert P_{C}\bigl(I- \lambda_{n}(I-W)\bigr)\phi_{n}-p_{n} \bigr\Vert = \lim_{n \rightarrow\infty} \bigl\Vert G(\psi_{n})-p_{n} \bigr\Vert =0. $$

(3.14)

From (3.13) and (3.14), we have

$$ \lim_{n \rightarrow\infty} \Vert \phi_{n}- \psi_{n}\Vert = 0. $$

(3.15)

Afterwards, we show that $\limsup_{n \rightarrow\infty} \langle u-p_{0},p_{n}-p_{0} \rangle\leq0$, where $p_{0}=P_{\mathcal{F}}u$. To show this inequality, take a subsequence $\{p_{n_{j}}\}$ of $\{p_{n}\}$ such that

$$ \limsup_{n \rightarrow\infty} \langle u-p_{0},p_{n}-p_{0} \rangle= \lim_{j \rightarrow\infty} \langle u-p_{0},p_{n_{j}}-p_{0} \rangle. $$

Without loss of generality, we may assume that $u_{n_{j}} \rightharpoonup \omega$ as $j \rightarrow\infty$. From (3.15), we have $v_{n_{j}} \rightharpoonup\omega$ as $j \rightarrow\infty$. By using the same method as [15] in Theorem 3.2, we have

$$ \omega\in \operatorname{EP}(\Psi_{1}) $$

(3.16)

and

$$ \omega\in \operatorname{EP}(\Psi_{2}). $$

(3.17)

Furthermore, we show that $\omega\in F(W)$. From Remark 2.9, we have $F(W)= F(P_{C}(I-\lambda_{n_{j}} (I-W)))$. Assume that $\omega \notin F(W)$, we have $\omega\neq P_{C}(I-\lambda_{n_{j}} (I-W))\omega$. From (3.13), we have $p_{n_{j}} \rightharpoonup\omega$ as $j \rightarrow\infty$. By (3.13), (3.14), the condition (iv), and Opial’s property, we have

$$\begin{aligned} \liminf_{j \rightarrow\infty} \Vert p_{n_{j}}-\omega \Vert < & \liminf_{j \rightarrow\infty} \bigl\Vert p_{n_{j}} - P_{C} \bigl(I-\lambda_{n_{j}} (I-W)\bigr)\omega\bigr\Vert \\ \leq& \liminf_{j \rightarrow\infty} \bigl(\bigl\Vert p_{n_{j}} - P_{C}\bigl(I-\lambda_{n_{j}}(I-W)\bigr)u_{n_{j}} \bigr\Vert \\ &{}+ \bigl\Vert P_{C}\bigl(I-\lambda_{n_{j}}(I-W) \bigr)u_{n_{j}} - P_{C}\bigl(I-\lambda_{n_{j}} (I-W) \bigr)p_{n_{j}} \bigr\Vert \\ &{}+ \bigl\Vert P_{C}\bigl(I-\lambda_{n_{j}}(I-W) \bigr)p_{n_{j}} - P_{C}\bigl(I-\lambda _{n_{j}} (I-W) \bigr)\omega\bigr\Vert \bigr) \\ \leq& \liminf_{j \rightarrow\infty} \bigl(\Vert u_{n_{j}} - p_{n_{j}} \Vert + \lambda_{n_{j}} \bigl\Vert (I-W)u_{n_{j}} - (I-W)p_{n_{j}} \bigr\Vert \\ &{}+ \Vert p_{n_{j}} - \omega \Vert + \lambda_{n_{j}} \bigl\Vert (I-W)p_{n_{j}} - (I-W)\omega\bigr\Vert \bigr) \\ =& \liminf_{j \rightarrow\infty} \Vert p_{n_{j}}-\omega \Vert . \end{aligned}$$

It is a contradiction. So we have

$$ \omega\in F(W). $$

(3.18)

After that, we show that $\omega\in F(G)$. Assume that $\omega\notin F(G)$, that is, $\omega\neq G(\omega)$. Since $p_{n_{j}} \rightharpoonup \omega$ as $j \rightarrow\infty$, (3.14), the condition (iv), and Opial’s property, we have

$$\begin{aligned} \liminf_{j \rightarrow\infty} \Vert p_{n_{j}}-\omega \Vert < & \liminf_{j \rightarrow\infty} \bigl\Vert p_{n_{j}} - G(\omega) \bigr\Vert \\ \leq& \liminf_{j \rightarrow\infty} \bigl(\bigl\Vert p_{n_{j}} - G( \psi _{n_{j}}) \bigr\Vert + \bigl\Vert G(\psi_{n_{j}}) - G(p_{n_{j}}) \bigr\Vert \\ &{}+ \bigl\Vert G(p_{n_{j}}) - G(\omega) \bigr\Vert \bigr) \\ \leq& \liminf_{j \rightarrow\infty} \bigl(\Vert \psi_{n_{j}} - p_{n_{j}} \Vert + \Vert p_{n_{j}} - \omega \Vert \bigr) \\ =& \liminf_{j \rightarrow\infty} \Vert p_{n_{j}}-\omega \Vert . \end{aligned}$$

It is a contradiction. So we have

$$ \omega\in F(G). $$

(3.19)

Therefore $\omega\in\mathcal{F}$. Since $p_{n_{j}} \rightharpoonup \omega$ as $j \rightarrow\infty$, we have

$$\begin{aligned} \limsup_{n \rightarrow\infty} \langle u-p_{0},p_{n}-p_{0} \rangle&= \lim_{j \rightarrow\infty} \langle u-p_{0},p_{n_{j}}-p_{0} \rangle \\ &= \langle u-p_{0},\omega-p_{0} \rangle\leq0. \end{aligned}$$

(3.20)

Finally, we show that the sequences $\{ p_{n} \}$, $\{ \phi_{n} \}$, and $\{ \psi_{n} \}$ converge strongly to $p_{0}=P_{\mathcal{F}}u$.

From the definition of $p_{n}$, (3.4), and $p_{0}=P_{\mathcal {F}}u$, we have

$$\begin{aligned} \Vert p_{n+1}-p_{0} \Vert ^{2} =& \bigl\Vert \alpha_{n}(u-p_{0})+\beta _{n}(p_{n}-p_{0})+ \gamma_{n} \bigl(P_{C}\bigl(I-\lambda_{n}(I-W) \bigr)\phi _{n}-p_{0} \bigr) \\ &{}+ \bigl.\bigl.\delta_{n} \bigl(G(\psi_{n})-p_{0} \bigr) \bigr\Vert \bigr.^{2} \\ \leq& \bigl\Vert \beta_{n}(p_{n}-p_{0})+ \gamma_{n} \bigl(P_{C}\bigl(I-\lambda _{n}(I-W) \bigr)\phi_{n}-p_{0} \bigr) \\ &{}+\bigl.\bigl. \delta_{n} \bigl(G(\psi_{n})-p_{0} \bigr) \bigr\Vert \bigr.^{2} + 2\alpha_{n}\langle u-p_{0} , p_{n+1}-p_{0} \rangle \\ \leq& (1-\alpha_{n} )\Vert p_{n}-p_{0} \Vert ^{2} + 2\alpha_{n}\langle u-p_{0} , p_{n+1}-p_{0} \rangle. \end{aligned}$$

From the condition (i), (3.20), and Lemma 2.6, we can conclude that the sequence $\{ p_{n} \}$ converges strongly to $p_{0}=P_{\mathcal{F}}u$. Consequently, we see that $\{\phi_{n}\}$ and $\{\psi_{n}\}$ also converge strongly to $p_{0}=P_{\mathcal{F}}u$. This completes the proof. □

From our main result, if we take $a=0$, we have the following corollary.

Corollary 3.2

Let C be a nonempty closed convex subset of a real Hilbert space H, let $\Psi_{1},\Psi_{2}:C\times C\rightarrow\mathbb{R}$ be bifunctions satisfying (J1)-(J4) and let $W:C\rightarrow C$ be a quasi-nonexpansive mapping. Let $D_{1},D_{2}:C\rightarrow H$ be $d_{1},d_{2}$-inverse strongly monotone mappings, respectively. Define the mapping $G:C\rightarrow C$ by $G(p) = P_{C}(I-\lambda_{A}D_{1}) (P_{C}(I-\lambda_{B}D_{2})p )$ for all $p \in C$. Assume $\mathcal{F}= \operatorname{EP}(\Psi_{1}) \cap \operatorname{EP}(\Psi _{2}) \cap F(G) \cap F(W) \neq\emptyset$. Suppose that $p_{1},u\in C$ and let $\{ p_{n} \}$, $\{ \phi_{n} \}$, and $\{ \psi_{n} \}$ be sequences generated by

$$ \textstyle\begin{cases} \Psi_{1} ( \phi_{n},\zeta ) + \frac{1}{g_{n}} \langle \zeta-\phi_{n}, \phi_{n} - p_{n} \rangle\geqslant0,\quad \forall\zeta\in C, \\ \Psi_{2} ( \psi_{n},\zeta ) + \frac{1}{h_{n}} \langle \zeta-\psi_{n}, \psi_{n} - p_{n} \rangle\geqslant0,\quad \forall\zeta\in C, \\ p_{n+1} = \alpha_{n}u + \beta_{n}p_{n} + \gamma_{n}P_{C} (I - \lambda_{n} (I-W) ) \phi_{n} + \delta_{n} G(\psi_{n}),\quad \forall n\in\mathbb{N} , \end{cases} $$

(3.21)

where the sequences $\lambda_{A} \in(0,2d_{1})$, $\lambda_{B} \in (0,2d_{2})$ and $\{ \alpha_{n} \}, \{ \beta_{n} \}, \{ \gamma_{n} \}, \{ \delta_{n} \} \subseteq[0,1]$ with $\alpha_{n}+\beta_{n}+\gamma _{n}+\delta_{n}=1$ for all $n \in\mathbb{N}$. Suppose the following conditions hold:

• (i) $\lim_{n \rightarrow\infty} \alpha_{n} =0$ and $\sum_{n=1}^{\infty} \alpha_{n} = \infty$,

• (ii) $0 < c \leq\beta_{n}, \gamma_{n}, \delta_{n} \leq d < 1$ for some $c, d > 0$ and for all $n \geq1$,

• (iii) $0 < e \leq g_{n},h_{n} \leq f$ for some $e, f > 0$ and for all $n \geq1$,

• (iv) $\sum_{n=1}^{\infty} \lambda_{n} < \infty$ and $0 < \lambda_{n} < 1$,

• (v) $\sum_{n=1}^{\infty} \vert \alpha_{n+1} - \alpha_{n} \vert < \infty$, $\sum_{n=1}^{\infty} \vert \beta_{n+1} - \beta_{n} \vert < \infty$, $\sum_{n=1}^{\infty} \vert \gamma_{n+1} - \gamma_{n} \vert < \infty$, $\sum_{n=1}^{\infty} \vert \lambda_{n+1} - \lambda_{n} \vert < \infty$, $\sum_{n=1}^{\infty} \vert g_{n+1} - g_{n} \vert < \infty$, $\sum_{n=1}^{\infty} \vert h_{n+1} - h_{n} \vert < \infty$.

Then $\{ p_{n} \}$, $\{ \phi_{n} \}$, and $\{ \psi_{n} \}$ converge strongly to $p_{0}=P_{\mathcal{F}}u$ and $(p_{0},\xi_{0} )$ is a solution of (1.5) where $\xi_{0}=P_{C}(I-\lambda_{B}D_{2})p_{0}$.

4 Application

In this section, we use our main result to obtain Theorem 4.7 and Theorem 4.8. Before we prove these theorems, we need the following definition and lemma. A mapping $W: C \rightarrow C$ is said to be nonspreading if

$$ 2\|Wp-W\zeta\|^{2}\leq\|Wp-\zeta\|^{2}+\|W \zeta-p\|^{2},\quad \forall p,\zeta\in C. $$

(4.1)

Such a mapping is defined by Kohsaka and Takahashi [16].

In 2009, Iemoto and Takahashi [17] proved that (4.1) is equivalent to

$$ \|Wp-W\zeta\|^{2}\leq\|p-\zeta\|^{2}+2\langle p-Wp,\zeta-W\zeta\rangle,\quad \forall p,\zeta\in C. $$

(4.2)

Remark 4.1

A nonspreading mapping W with $F(W)\neq\emptyset$ is quasi-nonexpansive mapping.

Example 4.2

Let $W:[-5,\infty) \rightarrow[-5,\infty)$ be defined by

$$ Wp=\frac{p-5}{2}, \quad \forall p \in[-5,\infty). $$

Since W is a nonspreading mapping and $F(W)=\{-5\}$, we have W is a quasi-nonexpansive mapping.

The following lemmas and definition are used to prove the results in this section.

Lemma 4.3

([8])

Let C be a nonempty closed convex subset of a real Hilbert space H and let $D_{1},D_{2}: C \rightarrow H$ be $d_{1},d_{2}$-inverse strongly monotone mappings, respectively, with $\operatorname{VIP} (C,D_{1} ) \cap \operatorname{VIP} (C,D_{2} ) \neq\emptyset$. Define a mapping $G: C \rightarrow C$ by

$$ G (p ) = P_{C} (I-\lambda_{A}D_{1} ) \bigl( ap+ (1-a )P_{C} (I-\lambda_{B}D_{2} )p \bigr), $$

for every $\lambda_{A}\in (0,2d_{1} )$, $\lambda_{B}\in (0,2d_{2} )$ and $a \in (0,1 )$. Then $F (G )=\operatorname{VIP} (C,D_{1} ) \cap \operatorname{VIP} (C,D_{2} )$.

Lemma 4.4

([16])

Let H be a Hilbert space, let C be a nonempty closed convex subset of H, and let W be a nonspreading mapping of C into itself. Then $F(W)$ is closed and convex.

In 2009, Kangtunyakarn and Suantai [18] introduced the S-mapping generated by $W_{1},W_{2},W_{3},\ldots,W_{N}$ and $\lambda_{1},\lambda _{2},\ldots,\lambda_{N}$ as follows.

Definition 4.5

Let C be a nonempty convex subset of a real Banach space. Let $\{ W_{i}\}_{i=1}^{N}$ be a finite family of (nonexpansive) mappings of C into itself. For each $j=1,2,\ldots,N$, let $\alpha_{j}=(\alpha_{1}^{j},\alpha _{2}^{j},\alpha_{3}^{j})\in I\times I\times I$, where $I\in[0,1]$ and $\alpha _{1}^{j}+\alpha_{2}^{j}+\alpha_{3}^{j}=1$. Define the mapping $S:C\rightarrow C$ as follows:

$$\begin{aligned}& U_{0} =I, \\& U_{1} =\alpha_{1}^{1}W_{1}U_{0}+ \alpha_{2}^{1}U_{0}+\alpha_{3}^{1}I, \\& U_{2} =\alpha_{1}^{2}W_{2}U_{1}+ \alpha_{2}^{2}U_{1}+\alpha_{3}^{2}I, \\& U_{3} =\alpha_{1}^{3}W_{3}U_{2}+ \alpha_{2}^{3}U_{2}+\alpha_{3}^{3}I, \\& \ldots, \\& U_{N-1} =\alpha_{1}^{N-1}W_{N-1}U_{N-2}+ \alpha_{2}^{N-1}U_{N-2}+\alpha _{3}^{N-1}I, \\& S =U_{N}=\alpha_{1}^{N}W_{N}U_{N-1}+ \alpha_{2}^{N}U_{N-1}+\alpha_{3}^{N}I. \end{aligned}$$

This mapping is called an S-mapping generated by $W_{1},W_{2},\ldots,W_{N}$ and $\alpha_{1},\alpha_{2},\ldots,\alpha_{N}$.

For every $i=1,2,\ldots N$, put $\alpha^{i}_{3}=0$ in Definition 4.5, then the S-mapping is reduced to the K-mapping generated by $\alpha _{1}^{1}, \alpha_{1}^{2},\ldots, \alpha_{1}^{N}$ where the K-mapping is defined by Kangtunyakarn and Suantai [19] as follows.

Lemma 4.6

([20])

Let C be a nonempty closed convex subset of a real Hilbert space. Let $\{W_{i}\}_{i=1}^{N}$ be a finite family of nonspreading mappings of C into C with $\bigcap_{i=1}^{N}F(W_{i})\neq\emptyset$, and let $\alpha _{j}=(\alpha_{1}^{j},\alpha_{2}^{j},\alpha_{3}^{j})\in I\times I\times I$, $j=1,2,\ldots,N$, where $I=[0,1]$, $\alpha_{1}^{j}+\alpha_{2}^{j}+\alpha_{3}^{j}=1$, $\alpha_{1}^{j}, \alpha_{3}^{j}\in(0,1)$ for all $j=1,2,\ldots,N-1$ and $\alpha_{1}^{N}\in(0,1]$, $\alpha_{3}^{N}\in[0,1)$, $\alpha_{2}^{j}\in[0,1)$ for all $j=1,2,\ldots,N$. Let S be the mapping generated by $W_{1},W_{2},\ldots,W_{N}$ and $\alpha_{1},\alpha_{2},\ldots,\alpha_{N}$. Then $F(S)=\bigcap_{i=1}^{N}F(W_{i})$ and S is a quasi-nonexpansive mapping.

By using these results, we obtain the following theorems.

Theorem 4.7

Let C be a nonempty closed convex subset of a real Hilbert space H, let $\Psi_{1},\Psi_{2}:C\times C\rightarrow\mathbb{R}$ be bifunctions satisfying (J1)-(J4) and let $W:C\rightarrow C$ be a quasi-nonexpansive mapping. Let $D_{1},D_{2}:C\rightarrow H$ be $d_{1},d_{2}$-inverse strongly monotone mappings, respectively. Assume $\mathcal{F}= \operatorname{EP}(\Psi_{1}) \cap \operatorname{EP}(\Psi_{2}) \cap F(W) \cap \operatorname{VIP}(C,D_{1}) \cap \operatorname{VIP}(C,D_{2}) \neq\emptyset$. Suppose that $p_{1},u\in C$ and let $\{ p_{n} \}$, $\{ \phi_{n} \}$, and $\{ \psi_{n} \}$ be sequences generated by

$$ \textstyle\begin{cases} \Psi_{1} ( \phi_{n},\zeta ) + \frac{1}{g_{n}} \langle \zeta-\phi_{n}, \phi_{n} - p_{n} \rangle\geqslant0,\quad \forall\zeta\in C, \\ \Psi_{2} ( \psi_{n},\zeta ) + \frac{1}{h_{n}} \langle \zeta-\psi_{n}, \psi_{n} - p_{n} \rangle\geqslant0,\quad \forall\zeta\in C, \\ p_{n+1} = \alpha_{n}u + \beta_{n}p_{n} + \gamma_{n}P_{C} (I - \lambda_{n} (I-W) ) \phi_{n} \\ \hphantom{p_{n+1} ={}}{}+ \delta_{n} P_{C} (I-\lambda_{A}D_{1}) ( ap_{n} + (1-a)P_{C} (I-\lambda_{B}D_{2}) p_{n} ) , \quad \forall n\in\mathbb{N} , \end{cases} $$

(4.3)

where the sequences $\lambda_{A} \in(0,2d_{1})$, $\lambda_{B} \in (0,2d_{2})$, and $\{ \alpha_{n} \}, \{ \beta_{n} \}, \{ \gamma_{n} \}, \{ \delta_{n} \} \subseteq[0,1]$ with $\alpha_{n}+\beta_{n}+\gamma _{n}+\delta_{n}=1$, for all $n \in\mathbb{N}$, and $a\in(0,1)$. Suppose the following conditions hold:

• (i) $\lim_{n \rightarrow\infty} \alpha_{n} =0$ and $\sum_{n=1}^{\infty} \alpha_{n} = \infty$,

• (ii) $0 < c \leq\beta_{n}, \gamma_{n}, \delta_{n} \leq d < 1$ for some $c, d > 0$ and for all $n \geq1$,

• (iii) $0 < e \leq g_{n},h_{n} \leq f$ for some $e, f > 0$ and for all $n \geq1$,

• (iv) $\sum_{n=1}^{\infty} \lambda_{n} < \infty$ and $0 < \lambda_{n} < 1$,

• (v) $\sum_{n=1}^{\infty} \vert \alpha_{n+1} - \alpha_{n} \vert < \infty$, $\sum_{n=1}^{\infty} \vert \beta_{n+1} - \beta_{n} \vert < \infty$, $\sum_{n=1}^{\infty} \vert \gamma_{n+1} - \gamma_{n} \vert < \infty$, $\sum_{n=1}^{\infty} \vert \lambda_{n+1} - \lambda_{n} \vert < \infty$, $\sum_{n=1}^{\infty} \vert g_{n+1} - g_{n} \vert < \infty$, $\sum_{n=1}^{\infty} \vert h_{n+1} - h_{n} \vert < \infty$.

Then $\{ p_{n} \}$, $\{ \phi_{n} \}$, and $\{ \psi_{n} \}$ converge strongly to $p_{0}=P_{\mathcal{F}}u$ and $(p_{0},\xi_{0} )$ be a solution of (1.6) where $\xi_{0}=P_{C}(I-\lambda_{B}D_{2})p_{0}$.

Proof

By using Theorem 3.1 and Lemma 4.3, we obtain the conclusion. □

Theorem 4.8

Let C be a nonempty closed convex subset of a real Hilbert space H, let $\Psi_{1},\Psi_{2}:C\times C\rightarrow\mathbb{R}$ be bifunctions satisfying (J1)-(J4). Let $\{W_{i}\}_{i=1}^{N}$ be a finite family of nonspreading mappings of C into C and let $\alpha_{j}=(\alpha _{1}^{j},\alpha_{2}^{j},\alpha_{3}^{j})\in I\times I\times I$, $j=1,2,\ldots,N$, where $I=[0,1]$, $\alpha_{1}^{j}+\alpha_{2}^{j}+\alpha_{3}^{j}=1$, $\alpha_{1}^{j}, \alpha_{3}^{j}\in (0,1)$ for all $j=1,2,\ldots,N-1$ and $\alpha_{1}^{N}\in(0,1]$, $\alpha _{3}^{N}\in[0,1)$, $\alpha_{2}^{j}\in[0,1)$ for all $j=1,2,\ldots,N$. Let S be the mapping generated by $W_{1},W_{2},\ldots,W_{N}$, and $\alpha_{1},\alpha _{2},\ldots,\alpha_{N}$. Let $D_{1},D_{2}:C\rightarrow H$ be $d_{1},d_{2}$-inverse strongly monotone mappings, respectively. Define the mapping $G:C\rightarrow C$ by $G(p) = P_{C}(I-\lambda_{A}D_{1}) (ap+(1-a)P_{C}(I-\lambda_{B}D_{2})p )$ for all $p \in C$ and $a \in [0,1 ]$. Assume $\mathcal{F}= \operatorname{EP}(\Psi_{1}) \cap \operatorname{EP}(\Psi_{2}) \cap F(G) \cap\bigcap_{i=1}^{N}F(W_{i}) \neq\emptyset$. Suppose that $p_{1},u\in C$ and let $\{ p_{n} \}$, $\{ \phi_{n} \}$, and $\{ \psi_{n} \}$ are sequences generated by

$$ \textstyle\begin{cases} \Psi_{1} ( \phi_{n},\zeta ) + \frac{1}{g_{n}} \langle \zeta-\phi_{n}, \phi_{n} - p_{n} \rangle\geqslant0,\quad \forall\zeta\in C, \\ \Psi_{2} ( \psi_{n},\zeta ) + \frac{1}{h_{n}} \langle \zeta-\psi_{n}, \psi_{n} - p_{n} \rangle\geqslant0,\quad \forall\zeta\in C, \\ p_{n+1} = \alpha_{n}u + \beta_{n}p_{n} + \gamma_{n}P_{C} (I - \lambda_{n} (I-S) ) \phi_{n} + \delta_{n} G(\psi_{n}),\quad \forall n\in\mathbb{N} , \end{cases} $$

(4.4)

where the sequences $\lambda_{A} \in(0,2d_{1})$, $\lambda_{B} \in (0,2d_{2})$, and $\{ \alpha_{n} \}, \{ \beta_{n} \}, \{ \gamma_{n} \}, \{ \delta_{n} \} \subseteq[0,1]$ with $\alpha_{n}+\beta_{n}+\gamma _{n}+\delta_{n}=1$ for all $n \in\mathbb{N}$. Suppose the following conditions hold:

• (i) $\lim_{n \rightarrow\infty} \alpha_{n} =0$ and $\sum_{n=1}^{\infty} \alpha_{n} = \infty$,

• (ii) $0 < c \leq\beta_{n}, \gamma_{n}, \delta_{n} \leq d < 1$ for some $c, d > 0$ and for all $n \geq1$,

• (iii) $0 < e \leq g_{n},h_{n} \leq f$ for some $e, f > 0$ and for all $n \geq1$,

• (iv) $\sum_{n=1}^{\infty} \lambda_{n} < \infty$ and $0 < \lambda_{n} < 1$,

• (v) $\sum_{n=1}^{\infty} \vert \alpha_{n+1} - \alpha_{n} \vert < \infty$, $\sum_{n=1}^{\infty} \vert \beta_{n+1} - \beta_{n} \vert < \infty$, $\sum_{n=1}^{\infty} \vert \gamma_{n+1} - \gamma_{n} \vert < \infty$, $\sum_{n=1}^{\infty} \vert \lambda_{n+1} - \lambda_{n} \vert < \infty$, $\sum_{n=1}^{\infty} \vert g_{n+1} - g_{n} \vert < \infty$, $\sum_{n=1}^{\infty} \vert h_{n+1} - h_{n} \vert < \infty$.

Then $\{ p_{n} \}$, $\{ \phi_{n} \}$, and $\{ \psi_{n} \}$ converge strongly to $p_{0}=P_{\mathcal{F}}u$ and $(p_{0},\xi_{0} )$ is a solution of (1.6) where $\xi_{0}=P_{C}(I-\lambda_{B}D_{2})p_{0}$.

Proof

By using Theorem 3.1 and Lemma 4.6, we obtain the conclusion. □

The following result is directly proven from Theorem 4.8. Therefore, we omit the proof.

Corollary 4.9

Let C be a nonempty closed convex subset of a real Hilbert space H, let $\Psi_{1},\Psi_{2}:C\times C\rightarrow\mathbb{R}$ be bifunctions satisfying (J1)-(J4). Let W be a nonspreading mappings of C into itself with $F(W) \neq\emptyset$. Let $D_{1},D_{2}:C\rightarrow H$ be $d_{1},d_{2}$-inverse strongly monotone mappings, respectively. Define the mapping $G:C\rightarrow C$ by $G(p) = P_{C}(I-\lambda_{A}D_{1}) (ap+(1-a)P_{C}(I-\lambda_{B}D_{2})p )$ for all $p \in C$ and $a \in [0,1 ]$. Assume $\mathcal{F}= \operatorname{EP}(\Psi_{1}) \cap \operatorname{EP}(\Psi_{2}) \cap F(G) \cap F(W) \neq\emptyset$. Suppose that $p_{1},u\in C$ and let $\{ p_{n} \}$, $\{ \phi_{n} \}$, and $\{ \psi_{n} \}$ be sequences generated by

$$ \textstyle\begin{cases} \Psi_{1} ( \phi_{n},\zeta ) + \frac{1}{g_{n}} \langle \zeta-\phi_{n}, \phi_{n} - p_{n} \rangle\geqslant0,\quad \forall\zeta\in C, \\ \Psi_{2} ( \psi_{n},\zeta ) + \frac{1}{h_{n}} \langle \zeta-\psi_{n}, \psi_{n} - p_{n} \rangle\geqslant0,\quad \forall\zeta\in C, \\ p_{n+1} = \alpha_{n}u + \beta_{n}p_{n} + \gamma_{n}P_{C} (I - \lambda_{n} (I-W) ) \phi_{n} + \delta_{n} G(\psi_{n}),\quad \forall n\in\mathbb{N} , \end{cases} $$

(4.5)

where the sequences $\lambda_{A} \in(0,2d_{1})$, $\lambda_{B} \in (0,2d_{2})$ and $\{ \alpha_{n} \}, \{ \beta_{n} \}, \{ \gamma_{n} \}, \{ \delta_{n} \} \subseteq[0,1]$ with $\alpha_{n}+\beta_{n}+\gamma _{n}+\delta_{n}=1$ for all $n \in\mathbb{N}$. Suppose the following conditions hold:

• (i) $\lim_{n \rightarrow\infty} \alpha_{n} =0$ and $\sum_{n=1}^{\infty} \alpha_{n} = \infty$,

• (ii) $0 < c \leq\beta_{n}, \gamma_{n}, \delta_{n} \leq d < 1$ for some $c, d > 0$ and for all $n \geq1$,

• (iii) $0 < e \leq g_{n},h_{n} \leq f$ for some $e, f > 0$ and for all $n \geq1$,

• (iv) $\sum_{n=1}^{\infty} \lambda_{n} < \infty$ and $0 < \lambda_{n} < 1$,

• (v) $\sum_{n=1}^{\infty} \vert \alpha_{n+1} - \alpha_{n} \vert < \infty$, $\sum_{n=1}^{\infty} \vert \beta_{n+1} - \beta_{n} \vert < \infty$, $\sum_{n=1}^{\infty} \vert \gamma_{n+1} - \gamma_{n} \vert < \infty$, $\sum_{n=1}^{\infty} \vert \lambda_{n+1} - \lambda_{n} \vert < \infty$, $\sum_{n=1}^{\infty} \vert g_{n+1} - g_{n} \vert < \infty$, $\sum_{n=1}^{\infty} \vert h_{n+1} - h_{n} \vert < \infty$.

Then $\{ p_{n} \}$, $\{ \phi_{n} \}$, and $\{ \psi_{n} \}$ converge strongly to $p_{0}=P_{\mathcal{F}}u$ and $(p_{0},\xi_{0} )$ is a solution of (1.6) where $\xi_{0}=P_{C}(I-\lambda _{B}D_{2})p_{0}$.

5 Example and numerical results

In this section, we give an example supporting Theorem 3.1.

Example 5.1

Let $\mathbb{R}$ be the set of real numbers and let the mapping $D_{1}, D_{2}: \mathbb{R} \rightarrow\mathbb{R}$ defined by $D_{1}p = \frac {p-2}{3}$ and $D_{2}p = \frac{p-2}{5}$, $\forall p \in\mathbb{R}$, respectively. Let the mapping $W: \mathbb{R} \rightarrow\mathbb{R}$ be defined by $Wp = \frac{p+2}{2}$, $\forall p \in\mathbb{R}$, let $\Psi _{1}, \Psi_{2}: \mathbb{R} \times\mathbb{R} \rightarrow\mathbb{R}$ be defined by

$$ \Psi_{1}(p,\zeta) = - ( p-\zeta ) ( -4+p+\zeta ),\quad \forall p, \zeta\in\mathbb{R} $$

and

$$ \Psi_{2}(p,\zeta) = -2(p-2)^{2}+(p-2) (\zeta-2)+( \zeta-2)^{2}, \quad \forall p,\zeta\in\mathbb{R}. $$

By the definition of $\Psi_{1}$, we have

$$\begin{aligned}& 0 \leq \Psi_{1} ( \phi_{n},\zeta ) + \frac{1}{g_{n}} \langle\zeta-\phi_{n}, \phi_{n}-p_{n} \rangle \\& \hphantom{0}= - ( \phi_{n}-\zeta ) ( -4+\phi_{n}+\zeta ) + \frac{1}{g_{n}} (\zeta-\phi_{n} ) (\phi_{n}-p_{n} ) \\& \hphantom{0}= - ( \phi_{n}-\zeta ) ( -4+\phi_{n}+\zeta ) + \frac{1}{g_{n}} \bigl( \zeta\phi_{n}-\zeta p_{n}- \phi_{n}^{2}+\phi _{n}p_{n} \bigr) \\& \quad \Leftrightarrow\quad 0 \leq -g_{n} ( \phi_{n}-\zeta ) ( -4+\phi_{n}+\zeta ) + \bigl( \zeta\phi_{n}-\zeta p_{n}-\phi_{n}^{2}+\phi_{n} p_{n} \bigr) \\& \hphantom{\quad \Leftrightarrow\quad 0}= 4g_{n}\phi_{n}-\phi_{n}^{2}-g_{n} \phi_{n}^{2}+\phi_{n}p_{n}+ ( -4g_{n}+\phi_{n}-p_{n} )\zeta+ g_{n} \zeta^{2}. \end{aligned}$$

Let $G(\zeta) = g_{n}\zeta^{2} + ( -4g_{n}+\phi_{n}-p_{n} )\zeta+ 4g_{n}\phi_{n}-\phi_{n}^{2}-g_{n}\phi_{n}^{2}+\phi_{n}p_{n}$, which is a quadratic function of ζ with coefficient $a = g_{n}$, $b = -4g_{n}+\phi_{n}-p_{n}$, and $c = 4g_{n}\phi_{n}-\phi _{n}^{2}-g_{n}\phi_{n}^{2}+\phi_{n}p_{n}$. Determine the discriminant Δ of G as follows:

$$\begin{aligned} \Delta&= b^{2} - 4ac \\ &= ( -4g_{n}+\phi_{n}-p_{n} )^{2}-4g_{n} \bigl( 4g_{n}\phi _{n}- \phi_{n}^{2}-g_{n}\phi_{n}^{2}+ \phi_{n}p_{n} \bigr) \\ &= 16g_{n}^{2} - 8g_{n}\phi_{n} - 16g_{n}^{2}\phi_{n} + \phi_{n}^{2} + 4g_{n}\phi_{n}^{2} + 4g_{n}^{2} \phi_{n}^{2} + 8g_{n}p_{n} - 2\phi _{n}p_{n} - 4g_{n}\phi_{n}p_{n} + p_{n}^{2} \\ &= ( -4g_{n} + \phi_{n} + 2g_{n} \phi_{n} - p_{n} )^{2}. \end{aligned}$$

We know that $G(\zeta) \geq0$, $\forall\zeta\in\mathbb{R}$. If it has at most one solution in $\mathbb{R}$, then $\Delta\leq0$. So we obtain

$$ \phi_{n} = \frac{4g_{n} + p_{n}}{1 + 2g_{n}}. $$

(5.1)

By using the same method as (5.1), we have

$$ \psi_{n} = \frac{6h_{n} + p_{n}}{1 + 3h_{n}}. $$

(5.2)

Let $p_{1},u \in\mathbb{R}$, and $\{p_{n}\}$ be generated by (3.1) as follows:

$$ \textstyle\begin{cases} \Psi_{1} ( \phi_{n},\zeta ) + \frac{1}{g_{n}} \langle \zeta-\phi_{n}, \phi_{n} - p_{n} \rangle\geqslant0,\quad \forall\zeta\in C, \\ \Psi_{2} ( \psi_{n},\zeta ) + \frac{1}{h_{n}} \langle \zeta-\psi_{n}, \psi_{n} - p_{n} \rangle\geqslant0,\quad \forall\zeta\in C, \\ p_{n+1} = \alpha_{n}u + \beta_{n}p_{n} + \gamma_{n}P_{C} (I - \lambda_{n} (I-W) ) \phi_{n} + \delta_{n} G(\psi_{n}),\quad \forall n\in\mathbb{N} , \end{cases} $$

where $a=0.5$, $\lambda_{A}= 1$, $\lambda_{B}=1$, $g_{n}=\frac{n}{3n+1}$, $h_{n}=\frac{n}{4n+1}$, $\alpha_{n}=\frac{1}{2n}$, $\beta_{n}=\frac {3n-1}{16n}$, $\gamma_{n}=\frac{10n-3}{16n}$, $\delta_{n}=\frac {3n-4}{16n}$, and $\lambda_{n}=\frac{1}{2n^{2}}$ for all $n \in\mathbb {N}$. By the definitions of $\Psi_{1}$, $\Psi_{2}$, G, and W, we have $\operatorname{EP}(\Psi_{1}) \cap \operatorname{EP}(\Psi_{2}) \cap F(G) \cap F(W)=\{2\}$. From Theorem 3.1, we can conclude that the sequences $\{ p_{n} \}$, $\{ \phi_{n} \}$, and $\{ \psi_{n} \}$ converge strongly to 2. From (5.1) and (5.2), we can rewrite (3.1) as follows:

$$ \textstyle\begin{cases} \phi_{n}= \frac{4g_{n} + p_{n}}{1 + 2g_{n}}, \\ \psi_{n} = \frac{6h_{n} + p_{n}}{1 + 3h_{n}}, \\ p_{n+1} = \frac{1}{2n}u + \frac{3n-1}{16n}p_{n} + \frac{10n-3}{16n} (I-\frac{1}{2n^{2}}(I-W) )\phi_{n} + \frac{3n-4}{16n}G(\psi _{n}),\quad \forall n \geq1 . \end{cases} $$

(5.3)

Table 1 shows the values of the sequences $\{ p_{n} \}$, $\{\phi_{n} \}$, and $\{\psi_{n} \}$ where $u=p_{1}=-1$ and $u=p_{1}=5$ and $n=300$.

Table 1 The values of $\pmb{\{\phi_{n}\}}$ , $\pmb{\{\psi_{n}\}}$ , and $\pmb{\{p_{n}\}}$ where $\pmb{n = 300}$

Conclusion

• 1. The sequences $\{p_{n}\}$, $\{\phi_{n}\}$, and $\{\psi_{n}\}$ in Table 1 and Figure 1 converge to 2, where $\{ 2 \} = \operatorname{EP}(\Psi_{1}) \cap \operatorname{EP}(\Psi_{2}) \cap F(G) \cap F(W)$.

  Figure 1

  

  The convergence comparison of the sequences $\pmb{\{p_{n}\}}$ , $\pmb{\{ \phi_{n}\}}$ , and $\pmb{\{\psi_{n}\}}$ with different initial values u and $\pmb{p_{1}}$ .

• 2. Theorem 3.1 ensures the convergence of $\{p_{n}\}$, $\{\phi _{n}\}$, and $\{\psi_{n}\}$ in Example 5.1.