Open Access

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

  • Kanyarat Cheawchan1,
  • Suthep Suantai2 and
  • Atid Kangtunyakarn1Email author
Fixed Point Theory and Applications20152015:216

https://doi.org/10.1186/s13663-015-0453-8

Received: 7 April 2015

Accepted: 3 November 2015

Published: 25 November 2015

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.

Keywords

quasi-nonexpansive mappingequilibrium problemvariational inequality problemfixed point problem

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. 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. 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. 1.

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

     
  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
  1. (i)

    \(\{\beta_{n}\}\) is strictly decreasing,

     
  2. (ii)

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

     
  3. (iii)

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

     
  4. (iv)

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

     
  5. (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. 1.

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

     
  2. 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. (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. (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. 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. 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 Opials 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. 1.

    \(\Vert p \pm\zeta \Vert ^{2} = \Vert p\Vert ^{2} \pm2 \langle p, \zeta \rangle+ \Vert \zeta \Vert ^{2}\),

     
  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:
  1. (J1)

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

     
  2. (J2)

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

     
  3. (J3)
    for each \(p,\zeta,\xi\in C\),
    $$\lim_{t \downarrow0} \Psi \bigl( t\xi+(1-t)p,\zeta \bigr) \leq\Psi (p, \zeta); $$
     
  4. (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. (1)

    \(W_{r}\) is single-valued;

     
  2. (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. (3)

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

     
  4. (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. (1)

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

     
  2. (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:
  1. (i)

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

     
  2. (ii)

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

     
  3. (iii)

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

     
  4. (iv)

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

     
  5. (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:
  1. (i)

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

     
  2. (ii)

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

     
  3. (iii)

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

     
  4. (iv)

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

     
  5. (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:
  1. (i)

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

     
  2. (ii)

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

     
  3. (iii)

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

     
  4. (iv)

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

     
  5. (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:
  1. (i)

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

     
  2. (ii)

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

     
  3. (iii)

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

     
  4. (iv)

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

     
  5. (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:
  1. (i)

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

     
  2. (ii)

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

     
  3. (iii)

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

     
  4. (iv)

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

     
  5. (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 \(\pmb{n = 300}\)

n

\(\boldsymbol{u=p_{1}=-1}\)

\(\boldsymbol{u=p_{1}=5}\)

\(\boldsymbol{\phi_{n}}\)

\(\boldsymbol{\psi_{n}}\)

\(\boldsymbol{p_{n}}\)

\(\boldsymbol{\phi_{n}}\)

\(\boldsymbol{\psi_{n}}\)

\(\boldsymbol{p_{n}}\)

1

0.0000

0.1250

−1.0000

4.000000

3.8750

5.0000

2

0.4339

0.5234

−0.4609

3.5661

3.4766

4.4609

3

0.7688

0.8360

0.0301

3.2312

3.1640

3.9699

4

1.0254

1.0771

0.4256

2.9746

3.9229

3.5744

5

1.2188

1.2595

0.7306

2.7812

2.7405

3.2694

150

1.9837

1.9845

1.9728

2.0163

2.0155

2.0272

296

1.9918

1.9922

1.9863

2.0082

2.0078

2.0137

297

1.9918

1.9922

1.9864

2.0082

2.0078

2.0136

298

1.9918

1.9922

1.9864

2.0082

2.0078

2.0136

299

1.9919

1.9923

1.9865

2.0081

2.0077

2.0135

300

1.9919

1.9923

1.9865

2.0081

2.0077

2.0135

Conclusion

  1. 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. 2.

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

     

Declarations

Acknowledgements

This paper was supported by the Thailand Research Fund under the research project RTA578007 and the Research Administration Division of King Mongkut’s Institute of Technology Ladkrabang.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok, Thailand
(2)
Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, Thailand

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