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An iterative scheme for split equality equilibrium problems and split equality hierarchical fixed point problem

Abstract

This paper deals with a split equality equilibrium problem for pseudomonotone bifunctions and a split equality hierarchical fixed point problem for nonexpansive and quasinonexpansive mappings. We suggest and analyze an iterative scheme where the stepsizes do not depend on the operator norms, the so-called simultaneous projected subgradient-proximal iterative scheme for approximating a common solution of the split equality equilibrium problem and the split equality hierarchical fixed point problem. Further, we prove a weak convergence theorem for the sequences generated by this scheme. Furthermore, we discuss some consequences of the weak convergence theorem. We present a numerical example to justify the main result.

Introduction

Let \(H_{1}\), \(H_{2}\), and \(H_{3}\) be real Hilbert spaces with their inner products and induced norms \(\langle \cdot ,\cdot \rangle \) and \(\|\cdot \|\). Let \(C_{1}\) and \(C_{2}\) be nonempty closed convex subsets of \(H_{1}\) and \(H_{2}\), respectively. Recall that a mapping \(U_{1}:H_{1} \to H_{1}\) is nonexpansive if \(\|U_{1}x_{1}-U_{1}y_{1}\|\leq \|x_{1}-y_{1}\|\) for all \(x_{1},y_{1} \in H_{1}\). Note that if \(\operatorname{Fix}(U_{1}):= \{ x_{1} \in H_{1}: U_{1}x_{1}=x_{1}\} \neq \emptyset \), then \(\operatorname{Fix}(U_{1})\) is closed and convex.

We consider the following split equality equilibrium problem (SEEP): Find \(x_{1}\in C_{1}\) and \(x_{2}\in C_{2}\) such that

$$\begin{aligned}& g_{1}(x_{1},y_{1}) \geq 0,\quad y_{1}\in C_{1}, \end{aligned}$$
(1.1)
$$\begin{aligned}& g_{2}(x_{2},y_{2}) \geq 0, \quad y_{2}\in C_{2}, \end{aligned}$$
(1.2)

and

$$ A_{1}x_{1}=A_{2}x_{2}, $$

where \(g_{1}:C_{1}\times C_{1}\to \mathbb{R}\) and \(g_{2}:C_{2}\times C_{2}\to \mathbb{R}\) are monotone bifunctions, and \(A_{1}:H_{1}\to H_{3}\) and \(A_{2}:H_{2}\to H_{3}\) are bounded linear operators. When looked separately, (1.1) is called the equilibrium problem (EP). EP (1.1) was introduced and studied by Blum and Otteli [3]. We denote the solution set of EP (1.1) by Sol(EP(1.1)). The solution set of SEEP (1.1)–(1.2) is denoted by \(\Omega =\{(x_{1}, x_{2}) \in C_{1} \times C_{1} : x_{1}\in \operatorname{Sol}(\mathrm{EP}( \mbox{1.1})), x_{2}\in \operatorname{Sol}(\mathrm{EP}\text{(1.2)}),\mbox{ and }A_{1}x_{1}=A_{2}x_{2} \}\). If \(H_{3}=H_{2}\) and \(A_{2}=I\) (the identity operator), then SEEP (1.1)–(1.2) is reduced to the split equilibrium problem (SEP), which was initially introduced by Moudafi [26] and studied by Kazmi and Rizvi [19] for monotone bifunctions. Recently, Hieu [14] studied the strong convergence of some projected subgradient-proximal iterative schemes for solving SEP for a pseudomonotone bifunction. For further related work, see [12, 15]. As particular cases, SEP includes the split variational inequalities [7] and split feasibility problem [6], which have a wide range of applications; see [4, 5, 7, 10, 11, 21, 31, 32].

SEEP (1.1)–(1.2) has been studied by many authors; see, for instance, Ma et al. [23, 24] and Ali et al. [2] for monotone bifunctions \(g_{1}\), \(g_{2}\). It is interesting to study SEEP (1.1)–(1.2) when both bifunctions \(g_{1}\), \(g_{2}\) are pseudomonotone.

Further, we consider the split equality hierarchical fixed point problem (SEHFPP) [8]: Find \(x_{1}\in \operatorname{Fix}(V_{1})\) and \(x_{2}\in \operatorname{Fix}(V_{2})\) such that

$$\begin{aligned}& \langle x_{1}-U_{1}x_{1},x_{1}-y_{1} \rangle \leq 0,\quad y_{1}\in \operatorname{Fix}(V_{1}), \end{aligned}$$
(1.3)
$$\begin{aligned}& \langle x_{2}-U_{2}x_{2},x_{2}-y_{2} \rangle \leq 0,\quad y_{2}\in \operatorname{Fix}(V_{2}), \end{aligned}$$
(1.4)

and

$$ A_{1}x_{1}=A_{2}x_{2}, $$

where \(U_{1}, V_{1}:C_{1} \to C_{1}\) and \(U_{2}, V_{2}:C_{2} \to C_{2}\) are nonexpansive mappings. When we look separately, (1.3) is called a hierarchical fixed point problem (HFPP), introduced and studied by Moudafi and Mainge [29]. Since then, HFPP has been studied by many authors; see, for example, [9, 1618, 20, 25, 29, 30, 33, 35]. The solution set of HFPP (1.3) is denoted by Sol(HFPP(1.3)). The solution set of SEHFPP (1.3)–(1.4) is denoted by \(\Gamma :=\{(x_{1},x_{2})\in \operatorname{Fix}(V_{1})\times \operatorname{Fix}(V_{2}): x_{1} \in \operatorname{Sol}(\mathrm{HFPP}\text{(1.3)}),x_{2}\in \operatorname{Sol}(\mathrm{HFPP}\text{(1.4)}), \mbox{ and } A_{1}x_{1}=A_{2}x_{2}\}\). If \(H_{3}=H_{2}\) and \(A_{2}=I\), then SEHFPP (1.3)–(1.4) reduces to a new class of problems called the split hierarchical fixed point problem. In particular, if we set \(U_{1}=I_{1}\) and \(U_{2}=I_{2}\) (the identity mappings), then SEHFPP (1.3)–(1.4) reduces to the split equality fixed point problem (SEFPP) [27]: Find \(x_{1} \in C_{1}\) and \(x_{2} \in C_{2}\) such that

$$ x_{1}\in \operatorname{Fix}(V_{1}), \qquad x_{2}\in \operatorname{Fix}(V_{2}), \quad \mbox{and}\quad A_{1}x_{1}=A_{2}x_{2}. $$
(1.5)

The solution set of SEFPP (1.5) is denoted by \(\Gamma _{1}\).

SEHFPP (1.3)–(1.4) was introduced and studied by Behzad et al. [8] for nonexpansive mappings \(U_{1}\), \(U_{2}\), \(V_{1}\), \(V_{2}\). SEHFPP (1.3)–(1.4) covers the split equality variational inequality problem over the fixed point sets, and so on; see [8]. Very recently, Alansari et al. [1] suggested an iterative scheme for solving a split equilibrium problem for a monotone bifunction, a pseudomonotone bifunction, and a hierarchical fixed point problem for nonexpansive and quasinonexpansive mappings.

In 2013, Moudafi and Al-Shemas [28] proved a weak convergence theorem for a simultaneous iterative algorithm to solve SEFPP (1.5). However, to employ this algorithm, we need to know a priori the norms (or at least estimates of the norms) of the bounded linear operators \(A_{1}\) and \(A_{2}\), which is in general not an easy work in practice. To overcome this difficulty, López et al. [22] presented a helpful iterative method for estimating the stepsizes, which do not need a priori knowledge of the operator norms for solving the split feasibility problems. In 2015, Zhao [36] extended the iterative method [22] for SEFPP (1.5). Very recently, Behzad et al. [8] have extended the iterative method [36] for SEHFPP (1.3)–(1.4).

Inspired by the works mentioned, in this paper, we consider SEEP (1.1)-(1.2) where the both bifunctions \(g_{1}\) and \(g_{2}\) are pseudomonotone, and SEHFPP (1.3)–(1.4) where the \(U_{1}\), \(U_{2}\) are quasinonexpansive mappings and \(V_{1}\), \(V_{2}\) are nonexpansive mappings in real Hilbert spaces. We propose an iterative scheme where the stepsizes do not depend on the operator norms for approximating a common solution of these problems. We further prove a weak convergence theorem for the proposed iterative scheme. We present a numerical example to justify the main result.

Preliminaries

Let the symbols → and denote strong and weak convergence, respectively.

Definition 2.1

A mapping \(U_{1}:C_{1}\to C_{1}\) is said to be:

  1. (i)

    quasinonexpansive if, for any \(p_{1} \in \operatorname{Fix}(U_{1})\),

    $$ \Vert U_{1}x_{1}-p_{1} \Vert \leq \Vert x_{1}-p_{1} \Vert , \quad x_{1} \in C_{1}; $$
  2. (ii)

    monotone if

    $$ \langle U_{1}x_{1}-U_{1}y_{1}, x_{1}-y_{1} \rangle \geq 0, \quad x_{1}, y_{1} \in C_{1}; $$

Lemma 2.1

([13])

Let \(V_{1}:C_{1}\to C_{1}\) be a nonexpansive mapping on \(C_{1}\). Then \(V_{1}\) is demiclosed on \(C_{1}\) in the sense that if \(\{x_{1}^{k}\}\) converges weakly to \(x_{1}\in C_{1}\) and \(\{x_{1}^{k}-V_{1}x_{1}^{k}\}\) converges strongly to 0, then \(x_{1} \in \operatorname{Fix}(V_{1})\).

Definition 2.2

A bifunction \(g_{1}:C_{1}\times C_{1}\to \mathbb{R}\) is said to be:

  1. (i)

    strongly monotone on \(C_{1}\) if there exists a constant \(\gamma _{1} >0\) such that \(g_{1}(x_{1},y_{1})+g_{1}(y_{1},x_{1}) \leq -\gamma \|x_{1}-y_{1}\|^{2}\), \(x_{1},y_{1} \in C_{1}\);

  2. (ii)

    monotone on \(C_{1}\) if \(g_{1}(x_{1},y_{1})+g_{1}(y_{1},x_{1}) \leq 0\), \(x_{1},y_{1} \in C_{1}\);

  3. (iii)

    pseudomonotone on \(C_{1}\) if \(g_{1}(x_{1},y_{1})\geq 0 \Rightarrow g_{1}(y_{1},x_{1})\leq 0\), \(x_{1},y_{1} \in C_{1}\).

Note that it is evident from the definition that a strongly monotone bifunction is monotone and a monotone bifunction is pseudomonotone.

Definition 2.3

([12])

Let \(g_{1}:C_{1}\times C_{1}\to \mathbb{R}\) be a bifunction, where \(g_{1}(x_{1}, \cdot )\) is a convex function for each \(x_{1}\in C_{1}\). Then, for \(\epsilon \geq 0\), the ϵ-subdifferential (ϵ-diagonal subdifferential) of \(g_{1}\) at \(x_{1}\), denoted by \(\partial _{\epsilon }g_{1}(x_{1},\cdot )(x_{1})\) or \(\partial _{\epsilon }g_{1}(x_{1},x_{1})\), is given by

$$ \partial _{\epsilon }g_{1}(x_{1},\cdot ) (x_{1})=\bigl\{ w_{1}\in H_{1}: g_{1}(x_{1},y_{1})-g_{1}(x_{1},x_{1})+ \epsilon \geq \langle w_{1}, y_{1}-x_{1} \rangle , y_{1}\in C_{1}\bigr\} . $$

Assumption 2.1

For each \(i=1,2\), the bifunction \(g_{i}:C_{i}\times C_{i}\longrightarrow \mathbb{R}\) satisfies the following assumptions:

  1. (i)

    \(g_{i}(x_{i},x_{i})=0\), \(x_{i} \in C_{i}\);

  2. (ii)

    \(g_{1}\) and \(g_{2}\) are pseudomonotone, respectively, on \(C_{1}\) with respect to \(x_{1}\in \operatorname{Sol}(\mathrm{EP}\text{(1.1)})\) and on \(C_{2}\) with respect to \(x_{2}\in \operatorname{Sol}(\mathrm{EP}\text{(1.2)})\);

  3. (iii)

    \(g_{i}\) satisfies the following condition, called the strict paramonotonicity property:

    $$\begin{aligned}& x_{1}\in \operatorname{Sol}(\mathrm{EP}\text{(1.1)}), y_{1}\in C_{1}, g_{1}(y_{1},x_{1})=0\quad \Rightarrow \quad y_{1}\in \operatorname{Sol}(\mathrm{EP}\text{(1.1)}); \\& x_{2}\in \operatorname{Sol}(\mathrm{EP}\text{(1.2)}), y_{2}\in C_{1}, g_{2}(y_{2},x_{2})=0\quad \Rightarrow \quad y_{2}\in \operatorname{Sol}(\mathrm{EP}\text{(1.2)}); \end{aligned}$$
  4. (iv)

    \(g_{i}\) is jointly weakly upper semicontinuous on \(C_{i}\times C_{i}\) in the sense that if \(x_{i},y_{i}\in C_{i}\) and \(\{x_{i}^{k}\}\), \(\{y_{i}^{k}\}\subseteq C_{i}\) converge weakly to \(x_{i}\) and \(y_{i}\), respectively, then \(g_{i}(x_{i}^{k}, y_{i}^{k})\to g_{i}(x_{i},y_{i})\) as \(k\to \infty \);

  5. (v)

    \(g_{i}(x_{i},\cdot )\) is convex, lower semicontinuous, and subdifferentiable on \(C_{i}\) for all \(x_{i}\in C_{i}\);

  6. (vi)

    If \(\{x_{i}^{k}\}\) is bounded sequence in \(C_{i}\) and \(\epsilon _{k}\to 0\), then the sequence \(\{w_{i}^{k}\}\) with \(w_{i}^{k}\in \partial _{\epsilon _{k}}g_{i}(x_{i}^{k},\cdot )(x_{i}^{k})\) is bounded.

Lemma 2.2

([34])

Let \(\{\delta _{k}\}\) and \(\{\gamma _{k}\}\) be nonnegative sequences satisfying

$$ \sum_{k=0}^{\infty }\delta _{k}< + \infty \quad \textit{and}\quad \gamma _{k+1} \leq \gamma _{k}+\delta _{k},\quad k=0,1,2,\ldots. $$

Then \(\{\gamma _{k}\}\) is a convergent sequence.

Simultaneous projected subgradient-proximal iterative scheme

We suggest the following simultaneous projected subgradient-proximal iterative scheme for solving SEEP (1.1)–(1.2) and SEHFPP (1.3)–(1.4).

Scheme 3.1

(Initialization)

For each \(i=1,2\), choose \(x_{i}^{0}\in C_{i}\). Take the sequences of real numbers \(\{\rho _{k}\}\), \(\{\beta _{k}\}\), \(\{\epsilon _{k}\}\), \(\{r_{k}\}\), \(\{\mu _{k}\}\), \(\{\delta _{k}\}\), and \(\{\sigma _{k}\}\) such that

  1. (i)

    \(\rho _{k}\geq \rho >0\), \(\beta _{k}\geq 0\), \(\epsilon _{k}>0\), \(\epsilon _{k}\to 0\) as \(k\to \infty \), \(r_{k}>r>0\), \(0< a<\delta _{k}<b<1\), and \(0< a^{\prime }<\sigma _{k}<b^{\prime }<1\).

  2. (ii)

    \(\sum_{k=0}^{\infty }\frac{\beta _{k}}{\rho _{k}}=+\infty \), \(\sum_{k=0}^{\infty } \frac{\beta _{k}\epsilon _{k}}{\rho _{k}}<+\infty \), and \(\sum_{k=0}^{\infty }\beta _{k}^{2}<+\infty \).

Step I. Choose \(w_{i}^{k}\in H_{i}\) such that \(w_{i}^{k} \in \partial _{\epsilon _{k}}g_{i}(x_{i}^{k},\cdot )(x_{i}^{k})\) and compute \(\alpha _{k}=\frac{\beta _{k}}{\eta _{k}}\) and \(\eta _{k}=\max \{\rho _{k}, \|w_{i}^{k}\|\}\).

Step II. Compute \(y_{i}^{k}=P_{C_{i}}(x_{i}^{k}-\alpha _{k}w_{i}^{k})\).

Step III. Compute \(t_{i}^{k}=(1-\delta _{k})x_{i}^{k}+\delta _{k}V_{i}((1-\sigma _{k})U_{i}y_{i}^{k}+ \sigma _{k}y_{i}^{k})\).

Step IV. \(x_{i}^{k+1}=P_{C_{i}}(t_{i}^{k}+\mu _{k}A_{i}^{*}(A_{i}t_{1}^{k}-A_{2}t_{2}^{k}))\) for all \(k \geq 0\), where the step size \(\mu _{k}\) is chosen in such a way that for some \(\epsilon > 0\),

$$ \mu _{k} \in (\epsilon , \gamma _{k}- \epsilon ),\quad k\in \Lambda ; $$
(3.1)

otherwise, \(\mu _{k}=\mu \) (\(\mu \geq 0\)), where \(\gamma _{k} := \frac{2\|A_{1}t_{1}^{k}-A_{2}t_{2}^{k}\|^{2}}{\|A_{1}^{*}(A_{1}t_{1}^{k}-A_{2}t_{2}^{k})\|^{2}+\|A_{2}^{*}(A_{1}t_{1}^{k}-A_{2}t_{2}^{k})\|^{2}}\), and the index set \(\Lambda :=\{k: A_{1}t_{1}^{k}-A_{2}t_{2}^{k}\neq 0\}\).

Remark 3.1

([36])

Condition (3.1) implies that \(\inf_{k\in \Lambda } \{\gamma _{k}-\mu _{k}\}>0\). Since \(\|A_{1}^{*}(A_{1}t_{1}^{k}-A_{2}t_{2}^{k})\| \leq \|A_{1}^{*}\|\|A_{1}t_{1}^{k}-A_{2}t_{2}^{k} \|\) and \(\|A_{2}^{*}(A_{1}t_{1}^{k}-A_{2}t_{2}^{k})\| \leq \|A_{2}^{*}\|\|A_{1}t_{1}^{k}-A_{2}t_{2}^{k} \|\), we observe that \(\{\gamma _{k}\}\) is bounded below by \(\frac{2}{\|A_{1}\|^{2}+\|A_{2}\|^{2}}\), and so \(\inf_{k\in \Lambda } \gamma _{k} >0\). Consequently, with an appropriate choice of \(\epsilon >0\) and \(\gamma _{n}\in (\epsilon , \inf_{n\in \Lambda } \mu _{n}-\epsilon )\) for \(k\in \Lambda \), we have \(\sup_{k\in \Lambda } \mu _{k} <+\infty \), and hence \(\{\mu _{k}\}\) is bounded.

Remark 3.2

([12])

For each \(i=1,2\), since \(g_{i}(x_{i}, \cdot )\) is a lower semicontinuous convex function and \(C_{i}\subset \operatorname{dom} g_{i}(x_{i}, \cdot )\) for every \(x_{i} \in C_{i}\), the \(\epsilon _{k}\)-diagonal subdifferential \(\partial _{\epsilon _{k}}g_{i}(x_{i}^{k},\cdot )(x_{i}^{k})\neq \emptyset \) for every \(\epsilon _{k}>0\). Moreover, \(\rho _{k}\geq \rho > 0\). Therefore each step of the scheme is well defined, implying that Scheme 3.1 is well defined.

Remark 3.3

([12])

For each \(i=1,2\), if \(g_{i}\) satisfies Assumption 2.1 ((i), (ii) and (iv)) then \(\operatorname{Sol}(\mathrm{EP}\text{(1.1)})\), \(\operatorname{Sol}(\mathrm{EP}\text{(1.2)})\) are closed and convex. For each \(i=1,2\), since \(A_{i}\) is a linear operator, the solution set Ω of SEEP (1.1)–(1.2) is closed and convex.

Weak convergence theorem

We now prove the following weak convergent theorem, which shows that the sequence \(\{(x_{1}^{k}, x_{2}^{k})\}\) generated by Scheme 3.1 converges weakly to \((q_{1}, q_{2})\in \Phi =\Omega \cap \Gamma \), a common solution of SEEP (1.1)–(1.2) and SEHFPP (1.3)–(1.4).

Assume that \(\Phi \neq \emptyset \).

Theorem 4.1

Let \(H_{1}\), \(H_{2}\), and \(H_{3}\) be real Hilbert spaces. For each \(i=1,2\), let \(C_{i}\subseteq H_{i}\) be a nonempty closed convex set; let \(A_{i}: H_{i}\to H_{3}\) be a bounded linear operator with its adjoint operator \(A_{i}^{*}\); let \(V_{i}: C_{i}\to C_{i}\) be a nonexpansive mapping, let \(U_{i}:C_{i}\to C_{i}\) be a continuous quasinonexpansive mapping such that \(I_{i}-U_{i}\) (\(I_{i}\) is the identity mapping on \(C_{i}\)) is monotone, and let \(g_{i}: C_{i}\times C_{i}\to \mathbb{R}\) be bifunctions satisfying Assumption 2.1. Assume that \(\operatorname{Fix}(U_{1})\cap \operatorname{Fix}(V_{1})\neq \emptyset \), \(\operatorname{Fix}(U_{2}) \cap \operatorname{Fix}(V_{2}))\neq \emptyset \), and \(\Theta =\Omega \cap (\operatorname{Fix}(U_{1})\cap \operatorname{Fix}(V_{1}), \operatorname{Fix}(U_{2}) \cap \operatorname{Fix}(V_{2})\neq \emptyset \). Then the iterative sequence \(\{(x_{1}^{k}, x_{2}^{k})\}\) generated by Scheme 3.1converges weakly to \((q_{1}, q_{2})\in \Phi \).

Proof

Let \((p_{1},p_{2})\in \Theta \). Then \((p_{1},p_{2})\in \Omega \), \(p_{1}\in \operatorname{Fix}(U_{1})\cap \operatorname{Fix}(V_{1})\), and \(p_{2} \in \operatorname{Fix}(U_{2})\cap \operatorname{Fix}(V_{2})\). For each \(i=1,2\), setting

$$ z_{i}^{k}=(1-\sigma _{k})Sy_{i}^{k}+\sigma _{k}y_{i}^{k} $$
(4.1)

and using the arguments used in the proof of [1, Theorem 3.1], we obtain that

$$\begin{aligned} \bigl\Vert z_{i}^{k}-p_{i} \bigr\Vert ^{2} \leq & \bigl\Vert y_{i}^{k}-p_{i} \bigr\Vert ^{2}-\sigma _{k}(1- \sigma _{k}) \bigl\Vert U_{i}y_{i}^{k}-y_{i}^{k} \bigr\Vert ^{2} \end{aligned}$$
(4.2)
$$\begin{aligned} \leq & \bigl\Vert y_{i}^{k}-p_{i} \bigr\Vert ^{2}, \end{aligned}$$
(4.3)
$$\begin{aligned}& \bigl\Vert t_{i}^{k}-p_{i} \bigr\Vert ^{2}\leq (1-\delta _{k}) \bigl\Vert x_{i}^{k}-p_{i} \bigr\Vert ^{2}+ \delta _{k} \bigl\Vert z_{i}^{k}-p_{i} \bigr\Vert ^{2}-\delta _{k}(1-\delta _{k}) \bigl\Vert V_{i}z_{i}^{k}-x_{i}^{k} \bigr\Vert ^{2}, \end{aligned}$$
(4.4)
$$\begin{aligned}& \lim_{k\to \infty } \bigl\Vert x_{i}^{k}-y_{i}^{k} \bigr\Vert =0, \end{aligned}$$
(4.5)

and

$$ \bigl\Vert t_{i}^{k}-p_{i} \bigr\Vert ^{2}\leq \bigl\Vert x_{i}^{k}-p_{i} \bigr\Vert ^{2}+2\delta _{k} \alpha _{k} \bigl\langle w_{i}^{k}, p_{i}-x_{i}^{k} \bigr\rangle +2\delta _{k} \beta _{k}^{2} - \delta _{k}(1-\delta _{k}) \bigl\Vert V_{i}z_{i}^{k}-x_{i}^{k} \bigr\Vert ^{2}. $$
(4.6)

Since \(x_{i}^{k}\in C_{i}\) and \(w_{i}^{k}\in \partial _{\epsilon _{k}}g_{i}(x_{i}^{k}, \cdot )(x_{i}^{k})\), we have

$$\begin{aligned} g_{i}\bigl(x_{i}^{k},p_{i} \bigr)+\epsilon _{k} =& g_{i}\bigl(x_{i}^{k},p_{i} \bigr)-g_{i}\bigl(x_{i}^{k},x_{i}^{k} \bigr)+ \epsilon _{k} \geq \bigl\langle w_{i}^{k}, p_{i}-x_{i}^{k}\bigr\rangle , \end{aligned}$$
(4.7)

and hence from (4.6) and (4.7) we have

$$\begin{aligned} \bigl\Vert t_{i}^{k}-p_{i} \bigr\Vert ^{2} \leq & \bigl\Vert x_{i}^{k}-p_{i} \bigr\Vert ^{2}+2\delta _{k} \alpha _{k} \bigl(g_{i}\bigl(x_{i}^{k},p_{i} \bigr)+\epsilon _{k}\bigr)+2\delta _{k}\beta _{k}^{2} \\ &{}-\delta _{k}(1-\delta _{k}) \bigl\Vert V_{i}z_{i}^{k}-x_{i}^{k} \bigr\Vert ^{2}. \end{aligned}$$
(4.8)

Now from the definitions of \(\alpha _{k}\) and \(\eta _{k}\) we obtain \(\alpha _{k}=\frac{\beta _{k}}{\eta _{k}}\leq \frac{\beta _{k}}{\rho _{k}}\). Hence from (4.8) we have

$$\begin{aligned} \bigl\Vert t_{i}^{k}-p_{i} \bigr\Vert ^{2} \leq & \bigl\Vert x_{i}^{k}-p_{i} \bigr\Vert ^{2}+2\delta _{k} \alpha _{k}g_{i}\bigl(x_{i}^{k},p_{i} \bigr)+ \frac{2\delta _{k}\beta _{k}\epsilon _{k}}{\rho _{k}}+2\delta _{k} \beta _{k}^{2} \\ &{}-\delta _{k}(1-\delta _{k}) \bigl\Vert V_{i}z_{i}^{k}-x_{i}^{k} \bigr\Vert ^{2}. \end{aligned}$$
(4.9)

Again, since \(p_{i} \in C_{i}\), we have

$$\begin{aligned}& \bigl\Vert x_{1}^{k+1}-p_{1} \bigr\Vert ^{2} \\& \quad = \bigl\Vert P_{C_{1}} \bigl(t_{1}^{k}+\mu _{k}A_{1}^{*} \bigl(A_{1}t_{1}^{k}-A_{2}t_{2}^{k} \bigr)\bigr)-(p_{1}) \bigr\Vert ^{2} \\ & \quad \leq \bigl\Vert t_{1}^{k}-p_{1} \bigr\Vert ^{2}-2\mu _{k} \bigl\langle A_{1}t_{1}^{k}-A_{1}p_{1},A_{1}t_{1}^{k}-A_{2}t_{2}^{k} \bigr\rangle +\mu _{k}^{2} \bigl\Vert A_{1}^{*}\bigl(A_{1}t_{1}^{k}-A_{2}t_{2}^{k} \bigr) \bigr\Vert ^{2} \\ & \quad = \bigl\Vert t_{1}^{k}-p_{1} \bigr\Vert ^{2}-\mu _{k} \bigl[ \bigl\Vert A_{1}t_{1}^{k}-A_{1}p_{1} \bigr\Vert ^{2} + \bigl\Vert A_{1}t_{1}^{k}-A_{2}t_{2}^{k} \bigr\Vert ^{2}- \bigl\Vert A_{2}t_{2}^{k}-A_{1}p_{1} \bigr\Vert ^{2} \bigr] \\ & \qquad {}+\mu _{k}^{2} \bigl\Vert A_{1}^{*} \bigl(A_{1}t_{1}^{k}-A_{2}t_{2}^{k} \bigr) \bigr\Vert ^{2}. \end{aligned}$$
(4.10)

Similarly, we have

$$\begin{aligned}& \bigl\Vert x_{2}^{k+1}-p_{2} \bigr\Vert ^{2} \\& \quad \leq \bigl\Vert t_{2}^{k}-p_{2} \bigr\Vert ^{2}-\mu _{k} \bigl[ \bigl\Vert A_{2}t_{2}^{k}-A_{2}p_{2} \bigr\Vert ^{2} + \bigl\Vert A_{1}t_{1}^{k}-A_{2}t_{2}^{k} \bigr\Vert ^{2}- \bigl\Vert A_{1}t_{1}^{k}-A_{2}p_{2} \bigr\Vert ^{2} \bigr] \\ & \qquad {} +\mu _{k}^{2} \bigl\Vert A_{2}^{*} \bigl(A_{1}t_{1}^{k}-A_{2}t_{2}^{k} \bigr) \bigr\Vert ^{2}. \end{aligned}$$
(4.11)

From (4.10), (4.11), and the fact that \(A_{1}p_{1}=A_{2}p_{2}\) we have

$$\begin{aligned}& \bigl\Vert x_{1}^{k+1}-p_{1} \bigr\Vert ^{2}+ \bigl\Vert x_{2}^{k+1}-p_{2} \bigr\Vert ^{2} \\& \quad \leq \bigl\Vert t_{1}^{k}-p_{1} \bigr\Vert ^{2}+ \bigl\Vert t_{2}^{k}-p_{2} \bigr\Vert ^{2}-\mu _{k} \bigl[ 2 \bigl\Vert A_{2}t_{2}^{k}-A_{2}p_{2} \bigr\Vert ^{2} \\ & \qquad {}-\mu _{k} \bigl( \bigl\Vert A_{1}^{*} \bigl(A_{1}t_{1}^{k}-A_{2}t_{2}^{k} \bigr) \bigr\Vert ^{2}+ \bigl\Vert A_{2}^{*} \bigl(A_{1}t_{1}^{k}-A_{2}t_{2}^{k} \bigr) \bigr\Vert ^{2} \bigr) \bigr]. \end{aligned}$$
(4.12)

From (4.9) and (4.12) we have

$$\begin{aligned}& \bigl\Vert x_{1}^{k+1}-p_{1} \bigr\Vert ^{2}+ \bigl\Vert x_{2}^{k+1}-p_{2} \bigr\Vert ^{2} \\ & \quad \leq \bigl\Vert x_{1}^{k}-p_{1} \bigr\Vert ^{2}+ \bigl\Vert x_{2}^{k}-p_{2} \bigr\Vert ^{2}+2\delta _{k} \alpha _{k} \bigl(g_{1}\bigl(x_{1}^{k},p_{1} \bigr)+g_{2}\bigl(x_{2}^{k},p_{2} \bigr) \bigr) \\ & \qquad {}-\mu _{k} \bigl[ 2 \bigl\Vert A_{2}t_{2}^{k}-A_{2}p_{2} \bigr\Vert ^{2} -\mu _{k} \bigl( \bigl\Vert A_{1}^{*}\bigl(A_{1}t_{1}^{k}-A_{2}t_{2}^{k} \bigr) \bigr\Vert ^{2}+ \bigl\Vert A_{2}^{*} \bigl(A_{1}t_{1}^{k}-A_{2}t_{2}^{k} \bigr) \bigr\Vert ^{2} \bigr) \bigr] \\ & \qquad {}-\delta _{k}(1-\delta _{k}) \bigl( \bigl\Vert V_{1}z_{1}^{k}-x_{1}^{k} \bigr\Vert ^{2}+ \bigl\Vert V_{2}z_{2}^{k}-x_{2}^{k} \bigr\Vert ^{2} \bigr)+\zeta _{k}, \end{aligned}$$
(4.13)

where \(\zeta _{k}=2\delta _{k}(\frac{\beta _{k}\epsilon _{k}}{\rho _{k}}+ \beta _{k}^{2})\).

Since \((p_{1},p_{2})\in \Omega \) and \(x_{i}^{k}\in C_{i}\) for \(i=1,2\), \(p_{i}\in C_{i}\), and hence \(g_{i}(p_{i}, x_{i}^{k})\geq 0\). By the pseudomonotonicity of \(g_{i}\) we have

$$ g_{i}\bigl(x_{i}^{k},p_{i} \bigr)\leq 0. $$
(4.14)

Hence, using condition (3.1) and \(\delta _{k} \in (0,1)\) in (4.13), we have

$$\begin{aligned} \bigl\Vert x_{1}^{k+1}-p_{1} \bigr\Vert ^{2}+ \bigl\Vert x_{2}^{k+1}-p_{2} \bigr\Vert ^{2} \leq & \bigl\Vert x_{1}^{k}-p_{1} \bigr\Vert ^{2}+ \bigl\Vert x_{2}^{k}-p_{2} \bigr\Vert ^{2}+\zeta _{k}. \end{aligned}$$
(4.15)

It follows from the conditions on \(\beta _{k}\), \(\epsilon _{k}\), and \(\rho _{k}\) that \(\sum_{k=0}^{\infty }\zeta _{k}<+\infty \). Hence it follows from Lemma 2.2 and (4.15) that the sequence \(\{ \|x_{1}^{k}-p_{1}\|^{2}+\|x_{2}^{k}-p_{2}\|^{2}\}\) is convergent, that is,

$$ \lim_{k\to \infty } \bigl( \bigl\Vert x_{1}^{k}-p_{1} \bigr\Vert ^{2}+ \bigl\Vert x_{2}^{k}-p_{2} \bigr\Vert ^{2} \bigr)\quad \mbox{exists,} $$
(4.16)

which implies that the sequences \(\{x_{1}^{k}\}\) and \(\{x_{2}^{k}\}\) are bounded. Therefore it follows from (4.5) and (4.3) that, for each \(i=1,2\), the sequences \(\{y_{i}^{k}\}\), \(\{z_{i}^{k}\}\) are bounded.

Since \(\delta _{k}\in (0,1)\), \(\sum_{k=0}^{\infty }\zeta _{k}<+\infty \), and \(\{\mu _{k}\}\) is bounded, from (4.13), (4.14), and (4.16) it follows that

$$ \lim_{k\to \infty } \bigl\Vert A_{1}^{*}\bigl(A_{1}t_{1}^{k}-A_{2}t_{2}^{k} \bigr) \bigr\Vert = \lim_{k\to \infty } \bigl\Vert A_{2}^{*}\bigl(A_{1}t_{1}^{k}-A_{2}t_{2}^{k} \bigr) \bigr\Vert =0. $$
(4.17)

Similarly, from (4.13) we obtain that

$$ \lim_{k\to \infty } \bigl\Vert V_{1}z_{1}^{k}-x_{1}^{k} \bigr\Vert = \lim_{k\to \infty } \bigl\Vert V_{2}z_{2}^{k}-x_{2}^{k} \bigr\Vert =0. $$
(4.18)

Now from \(\sum_{k=0}^{\infty }\zeta _{k}<+\infty \), (4.13), (4.14), and (4.16)–(4.18) it follows that

$$ \lim_{k\to \infty } \bigl\Vert A_{1}t_{1}^{k}-A_{2}t_{2}^{k} \bigr\Vert =0. $$
(4.19)

Again, since \(\delta _{k}\in (0,1)\), from conditions (3.1), (4.13), and (4.17)–(4.19) it follows that

$$\begin{aligned}& 2\delta _{k}\alpha _{k} \bigl(g_{1}\bigl(x_{1}^{k},p_{1} \bigr)+g_{2}\bigl(x_{2}^{k},p_{2} \bigr) \bigr) \\& \quad \leq \bigl\Vert x_{1}^{k}-p_{1} \bigr\Vert ^{2}- \bigl\Vert x_{1}^{k+1}-p_{1} \bigr\Vert ^{2}+ \bigl\Vert x_{2}^{k}-p_{2} \bigr\Vert ^{2}- \bigl\Vert x_{2}^{k+1}-p_{2} \bigr\Vert ^{2}+\zeta _{k}. \end{aligned}$$
(4.20)

Hence, for every m, from (4.14) and (4.20) it follows that

$$\begin{aligned} 0 \leq & \sum_{k=0}^{m}2 \delta _{k}\alpha _{k} \bigl(g_{1} \bigl(x_{1}^{k},p_{1}\bigr)+g_{2} \bigl(x_{2}^{k},p_{2}\bigr) \bigr) \\ \leq & \bigl\Vert x_{1}^{0}-p_{1} \bigr\Vert ^{2}- \bigl\Vert x_{1}^{m+1}-p_{1} \bigr\Vert ^{2}+ \bigl\Vert x_{2}^{0}-p_{2} \bigr\Vert ^{2}- \bigl\Vert x_{2}^{m+1}-p_{2} \bigr\Vert ^{2}+4\sum_{k=0}^{m} \frac{\beta _{k}\epsilon _{k}}{\rho _{k}}+4\sum_{k=0}^{m} \beta _{k}^{2}. \end{aligned}$$

By taking the limit as \(m\to \infty \) we have

$$ 0 \leq 2 \sum_{k=0}^{\infty }\delta _{k}\alpha _{k} \bigl(g_{1} \bigl(x_{1}^{k},p_{1}\bigr)+g_{2} \bigl(x_{2}^{k},p_{2}\bigr) \bigr)< +\infty , $$

which implies

$$ \sum_{k=0}^{\infty }\delta _{k}\alpha _{k}g_{i}\bigl(x_{i}^{k},p_{i} \bigr)< + \infty $$
(4.21)

for \(i=1,2\). For \(i=1,2\), the boundedness of the sequence \(\{x_{i}^{k}\}\) and Assumption 2.1(vi) imply that the sequence \(\{w_{i}^{k}\}\) is bounded. Further, using the conditions on the parameters, we have \(\alpha _{k}= \frac{\beta _{k}}{\rho _{k}\max \{1,\frac{\|w^{k}\|}{\rho _{k}}\|\}} \geq \frac{\beta _{k}\rho }{\rho _{k}w}\). Since \(\delta _{k}\in (a,b)\subset (0,1)\), from (4.21) it follows that

$$ 0\leq \frac{2\rho a}{w}\sum_{k=0}^{\infty } \frac{\beta _{k}}{\rho _{k}}\bigl(-g_{i}\bigl(x_{i}^{k},p_{i} \bigr)\bigr)\leq 2a\sum_{k=0}^{\infty }\alpha _{k}\bigl(-g_{i}\bigl(x_{i}^{k},p_{i} \bigr)\bigr)< +\infty . $$
(4.22)

Since \(\sum_{k=0}^{\infty }\frac{\beta _{k}}{\rho _{k}}=+\infty \), from (4.14) and (4.22) it follows that

$$ \limsup _{k\to \infty } g_{1}\bigl(x_{1}^{k},p_{1} \bigr)=\limsup _{k\to \infty } g_{2}\bigl(x_{2}^{k},p_{2} \bigr)=0. $$
(4.23)

Further, from the equation in Step III of Scheme 3.1 and (4.18) it follows that

$$ \lim_{k\to \infty } \bigl\Vert t_{1}^{k}-x_{1}^{k} \bigr\Vert =\lim_{k \to \infty } \bigl\Vert t_{2}^{k}-x_{2}^{k} \bigr\Vert =0. $$
(4.24)

Since

$$ \bigl\Vert y_{i}^{k}-p_{i} \bigr\Vert ^{2}\leq \bigl\Vert x_{i}^{k}-p_{i} \bigr\Vert ^{2}+2\bigl\langle y_{i}^{k}-x_{i}^{k},y_{i}^{k}-p_{i} \bigr\rangle \quad (i=1,2) $$
(4.25)

and \(\{y_{1}^{k}\}\), \(\{y_{1}^{k}\}\) are bounded, from (4.2), (4.4), and (4.12) it follows that

$$\begin{aligned}& \delta _{k}\sigma _{k}(1-\sigma _{k}) \bigl( \bigl\Vert U_{1}y_{1}^{k}-y_{1}^{k} \bigr\Vert ^{2}+ \bigl\Vert U_{2}y_{2}^{k}-y_{2}^{k} \bigr\Vert ^{2} \bigr) \\& \quad \leq \bigl\Vert x_{1}^{k}-p_{1} \bigr\Vert ^{2}- \bigl\Vert x_{1}^{k+1}-p_{1} \bigr\Vert ^{2}+ \bigl\Vert x_{2}^{k}-p_{2} \bigr\Vert ^{2}- \bigl\Vert x_{2}^{k+1}-p_{2} \bigr\Vert ^{2} \\& \qquad {}+2\delta _{k} \bigl[ \bigl\Vert y_{1}^{k}-x_{1}^{k} \bigr\Vert \bigl\Vert y_{1}^{k}-p_{1} \bigr\Vert + \bigl\Vert y_{2}^{k}-x_{2}^{k} \bigr\Vert \bigl\Vert y_{2}^{k}-p_{2} \bigr\Vert \bigr] \\& \qquad {}-\delta _{k}(1-\delta _{k}) \bigl[ \bigl\Vert V_{1}z_{1}^{k}-x_{1}^{k} \bigr\Vert ^{2}+ \bigl\Vert V_{1}z_{2}^{k}-x_{2}^{k} \bigr\Vert ^{2} \bigr] \\& \qquad {}-\mu _{k} \bigl[ 2 \bigl\Vert A_{2}t_{2}^{k}-A_{2}p_{2} \bigr\Vert ^{2} -\mu _{k} \bigl( \bigl\Vert A_{1}^{*}\bigl(A_{1}t_{1}^{k}-A_{2}t_{2}^{k} \bigr) \bigr\Vert ^{2} \\& \qquad {}+ \bigl\Vert A_{2}^{*}\bigl(A_{1}t_{1}^{k}-A_{2}t_{2}^{k} \bigr) \bigr\Vert ^{2} \bigr) \bigr]. \end{aligned}$$
(4.26)

Again, since \(\delta _{k}\in (a,b)\subset (0,1)\) and \(\sigma _{k}\in (a^{\prime },b^{\prime })\subset (0,1)\), from (4.5) and (4.16)–(4.19) it follows that

$$ \lim_{k\to \infty } \bigl\Vert U_{1}y_{1}^{k}-y_{1}^{k} \bigr\Vert ^{2}=\lim_{k\to \infty } \bigl\Vert U_{2}y_{2}^{k}-y_{2}^{k} \bigr\Vert ^{2}=0. $$
(4.27)

For each \(i=1,2\), from the inequality

$$\begin{aligned} \bigl\Vert V_{i}z_{i}^{k}-y_{i}^{k} \bigr\Vert ^{2} \leq & \bigl\Vert V_{i}z_{i}^{k}-x_{i}^{k} \bigr\Vert ^{2}+2 \bigl\langle x_{i}^{k}-y_{i}^{k}, V_{i}z_{i}^{k}-y_{i}^{k} \bigr\rangle \\ \leq & \bigl\Vert V_{i}z_{i}^{k}-x_{i}^{k} \bigr\Vert ^{2}+2 \bigl\Vert x_{i}^{k}-y_{i}^{k} \bigr\Vert \bigl\Vert V_{i}z_{i}^{k}-y_{i}^{k} \bigr\Vert , \end{aligned}$$
(4.28)

the boundedness of the sequences \(\{y_{i}^{k}\}\) and \(\{z_{i}^{k}\}\), (4.5), and(4.18) it follows that

$$ \lim_{k\to \infty } \bigl\Vert V_{i}z_{i}^{k}-y_{i}^{k} \bigr\Vert ^{2}=0. $$
(4.29)

Since

$$ \bigl\Vert U_{i}y_{i}^{k}-V_{i}z_{i}^{k} \bigr\Vert \leq \bigl\Vert U_{i}y_{i}^{k}-y_{i}^{k} \bigr\Vert + \bigl\Vert y_{i}^{k}-V_{i}z_{i}^{k} \bigr\Vert , $$
(4.30)

from (4.27), (4.29), and (4.30) it follows that

$$ \lim_{k\to \infty } \bigl\Vert U_{i}y_{i}^{k}-V_{i}z_{i}^{k} \bigr\Vert =0. $$
(4.31)

The equality

$$\begin{aligned} \bigl\Vert z_{i}^{k}-y_{i}^{k} \bigr\Vert =& (1-\sigma _{k}) \bigl\Vert U_{i}y_{i}^{k}-y_{i}^{k} \bigr\Vert \end{aligned}$$

implies that

$$ \lim_{k\to \infty } \bigl\Vert z_{i}^{k}-y_{i}^{k} \bigr\Vert =0. $$
(4.32)

The inequality

$$ \bigl\Vert V_{i}z_{i}^{k}-z_{i}^{k} \bigr\Vert \leq \bigl\Vert V_{i}z_{i}^{k}-y_{i}^{k} \bigr\Vert + \bigl\Vert y_{i}^{k}-z_{i}^{k} \bigr\Vert $$
(4.33)

implies that

$$ \lim_{k\to \infty } \bigl\Vert V_{i}z_{i}^{k}-z_{i}^{k} \bigr\Vert =0. $$
(4.34)

Now, since the sequence \(\{x_{i}^{k}\}\) is bounded in \(C_{i}\) for \(i=1,2\),, without the loss of generality, we can assume that there exists a subsequence \(\{x_{i}^{k_{l}}\}\) of \(\{x_{i}^{k}\}\) such that \(x_{i}^{k_{l}}\rightharpoonup q_{i}\in C_{i}\) as \(l\to \infty \) and \(\limsup_{k\to \infty } g_{i}(x_{i}^{k},p_{i})=\lim_{l \to \infty }g_{i}(x_{i}^{k_{l}},p_{i})\). From (4.5), (4.24), and (4.32) it follows that the sequences \(\{x_{i}^{k}\}\), \(\{y_{i}^{k}\}\), \(\{t_{i}^{k}\}\), and \(\{z_{i}^{k}\}\) have the same asymptotic behavior, and hence there are subsequences \(\{y_{i}^{k_{l}}\}\) of \(\{y_{i}^{k}\}\), \(\{t_{i}^{k_{l}}\}\) of \(\{t_{i}^{k}\}\), and \(\{z_{i}^{k_{l}}\}\) of \(\{z_{i}^{k}\}\) such that \(y_{i}^{k_{l}}\rightharpoonup q_{i}\), \(t_{i}^{k_{l}}\rightharpoonup q_{i}\), and \(z_{i}^{k_{l}}\rightharpoonup q_{i}\) as \(l\to \infty \). Since \(A_{i}\) is continuous for \(i=1,2\), \(A_{i}t_{i}^{k_{l}}\rightharpoonup A_{i}q_{i}\). Further, for \(i=1,2\), it follows from the demiclosedness of \(I_{i}-V_{i}\) on \(C_{i}\) and (4.34) that \(q_{i}\in \operatorname{Fix}(V_{i})\). We now show that \((q_{1},q_{2})\in \Gamma \). From (4.1) it follows that

$$ \frac{z_{i}^{k}-V_{i}z_{i}^{k}}{\sigma _{k}}=(I_{i}-U_{i})y_{i}^{k}+ \frac{1}{\sigma _{k}} \bigl(U_{i}y_{i}^{k}-V_{i}z_{i}^{k} \bigr). $$
(4.35)

Therefore, for all \(z_{i}\in \operatorname{Fix}(V_{i})\), using (4.1) and the monotonicity of \((I_{i}-U_{i})\), we estimate

$$\begin{aligned}& \biggl\langle \frac{z_{i}^{k}-V_{i}z_{i}^{k}}{\sigma _{k}}, y_{i}^{k}-z_{i} \biggr\rangle \\& \ \quad =\bigl\langle (I_{i}-U_{i})y_{i}^{k}-(I_{i}-U_{i})z_{i},y_{i}^{k}-z_{i} \bigr\rangle +\bigl\langle z_{i}-U_{i}z_{i},y_{i}^{k}-z_{i} \bigr\rangle \\& \qquad {}+\frac{1}{\sigma _{k}}\bigl\langle U_{i}y_{i}^{k}-V_{i}z_{i}^{k},y_{i}^{k}-z_{i} \bigr\rangle \\& \quad \geq \bigl\langle z_{i}-U_{i}z_{i},y_{i}^{k}-z_{i} \bigr\rangle + \frac{1}{\sigma _{k}}\bigl\langle U_{i}y_{i}^{k}-V_{i}z_{i}^{k},y_{i}^{k}-z_{i} \bigr\rangle . \end{aligned}$$
(4.36)

Since \(\{y_{i}^{k}\}\) is bounded and \(\sigma _{k}\in (a^{\prime },b^{\prime })\subset (0,1)\), from (4.31), (4.34), and (4.36) it follows that

$$ {\limsup _{k\to \infty }}\bigl\langle z_{i}-U_{i}z_{i},y_{i}^{k}-z_{i} \bigr\rangle \leq 0,\quad z_{i}\in \operatorname{Fix}(V_{i}). $$
(4.37)

Replacing k with \(k_{l}\) in (4.37) and then taking the limit as \(l\to \infty \), we have

$$ \bigl\langle (I_{i}-U_{i})z_{i},q_{i}-z_{i} \bigr\rangle \leq 0,\quad z_{i}\in \operatorname{Fix}(V_{i}). $$
(4.38)

Since \(\operatorname{Fix}(V_{i})\) is convex, \(\lambda z_{i}+(1-\lambda )q_{i}\in \operatorname{Fix}(V_{i})\) for \(\lambda \in (0,1)\), and hence

$$ \bigl\langle (I_{i}-U_{i}) \bigl( \lambda z_{i}+(1-\lambda )q_{i}\bigr),q_{i}-z_{i} \bigr\rangle \leq 0,\quad z_{i}\in \operatorname{Fix}(V_{i}). $$
(4.39)

Since \((I_{i}-U_{i})\) is continuous, by taking the limit as \(\lambda \to 0_{+}\), we have

$$ \bigl\langle (I_{i}-U_{i})q_{i},q_{i}-z_{i} \bigr\rangle \leq 0,\quad z_{i}\in \operatorname{Fix}(V_{i}), $$
(4.40)

that is, \(q_{1}\in \operatorname{Sol}(\mathrm{HFPP}(\mbox{1.3}))\) and \(q_{1}\in \operatorname{Sol}(\mathrm{HFPP}(\mbox{1.3}))\). Further, since \(\|\cdot \|^{2}\) is weakly lower semicontinuous, from (4.19) it follows that

$$ \Vert A_{1}q_{1}-A_{2}q_{2} \Vert ^{2}\leq \liminf _{k\to \infty } \bigl\Vert A_{1}t_{1}^{k_{l}}-A_{2}t_{2}^{k_{l}} \bigr\Vert ^{2}=0, $$
(4.41)

that is, \(A_{1}q_{1}=A_{2}q_{2}\). Hence \((q_{1},q_{2}) \in \Gamma \). Next, we show that \((q_{1},q_{2})\in \Omega \). Since \(x_{i}^{k_{l}}\rightharpoonup q_{i}\) and \(\limsup_{k\to \infty }g_{i}(x_{i}^{k},p_{i})=\lim_{l \to \infty }g_{i}(x_{i}^{k_{l}},p_{i})\), by the weak upper semicontinuity of \(g_{i}(\cdot ,p_{i})\) and (4.23) we have

$$ g_{i}(q_{i},p_{i})\geq \limsup _{l\to \infty }g_{i}\bigl(x_{i}^{k_{l}},p_{i} \bigr)= \lim_{l\to \infty }g_{i}\bigl(x_{i}^{k_{l}},p_{i} \bigr)=\limsup _{k\to \infty }g_{i}\bigl(x_{i}^{k},p_{i} \bigr)=0. $$
(4.42)

Since \((p_{1},p_{2})\in \Omega \) and \(q_{i}\in C_{i}\), we have \(g_{i}(p_{i},q_{i})\geq 0\), and hence from Assumption 2.1(ii) it follows that \(g_{i}(q_{i},p_{i})\leq 0\). Consequently, \(g_{i}(q_{i},p_{i})=0\), and therefore by Assumption 2.1(iv) we have \(q_{1}\in \operatorname{Sol}(\mathrm{EP}(\mbox{1.1}))\) and \(q_{2}\in \operatorname{Sol}(\mathrm{EP}(\mbox{1.2}))\). Hence \((q_{1},q_{2}) \in \Omega \), and thus \((q_{1},q_{2}) \in \Phi \).

From (4.16) it follows that \(\lim_{k \to \infty }\|x_{i}^{k}- p_{i}\|\) exists for \(i=1,2\). Therefore since the Hilbert space \(H_{i}\) satisfies the Opial condition, it follows that the sequence \(\{x_{i}^{k}\}\) has only one weak cluster point, and hence \(\{(x_{1}^{k}, x_{2}^{k})\}\) converges weakly to \((q_{1}, q_{2})\in \Phi \). □

Consequences

Now, we give some consequences of Theorem 4.1.

(I). The following theorem shows that the sequence \(\{(x_{1}^{k}, x_{2}^{k})\}\) generated by Scheme 3.1 with \(U_{i}=I_{i}~(i=1,2)\) converges weakly to \((q_{1}, q_{2})\in \Phi _{1}=\Omega \cap \Gamma _{1}\), a common solution of SEEP (1.1)–(1.2) and SEFPP (1.5).

Assume that \(\Phi _{1}\neq \emptyset \).

Theorem 5.1

Let \(H_{1}\), \(H_{2}\), and \(H_{3}\) be real Hilbert spaces. For \(i=1,2\), let \(C_{i}\subseteq H_{i}\) be a nonempty closed convex set, let \(A_{i}: H_{i}\to H_{3}\) be a bounded linear operator with its adjoint operator \(A_{i}^{*}\), let \(V_{i}: C_{i}\to C_{i}\) be a nonexpansive mapping, and let \(g_{i}: C_{i}\times C_{i}\to \mathbb{R}\) be a bifunction satisfying Assumption 2.1. Assume that \(\operatorname{Fix}(V_{1})\neq \emptyset \), \(\operatorname{Fix}(V_{2}))\neq \emptyset \), and \(\Theta _{1}=\Omega \cap (\operatorname{Fix}(V_{1}), \operatorname{Fix}(V_{2})\neq \emptyset \). Then the iterative sequence \(\{(x_{1}^{k}, x_{2}^{k})\}\) generated by Scheme 3.1with \(U_{i}=I_{i}\) (\(i=1,2\)) converges weakly to \((q_{1}, q_{2})\in \Phi _{1}\).

(II). The following theorem shows that the sequence \(\{x_{1}^{k}\}\) generated by Scheme 3.1 with \(H_{1}=H_{2}\), \(U_{1}=U_{2}\), \(V_{1}=V_{2}\), \(C_{1}=C_{2}=Q_{2}=Q_{1}\), and \(A_{i}=B_{i}=I_{i}~(i=1,2)\) converges weakly to \(q_{1}\in \Phi _{2}=\operatorname{Sol}(\mathrm{EP}\text{(1.1)}) \cap \operatorname{Sol}(\mathrm{HFPP}( \mbox{1.3}))\), a common solution of EP (1.1) and HFPP (1.3).

Assume that \(\Phi _{2}\neq \emptyset \).

Theorem 5.2

Let \(H_{1}\) and \(H_{3}\) be real Hilbert spaces. Let \(C_{1}\subseteq H_{1}\) be a nonempty closed convex set, let \(V_{1}: C_{1}\to C_{1}\) be a nonexpansive mapping, let \(U_{1}:C_{1}\to C_{1}\) be a continuous quasi-onexpansive mapping such that \(I_{1}-U_{1}\) (\(I_{1}\) is the identity mapping on \(C_{1}\)) is monotone, and let \(g_{1}: C_{1}\times C_{1}\to \mathbb{R}\) be a bifunction satisfying Assumption 2.1. Assume that \(\operatorname{Fix}(U_{1})\cap \operatorname{Fix}(V_{1})\neq \emptyset \) and \(\Theta _{2}=\operatorname{Sol}(\mathrm{EP}\textit{(1.1)})\cap \operatorname{Fix}(U_{1})\cap \operatorname{Fix}(V_{1}) \neq \emptyset \). Then the iterative sequence \(\{x_{1}^{k}\}\) generated by Scheme 3.1with \(H_{1}=H_{2}\), \(U_{1}=U_{2}\), \(V_{1}=V_{2}\), \(C_{1}=C_{2}=Q_{2}=Q_{1}\), and \(A_{i}=B_{i}=I_{i}~(i=1,2)\) converges weakly to \(q_{1} \in \Phi _{2}\).

Numerical example

Finally, we give a numerical example for Scheme 3.1.

Example 6.1

Let \(H_{1}=H_{2}=H_{3}=\mathbb{R}\), the set of all real numbers, with the inner product defined by \(\langle x,y \rangle =xy\), \(x,y\in \mathbb{R}\), and induced usual norm \(|\cdot |\). Let \(C_{1}=[-\pi ,0]\) and \(C_{2}=[0,\pi ]\), let \(g_{1}:C_{1}\times C_{1}\to \mathbb{R}\) and \(g_{2}:C_{2}\times C_{2}\to \mathbb{R}\) be defined by \(g_{1}(x_{1},y_{1})=2x_{1}y_{1}(y_{1}-x_{1})+x_{1}y_{1}|y_{1}-x_{1}|\), \(x_{1},y_{1} \in C_{1}\), and \(g_{2}(x_{2},y_{2})=x_{2}^{2}(y_{2}-x_{2})\), \(x_{2},y_{2}\in C_{2}\). Let the mappings \(A_{1}:\mathbb{R} \to \mathbb{R}\) and \(A_{2}:\mathbb{R} \to \mathbb{R}\) be defined by \(A_{1}(x_{1})=2x_{1}\), \(x_{1}\in \mathbb{R}\), and \(A_{2}(x_{2})=-2x_{2}\), \(x_{2}\in \mathbb{R}\). Let the mappings \(V_{1}: C_{1}\to C_{1}\) and \(U_{1}: C_{1}\to C_{1}\) be defined by \(V_{1}x_{1}=\frac{x_{1}}{2}\), \(U_{1}x_{1}= x_{1} \cos x_{1}\), \(x_{1}\in C_{1}\), and \(V_{2}: C_{2}\to C_{2}\) and \(U_{2}: C_{2}\to C_{2}\) be defined by \(V_{2}x_{2}=\frac{x_{2}}{3}\), \(U_{2}x_{2}= -x_{2} \cos x_{2}\), \(x_{2}\in C_{2}\). Setting \(\delta _{k}=\frac{1}{2k}\), \(\sigma _{k}=\frac{1}{2k}\), \(\rho _{k}=1\), \(\epsilon _{k}=0\), \(\alpha _{k}=\frac{1}{2}\), \(\beta _{k}=\frac{1}{k}\), \(k\geq 1\). Then the sequences \(\{x_{1}^{k}\}\) and \(\{x_{2}^{k}\}\) generated by Scheme 3.1 converge to \(q_{1}=0\) and \(q_{2}=0\), respectively, so that \((q_{1}, q_{2})=(0,0)\in \Phi \).

Proof

It is easy to prove that the bifunctions \(g_{1}\) and \(g_{2}\) are pseudomonotone on \(C_{1}\) and \(C_{2}\), respectively. Note that \(g_{1}(x_{1},\cdot )\) and \(g_{1}(x_{2},\cdot )\) are convex for \(x_{1}\in C_{1}\) and \(x_{2}\in C_{2}\) and \(\partial g_{1}(x,\cdot )x_{1}=[x_{1}^{2},3x_{1}^{2}]\) and \(\partial g_{2}(x_{2},\cdot )x_{2}=[x_{2}^{2}]\) by taking \(\epsilon _{k}=0\) for all \(k\in \mathbb{N}\). \(A_{1}\) and \(A_{2}\) are bounded linear operators on \(\mathbb{R}\) with adjoint operators \(A_{1}^{*}\) and \(A_{2}^{*}\), \(\|A_{1}\|=\|A_{1}^{*}\|=2\), \(\|A_{2}\|=\|A_{2}^{*}\|=2\), and hence \(\mu _{k} \in (\epsilon , \frac{1}{9}-\epsilon )\). Therefore, for \(\epsilon =\frac{1}{100}\), we choose \(\mu _{k}=0.02+\frac{0.02}{k}\) for all k. The mappings \(V_{1}\) and \(V_{2}\) are nonexpansive with \(\operatorname{Fix}(V_{1})=\{0\}\) and \(\operatorname{Fix}(V_{2})=\{0\}\). Further, \(U_{1}\) and \(U_{2}\) are continuous with \(\mathrm{Fix} (U_{1})=\{0\}\) and \(\mathrm{Fix} (U_{2})=\{0\}\), and \((I-U_{1})\) and \((I-U_{2})\) are monotone. The mappings \(U_{1}\) and \(U_{2}\) are quasinonexpansive but not nonexpansive. After computation, we obtain \(\Gamma =\operatorname{Sol}(\mathrm{SEHFPP}\text{(1.3)--(1.4)})=\{0\}\) and \(\Omega =\{0\}\). Therefore \(\Phi =\Omega \cap \Gamma =\{0\}\neq \emptyset \). After simplification, Scheme 3.1 is reduced to the following:

$$\begin{aligned} \textstyle\begin{cases} w_{1}^{k}\in H_{1}, w_{2}^{k}\in H_{2}\quad \mbox{such that } w_{1}^{k} \in \partial _{\epsilon _{k}}g_{1}(x_{1}^{k},\cdot )(x_{1}^{k})=[(x_{1}^{k})^{2},3(x_{1}^{k})^{2}] \\ \mbox{and}\quad w_{2}^{k} \in \partial _{\epsilon _{k}}g_{2}(x_{2}^{k}, \cdot )(x_{2}^{k})=[(x_{2}^{k})^{2}] ; \\ y_{1}^{k}= \textstyle\begin{cases} 0&\mbox{if } x_{1}< 0, \\ 1&\mbox{if } x_{1}>1, \\ x_{1}^{k}-\alpha _{k}w_{1}^{k}&\mbox{otherwise}; \end{cases}\displaystyle \\ y_{2}^{k}= \textstyle\begin{cases} 0&\mbox{if } x_{2}< 0, \\ 1&\mbox{if } x_{2}>1, \\ x_{2}^{k}-\alpha _{k}w_{2}^{k}&\mbox{otherwise}; \end{cases}\displaystyle \\ t_{1}^{k}=(1-\delta _{k})x_{1}^{k}+ \delta _{k}V_{1}\bigl((1-\sigma _{k})y_{1}^{k} \cos y_{1}^{k}+\sigma _{k}y_{1}^{k} \bigr); \\ t_{2}^{k}=(1-\delta _{k})x_{2}^{k}+ \delta _{k}V_{2}\bigl(-(1-\sigma _{k})y_{2}^{k} \cos y_{2}^{k}+\sigma _{k}y_{2}^{k} \bigr); \\ x_{1}^{k+1}=P_{C_{1}}\bigl(t_{1}^{k}+ \mu _{k}A_{1}^{*}\bigl(A_{1}t_{1}^{k}-A_{2}t_{2}^{k} \bigr)\bigr); \\ x_{2}^{k+1}=P_{C_{2}}\bigl(t_{2}^{k}+ \mu _{k}A_{2}^{*}\bigl(A_{2}t_{1}^{k}-A_{2}t_{2}^{k} \bigr)\bigr). \end{cases}\displaystyle \end{aligned}$$
(6.1)

Finally, using the software Matlab 7.8.0, we have Fig. 1, which shows that \(\{x_{1}^{k}\}\) and \(\{x_{2}^{k}\}\) converge to \(q_{1}=0\) and \(q_{2}=0\), respectively, so that \((q_{1}, q_{2})=(0,0)\in \Phi \).

Figure 1
figure1

Convergence for initial values \(x_{1}^{0} = -3\), \(x_{2}^{0} = 3\)

 □

Conclusion

We have proved a weak convergence theorem for an iterative scheme called the simultaneous projected subgradient-proximal iterative scheme, where the stepsizes do not depend on the operator norms, for solving the split equality equilibrium problem SEEP (1.1)–(1.2) for pseudomonotone bifunctions and the split equality hierarchical fixed point problem SEHFPP (1.3)–(1.4) for nonexpansive and quasinonexpansive mappings. Further, we have discussed some consequences of Theorem 4.1. Finally, we presented a numerical example to justify Theorem 4.1. Further research is needed to extend the presented work to the setting of Banach spaces.

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Acknowledgements

The author is thankful to Prof. Kaleem Raza Kazmi for his several useful suggestions toward the improvement of this paper. The author is also very thankful to the referees for their critical and helpful comments.

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Department of Mathematics, King Abdulaziz University, Jeddah, Kingdom of Saudi Arabia

Funding

This work was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant No. (J:7-247-1441). Therefore the author acknowledges DSR for technical and financial support.

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Alansari, M. An iterative scheme for split equality equilibrium problems and split equality hierarchical fixed point problem. Adv Differ Equ 2021, 226 (2021). https://doi.org/10.1186/s13662-021-03384-y

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MSC

  • 47H09
  • 47J20
  • 49J30
  • 90C25

Keywords

  • Split equality equilibrium problem
  • Split equality hierarchical fixed point problem
  • Simultaneous hybrid projected subgradient-proximal iterative scheme
  • Quasinonexpansive mapping
  • Pseudomonotone bifunction
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