1 Introduction

The class of convex feasibility problems (CFP) has been widely studied in the current literature as it encompasses a variety of problems arising in mathematical and physical sciences. Numerous iterative algorithms have been studied to obtain an approximate solution for the CFP in Hilbert spaces. However, the class of projection algorithms is prominent among various iterative algorithms to solve the CFP. It is remarked that the class of CFP is closely related to the theory of convex optimization and hence monotone operator theory. As a consequence, CFP found valuable applications in the field of partial differential equations, image recovery problem, approximation theory, signal and image processing through projection algorithms, control problems, evolution equations and inclusions, see for instance [6, 15, 16] and the references cited therein.

The class of CFP has been generalized in several ways. One of the elegant modifications and generalizations of the CFP is the split convex feasibility problems (SCFP) proposed by Censor and Elfving [12]. The mathematical formulation of the SCFP opens up an interesting framework to model the medical image reconstruction problem and the intensity-modulated radiation therapy [8, 11]. As a consequence, the SCFP has been studied extensively in the current literature with possible real-world applications, see for example [9, 11, 13, 14, 19] and the references cited therein. Recall that a SCFP deals with a model aiming to find a point

$$\begin{aligned} \bar{x}\in C\subseteq \mathcal{H}_{1} \end{aligned}$$
(1)

such that

$$\begin{aligned} \bar{y}=\hbar \bar{x}\in Q\subseteq \mathcal{H}_{2}, \end{aligned}$$
(2)

where \(\hbar:\mathcal{H}_{1}\rightarrow \mathcal{H}_{2}\) is a bounded linear operator between two real Hilbert spaces \(\mathcal{H}_{1}\) and \(\mathcal{H}_{2}\).

Since the introduction of SCFP, various important instances of SCFP have been introduced and analyzed such as the split variational inequality problem [13], the split common null point problem (SCNPP) [9], the split common FPP [14], and the split equilibrium problem [7]. We are interested in studying the SCNPP, one of the important instances of SCFP, defined as follows:

Given two multivalued operators \(A_{1}:\mathcal{H}_{1}\rightarrow 2^{\mathcal{H}_{1}}\) and \(A_{2}:\mathcal{H}_{2}\rightarrow 2^{\mathcal{H}_{2}}\), the SCNPP problem deals with a model aiming to find a point

$$\begin{aligned} \bar{x}\in \mathcal{H}_{1} \quad\text{such that } 0\in A_{1}(\bar{x}) \text{ and } 0\in A_{2}(\hbar \bar{x}). \end{aligned}$$
(3)

In 2012, Byrne et al. [9] suggested the following iterative schemes to solve the SCNPP (3) associated with two maximal monotone operators \(A_{1}\) and \(A_{2}\):

$$\begin{aligned} x_{k+1}=J_{m}^{A_{1}} \bigl(x_{k}+\delta \hbar \bigl(J_{m}^{A_{2}}-\mathrm{Id} \bigr)\hbar x_{k} \bigr),\quad k \in \mathbb{N}, \end{aligned}$$
(4)

and

$$\begin{aligned} \textstyle\begin{cases} x_{0},\quad v \in \mathcal{H}_{1}; \\ x_{k+1}=\beta _{k}v+(1-\beta _{k})J^{A_{1}}_{m}(x_{k}+\delta \hbar ^{*}(J^{A_{2}}_{m}-\mathrm{Id}) \hbar x_{k}), \quad k \in \mathbb{N}, \end{cases}\displaystyle \end{aligned}$$
(5)

where \(\hbar ^{*}\) denotes the adjoint operator of ħ, Id denotes the identity operator and \(J_{m}^{A_{1}},J_{m}^{A_{2}}\) denote the corresponding resolvents of \(A_{1},A_{2}\), respectively. The set of solutions of the SCNPP (3) is denoted by \(\Omega:=\{\bar{x} \in A^{-1}_{1}(0):\hbar \bar{x} \in A^{-1}_{2}(0) \}\). It is remarked that the scheme (4) exhibits weak convergence, while the scheme (5) exhibits strong convergence under suitable sets of constraints.

In 1994, Blum and Oettli [7] proposed, in a mathematical formulation, an EP with respect to a (monotone) bifunction g defined on a nonempty subset C of a real Hilbert space \(\mathcal{H}_{1}\) that aims to find a point \(\bar{x}\in C\) such that

$$\begin{aligned} g ( \bar{x},\bar{y} ) \geq 0 \quad\text{for all } \bar{y}\in C. \end{aligned}$$
(6)

The set of equilibrium points or solutions of the problem (6) is denoted by \(EP(g)\).

In 2006, Tada and Takahashi [29] suggested a hybrid algorithm for the analysis of monotone EP and FPP in Hilbert spaces. Nevertheless, the iterative algorithm proposed in [29] fails for the case of pseudomonotone EP. In order to address this issue, Anh [2] suggested a hybrid extragradient method, based on the seminal work of Korpelevich [23], in Hilbert spaces. Inspired by the work of Anh [2], Hieu et al. [20] suggested a parallel hybrid extragradient framework to address pseudomonotone EP together with the FPP. In 2015, Takahashi et al. [30] constructed a common solution of the zero point problem and FPP. Therefore, it is natural to study the pseudomonotone EP and the SCNPP with the FPP associated with a more general class of demicontractive operators.

It is remarked that the computational performance of an iterative algorithm can be enhanced by employing different techniques. The parallel architecture of an iterative algorithm reduces the computational cost, whereas the inertial extrapolation technique [26] provides fast convergence characteristics of the algorithm. The latter technique has successfully been combined with different classical iterative algorithms, see for example [1, 3–5, 10, 18, 21, 22, 25, 32] and the references cited therein. We, therefore, study the convergence analysis of a variant of parallel hybrid extragradient iterative algorithm embedded with the inertial extrapolation technique in Hilbert spaces.

The rest of the paper is organized as follows: Sect. 2 contains some relevant preliminary concepts and results for (split) monotone operator theory, EP theory, and fixed point theory. Section 3 comprises strong convergence results, whereas Sect. 4 provides numerical results concerning the viability of the proposed algorithm with respect to various real world applications.

2 Preliminaries

We first define some necessary notions from fixed point theory. Let \(T:C\rightarrow C\) be an operator defined on a nonempty subset C of a real Hilbert space \(\mathcal{H}_{1}\), then T is known as nonexpansive if \(\Vert Tx-Ty\Vert \leq \Vert x-y\Vert \) for all \(x,y\in C\). Further, T is known as firmly nonexpansive if

$$\begin{aligned} \Vert Tx-Ty \Vert ^{2}\leq \Vert x-y \Vert ^{2}- \bigl\Vert (\mathrm{Id}-T)x-(\mathrm{Id}-T)y \bigr\Vert ^{2}\quad \text{for all }x,y\in C. \end{aligned}$$

Moreover, the operator T is defined as -demicontractive if \(\mathrm{Fix}(T)\neq \emptyset \), and there exists such that

where \(\mathrm{Fix}(T)= \{ x\in C:x-Tx=0 \} \) denotes the set of all fixed points of the operator T. Note that the operator \(\mathrm{Id}-T\) is said to be demiclosed at the origin if for any sequence \((x_{k})\) in a nonempty closed and convex subset C of \(\mathcal{H}_{1}\) converges weakly to some x and if \(((\mathrm{Id}-T)x_{k})\) converges strongly to 0, then \((\mathrm{Id}-T)(x)=0\). It is remarked that, for each \(x\in \mathcal{H}_{1}\), there exists unique \(P_{C}x\in C\) satisfying \(\Vert x-P_{C}x\Vert \leq \Vert x-z\Vert \text{ for all } z\in C\). Such an operator \(P_{C}:\mathcal{H}_{1}\rightarrow C\) is coined as metric projection and satisfies \(\langle x-P_{C}x,P_{C}x-y\rangle \geq 0,\text{ for all } x\in \mathcal{H}_{1}\text{ and }y\in C\).

We now state a brief introductory material covering monotone operator theory from the celebrated monograph of Bauschke and Combettes [6].

For a set-valued operator \(A_{1}:\mathcal{H}_{1}\rightarrow 2^{\mathcal{H}_{1}}\), the following sets \(\operatorname{dom}(A_{1})=\{x\in \mathcal{H}_{1} | A_{1}x\neq \emptyset \}\), \(\operatorname{ran}(A_{1})=\{u\in \mathcal{H}_{1} | (\exists x\in \mathcal{H}_{1}) u\in A_{1}x\}\), \(\operatorname{gra}(A_{1})=\{(x,u)\in \mathcal{H}_{1}\times \mathcal{H}_{1} | u\in A_{1}x\}\), and \(\operatorname{zer}(A_{1})=\{x\in \mathcal{H}_{1} | 0\in A_{1}x\}\) define and denote the domain, range, graph, and zeros of \(A_{1}\), respectively. The inverse operator \(A_{1}^{-1}\) of \(A_{1}\) can be defined as

$$\begin{aligned} x\in A_{1}^{-1}(y)\quad\text{if and only if}\quad y\in A_{1}(x). \end{aligned}$$

The set-valued operator \(A_{1}\) is said to be monotone if \(\langle x-y,u-v\rangle \geq 0\) for all \((x,u),(y,v)\in \operatorname{gra}(A_{1})\). A monotone operator \(A_{1}\) is coined as maximal monotone operator if there is no proper monotone extension of \(A_{1}\), equivalently if \(\operatorname{ran}(\mathrm{Id}+mA_{1})=\mathcal{H}_{1}\) for all \(m>0\). An important notion associated with the monotone operator \(A_{1}\) is the well-defined resolvent operator \(J_{m}^{A_{1}}=(\mathrm{Id}+mA_{1})^{-1}\). Such an operator is single-valued and satisfies nonexpansiveness as well as \(\mathrm{Fix}(J_{m}^{A_{1}})=A_{1}^{-1}(0)\) for all \(m>0\).

The rest of this section is organized with celebrated results required in the sequel.

Assumption 2.1

Let \(g:C\times C\rightarrow \mathbb{R}\cup \{+\infty \}\) be a bifunction satisfying the following assumptions:

(A1): g is pseudomonotone, i.e., \(g(x,y)\leq 0\Rightarrow g(x,y)\geq 0\text{ for all } x,y\in C\);

(A2): g is Lipschitz-type continuous, i.e., there exist two nonnegative constants \(d_{1},d_{2}\) such that

$$\begin{aligned} g(x,y)+g(y,z)\geq g(x,z)-d_{1} \Vert x-y \Vert ^{2}-d_{2} \Vert y-z \Vert ^{2} \quad\text{for all } x,y,z\in C; \end{aligned}$$

(A3): g is weakly continuous on \(C\times C\) implies that, if \(x,y \in C\) and \((x_{k}), (y_{k})\) are two sequences in C converging weakly to x and y, respectively, then \(g(x_{k},y_{k})\) converges to \(g(x,y)\);

(A4): For each fixed \(x\in C\), \(g(x,.)\) is convex and subdifferentiable on C.

It is remarked that the monotonicity of g, i.e., \(g(x,y)+g(y,x)\leq 0,\text{ for all }x,y\in C\) implies the pseudomonotonicity but the converse is not true in general. The following lemmas are helpful in proving the strong convergence results in the next section.

Lemma 2.2

Let \(x,y\in \mathcal{H}_{1}\) and \(\beta \in \mathbb{R}\), then

  1. 1.

    \(\Vert x+y\Vert ^{2} \leq \Vert x \Vert ^{2}+2 \langle y, x+y \rangle \);

  2. 2.

    \(\Vert x-y\Vert ^{2} \leq \Vert x \Vert ^{2}-\Vert y \Vert ^{2}-2 \langle x-y, y \rangle \);

  3. 3.

    \(\|\beta x+(1-\beta )y\|^{2}=\beta \|x\|^{2}+(1-\beta )\|y\|^{2}- \beta (1-\beta )\|x-y\|^{2}\).

Lemma 2.3

([24])

Let C be a nonempty closed and convex subset of a real Hilbert space \(\mathcal{H}_{1}\). For every \(x,y,z\in \mathcal{H}_{1}\) and \(\gamma \in \mathbb{R}\), the set

$$\begin{aligned} D= \bigl\{ v\in C: \Vert y-v \Vert ^{2}\leq \Vert x-v \Vert ^{2}+\langle z,v \rangle +\gamma \bigr\} \end{aligned}$$

is closed and convex.

Lemma 2.4

([31])

Let C be a nonempty closed and convex subset of a real Hilbert space \(\mathcal{H}_{1}\), and let \(h:C \rightarrow \mathbb{R}\) be a convex and subdifferentiable function on C. Then x̄ is the solution of convex problem \(\min \{h(x):x \in C\}\) if and only if \(0 \in \partial h(\bar{x})+N_{C}(\bar{x})\), where \(\partial h(\cdot )\) denotes the subdifferential of h and \(N_{C}(\bar{x})\) is the normal cone of C at x̄.

3 Algorithm and convergence analysis

We enlist standard necessary hypotheses for the main result of this section. Note that, for a finite family of pseudomonotone bifunctions \(g_{i}\), we can compute the same Lipschitz coefficients \((d_{1},d_{2})\) by employing Assumption 2.1(A2) as follows:

$$\begin{aligned} g_{i}(x,z)-g_{i}(x,y)-g_{i}(y,z)\leq d_{1,i} \Vert x-y \Vert ^{2}+d_{2,i} \Vert y-z \Vert ^{2} \leq d_{1} \Vert x-y \Vert ^{2}+d_{2} \Vert y-z \Vert ^{2}, \end{aligned}$$

where \(d_{1}=\max_{1\leq i \leq M}\{d_{1,i}\}\) and \(d_{2}=\max_{1\leq i \leq M}\{d_{2,i}\}\). Therefore, \(g_{i}(x,y)+g_{i}(y,z)\geq g_{i}(x,z)-d_{1}\Vert x-y \Vert ^{2}-d_{2} \Vert y-z\Vert ^{2}\).

Let \(\mathcal{H}_{1}\), \(\mathcal{H}_{2}\) be two real Hilbert spaces, and let \(C \subseteq \mathcal{H}_{1}\) be a nonempty, closed, and convex subset. Then

  1. (H1)

    Let \(A_{1}:\mathcal{H}_{1} \rightarrow 2^{\mathcal{H}_{1}}\), \(A_{2}:\mathcal{H}_{2} \rightarrow 2^{\mathcal{H}_{2}}\) be two maximal monotone operators, and for \(m,n>0\), let \(J^{A_{1}}_{m}\), \(J^{A_{2}}_{n}\) be the resolvents of \(A_{1}\) and \(A_{2}\), respectively;

  2. (H2)

    Let \(\hbar: \mathcal{H}_{1} \rightarrow \mathcal{H}_{2}\) be a bounded linear operator such that \(\hbar ^{\ast }\) is the adjoint operator of ħ;

  3. (H3)

    Let \(g_{i}:C\times C\rightarrow \mathbb{R}\cup \{+\infty \}\) be a finite family of bifunctions satisfying Assumption 2.1;

  4. (H4)

    Let \(S_{j}:\mathcal{H}_{1} \rightarrow \mathcal{H}_{1}\) be a finite family of -demicontractive operators;

  5. (H5)

    Assume that \(\Gamma:= \Omega \cap (\bigcap^{M}_{i=1}EP(g_{i}) ) \cap ( \bigcap^{N}_{j=1}\mathrm{Fix}(S_{j}) ) \neq \emptyset \).

Theorem 3.1

If \(\Gamma \neq \emptyset \), then the sequence \((x_{k})\) generated by Algorithm 1 converges strongly to an element in Γ, provided the following conditions hold:

  1. (C1)

    \(\sum^{\infty }_{k=1}\Theta _{k}\|x_{k}-x_{k-1}\|<\infty \);

  2. (C2)

    \(0 < a^{\ast } < \liminf_{k \rightarrow \infty } \alpha _{k} \leq \limsup_{k \rightarrow \infty } \alpha _{k} \leq b^{\ast } < 1\) and ;

  3. (C3)

    \(\liminf_{k \rightarrow \infty } \beta _{k} > 0\);

  4. (C4)

    \(\liminf_{k \rightarrow \infty } m_{k} > 0\), \(\liminf_{k \rightarrow \infty } n_{k} > 0\).

Algorithm 1
figure a

Accelerated projection based parallel hybrid extragradient algorithm (Alg. 1)

Full size image

Remark 3.2

We remark here that the condition (C1) is easily implementable in a numerical computation since the values of \(\|x_{k}-x_{k-1}\|\) are known before choosing \(\Theta _{k}\). The parameter \(\Theta _{k}\) can be taken as \(0 \leq \Theta _{k} \leq \widehat{\Theta _{k}}\),

$$\begin{aligned} \widehat{\Theta _{k}} = \textstyle\begin{cases} \min \{\frac{\nu _{k}}{ \Vert x_{k}-x_{k-1} \Vert }, \Theta \} &\text{if } x_{k} \neq x_{k-1}; \\ \Theta & \text{otherwise}, \end{cases}\displaystyle \end{aligned}$$

where \(\{ \nu _{k}\}\) is a positive sequence such that \(\sum^{\infty }_{k = 1}\nu _{k} < \infty \) and \(\Theta \in [0,1)\).

We use the following result for the analysis of Algorithm 1.

Lemma 3.3

([27])

Suppose that \(\bar{x} \in EP(g_{i})\), and \(x_{k}\), \(b_{k}\), \(u^{i}_{k}\), \(w^{i}_{k}\), \(i \in \{1,2,\ldots,M\}\) are defined in Step 1 of Algorithm 1. Then we have

$$\begin{aligned} \bigl\Vert v^{i}_{k}-\bar{x} \bigr\Vert ^{2} \leq \Vert b_{k}-\bar{x} \Vert ^{2}-(1-2 \gamma d_{1}) \bigl\Vert u^{i}_{k}-b_{k} \bigr\Vert ^{2}-(1-2\gamma d_{2}) \bigl\Vert u^{i}_{k}-v^{i}_{k} \bigr\Vert ^{2}. \end{aligned}$$

Proof of Theorem 3.1

The proof is divided into the following steps.

Step 1. We show that the sequence \(( x_{k})\) defined in Algorithm 1 is well defined.

We know that Γ is closed and convex. Moreover, from Lemma 2.3 we have that \(C_{k+1}\) is closed and convex for each \(k\geq 1\). Hence the projection \(P_{C_{k+1}}x_{1}\) is well defined. For any \(\bar{x}\in \Gamma \), observe that

$$\begin{aligned} \Vert b_{k}-\bar{x} \Vert ^{2}&= \bigl\Vert (x_{k}-\bar{x})+\Theta _{k}(x_{k}-x_{k-1}) \bigr\Vert ^{2} \\ &\leq \Vert x_{k}-\bar{x} \Vert ^{2}+\Theta ^{2}_{k} \Vert x_{k}-x_{k-1} \Vert ^{2}+2\Theta _{k}\langle x_{k}- \bar{x},x_{k}-x_{k-1}\rangle. \end{aligned}$$
(7)

Further

(8)

Furthermore,

$$\begin{aligned} \Vert y_{k}-\bar{x} \Vert ^{2}={}& \bigl\Vert (1- \beta _{k}) (\bar{w}_{k}- \bar{x})+\beta _{k} \bigl(J^{A_{1}}_{m_{k}} \bigl(\bar{w}_{k}+\delta \hbar ^{\ast } \bigl(J^{A_{2}}_{n_{k}}-\mathrm{Id} \bigr) \hbar \bar{w}_{k} \bigr)-\bar{x} \bigr) \bigr\Vert ^{2} \\ \leq {}& (1-\beta _{k}) \Vert \bar{w}_{k}-\bar{x} \Vert ^{2} \\ &{}+ \beta _{k} \bigl\Vert J^{A_{1}}_{m_{k}} \bigl( \bar{w}_{k}+\delta \hbar ^{\ast } \bigl(J^{A_{2}}_{n_{k}}- \mathrm{Id} \bigr) \hbar \bar{w}_{k} \bigr)-\bar{x} \bigr\Vert ^{2}. \end{aligned}$$
(9)

Since \(J^{A_{1}}_{m_{k}}\) is nonexpansive, therefore the expression \(\Vert J^{A_{1}}_{m_{k}}(\bar{w}_{k}+\delta \hbar ^{\ast }(J^{A_{2}}_{n_{k}}-\mathrm{Id}) \hbar \bar{w}_{k})-\bar{x}\Vert ^{2}\) simplifies as follows:

$$\begin{aligned} & \bigl\Vert J^{A_{1}}_{m_{k}} \bigl(\bar{w}_{k}+ \delta \hbar ^{\ast } \bigl(J^{A_{2}}_{n_{k}}-\mathrm{Id} \bigr) \hbar \bar{w}_{k} \bigr)-J^{A_{1}}_{m_{k}}\bar{x} \bigr\Vert ^{2} \\ & \quad\leq \bigl\Vert \bar{w}_{k}+\delta \hbar ^{\ast } \bigl(J^{A_{2}}_{n_{k}}-\mathrm{Id} \bigr) \hbar \bar{w}_{k}-\bar{x} \bigr\Vert ^{2} \\ & \quad\leq \Vert \bar{w}_{k}-\bar{x} \Vert ^{2}+ \delta ^{2} \bigl\Vert \hbar ^{\ast } \bigl(J^{A_{2}}_{n_{k}}- \mathrm{Id} \bigr)\hbar \bar{w}_{k} \bigr\Vert ^{2}+2 \delta \bigl\langle \bar{w}_{k}-\bar{x},\hbar ^{\ast } \bigl(J^{A_{2}}_{n_{k}}-\mathrm{Id} \bigr) \hbar \bar{w}_{k} \bigr\rangle \\ & \quad\leq \Vert \bar{w}_{k}-\bar{x} \Vert ^{2}+\delta ^{2} \Vert \hbar \Vert ^{2} \bigl\Vert \bigl(J^{A_{2}}_{n_{k}}-\mathrm{Id} \bigr)\hbar \bar{w}_{k} \bigr\Vert ^{2}+2 \delta \bigl\langle \hbar \bar{w}_{k}- \hbar \bar{x}, \bigl(J^{A_{2}}_{n_{k}}-\mathrm{Id} \bigr) \hbar \bar{w}_{k}) \bigr\rangle . \end{aligned}$$
(10)

Using \(J^{A_{2}}_{n_{k}}\) as firmly nonexpansive, we simplify the expression \(\lambda _{k}=2\delta \langle \hbar \bar{w}_{k}-\hbar \bar{x},(J^{A_{2}}_{n_{k}}-\mathrm{Id}) \hbar \bar{w}_{k} \rangle \) as follows:

$$\begin{aligned} \lambda _{k}&=2\delta \bigl\langle \hbar \bar{w}_{k}- \hbar \bar{x}+ \bigl(J^{A_{2}}_{n_{k}}( \hbar \bar{w}_{k})- \hbar \bar{w}_{k} \bigr)- \bigl(J^{A_{2}}_{n_{k}}(\hbar \bar{w}_{k})-\hbar \bar{w}_{k} \bigr), J^{A_{2}}_{n_{k}}( \hbar \bar{w}_{k})- \hbar \bar{w}_{k} \bigr\rangle \\ &= 2\delta \bigl( \bigl\langle J^{A_{2}}_{n_{k}}(\hbar \bar{w}_{k})-\hbar \bar{x},J^{A_{2}}_{n_{k}}(\hbar \bar{w}_{k})-\hbar \bar{w}_{k} \bigr\rangle - \bigl\Vert \bigl(J^{A_{2}}_{n_{k}}-\mathrm{Id} \bigr)\hbar \bar{w}_{k} \bigr\Vert ^{2} \bigr) \\ &\leq -2\delta \Vert \bigl(J^{A_{2}}_{n_{k}}-\mathrm{Id} \bigr) \hbar \bar{w}_{k})\Vert ^{2}. \end{aligned}$$
(11)

Utilizing (10), (11), and Lemma 3.3, we then obtain from (9) that

$$\begin{aligned} \Vert y_{k}-\bar{x} \Vert ^{2}\leq {}& (1-\beta _{k}) \Vert \bar{w}_{k}- \bar{x} \Vert ^{2}+ \beta _{k} \bigl( \Vert \bar{w}_{k}-\bar{x} \Vert ^{2}+ \delta ^{2} \Vert \hbar \Vert ^{2}\bigl\Vert \bigl(J^{A_{2}}_{n_{k}}-\mathrm{Id} \bigr)\hbar \bar{w}_{k} \bigr) \bigr\Vert ^{2} \\ &{}-2\delta \bigl\Vert \bigl(J^{A_{2}}_{n_{k}}-\mathrm{Id} \bigr) \hbar \bar{w}_{k} \bigr\Vert ^{2}), \\ \leq {}&(1-\beta _{k}) \Vert \bar{w}_{k}-\bar{x} \Vert ^{2}+ \beta _{k} \bigl( \Vert \bar{w}_{k}- \bar{x} \Vert ^{2}-\delta \bigl(2-\delta \Vert \hbar \Vert ^{2} \bigr) \bigl\Vert \bigl(J^{A_{2}}_{n_{k}}- \mathrm{Id} \bigr) \hbar \bar{w}_{k} \bigr\Vert ^{2} \bigr) \\ \leq {}& \Vert \bar{w}_{k}-\bar{x} \Vert ^{2} \\ \leq {}& \Vert x_{k}-\bar{x} \Vert ^{2}+\Theta ^{2}_{k} \Vert x_{k}-x_{k-1} \Vert ^{2}+2 \Theta _{k}\langle x_{k}-\bar{x}, x_{k}-x_{k-1} \rangle. \end{aligned}$$
(12)

It follows from (12) that

$$\begin{aligned} \Vert y_{k}-\bar{x} \Vert \leq \Vert x_{k}-\bar{x} \Vert ^{2}+\Theta ^{2}_{k} \Vert x_{k}-x_{k-1} \Vert ^{2}+2\Theta _{k} \Vert x_{k}-\bar{x} \Vert \Vert x_{k}-x_{k-1} \Vert . \end{aligned}$$
(13)

The above estimate (13) infers that \(\Gamma \subset C_{k+1}\). Hence, we conclude that Algorithm 1 is well defined.

Step 2. We show that the limit \(\lim_{k\rightarrow \infty }\Vert x_{k}-x_{1}\Vert \) exists.

Note that, for \(x_{k+1}=P_{C_{k+1}}x_{1}\), we have \(\Vert x_{k+1}-x_{1}\Vert \leq \Vert x^{\ast }-x_{1}\Vert \) for all \(x^{\ast }\in C_{k+1}\). In particular \(\Vert x_{k+1}-x_{1}\Vert \leq \Vert \bar{x}-x_{1}\Vert \) for all \(\bar{x}\in \Gamma \subset C_{k+1}\) This proves that the sequence \((\Vert x_{k}-x_{1}\Vert )\) is bounded. On the other hand, from \(x_{k}=P_{C_{k}}x_{1}\) and \(x_{k+1}=P_{C_{k+1}}x_{1}\in C_{k+1}\), we have

$$\begin{aligned} \Vert x_{k}-x_{1} \Vert \leq \Vert x_{k+1}-x_{1} \Vert . \end{aligned}$$

This implies that \((\Vert x_{k}-x_{1}\Vert )\) is nondecreasing, and hence

$$\begin{aligned} \lim_{k\rightarrow \infty } \Vert x_{k}-x_{1} \Vert \quad \text{exists.} \end{aligned}$$
(14)

Step 3. We show that \(\bar{x_{*}}\in \Gamma \).

First, observe that

$$\begin{aligned} \Vert x_{k+1}-x_{k} \Vert ^{2} &= \Vert x_{k+1}-x_{1}+x_{1}-x_{k} \Vert ^{2} \\ &= \Vert x_{k+1}-x_{1} \Vert ^{2}+ \Vert x_{k}-x_{1} \Vert ^{2}-2 \langle x_{k}-x_{1},x_{k+1}-x_{1} \rangle \\ &= \Vert x_{k+1}-x_{1} \Vert ^{2}+ \Vert x_{k}-x_{1} \Vert ^{2}-2 \langle x_{k}-x_{1},x_{k+1}-x_{k}+x_{k}-x_{1} \rangle \\ &= \Vert x_{k+1}-x_{1} \Vert ^{2}- \Vert x_{k}-x_{1} \Vert ^{2}-2 \langle x_{k}-x_{1},x_{k+1}-x_{k} \rangle \\ &\leq \Vert x_{k+1}-x_{1} \Vert ^{2}- \Vert x_{k}-x_{1} \Vert ^{2}. \end{aligned}$$

Taking limsup on both sides of the above estimate and utilizing (14), we have

\(\limsup_{k\rightarrow \infty } \Vert x_{k+1}-x_{k} \Vert ^{2}=0\). That is,

$$\begin{aligned} \lim_{k\rightarrow \infty } \Vert x_{k+1}-x_{k} \Vert =0. \end{aligned}$$
(15)

By the definition of \((b_{k})\) and (C1), we have

$$\begin{aligned} \lim_{k\rightarrow \infty } \Vert b_{k}-x_{k} \Vert = \lim_{k \rightarrow \infty }\Theta _{k} \Vert x_{k}-x_{k-1} \Vert =0. \end{aligned}$$
(16)

Consider the following triangular inequality:

$$\begin{aligned} \Vert b_{k}-x_{k+1} \Vert \leq \Vert b_{k}-x_{k} \Vert + \Vert x_{k}-x_{k+1} \Vert . \end{aligned}$$

From (15) and (16), we have

$$\begin{aligned} \lim_{k \rightarrow \infty } \Vert b_{k}-x_{k+1} \Vert =0. \end{aligned}$$
(17)

Since \(x_{k+1} \in C_{k+1}\), therefore, we have

$$\begin{aligned} \Vert y_{k}-x_{k+1} \Vert \leq \Vert x_{k}-x_{k+1} \Vert +2\Theta _{k} \Vert x_{k}-x_{k-1} \Vert +2\langle x_{k}-x_{k+1},x_{k}-x_{k-1} \rangle. \end{aligned}$$

Utilizing (15) and (C1), the above estimate implies that

$$\begin{aligned} \lim_{k\rightarrow \infty } \Vert y_{k}-x_{k+1} \Vert =0. \end{aligned}$$
(18)

From (15), (18), and the following triangular inequality

$$\begin{aligned} \Vert y_{k}-x_{k} \Vert \leq \Vert y_{k}-x_{k+1} \Vert + \Vert x_{k+1}-x_{k} \Vert , \end{aligned}$$

we get

$$\begin{aligned} \lim_{k \rightarrow \infty } \Vert y_{k}-x_{k} \Vert =0. \end{aligned}$$
(19)

Consider the following re-arranged variant of the estimate (12) by applying Lemma 3.3:

$$\begin{aligned} &(1-2\gamma d_{1}) \bigl\Vert u^{i}_{k}-b_{k} \bigr\Vert ^{2}-(1-2\gamma d_{2}) \bigl\Vert u^{i}_{k}-v^{i}_{k} \bigr\Vert ^{2} \\ &\quad \leq \bigl( \Vert x_{k}-\bar{x} \Vert + \Vert y_{k}-\bar{x} \Vert \bigr) \Vert x_{k}-y_{k} \Vert +\Theta ^{2}_{k} \Vert x_{k}-x_{k-1} \Vert ^{2}+2\Theta _{k} \Vert x_{k}-\bar{x} \Vert \Vert x_{k}-x_{k-1} \Vert . \end{aligned}$$

Letting \(k \rightarrow \infty \), using (C1) and (19), we have

$$\begin{aligned} (1-2\gamma d_{1})\lim_{k \rightarrow \infty } \bigl\Vert u^{i}_{k}-b_{k} \bigr\Vert ^{2}-(1-2 \gamma d_{2})\lim_{k \rightarrow \infty } \bigl\Vert u^{i}_{k}-v^{i}_{k} \bigr\Vert ^{2} = 0. \end{aligned}$$
(20)

This implies that

$$\begin{aligned} \lim_{k \rightarrow \infty } \bigl\Vert u^{i}_{k}-b_{k} \bigr\Vert ^{2} = \lim_{k \rightarrow \infty } \bigl\Vert u^{i}_{k}-v^{i}_{k} \bigr\Vert ^{2} =0. \end{aligned}$$
(21)

Again, consider the following re-arranged variant of the estimate (13):

$$\begin{aligned} a^{\ast } \bigl(1-b^{\ast } \bigr) \bigl\Vert ( \mathrm{Id}-S_{j}) \bar{v}_{k} \bigr\Vert ^{2} \leq{} & \bigl( \Vert x_{k}-\bar{x} \Vert + \Vert y_{k}-\bar{x} \Vert \bigr) \Vert x_{k}-y_{k} \Vert +\Theta ^{2}_{k} \Vert x_{k}-x_{k-1} \Vert ^{2} \\ &{}+2\Theta _{k} \Vert x_{k}-\bar{x} \Vert \Vert x_{k}-x_{k-1} \Vert . \end{aligned}$$

Letting \(k \rightarrow \infty \) and utilizing (C1), (C2), and (19), we have

$$\begin{aligned} \lim_{k \rightarrow \infty } \bigl\Vert (\mathrm{Id}-S_{j}) \bar{v}_{k} \bigr\Vert = 0. \end{aligned}$$
(22)

This implies that

$$\begin{aligned} \lim_{k \rightarrow \infty } \Vert \bar{w_{k}}- \bar{v}_{k} \Vert = \lim_{k \rightarrow \infty }a^{\ast } \bigl\Vert (\mathrm{Id}-S_{j})\bar{v}_{k} \bigr\Vert =0. \end{aligned}$$
(23)

Utilizing (16), (21), (23), and the following triangle inequalities, we have

  1. (i)

    \(\Vert \bar{v_{k}}-b_{k}\Vert \leq \Vert \bar{v_{k}}-\bar{u_{k}} \Vert +\Vert \bar{u_{k}}-b_{k}\Vert \rightarrow 0\);

  2. (ii)

    \(\Vert \bar{v_{k}}-x_{k}\Vert \leq \Vert \bar{v_{k}}-b_{k}\Vert + \Vert b_{k}-x_{k}\Vert \rightarrow 0\);

  3. (iii)

    \(\Vert \bar{w_{k}}-b_{k}\Vert \leq \Vert \bar{w_{k}}-\bar{v_{k}} \Vert +\Vert \bar{v_{k}}-b_{k}\Vert \rightarrow 0\);

  4. (iv)

    \(\Vert \bar{w_{k}}-x_{k}\Vert \leq \Vert \bar{w_{k}}-b_{k}\Vert + \Vert b_{k}-x_{k}\Vert \rightarrow 0\).

From (10), (11), and Lemma 2.2, we have

$$\begin{aligned} \Vert y_{k}-\bar{x} \Vert ^{2} ={}&\Vert (1-\beta _{k}) \bar{w}_{k}+ \beta _{k} \bigl(J^{A_{1}}_{m_{k}} \bigl(\bar{w}_{k}+\delta \hbar ^{\ast } \bigl(J^{A_{2}}_{n_{k}}-\mathrm{Id} \bigr) \hbar \bar{w}_{k} \bigr) \bigr)-\bar{x})\Vert ^{2} \\ \leq {}&(1-\beta _{k}) \Vert \bar{w}_{k}-\bar{x} \Vert ^{2}+ \beta _{k} \bigl( \Vert \bar{w}_{k}- \bar{x} \Vert ^{2}-\delta \bigl(2-\delta \Vert \hbar \Vert ^{2} \bigr) \bigl\Vert \bigl(J^{A_{2}}_{n_{k}}- \mathrm{Id} \bigr) \hbar \bar{w}_{k} \bigr\Vert ^{2} \bigr) \\ \leq {}& \Vert \bar{w}_{k}-\bar{x} \Vert ^{2}-\beta _{k}\delta \bigl(2- \delta \Vert \hbar \Vert ^{2} \bigr) \bigl\Vert \bigl(J^{A_{2}}_{n_{k}}-\mathrm{Id} \bigr)\hbar \bar{w}_{k} \bigr\Vert ^{2} \\ \leq{} & \Vert x_{k}-\bar{x} \Vert ^{2}+2\Theta _{k}\langle x_{k}-x_{k-1}, b_{k}- \bar{x}\rangle \\ &{}-\beta _{k}\delta \bigl(2-\delta \Vert \hbar \Vert ^{2} \bigr) \bigl\Vert \bigl(J^{A_{2}}_{n_{k}}- \mathrm{Id} \bigr) \hbar \bar{w}_{k} \bigr\Vert ^{2}. \end{aligned}$$
(24)

Rearranging the above estimate, we have

$$\begin{aligned} &\beta _{k}\delta \bigl(2-\delta \Vert \hbar \Vert ^{2} \bigr) \bigl\Vert \bigl(J^{A_{2}}_{n_{k}}-\mathrm{Id} \bigr) \hbar \bar{w}_{k} \bigr\Vert ^{2} \\ & \quad\leq \bigl( \Vert x_{k}-\bar{x} \Vert + \Vert y_{k}-\bar{x} \Vert \bigr) \Vert x_{k}-y_{k} \Vert +2\Theta _{k}\langle x_{k}-x_{k-1},b_{k}- \bar{x}\rangle. \end{aligned}$$
(25)

By using (C1), (C3), (19), and \(\delta \in (0,\frac{2}{\Vert \hbar \Vert ^{2}})\), estimate (25) implies that

$$\begin{aligned} \lim_{k\rightarrow \infty } \bigl\Vert \bigl(J^{A_{2}}_{n_{k}}- \mathrm{Id} \bigr)\hbar \bar{w}_{k} \bigr\Vert =0. \end{aligned}$$
(26)

Note that \(J^{A_{1}}_{m_{k}}\) is firmly nonexpansive, it follows that

$$\begin{aligned} \Vert y_{k}-\bar{x} \Vert ^{2} ={}& \bigl\Vert (1- \beta _{k})\bar{w}_{k}+ \beta _{k} \bigl(J^{A_{1}}_{m_{k}} \bigl(\bar{w}_{k}+\delta \hbar ^{\ast } \bigl(J^{A_{2}}_{n_{k}}-\mathrm{Id} \bigr) \hbar \bar{w}_{k} \bigr) \bigr)-\bar{x} \bigr\Vert ^{2} \\ \leq {}& (1-\beta _{k}) \Vert \bar{w}_{k}-\bar{x} \Vert ^{2} \\ &{}+\beta _{k} \bigl\Vert J^{A_{1}}_{m_{k}} \bigl( \bar{w}_{k}+\delta \hbar ^{\ast } \bigl(J^{A_{2}}_{n_{k}}- \mathrm{Id} \bigr) \hbar \bar{w}_{k} \bigr)-\bar{x} \bigr\Vert ^{2}. \end{aligned}$$
(27)

Utilizing (10) and (11), the expression \(J^{A_{1}}_{m_{k}}(\bar{w}_{k}+\delta \hbar ^{\ast }(J^{A_{2}}_{n_{k}}-\mathrm{Id}) \hbar \bar{w}_{k})\) from the above estimate simplifies as follows:

$$\begin{aligned} \bigl\Vert J^{A_{1}}_{m_{k}} \bigl(\bar{w}_{k}+\delta \hbar ^{\ast } \bigl(J^{A_{2}}_{n_{k}}-\mathrm{Id} \bigr) \hbar \bar{w}_{k} \bigr)-J^{A_{1}}_{m_{k}}\bar{x} \bigr\Vert ^{2} &\leq \bigl\Vert \bar{w}_{k}+\delta \hbar ^{\ast } \bigl(J^{A_{2}}_{n_{k}}-\mathrm{Id} \bigr)\hbar \bar{w}_{k}- \bar{x} \bigr\Vert ^{2} \\ & \leq \Vert \bar{w}_{k}-\bar{x} \Vert ^{2}. \end{aligned}$$
(28)

Setting \(\xi _{k}=J^{A_{1}}_{m_{k}}(\bar{w}_{k}+\delta \hbar ^{\ast }(J^{A_{2}}_{n_{k}}-\mathrm{Id}) \hbar \bar{w}_{k})\) in (27), it follows that

$$\begin{aligned} \Vert \xi _{k}-\bar{x} \Vert ^{2}={}&\bigl\Vert J^{A_{1}}_{m_{k}}\bar{w}_{k}+ \delta \hbar ^{\ast } \bigl(J^{A_{2}}_{n_{k}}-\mathrm{Id} \bigr)\hbar \bar{w}_{k})-J^{A_{1}}_{m_{k}} \bar{x}\bigr\Vert ^{2} \\ \leq {}& \bigl\langle J^{A_{1}}_{m_{k}} \bigl(\bar{w}_{k}+ \delta \hbar ^{\ast } \bigl(J^{A_{2}}_{n_{k}}-\mathrm{Id} \bigr) \hbar \bar{w}_{k} \bigr)-J^{A_{1}}_{m_{k}}\bar{x}, \bar{w}_{k}+\delta \hbar ^{ \ast } \bigl(J^{A_{2}}_{n_{k}}- \mathrm{Id} \bigr)\hbar \bar{w}_{k}-\bar{x} \bigr\rangle \\ ={}& \bigl\langle \xi _{k}-\bar{x},\bar{w}_{k}+\delta \hbar ^{\ast } \bigl(J^{A_{2}}_{n_{k}}-\mathrm{Id} \bigr) \hbar \bar{w}_{k}-\bar{x} \bigr\rangle \\ ={}&\frac{1}{2} \bigl( \Vert \xi _{k}-\bar{x} \Vert ^{2}+ \bigl\Vert \bar{w}_{k}+ \delta \hbar ^{\ast } \bigl(J^{A_{2}}_{n_{k}}-\mathrm{Id} \bigr)\hbar \bar{w}_{k}-\bar{x} \bigr\Vert ^{2} \\ &{}- \bigl\Vert \xi _{k}-\bar{w}_{k}-\delta \hbar ^{\ast } \bigl(J^{A_{2}}_{n_{k}}-\mathrm{Id} \bigr) \hbar \bar{w}_{k} \bigr\Vert ^{2} \bigr) \\ \leq {}&\frac{1}{2} \bigl( \Vert \xi _{k}-\bar{x} \Vert ^{2}+ \Vert \bar{w}_{k}- \bar{x} \Vert ^{2}- \bigl\Vert \xi _{k}-\bar{w}_{k}-\delta \hbar ^{\ast } \bigl(J^{A_{2}}_{n_{k}}-\mathrm{Id} \bigr) \hbar \bar{w}_{k} \bigr\Vert ^{2} \bigr) \\ ={}&\frac{1}{2} \bigl( \Vert \xi _{k}-\bar{x} \Vert ^{2}+ \Vert \bar{w}_{k}- \bar{x} \Vert ^{2}- \Vert \xi _{k}-\bar{w}_{k} \Vert ^{2}-\delta ^{2} \bigl\Vert \hbar ^{\ast } \bigl(J^{A_{2}}_{n_{k}}- \mathrm{Id} \bigr)\hbar \bar{w}_{k} \bigr\Vert ^{2} \\ &{}+2\delta \bigl\langle \xi _{k}-\bar{w}_{k},\hbar ^{\ast } \bigl(J^{A_{2}}_{n_{k}}-\mathrm{Id} \bigr) \hbar \bar{w}_{k} \bigr\rangle \bigr) \\ \leq {}&\frac{1}{2} \bigl( \Vert \xi _{k}-\bar{x} \Vert ^{2}+ \Vert \bar{w}_{k}- \bar{x} \Vert ^{2}- \Vert \xi _{k}-\bar{w}_{k} \Vert ^{2}-\delta ^{2} \bigl\Vert \hbar ^{\ast } \bigl(J^{A_{2}}_{n_{k}}- \mathrm{Id} \bigr)\hbar \bar{w}_{k} \bigr\Vert ^{2} \\ &{}+2\delta \Vert \xi _{k}-\bar{w}_{k} \Vert \bigl\Vert \hbar ^{\ast } \bigl(J^{A_{2}}_{n_{k}}-\mathrm{Id} \bigr) \hbar \bar{w}_{k} \bigr\Vert \bigr). \end{aligned}$$
(29)

This implies that

$$\begin{aligned} \Vert \xi _{k}-\bar{x} \Vert ^{2}\leq \Vert \bar{w}_{k}-\bar{x} \Vert ^{2}- \Vert \xi _{k}-\bar{w}_{k} \Vert ^{2} +2\delta \Vert \xi _{k}-\bar{w}_{k} \Vert \bigl\Vert \hbar ^{\ast } \bigl(J^{A_{2}}_{n_{k}}-\mathrm{Id} \bigr)\hbar \bar{w}_{k} \bigr\Vert . \end{aligned}$$
(30)

So, we have

$$\begin{aligned} \Vert y_{k}-\bar{x} \Vert ^{2} \leq {}& (1-\beta _{k}) \Vert \bar{w}_{k}- \bar{x} \Vert ^{2}+ \beta _{k} \Vert \xi _{k}-\bar{x} \Vert ^{2} \\ \leq {}& (1-\beta _{k}) \Vert \bar{w}_{k}-\bar{x} \Vert ^{2}+\beta _{k} \bigl( \Vert \bar{w}_{k}- \bar{x} \Vert ^{2}- \Vert \xi _{k}-\bar{w}_{k} \Vert ^{2} \\ &{}+2\delta \Vert \xi _{k}-\bar{w}_{k} \Vert \bigl\Vert \hbar ^{\ast } \bigl(J^{A_{2}}_{n_{k}}-\mathrm{Id} \bigr) \hbar \bar{w}_{k} \bigr\Vert \bigr). \end{aligned}$$
(31)

After simplification, we have

$$\begin{aligned} \beta _{k} \Vert \xi _{k}-\bar{w}_{k} \Vert ^{2} \leq {}& \Vert \bar{w}_{k}- \bar{x} \Vert ^{2}- \Vert y_{k}-\bar{x} \Vert ^{2}-2\beta _{k}\delta \Vert \xi _{k}-\bar{w}_{k} \Vert \bigl\Vert \hbar ^{\ast } \bigl(J^{A_{2}}_{n_{k}}- \mathrm{Id} \bigr) \hbar \bar{w}_{k} \bigr\Vert ) \\ \leq{} & \Vert x_{k}-\bar{x} \Vert ^{2}- \Vert y_{k}-\bar{x} \Vert ^{2}-2 \beta _{k}\delta \Vert \xi _{k}-\bar{w}_{k} \Vert \bigl\Vert \hbar ^{\ast } \bigl(J^{A_{2}}_{n_{k}}-\mathrm{Id} \bigr) \hbar \bar{w}_{k} \bigr\Vert ) \\ \leq{} & \bigl( \Vert x_{k}-\bar{x} \Vert + \Vert y_{k}-\bar{x} \Vert \bigr) \Vert x_{k}-y_{k} \Vert \\ &{}-2\beta _{k}\delta \Vert \xi _{k}- \bar{w}_{k} \Vert \bigl\Vert \hbar ^{ \ast } \bigl(J^{A_{2}}_{n_{k}}-\mathrm{Id} \bigr)\hbar \bar{w}_{k} \bigr\Vert ). \end{aligned}$$
(32)

Making use of (19), (26), and (C3), we have the following estimate:

$$\begin{aligned} \lim_{k \rightarrow \infty } \Vert \xi _{k}-\bar{w}_{k} \Vert =0. \end{aligned}$$
(33)

This implies that

$$\begin{aligned} \lim_{k \rightarrow \infty } \Vert \xi _{k}-\bar{v}_{k} \Vert =0. \end{aligned}$$
(34)

Reasoning as above, we get from the definition of \((b_{k})\), (C1), and (34), that

$$\begin{aligned} \lim_{k\rightarrow \infty } \Vert \xi _{k}-x_{k} \Vert =0. \end{aligned}$$
(35)

Since \((x_{k})\) is bounded, then there exists a subsequence \((x_{k_{t}})\) of \((x_{k})\) such that \(x_{k_{t}} \rightharpoonup \bar{x_{*}} \in \mathcal{H}_{1}\) as \(t \rightarrow \infty \). Therefore \(\xi _{k_{t}}\rightharpoonup \bar{x_{*}}\) and \(\bar{w}_{k_{t}} \rightharpoonup \bar{x_{*}}\) as \(t \rightarrow \infty \). In order to show that \(\bar{x_{*}} \in \Omega \), we assume that \((r,s) \in \operatorname{gra}(A_{1})\). Since \(\xi _{k_{t}}=J^{A_{1}}_{m_{k_{t}}}(\bar{w}_{k_{t}}+\delta \hbar ^{ \ast }(J^{A_{2}}_{n_{k_{t}}}-\mathrm{Id})\hbar \bar{w}_{k_{t}})\), we have

$$\begin{aligned} \bar{w}_{k_{t}}+\delta \hbar ^{\ast } \bigl(J^{A_{2}}_{n_{k_{t}}}- \mathrm{Id} \bigr)\hbar \bar{w}_{k_{t}} \in \xi _{k_{t}}+m_{k_{t}}A_{1}( \xi _{k_{t}}). \end{aligned}$$

This implies that

$$\begin{aligned} \frac{1}{m_{k_{t}}}(\bar{w}_{k_{t}}-\xi _{k_{t}})+ \frac{1}{m_{k_{t}}} \delta \hbar ^{\ast } \bigl(J^{A_{2}}_{n_{k_{t}}}- \mathrm{Id} \bigr)\hbar \bar{w}_{k_{t}} \in A_{1}(\xi _{k_{t}}). \end{aligned}$$

From the monotonicity of \(A_{1}\), we have

$$\begin{aligned} \biggl\langle r-\xi _{k_{t}},s- \biggl(\frac{1}{m_{k_{t}}}( \bar{w}_{k_{t}}-\xi _{k_{t}})+ \frac{1}{m_{k_{t}}} \bigl(\delta \hbar ^{\ast } \bigl(J^{A_{2}}_{n_{k_{t}}}-\mathrm{Id} \bigr) \hbar \bar{w}_{k_{t}} \bigr) \biggr) \biggr\rangle \geq 0. \end{aligned}$$

From the above estimate, we also have

$$\begin{aligned} \langle r-\xi _{k_{t}},s\rangle \geq {}& \biggl\langle r-\xi _{k_{t}}, \frac{1}{m_{k_{t}}}(\bar{w}_{k_{t}}-\xi _{k_{t}})+ \frac{1}{m_{k_{t}}} \bigl( \delta \hbar ^{\ast } \bigl(J^{A_{2}}_{n_{k_{t}}}- \mathrm{Id} \bigr)\hbar \bar{w}_{k_{t}} \bigr) \biggr\rangle \\ ={}& \biggl\langle r-\xi _{k_{t}},\frac{1}{m_{k_{t}}}( \bar{w}_{k_{t}}- \xi _{k_{t}}) \biggr\rangle \\ &{} + \biggl\langle r-\xi _{k_{t}},\frac{1}{m_{k_{t}}} \bigl(\delta \hbar ^{\ast } \bigl(J^{A_{2}}_{n_{k_{t}}}-\mathrm{Id} \bigr) \hbar \bar{w}_{k_{t}} \bigr) \biggr\rangle . \end{aligned}$$
(36)

Since \(\xi _{k_{t}} \rightharpoonup \bar{x_{*}}\), we obtain

$$\begin{aligned} \lim_{t \rightarrow \infty }\langle r-\xi _{k_{t}},v\rangle = \langle r- \bar{x_{*}},s\rangle. \end{aligned}$$

By making use of (33), (34), and (36), it follows that

$$\begin{aligned} \langle r-\bar{x_{*}},s\rangle \geq 0. \end{aligned}$$

This implies that \(0 \in A_{1}(\bar{x_{*}})\). Since ħ is a bounded linear operator, we have \(\hbar \bar{w}_{k_{t}} \rightharpoonup \hbar \bar{x_{*}}\) as \(t \rightarrow \infty \). Moreover, from (26) it then follows from the demiclosedness principle that \(0 \in A_{2}(\bar{x_{*}})\), and hence \(\bar{x_{*}} \in \Omega \).

Step 4. We show that \(\bar{x_{*}} \in \bigcap^{M}_{i=1}EP(g_{i})\).

Observe that the following relation

$$\begin{aligned} u^{i}_{k}=\arg \min \biggl\{ \gamma g_{i}(b_{k},y)+ \frac{1}{2} \Vert b_{k}-y \Vert ^{2}:y \in C \biggr\} \end{aligned}$$

implies via Lemma 2.4 that

$$\begin{aligned} 0 \in \partial _{2} \biggl\{ \gamma g_{i}(b_{k},y)+ \frac{1}{2} \Vert b_{k}-y \Vert ^{2} \biggr\} \bigl(u^{i}_{k} \bigr)+N_{C} \bigl(u^{i}_{k} \bigr). \end{aligned}$$

This infers the existence of \(\bar{x_{*}} \in \partial _{2} g_{i}(b_{k},u^{i}_{k})\) and \(\bar{p} \in N_{C}(u^{i}_{k})\) such that

$$\begin{aligned} \gamma \bar{x_{*}}+b_{k}-u^{i}_{k}+ \bar{p}. \end{aligned}$$
(37)

Since \(\bar{p} \in N_{C}(u^{i}_{k})\) and \(\langle \bar{p},u-u^{i}_{k} \rangle \leq 0\) for all \(u \in C\), by using (37), we have

$$\begin{aligned} \gamma \bigl\langle \bar{x_{*}}, u-u^{i}_{k} \bigr\rangle \geq \bigl\langle u^{i}_{k}-b_{k},u-u^{i}_{k} \bigr\rangle ,\quad \forall u \in C. \end{aligned}$$
(38)

Since \(\bar{x_{*}} \in \partial _{2}g_{i}(b_{k},u^{i}_{k})\),

$$\begin{aligned} g_{i}(b_{k},y)-g_{i} \bigl(b_{k},u^{i}_{k} \bigr)\geq \bigl\langle \bar{x_{*}},u-u^{i}_{k} \bigr\rangle , \quad\forall u \in C. \end{aligned}$$
(39)

Utilizing (38) and (39), we obtain

$$\begin{aligned} \gamma \bigl(g_{i}(b_{k},y)-g_{i} \bigl(b_{k},u^{i}_{k} \bigr) \bigr)\geq \bigl\langle u^{i}_{k}-b_{k},u-u^{i}_{k} \bigr\rangle , \quad\forall u \in C. \end{aligned}$$
(40)

Since \(b_{k} \rightharpoonup \bar{x_{*}}\) and \(\Vert b_{k}-u^{i}_{k}\Vert \rightarrow 0\) as \(k \rightarrow \infty \), this implies \(u^{i}_{k} \rightharpoonup \bar{x_{*}}\). By using (A3) and (40), letting \(k \rightarrow \infty \), we deduce that \(g_{i}(\bar{x_{*}},u) \geq 0\) for all \(u \in C\) and \(i \in \{1,2,\ldots,M\}\). Therefore, \(\bar{x_{*}} \in \bigcap^{M}_{i=1}EP(g_{i})\).

Step 5. We show that \(\bar{x_{*}}=\bigcap^{N}_{j=1}\mathrm{Fix}(T_{j})\).

Since \(x_{k_{t}} \rightharpoonup \bar{x_{*}}\) and \(\Vert \bar{v_{k}}-x_{k}\Vert \rightarrow 0\) as \(t \rightarrow \infty \), this implies \(\bar{v_{k}} \rightharpoonup \bar{x_{*}}\). Therefore, utilizing the demiclosedness principle along with estimate (22), we have \(\bar{x_{*}} \in \bigcap^{N}_{j=1}\mathrm{Fix}(T_{j})\). Hence \(\bar{x_{*}} \in \Gamma \).

Step 6. We show that \(x_{k}\rightarrow \bar{x}=P_{\Gamma }x_{1}\).

Note that \(\bar{x}=P_{\Gamma }x_{1}\) and \(\bar{x_{*}} \in \Gamma \) implies that \(x_{k+1}=P_{\Gamma }x_{1}\) and \(\bar{x} \in \Gamma \in C_{k+1}\). This infers that \(\Vert x_{k+1}-x_{1} \Vert \leq \Vert \bar{x}-x_{1} \Vert \). On the other hand, we have

$$\begin{aligned} \Vert \bar{x}-x_{1} \Vert \leq \Vert \bar{x_{*}}-x_{1} \Vert \leq \liminf_{k\rightarrow \infty } \Vert x_{k}-x_{1} \Vert \leq \limsup_{k\rightarrow \infty } \Vert x_{k}-x_{1} \Vert \leq \Vert \bar{x}-x_{1} \Vert . \end{aligned}$$

That is,

$$\begin{aligned} \Vert \bar{x_{*}}-x_{1} \Vert =\lim _{k\rightarrow \infty } \Vert x_{k}-x_{1} \Vert = \Vert \bar{x}-x_{1} \Vert . \end{aligned}$$

Therefore, we conclude that \(\lim_{k\rightarrow \infty }x_{k}=\bar{x_{*}}=P_{\Gamma }x_{1}\). This completes the proof. □

If we take \(A_{2}=0\) in hypothesis (H1), then we have the following results.

Corollary 3.4

Assume that \(\Gamma:= \{x \in A^{-1}_{1}(0) \cap (\bigcap^{M}_{i=1}EP(g_{i}) ) \cap (\bigcap^{N}_{j=1}\mathrm{Fix}(S_{i})\} ) \neq \emptyset \). Then the sequence \((x_{k})\) defined as

$$\begin{aligned} \textstyle\begin{cases} b_{k}=x_{k}+\Theta _{k}(x_{k}-x_{k-1}); \\ u^{i}_{k}=\arg \min \{\gamma g_{i}(b_{k},y)+\frac{1}{2} \Vert b_{k}-u \Vert ^{2}:u \in C\},\quad i=1,2,\ldots,M; \\ v^{i}_{k}=\arg \min \{\gamma g_{i}(u^{i}_{k},y)+\frac{1}{2} \Vert b_{k}-u \Vert ^{2}:u \in C\}, \quad i=1,2,\ldots,M; \\ i_{k}=\arg \max \{ \Vert v^{i}_{k}-x_{k} \Vert : i = 1,2, \ldots,M\}, \quad \bar{v_{k}}=v^{i_{k}}_{k}; \\ w^{j}_{k}=(1-\alpha _{k})\bar{v}_{k}+\alpha _{k}S_{j}\bar{v}_{k}; \\ j_{k}=\arg \max \{ \Vert w^{j}_{k}-x_{k} \Vert : j = 1,2, \ldots,N\},\quad\bar{w_{k}}=w^{j_{k}}_{k}; \\ y_{k}=(1-\beta _{k})\bar{w}_{k}+\beta _{k}J^{A_{1}}_{m_{k}} \bar{w}_{k}; \\ C_{k+1}=\{z \in C_{k}: \Vert y_{k}-z \Vert ^{2} \leq \Vert x_{k}-z \Vert ^{2}+\Theta ^{2}_{k} \Vert x_{k}-x_{k-1} \Vert ^{2} \\ \phantom{C_{k+1}=}{}+2 \Theta _{k} \langle x_{k}-z,x_{k}-x_{k-1} \rangle \}; \\ x_{k+1}=P_{C_{k+1}}x_{1},\quad \forall k \geq 1, \end{cases}\displaystyle \end{aligned}$$
(41)

converges strongly to an element in Γ provided that conditions (C1)–(C4) hold.

4 Numerical experiment and results

This section shows the effectiveness of our algorithm by the following example and numerical results.

Example 4.1

Let \(\mathcal{H}_{1} = \mathcal{H}_{2} = \mathbb{R}\) with the inner product defined by \(\langle x, y\rangle = xy\) for all \(x, y \in \mathbb{R}\) and induced usual norm \(|\cdot |\). We define three operators \(\hbar, A_{1}, A_{2}: \mathbb{R}\rightarrow \mathbb{R}\) as \(\hbar (x)=3x\), \(A_{1}x=2x\), and \(A_{2}x=3x\) for all \(x \in \mathbb{R}\). It is clear that ħ is a bounded linear operator and \(A_{1}, A_{2}\) are maximal monotone operators such that \(\Omega:= \{\hat{x} \in A^{-1}_{1}0:\hbar \hat{x} \in A^{-1}_{2}0\}=0\). For each \(i \in \{1,2,\ldots,M\}\), let the family of pseudomonotone bifunctions \(g_{i}(x,y): C\times C \rightarrow \mathbb{R}\) on \(C = [0,1] \subset \mathbb{R}\) be defined by \(g_{i}(x,y)= T_{i}(x)(y-x)\), where

$$\begin{aligned} T_{i}(x)= \textstyle\begin{cases} 0& 0 \leq x \leq \mu _{i}; \\ \sin (x-\mu _{i})+\exp ( x-\mu _{i})-1,& \mu _{i} \leq x \leq 1, \end{cases}\displaystyle \end{aligned}$$

where \(0 < \mu _{1} < \mu _{2} <\cdots<\mu _{M} < 1\). Note that \(EP(g_{i})=[0,\mu _{i}]\) if and only if \(0 \leq x \leq \mu _{i}\) and \(y \in [0,1]\). Consequently, \(\bigcap^{M}_{i=1}EP(g_{i})=[0,\mu _{1}]\). For each \(j \in \{1,2,\ldots,N\}\), let the family of operators \(S_{j}: \mathbb{R} \rightarrow \mathbb{R}\) be defined by

$$\begin{aligned} S_{j}(x)=\textstyle\begin{cases} -\frac{x}{j}& \text{if }x \in [0,\infty ); \\ x& \text{if }x \in (-\infty,0). \end{cases}\displaystyle \end{aligned}$$

It is also clear that \(S_{j}\) defines a finite family of \(\frac{1-j^{2}}{(1+j)^{2}}\)-demicontractive operators with \(\bigcap^{N}_{j=1}\mathrm{Fix}(S_{j})=\{0\}\). Hence \(\Gamma = \Omega \cap (\bigcap^{M}_{i=1}EP(g_{i})) \cap (\bigcap^{N}_{j=1}\mathrm{Fix}(S_{j})) = 0\). In order to compute the numerical values of \((x_{k+1})\), we choose: \(\Theta = 0.5\), \(\gamma =\frac{1}{8}\), \(\alpha _{k} =\frac{1}{100k+1}\), \(\beta _{k} =\frac{1}{100k+1}\), \(\delta =\frac{1}{9}\), \(L=3\), and \(m=0.01\). Since

$$\begin{aligned} \textstyle\begin{cases} \min \{\frac{1}{k^{2} \Vert x_{k}-x_{k-1} \Vert },0.5\}& \text{if }x_{k}\neq x_{k-1}; \\ 0.5& \text{otherwise}. \end{cases}\displaystyle \end{aligned}$$

Observe that the expression

$$\begin{aligned} u_{k}^{i}=\arg \min \biggl\{ \gamma T_{i}(b_{k}) (y-b_{k})+\frac{1}{2}(y-x_{k})^{2}, \forall y \in [0,1] \biggr\} \end{aligned}$$

in Algorithm 1 is equivalent to the following relation \(u_{k}^{i}=b_{k}-\gamma T_{i}(b_{k}) \text{ for all } i \in \{1,2, \ldots, M\}\). Similarly, \(v_{k}^{i}=b_{k}-\gamma T_{i}(u^{i}_{k}) \text{ for all } i \in \{1,2, \ldots,M\}. \) Hence, we can compute the intermediate approximation \(\bar{v}_{k}\) which is farthest from \(b_{k}\) among \(v_{k}^{i}\) for all \(i \in \{1,2,\ldots,M\}\). We compare the parallel hybrid accelerated extragradient algorithm defined in Algorithm 1 (i.e., \(\Theta _{k}\neq 0\)) and its variant with \(\Theta _{k}=0\). The stopping criteria are defined as Error=\(E_{k}=\|x_{k}-x_{k-1} \|<10^{-5}\). The values of Algorithm 1 and its variant are listed in Table 1.

Table 1 Numerical results for Example 4.1
Full size table

The error plotting \(E_{k}\) and \((x_{k})\) of Algorithm 1 with \(\Theta _{k} \neq 0\) and \(\Theta _{k}=0\) for each choice in Table 1 is illustrated in Fig. 1.

Figure 1
figure 1

Comparison of Alg. 1, \(\Theta _{k} \neq 0\) and \(\Theta _{k} = 0\) for \(N=20\)

Full size image

We can see from Table 1 and Fig. 1 that Algorithm 1 performs faster and better in view of the error analysis, time consumption, and the number of iterations required for the convergence towards the desired solution in comparison with the variant of Algorithm 1 with \(\Theta _{k} = 0\).

5 Applications

In this section, we discuss some important instances of the main result in Sect. 3 as applications.

5.1 Split feasibility problems

Let \(\mathcal{H}_{1}\) and \(\mathcal{H}_{2}\) be two real Hilbert spaces and \(\hbar: \mathcal{H}_{1} \rightarrow \mathcal{H}_{2}\) be a bounded linear operator. Let C and Q be nonempty, closed, and convex subsets of \(\mathcal{H}_{1}\) and \(\mathcal{H}_{2}\), respectively. The split feasibility problem (SFP) is the problem of finding \(\bar{x} \in C\) such that \(\hbar \bar{x} \in Q\). We represent the solution set by \(\omega:= C \cap \hbar ^{-1}(Q) = \{\bar{x} \in C: \hbar \bar{x} \in Q\}\). This problem is essentially due to Censor and Elfving [12] to solve the inverse problems and their application to medical image reconstruction, radiation therapy, and modeling and simulation in a finite dimensional Hilbert space. Recall the indicator function of C

$$\begin{aligned} b_{C}(\bar{x}):=\textstyle\begin{cases} 0,& \bar{x} \in C; \\ \infty,& \text{otherwise}. \end{cases}\displaystyle \end{aligned}$$

The proximal operator of \(b_{C}\) is the metric projection on C

$$\begin{aligned} \mathrm{prox}_{b_{C}}&=\arg \min_{\bar{p} \in C} \Vert \bar{p}- \bar{x} \Vert \\ &=P_{C}(\bar{x}). \end{aligned}$$

Let \(P_{Q}\) be the projection of \(\mathcal{H}_{2}\) onto a nonempty, closed, and convex subset Q. Take \(f(\bar{x})=\frac{1}{2}\|\hbar \bar{x}-P_{Q}\hbar \bar{x}\|^{2}\) and \(g(\bar{x})=b_{C}(\bar{x})\). Then we have the following result.

Corollary 5.1

Assume that \(\Gamma =\omega \cap (\bigcap^{M}_{i=1}EP(g_{i}) )\cap ( \bigcap^{N}_{i=1}\mathrm{Fix}(S_{j}) )\neq \emptyset \) via hypotheses (H1)–(H5). For given \(x_{0},x_{1} \in \mathcal{H}_{1}\), let the iterative sequence \((x_{k})\) be generated by

$$\begin{aligned} \textstyle\begin{cases} b_{k}=x_{k}+\Theta _{k}(x_{k}-x_{k-1}); \\ u^{i}_{k}=\arg \min \{\gamma g_{i}(b_{k},y)+\frac{1}{2} \Vert b_{k}-y \Vert ^{2}:u \in C\}, \quad i=1,2,\ldots,M; \\ v^{i}_{k}=\arg \min \{\gamma g_{i}(u^{i}_{k},y)+\frac{1}{2} \Vert b_{k}-y \Vert ^{2}:u \in C\}, \quad i=1,2,\ldots,M; \\ i_{k}=\arg \max \{ \Vert v^{i}_{k}-x_{k} \Vert : i = 1,2, \ldots,M\},\quad\bar{v_{k}}=v^{i_{k}}_{k}; \\ w^{j}_{k}=(1-\alpha _{k})\bar{v_{k}}+\alpha _{k}S_{j} \bar{v_{k}}; \\ j_{k}=\arg \max \{ \Vert w^{j}_{k}-x_{k} \Vert : j = 1,2, \ldots,N\},\quad \bar{w_{k}}=w^{j_{k}}_{k}; \\ y_{k}=(1-\beta _{k})\bar{w}_{k}+\beta _{k}(P_{C}(\bar{w}_{k}+ \delta \hbar ^{\ast }(P_{Q}-\mathrm{Id})\hbar \bar{w}_{k})); \\ C_{k+1}=\{z \in C_{k}: \Vert y_{k}-z \Vert ^{2} \leq \Vert x_{k}-z \Vert ^{2}+\Theta ^{2}_{k} \Vert x_{k}-x_{k-1} \Vert ^{2} \\ \phantom{C_{k+1}=}{}+2 \Theta _{k} \langle x_{k}-z,x_{k}-x_{k-1} \rangle \}; \\ x_{k+1}=P_{C_{k+1}}x_{1}, \quad \forall k \geq 1, \end{cases}\displaystyle \end{aligned}$$
(42)

where \(0 < \gamma < \min (\frac{1}{2d_{1}},\frac{1}{2d_{2}})\). Assume that conditions (C1)–(C4) hold, then the sequence \((x_{k})\) generated by (42) converges strongly to an element in Γ.

5.2 Split variational inequality problems

Let \(A:C \rightarrow \mathcal{H}\) be a nonlinear monotone operator defined on a nonempty, closed, and convex subset C of a real Hilbert space \(\mathcal{H}\). The classical variational inequality problem aims to find a point \(\bar{x} \in C\) such that

$$\begin{aligned} \langle A\bar{x},\bar{y}-\bar{x} \rangle \geq 0\quad \forall \bar{y} \in C. \end{aligned}$$

The solution set of the above problem is denoted by \(VI(C,A)\). Let the operator \(\Pi _{A}\subset \mathcal{H} \times \mathcal{H}\) be defined by

$$\begin{aligned} \Pi _{A}\bar{x}= \textstyle\begin{cases} A(\bar{x})+N_{C}(\bar{x}) & \text{if }\bar{x} \in C, \\ \emptyset& \text{if }\bar{x} \notin C, \end{cases}\displaystyle \end{aligned}$$

where \(N_{C}(\bar{x}):=\{z \in \mathcal{H}:\langle \bar{y}-\bar{x}, z \rangle \leq 0 \text{ for all } \bar{y} \in C\}\). It follows from [28] that \(\Pi _{A}\) is a maximal monotone such that \(0 \in \Pi _{A}(\bar{x})\iff \bar{x} \in VI(C,A)\iff \bar{x}=P_{C}( \bar{x}-\lambda A(\bar{x}))\). As an application, we have the following result.

Corollary 5.2

Let \(\{\mathcal{H}_{n}\}_{n=1}^{2}\) be real Hilbert spaces, and let \(\{C_{n}\}_{n=1}^{2}\) be nonempty, closed, and convex subsets of \(\mathcal{H}_{n}\), respectively. Let \(A_{n}:C_{n} \rightarrow \mathcal{H}_{n}\) for \(n=1,2\) be single-valued monotone and hemicontinuous operators, and let \(\hbar:\mathcal{H}_{1} \rightarrow \mathcal{H}_{2}\) be a bounded linear operator such that \(\Gamma =VI(C_{1},A_{1})\cap \hbar ^{-1} (VI(C_{2},A_{2})) \cap ( \bigcap^{M}_{i=1}EP(g_{i}) )\cap (\bigcap^{N}_{j=1}\mathrm{Fix}(S_{j}) ) \neq \emptyset \) via hypotheses (H1)–(H5). For given \(x_{0},x_{1} \in \mathcal{H}_{1}\), let the iterative sequences \((x_{k})\) be generated by

$$\begin{aligned} \textstyle\begin{cases} b_{k}=x_{k}+\Theta _{k}(x_{k}-x_{k-1}); \\ u^{i}_{k}=\arg \min \{\gamma g_{i}(b_{k},y)+\frac{1}{2} \Vert b_{k}-y \Vert ^{2}:u \in C\}, \quad i=1,2,\ldots,M; \\ v^{i}_{k}=\arg \min \{\gamma g_{i}(u^{i}_{k},y)+\frac{1}{2} \Vert b_{k}-y \Vert ^{2}:u \in C\}, \quad i=1,2,\ldots,M; \\ i_{k}=\arg \max \{ \Vert v^{i}_{k}-x_{k} \Vert : i = 1,2, \ldots,M\},\quad \bar{v_{k}}=v^{i_{k}}_{k}; \\ w^{j}_{k}=(1-\alpha _{k})\bar{v_{k}}+\alpha _{k}S_{j} \bar{v_{k}}; \\ j_{k}=\arg \max \{ \Vert w^{j}_{k}-x_{k} \Vert : j = 1,2, \ldots,N\},\quad \bar{w_{k}}=w^{j_{k}}_{k}; \\ y_{k}=(1-\beta _{k})\bar{w}_{k}+\beta _{k}(P_{C_{1}}^{(A_{1}, \lambda )}(\bar{w}_{k}+\delta \hbar ^{\ast }(P_{C_{2}}^{(A_{2}, \lambda )}-\mathrm{Id})\hbar \bar{w}_{k})); \\ C_{k+1}=\{z \in C_{k}: \Vert y_{k}-z \Vert ^{2} \leq \Vert x_{k}-z \Vert ^{2}+\Theta ^{2}_{k} \Vert x_{k}-x_{k-1} \Vert ^{2} \\ \phantom{C_{k+1}=}{}+2 \Theta _{k} \langle x_{k}-z,x_{k}-x_{k-1} \rangle \}; \\ x_{k+1}=P_{C_{k+1}}x_{1}, \quad \forall k \geq 1, \end{cases}\displaystyle \end{aligned}$$
(43)

where \(0 < \gamma < \min (\frac{1}{2d_{1}},\frac{1}{2d_{2}})\) and \(P_{C_{1}}^{(A_{1},\lambda )}\), \(P_{C_{2}}^{(A_{2},\lambda )}\) denote \(P_{C}(\mathrm{Id}-\lambda A)\). Assume that conditions (C1)–(C4) hold, then the sequence \((x_{k})\) generated by (43) converges strongly to an element in Γ.

5.3 Split optimization problems

Let \(\phi: \mathcal{H}_{1} \rightarrow (-\infty,\infty ]\) be a proper, convex, and lower semicontinuous (pcls) function, then the set of minimizers associated with ϕ is defined as

$$\begin{aligned} \operatorname{argmin} \phi:= \bigl\{ x^{\ast } \in \mathcal{H}_{1}: \phi \bigl(x^{\ast } \bigr) \leq \phi (\bar{z}) \text{ for all } \bar{z} \in \mathcal{H}_{1} \bigr\} . \end{aligned}$$

Recall that ∂ϕ of the pcls function ϕ is a maximal monotone operator and the corresponding resolvent operator of ∂ϕ is called the proximity operator (see [17]). Hence argmin \(\phi = (\partial \phi )^{-1}(0)\). Hence, we have the following application.

Corollary 5.3

Let \(\mathcal{H}_{1}\), \(\mathcal{H}_{2}\) be two Hilbert spaces, and let \(C \subseteq \mathcal{H}_{1}\) be nonempty, closed, and convex subset of \(\mathcal{H}_{1}\). Let \(\phi _{1}\) and \(\phi _{2}\) be pcls functions on \(\mathcal{H}_{1}\) and \(\mathcal{H}_{2}\), respectively. Assume that \(\Gamma = \{x \in \arg \min \phi _{1}:\hbar x \in \arg \min \phi _{2} \} \cap (\bigcap^{M}_{i=1}EP(g_{i}))\cap (\bigcap_{i=1}^{N}\mathrm{Fix}(S_{j})) \neq \emptyset \) via hypotheses (H1)–(H5). For given \(x_{0},x_{1} \in \mathcal{H}_{1}\), let the iterative sequence \((x_{k})\) be generated by

$$\begin{aligned} \textstyle\begin{cases} b_{k}=x_{k}+\Theta _{k}(x_{k}-x_{k-1}); \\ u^{i}_{k}=\arg \min \{\gamma g_{i}(b_{k},y)+\frac{1}{2} \Vert b_{k}-y \Vert ^{2}:u \in C\}, \quad i=1,2,\ldots,M; \\ v^{i}_{k}=\arg \min \{\gamma g_{i}(u^{i}_{k},y)+\frac{1}{2} \Vert b_{k}-y \Vert ^{2}:u \in C\}, \quad i=1,2,\ldots,M; \\ i_{k}=\arg \max \{ \Vert v^{i}_{k}-x_{k} \Vert : i = 1,2, \ldots,M\},\quad \bar{v_{k}}=v^{i_{k}}_{k}; \\ w^{j}_{k}=(1-\alpha _{k})\bar{v_{k}}+\alpha _{k}S_{j} \bar{v_{k}}; \\ j_{k}=\arg \max \{ \Vert w^{j}_{k}-x_{k} \Vert : j = 1,2, \ldots,N\},\quad \bar{w_{k}}=w^{j_{k}}_{k}; \\ y_{k}=(1-\beta _{k})b_{k}+\beta _{k}(J^{\partial g_{1}}_{m_{k}}(b_{k}+ \delta \hbar ^{\ast }(J^{\partial g_{2}}_{n_{k}}-\mathrm{Id})\hbar b_{k})); \\ C_{k+1}=\{z \in C_{k}: \Vert \bar{y}_{k}-z \Vert ^{2} \leq \Vert x_{k}-z \Vert ^{2}+\Theta ^{2}_{k} \Vert x_{k}-x_{k-1} \Vert ^{2} \\ \phantom{C_{k+1}=}{}+2 \Theta _{k} \langle x_{k}-z,x_{k}-x_{k-1}\rangle \}; \\ x_{k+1}=P_{C_{k+1}}x_{1}, \quad\forall k \geq 1, \end{cases}\displaystyle \end{aligned}$$
(44)

where \(0 < \gamma < \min (\frac{1}{2d_{1}},\frac{1}{2d_{2}})\). Assume that conditions (C1)–(C4) hold, then the sequence \((x_{k})\) generated by (44) converges strongly to an element in Γ.

6 Conclusions

In this paper, we have analyzed an accelerated projection based parallel hybrid extragradient algorithm for pseudomonotone equilibrium, fixed point, and split null point problems in Hilbert spaces. The convergence analysis of the algorithm is established under the suitable set of conditions. A suitable numerical example has been incorporated to exhibit the effectiveness of the algorithm. Moreover, some well-known instances, as applications, of the main result that can pave a way for an important future research direction are also discussed.