A new double inertial subgradient extragradient algorithm for solving split pseudomonotone equilibrium problems and fixed point problems

Abstract

The purpose of this article is to suggest a modified subgradient extragradient method that includes double inertial extrapolations and viscosity approach for finding the common solution of split equilibrium problem and fixed point problem. The strong convergence result of the suggested method is obtained under some standard assumptions on the control parameters. Our method does not require solving two strongly convex optimization problems in the feasible sets per iteration, and the step-sizes do not depend on bifunctional Lipschitz-type constants. Furthermore, unlike several methods in the literature, our method does not depend on the prior knowledge of the operator norm of the bounded linear operator. Instead, the step-sizes are self adaptively updated. We apply our method to solve split variational inequality problem. Lastly, we conduct some numerical test to compare our method with some well known methods in the literature.

1 Introduction

Let C and Q be two nonempty, convex and closed subsets of two Hilbert spaces $H_1$ and $H_2$, respectively. The split equilibrium problem (SEP) is generated as follows: Find ${\bar{x}}\in C$ such that

$$\begin{aligned} f({\bar{x}},x)\ge 0,\forall x\in C, \end{aligned}$$

(1.1)

and ${\bar{y}}=T{\bar{x}}\in Q$ solves

$$\begin{aligned} g({\bar{y}},y)\ge 0,\forall y\in Q, \end{aligned}$$

(1.2)

where $f:C\times C\rightarrow {\mathbb {R}}$ and $g:Q\times Q\rightarrow {\mathbb {R}}$ are equilibrium bifunctions, and $T:H_1\rightarrow H_2$ is a bounded linear operator. This problem was introduced and studied by Moudafi [18]. It is important to note that SEP properly includes multiple set split feasibility problem. Thus, it contains the split variational inequality problem, which is a generalization of split zero problemand split feasibility problem [1, 16, 23, 25, 26]. If $g=0$ and $T=0$, then SEP becomes the following classical equilibrium problem (EP): Find ${\bar{x}}\in C$ such that

$$\begin{aligned} f({\bar{x}},x)\ge 0,\forall x\in C. \end{aligned}$$

(1.3)

The solution set of the EP (1.3) is denoted by SOL(f, C). It is well known that the EP (1.3) has applications in several mathematical problems, such as variational inequity problem, optimization problem, Nash equilibrium problem, maximax problem, fixed point problem and saddle point problem (see, [4, 12, 19] and the reference therein ). One of the widely used methods for solving EP (1.3), when the bifunction f is monotone, is the maximal point method [17]. It is worthy to note that the weak convergence of this method cannot be guaranteed for a weaker assumption, such as pseudomonotone bifunction. To surmount this limitation, in [28], Tran et al. introduced the extragradient method for EP (1.3) as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} x_0\in C\\ y_m=\text {argmin}_{y\in C}\{\lambda g(x_m,y) + \frac{1}{2}\Vert y-x_m\Vert ^2\}\\ x_{m+1}=\text {argmin}_{y\in C}\{\lambda g(y_m,y) + \frac{1}{2}\Vert y-x_m\Vert ^2\},\\ \end{array}\right. } \end{aligned}$$

(1.4)

where f is a pseudomonotone and Lipschitz-type continuous with positive constants $L_1$ and $L_2$ such that $0<\lambda <\min \{\frac{1}{2L_1}, \frac{1}{2L_2}\}.$ The authors showed that the method (1.4) weakly converges to a solution of the EP (1.3).

In recent years, the inertial-type algorithms have been studied by many researchers. This concept emanated from an implit discretization method (heavy ball method) of the second-order dynamical in time [2, 3] and it has been shown by many authors that the inclusion of the inertial terms in iterative methods enhance their speed of convergence (see [5, 6, 14, 29] and the references therein). In [29], Vinh and Muu, introduced an inertial extragradient for solving EP as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} x_0,x_1\in C\\ w_m=x_m+\theta _{m}(x_m-x_{m-1}),\\ y_m=\text {argmin}_{y\in C}\{\lambda g(w_m,y) + \frac{1}{2}\Vert y-w_m\Vert ^2\},\\ x_{m+1}=\text {argmin}_{y\in C}\{\lambda g(y_m,y) + \frac{1}{2}\Vert y-w_m\Vert ^2\},\\ \end{array}\right. } \end{aligned}$$

(1.5)

where f is pseudomonotone and Lipschitz-type continuous with positive constants $L_1$ and $L_2$ such that $0<\lambda <\min \{\frac{1}{2L_1}, \frac{1}{2L_2}\}$, and $\theta _{m}$ is a suitable parameter. It is not hard to see that if $\theta _{m}=0,\forall m\in {\mathbb {N}}$, then (1.5) reduces to (1.4). Moreover, they proved the strong convergence results of the following algorithm for solving EP (1.3):

$$\begin{aligned} {\left\{ \begin{array}{ll} x_0,x_1\in C\\ w_m=x_m+\theta _{m}(x_m-x_{m-1}),\\ y_m=\text {argmin}_{y\in C}\{\lambda g(w_m,y) + \frac{1}{2}\Vert y-w_m\Vert ^2\},\\ z_{m}=\text {argmin}_{y\in C}\{\lambda g(y_m,y) + \frac{1}{2}\Vert y-w_m\Vert ^2\},\\ x_{m+1}=(1-\beta _{m}-\gamma _{m})w_m+\beta _{m}z_m, \end{array}\right. } \end{aligned}$$

(1.6)

where f is pseudomonotone and Lipschitz-type continuous with positive constants $L_1$ and $L_2$ such that $0<\lambda <\min \{\frac{1}{2L_1}, \frac{1}{2L_2}\},$ $\{\beta _{m}\},\{\gamma _{m}\}\subset (0,1)$ such that $\lim \limits _{m\rightarrow \infty }\gamma _{m}=0$, $\sum \limits _{m=0}^{\infty }\gamma _{m}=\infty$, and $\liminf _{m\rightarrow \infty }\beta _{m}(1-\beta _{m}-\gamma _{m})>0$.

On the other hand, the SEP is a combination of a pair of EPs, and aims at finding a solution ${\bar{x}}$ of an EP such that its image ${\bar{y}} = T{\bar{x}}$ under a given bounded linear operator T also solves another equilibrium problem. The solution set of SEP is denoted by

$$\begin{aligned} \Gamma = \{{\bar{x}} \in SOL(f,C) : T{\bar{x}} \in SOL(g,Q)\}. \end{aligned}$$

(1.7)

For some years now, many methods have focused on solving the SEP (see, [10, 21, 26] and the references therein). For instance, He [13] proposed the following proximal point method for solving the SEP (1.1)–(1.2):

$$\begin{aligned} {\left\{ \begin{array}{ll} x_0\in C,\\ y_m\in C\,\,\,\,\text {such that}\,\,\,f(y_m,y) + \frac{1}{r_m}\langle y-y_m,y_m-x_m\rangle \ge 0,\forall y\in C\\ u_m\in Q\,\,\,\,\text {such that}\,\,\,g(u_m,v) + \frac{1}{r_m}\langle v-u_m,u_m-Ty_m\rangle \ge 0,\forall v\in Q,\\ x_{m+1}=P_C(y_m+\eta T^*(u_m-Ty_m)), \end{array}\right. } \end{aligned}$$

(1.8)

where C and Q are two nonempty convex and closed subsets of the real Hilbert spaces $H_1$ and $H_2$, respectively, $f:C\times C\rightarrow {\mathbb {R}}$ and $g:Q\times Q\rightarrow {\mathbb {R}}$ are two monotone bifunctions, $\eta \in \left( 0,\frac{1}{\Vert T\Vert ^2}\right)$, $\{r_m\}\subset (0,+\infty )$ such that $\liminf _{m\rightarrow \infty }r_m>0$, and $T^*$ is the adjoint the bounded linear operator $T:H_1\rightarrow H_2$. The author showed that the sequence (1.8) converges weakly to the solution of the SEP (1.1)–(1.2). In [16], Kim and Dinh introduced the following extragradient method for solving the SEP (1.1)–(1.2):

$$\begin{aligned} {\left\{ \begin{array}{ll} x_0\in C\\ y_m=\text {argmin}_{y\in C}\{\lambda _m f(x_m,y) + \frac{1}{2}\Vert y-x_m\Vert ^2\},\\ z_{m}=\text {argmin}_{y\in C}\{\lambda _m f(y_m,y) + \frac{1}{2}\Vert y-x_m\Vert ^2\},\\ u_{m}=\text {argmin}_{u\in Q}\{\mu _m g(Tz_m,y) + \frac{1}{2}\Vert u-Tz_m\Vert ^2\},\\ v_{m}=\text {argmin}_{u\in Q}\{\mu _m g(u_m,u) + \frac{1}{2}\Vert u-Tz_m\Vert ^2\},\\ x_{m+1}=P_C(z_m+\eta T^*(v_m-Tz_m)), \end{array}\right. } \end{aligned}$$

(1.9)

where the bifunctions $f:H_1\times H_1\rightarrow {\mathbb {R}}$ and $g:H_2\times H_2\rightarrow {\mathbb {R}}$ are pseudomonotone and Lipschitz-type continuous with positive constants $L_1$ and $L_2$ such that $\{\lambda _m\}\{\mu _m\}\subset [\xi ,{\bar{\xi }}]$ with $0<\xi \le {\bar{\xi }}<\min \{\frac{1}{2L_1}, \frac{1}{2L_2}\},$ and $\eta \in \left( 0,\frac{1}{\Vert T\Vert ^2}\right)$. They proved that their method weakly converges to a solution of the SEP (1.1)–(1.2).

It important to outline the limitations of the above methods (1.4)–(1.9) as follows:

• (1) The extragradient methods (1.4)–(1.6) require solving two strongly convex optimization problems in the feasible sets C per iteration, and $\lambda$ depend on bifunctional Lipschitz-type constants. Also, (1.9) require solving four strongly convex optimization problems in the feasible sets C and Q per iteration, and $\lambda _m$ and $\mu _m$ depend on bifunctional Lipschitz-type constants.

• (2) The step-size $\eta$ of the extragradient methods (1.8) and (1.9) depends on the prior knowledge of the operator norm $\Vert T\Vert$ of the bounded linear operator.

• (3) The method (1.9) is not used to solve the problem (1.1)–(1.2) under the setting of $g:Q\times Q\rightarrow {\mathbb {R}}$, since one can not guarantee if $Tz_m$ belongs to the considered closed convex set Q.

The above mentioned drawbacks limit the applicability of the concerned methods and also affects their computational efficiency. In order to overcome the difficulty of solving two strongly convex optimization problems in the feasible sets per iteration, the subgradient extragradient method (SEM) in [8, 9] was reformulated by Rehman et al. [22] to solve EP (1.3). The main benefit of the SEM over the extragradient method is the fact that, the second convex optimization problem is onto a half-space and has a closed form solution. So, compared to the extragradient approach, its computational complexity is less expensive.

Motivated by the above results, we will continue with the development of an efficient method for solving SEP (1.1)–(1.2). We suggest a modified subgradient extragradient method that includes double inertial extrapolations and viscosity approach for finding the common solution of SEP and fixed point problem (FPP). The strong convergence result of the suggested method is obtained under some standard assumptions on the control parameters. Our method does not require solving two strongly convex optimization problems in the feasible sets per iteration, and the step-sizes do not depend on bifunctional Lipschitz-type constants. Furthermore, unlike several existing methods, our method does not depend on the prior knowledge of the operator norm of the bounded linear operator. Instead, the step-sizes are self adaptively updated. We apply our method to solve split variational inequality problem. Lastly, we conduct some numerical test to compare our method with some well known methods in the literature.

2 Preliminaries

In this section, we begin by recalling some known and useful results which are needed in the sequel. Let H be a real Hilbert space. We denotes strong and weak convergence by “$\rightarrow$” and “$\rightharpoonup$”, respectively.

Let $S:H_1\rightarrow H_1$ be a mapping. Then, a point $x\in H_1$ is called the fixed point of the mapping S if $Sx=x$. The theory of fixed point plays vital roles in many areas of applied sciences and engineering such as game theory, image processing problem, physics, chemistry and so on. We denoted by the set of all fixed point of S by $F(S)=\{x\in H_1:Sx=x\}.$ For any $x,y \in H$ and $\alpha \in [0,1],$ it is well-known that

$$\begin{aligned} \Vert x-y\Vert ^2&= \Vert x\Vert ^2 -2\langle x,y\rangle +\Vert y\Vert ^2. \end{aligned}$$

(2.1)

$$\begin{aligned} \Vert x+y\Vert ^2&= \Vert x\Vert ^2 +2\langle x,y\rangle +\Vert y\Vert ^2. \end{aligned}$$

(2.2)

$$\begin{aligned} \Vert x-y\Vert ^2&\le \Vert x\Vert ^2 +2 \langle y, x-y \rangle . \end{aligned}$$

(2.3)

$$\begin{aligned} \Vert \alpha x + (1- \alpha )y\Vert ^2&= \alpha \Vert x\Vert ^2 + (1- \alpha )\Vert y\Vert ^2- \alpha (1- \alpha )\Vert x-y\Vert ^2. \end{aligned}$$

(2.4)

$$\begin{aligned} \Vert \alpha x + \beta y + \gamma z\Vert ^2&= \alpha \Vert x\Vert ^2 + \beta \Vert y\Vert ^2 + \gamma \Vert z\Vert ^2- \alpha \beta \Vert x-y\Vert ^2\nonumber \\&\quad -\alpha \gamma \Vert x-z\Vert ^2 -\gamma \beta \Vert y-z\Vert ^2. \end{aligned}$$

(2.5)

Definition 2.1

[7, 8, 13] Let $T: H \rightarrow H$ be an operator. Then the operator T is called

• (a) L-Lipschitz continuous if there exists $L>0$ such that

  $$\begin{aligned} \Vert Tx-Ty\Vert \le L\Vert x-y\Vert , \end{aligned}$$

  for all $x,y\in H.$ If $L=1,$ then T is called nonexpansive, and T is quasinonexpansive, if

  $$\begin{aligned} \Vert x-Ty\Vert =\Vert Tx-Ty\Vert \le \Vert x-y\Vert , \end{aligned}$$

  for all $x\in F(T)$ and $y\in H.$;

• (b) monotone if

  $$\begin{aligned} \langle Tx-Ty, x-y\rangle \ge 0, ~~\forall x,y\in H; \end{aligned}$$

• (c) pseudomonotone if

  $$\begin{aligned} \langle Tx, y-x\rangle \ge 0 \Rightarrow \langle Ty, y-x\rangle \ge 0, ~~\forall x,y\in H; \end{aligned}$$

• (d) $\beta$- strongly monotone if there exists $\beta >0,$ such that

  $$\begin{aligned} \langle Tx-Ty, x-y\rangle \ge \beta \Vert x-y\Vert ^2, ~~\forall ~ x,y\in H; \end{aligned}$$

• (e) firmly nonexpansive

  $$\begin{aligned} \Vert Tx-Ty\Vert ^2 \le \langle Tx-Ty, x-y\rangle ~~\forall ~ x,y\in H; \end{aligned}$$

  or equivalently

  $$\begin{aligned} \Vert Tx-Ty\Vert ^2 \le \Vert x-y\Vert ^2 -\Vert (I-T)x- (I-T)y\Vert ^2 ~~\forall ~ x,y\in H; \end{aligned}$$

• (f) directed (also called to be firmly quasi-nonexpansive) if $F(T)\ne \emptyset$ and

  $$\begin{aligned} \langle x-y, Tx-Ty\rangle \ge \Vert Tx-p\Vert ^2 ~~\forall ~ x, y\in H ~~\text {and}~~p\in F(T); \end{aligned}$$

• (g) sequentially weakly continuous if for each sequence $\{x_n\},$ we obtain $\{x_n\}$ converges weakly to x implies that $Tx_n$ converges weakly to Tx.

Definition 2.2

[7, 11, 13, 16, 21] Let $g: C\times C \rightarrow {\mathbb {R}}$ is said to be:

• (a) Strongly monotone on C if there exists a constant $\tau >0$ such that

  $$\begin{aligned} g(x,y) + g(y,x)\le -\tau \Vert x-y\Vert ^2 \end{aligned}$$

  (2.6)

  for all $x,y\in C;$

• (b) Monotone on C if

  $$\begin{aligned} g(x,y) + g(y,x)\le 0 \end{aligned}$$

  (2.7)

  for all $x,y\in C;$

• (c) Strongly pseudomonote on C if there exists a constant $\gamma >0$ such that

  $$\begin{aligned} g(x,y)\ge 0\Rightarrow g(x,y)\le -\gamma \Vert x-y\Vert ^2, ~~\forall x,y\in C; \end{aligned}$$

• (d) Psuedomonotone on C if

  $$\begin{aligned} g(x,y)\ge 0 \Rightarrow g(y,x)\le 0, ~~\forall ~ x,y\in C; \end{aligned}$$

• (e) Satisfying a Lipschitz-like condition if there exist two positive constant $L_1, L_2$ such that

  $$\begin{aligned} g(x,y)+g(y,z)\ge g(x,z)-L_1\Vert x-y\Vert ^2-L_2\Vert y-z\Vert ^2~~\forall ~ x,y,z\in C.\nonumber \\ \end{aligned}$$

  (2.8)

Let C be a nonempty, closed and convex subset of H. For any $u\in H,$ there exists a unique point $P_{C}u\in C$ such that

$$\begin{aligned} \Vert u-P_Cu\Vert \le \Vert u-y\Vert ~~\forall y\in C. \end{aligned}$$

The operator $P_C$ is called the metric projection of H onto C. It is well-known that $P_C$ is a nonexpansive mapping and that $P_C$ satisfies

$$\begin{aligned} \langle x-y, P_Cx - P_Cy\rangle \ge \Vert P_Cx-P_Cy\Vert ^2, \end{aligned}$$

(2.9)

for all $x,y \in H.$ Furthermore, $P_C$ is characterized by the property

$$\begin{aligned} \Vert x-y\Vert ^2 \ge \Vert x-P_Cx\Vert ^2 + \Vert y-P_Cx\Vert ^2 \end{aligned}$$

and

$$\begin{aligned} \langle x- P_{C}x, y-P_{C}x\rangle \le 0, \end{aligned}$$

(2.10)

for all $x\in H$ and $y\in C.$ A subset C of H is called proximal if for each $x\in H,$ there exists $y\in C$ such that

$$\begin{aligned} \Vert x-y\Vert = d(x,C). \end{aligned}$$

The Hausdorff metric on H is as follows

$$\begin{aligned} {\mathbb {H}}(A,B):=\max \bigg \{\sup _{x\in A}d(x,B), \sup _{y\in B}d(y,A)\bigg \} \end{aligned}$$

for all subsets A and B of H.

The normal cone $N_C$ to C at a point $x\in C$ is defined by $N_C(x)=\{z\in H:\langle z, x-y\rangle \ge 0~\forall y\in C\}.$

Lemma 2.3

[11] Let C be a convex subset of a real Hilbert space H and $\phi :C \rightarrow {\mathbb {R}}$ be a subdifferential function on C. Then $x^*$ is a solution to the convex problem: $\text {minimize}\{\phi (x): x\in C\}$ if and only if $0\in \partial \phi (x^*) + N_C(x^*),$ where $\partial \phi (x^*)$ denotes the subdifferential of $\phi$ and $N_C(x^*)$ is the normal cone of C at $x^*.$

Lemma 2.4

[24] Let $\{a_m\}$ be a sequence of positive real numbers, $\{\alpha _{m}\}$ be a sequence of real numbers in (0, 1) such that $\sum _{m=1}^{\infty }\alpha _m=\infty$ and $\{d_m\}$ be a sequence of real numbers. Suppose that

$$\begin{aligned} a_{m+1} \le (1-\alpha _m) a_m + \alpha _{m}d_m, m\ge 1. \end{aligned}$$

If $\limsup _{k\rightarrow \infty }d_{m_k} \le 0$ for all subsequences $\{a_{m_k}\}$ of $\{a_{m}\}$ satisfying the condition

$$\begin{aligned} \liminf _{k\rightarrow \infty }\{a_{m_{k}+1}- a_{m_k}\}\ge 0, \end{aligned}$$

then, $\lim \limits _{m\rightarrow \infty }a_{m} =0.$

3 Proposed algorithm

In this section, we present our proposed method for solving a problem (1.1)–(1.2).

Assumption 3.1

Condition A. Suppose $f: H_2\times H_2 \rightarrow {\mathbb {R}}, g:H_1\times H_1\rightarrow {\mathbb {R}}$ satisfies the following conditions:

• (1) f is pseudomonotone on Q and satisfies the Lipschitz-type condition (2.8) on $H_2$ with positive constants $L_1, L_2$;

• (2) g is pseudomonotone on C and satisfies the Lipschitz-type condition (2.8) on $H_1$ with positive constants $L_3, L_4$;

• (3) $g(\cdot , u)$ and $f(\cdot , y)$ are sequentially weakly upper semi-continuous on C and Q for each fixed $y\in C$ and $u\in Q;$

• (4) $g(x, \cdot )$ and $f(y, \cdot )$ are convex, lower semi-continuous on $H_1$ and $H_2$ for every fixed $x\in H_1$ and $y\in H_2.$

In addition, we suppose that

• (1) $T: H_1 \rightarrow H_2$ is a bounded linear operator with the adjoint operator $T^*;$

• (2) $h: H_1\rightarrow H_1$ is a contraction with constant $\rho \in [0,1);$

• (3) $S:H_1\rightarrow H_1$ is a quasinonexpansive mapping;

• (4) The solution set $\Omega =\Gamma \cap F(S)\ne \emptyset ,$ where $\Gamma$ is as defined in (1.7).

Condition B. Suppose that $\{\alpha _{m}\}, \{\eta _{m}\},\{\beta _m\}, \{\epsilon _m\}$, $\{\delta _m\}$ and $\{\eta _{m}\}$ are positive sequences such that

• (1) $\{\alpha _{m}\}\subset (0,1),$ $\lim _{m\rightarrow \infty }\alpha _{m} =0,$ such that $\sum _{m=1}^{\infty }\alpha _{m}=\infty ,$ $\lim _{m\rightarrow \infty } \frac{\epsilon _m}{\alpha _m}=\frac{\delta _m}{\alpha _m}=0.$

• (2) $\{\beta _m\},\{\eta _{m}\} \subset [a,b]\subset (0,1)$ such that $\alpha _{m}+\eta _{m}+\beta _{m}=1$.

• (3) $\lambda _1,\psi ,\theta ,\tau _1>0$, $\phi ,\nu \in (0,1)$.

Remark 3.3

The sequence $\{\lambda _{m}\}$ and $\{\tau _m\}$ are generated by (3.3) and (3.4) are monotonically non-increasing and

$$\begin{aligned}&\lim _{m\rightarrow \infty }\lambda _{m}=\lambda \ge \frac{\nu }{2\max \{L_1, L_2\}}. \end{aligned}$$

(3.5)

$$\begin{aligned}&\lim _{m\rightarrow \infty }\tau _{m}=\tau \ge \frac{\phi }{2\max \{L_3, L_4\}}. \end{aligned}$$

(3.6)

Proof

Clearly, $\{\lambda _{m}\}$ is monotonically non-increasing. Also, since f satisfies condition A (1), we have

$$\begin{aligned}&\frac{\nu [\Vert Tw_m-y_m\Vert ^2 +\Vert z_m -y_m\Vert ^2]}{2(f(Tw_m , z_m)-f(Tw_m,y_m)-f(y_n, z_n))\}}\nonumber \\&\quad \ge \frac{\nu [\Vert Tw_m-y_m\Vert ^2 +\Vert z_m -y_m\Vert ^2]}{2(L_1\Vert Tw_m-y_m\Vert ^2+ L_2\Vert y_m -z_m\Vert ^2)}\nonumber \\&\quad \ge \frac{\nu }{2\max \{L_1, L_2\}}. \end{aligned}$$

(3.7)

Hence $\{\lambda _{m}\}$ is bounded below by $\frac{\nu }{2\max \{L_1, L_2\}}.$ This implies that there exists

$$\begin{aligned} \lim _{m\rightarrow \infty }\lambda _{m}=\lambda \ge \frac{\mu }{2\max \{L_1, L_2\}}. \end{aligned}$$

Using similar approach, we can show that (3.6) holds. $\square$

Remark 3.4

The choice of the step size $\{\gamma _m\}$ in Algorithms 3.2 do not require the prior knowledge of the operator norm $\Vert T\Vert .$ In addition, the step size is well defined. To see this, observe that From Algorithm 3.2, we have $z_m= \text {argmin}_{y\in Q_m}\{\lambda _{m} f(y_m, y)+ \frac{1}{2}\Vert y-Tw_m\Vert ^2\},$ and using Lemma 2.3, we get

$$\begin{aligned} \lambda _{m}f(y_m, y)-f(y_m, z_m)\ge \langle Tw_m -z_m, y-z_m\rangle , ~~\forall y\in Q_m. \end{aligned}$$

(3.8)

Let $p\in \text {SOL}(g,C),$ since $Tp\in \text {SOL}(f,Q)\subset Q\subset Q_m,$ and taking $y:=Tp,$ we get

$$\begin{aligned} \lambda _{m}(f(y_m, Tp)-f(y_m, z_m))\ge \langle Tw_m -z_m, Tp-z_m\rangle . \end{aligned}$$

(3.9)

Since $Tp, y_m \in Q,$ we have $f(Tp, y_m)\ge 0,$ and by the pseudomonotonicity of f,  we get $f(y_m, Tp)\le 0.$ Thus, (3.9) becomes

$$\begin{aligned} -\lambda _{m}f(y_m, z_m)\ge \langle Tw_m -z_m, Tp-z_m\rangle . \end{aligned}$$

(3.10)

In addition, from $\chi _m\in \partial f(Tw_m, \cdot )(y_m),$ we obtain

$$\begin{aligned} \lambda _{m}(f(Tw_m, z_m)-f(Tw_m, y_m))\ge \lambda _{m}\langle \chi _m, y-y_m\rangle . \end{aligned}$$

(3.11)

Using the definition of $Q_m,$ we get

$$\begin{aligned} \langle Tw_m -\lambda _{m}\chi _m -y_m, z_m-y_m \rangle \le 0, \end{aligned}$$

(3.12)

it follows that

$$\begin{aligned} \lambda _{m}\langle \chi _m, z_m-y_m\rangle \ge \langle Tw_m-y_m, z_m -y_m\rangle . \end{aligned}$$

(3.13)

From (3.13) and (3.11), we get

$$\begin{aligned} \lambda _{m}(f(Tw_m , z_m)-f(Tw_m,y_m))\ge \langle Tw_m-y_m, z_m -y_m\rangle . \end{aligned}$$

(3.14)

Also, adding (3.14) and (3.10), we obtain

$$\begin{aligned}&\lambda _{m}(f(Tw_m , z_m)-f(Tw_m,y_m)-f(y_m, z_m))\nonumber \\&\quad \ge \langle Tw_m-y_m, z_m -y_n\rangle + \langle Tw_n -z_n, Tp-z_m\rangle \nonumber \\&\quad \Rightarrow 2\lambda _{m}(f(Tw_m , z_m)-f(Tw_m,y_m)-f(y_m, z_m))\nonumber \\&\quad \ge 2\langle Tw_m-y_m, z_m -y_m\rangle +2 \langle Tw_m-z_m, Tp-z_m\rangle \nonumber \\&\quad =\Vert Tw_m-y_m\Vert ^2 + \Vert z_m-y_m\Vert ^2-\Vert z_m-Tw_m\Vert ^2 \nonumber \\&\qquad +\Vert Tw_m-z_m\Vert ^2+\Vert z_m-Tp\Vert ^2 -\Vert Tw_m -Tm\Vert ^2\nonumber \\&\quad =\Vert Tw_m-y_m\Vert ^2 + \Vert z_m-y_m\Vert ^2+\Vert z_m-Tp\Vert ^2 -\Vert Tw_m -Tp\Vert ^2. \end{aligned}$$

(3.15)

From (3.15), we get

$$\begin{aligned} \Vert z_m-Tp\Vert ^2&\le \Vert Tw_m -Tp\Vert ^2- \Vert Tw_m-y_m\Vert ^2 - \Vert z_m-y_m\Vert ^2 \nonumber \\&\quad + 2\lambda _{m}(f(Tw_m , z_m)-f(Tw_m,y_m)-f(y_m, z_m)), \end{aligned}$$

(3.16)

and using the step size (3.3), and the fact that $\nu \in (0,1),$ since $\lim _{m \rightarrow \infty }\lambda _{m+1} = \lim _{m \rightarrow \infty }\lambda _{m}>0,$ we $\lim _{n \rightarrow \infty }(1- \frac{\nu \lambda _{m}}{\lambda _{m+1}^2} )= 1-\nu >0,$ we obtain

$$\begin{aligned} \Vert z_m-Tp\Vert ^2&\le \Vert Tw_m -Tp\Vert ^2-(1-\frac{\nu \lambda _{m}}{\lambda _{m+1}})[\Vert Tw_m-y_m\Vert ^2+\Vert z_m-y_m\Vert ^2]\nonumber \\&\le \Vert Tw_m -Tp\Vert ^2. \end{aligned}$$

(3.17)

$$\begin{aligned} \Vert z_m -Tp\Vert&\le \Vert Tw_m -Tp\Vert . \end{aligned}$$

(3.18)

Using the Cauchy-Schwarz inequality and (3.17), we have

$$\begin{aligned} \Vert T^*(Tw_m-z_m)\Vert \Vert w_m-p\Vert&\ge \langle T^*(Tw_m-z_m), w_m -p \rangle \nonumber \\&=\langle Tw_m-z_m, Tw_m -Tp \rangle \nonumber \\&=\frac{1}{2}[\Vert Tw_m-z_m\Vert ^2 + \Vert Tw_m-Tp\Vert ^2 -\Vert z_m -Tp\Vert ^2]\nonumber \\&\ge \frac{1}{2}\Vert Tw_m-z_m\Vert ^2. \end{aligned}$$

(3.19)

Since $z_m \ne Tw_m,$ we have $\Vert Tw_m-z_m\Vert \ge 0,$ thus, we obtain that $\Vert T^*(Tw_m-z_m)\Vert \Vert w_m-p\Vert >0.$ Hence, we have $\Vert T^*(Tw_m-z_m)\Vert \ne 0$ and so $\gamma _m$ is well defined.

4 Convergence analysis

Lemma 4.1

Let $\{x_m\}$ be a sequence generated by Algorithm 3.2, under Assumption 3.1, we obtain that $\{x_n\}$ is bounded.

Proof

Let $p\in SOL(g,C),$ using (3.1) and the fact that $0\le \theta _m \le {\overline{\theta }}_m,$ we have

$$\begin{aligned} \theta _m\Vert x_m-x_{m-1}\Vert \le {\overline{\theta }}_m\Vert x_m -x_{m-1}\Vert \le \epsilon _m. \end{aligned}$$

Therefore, it follows from $\lim _{m \rightarrow \infty }\frac{\epsilon _m}{\alpha _{m}}=0,$ that

$$\begin{aligned} \lim \limits _{m\rightarrow \infty }\frac{\theta _m}{\alpha _{m}}\Vert x_m-x_{m-1}\Vert \le \lim \limits _{m\rightarrow \infty }\frac{\epsilon _m}{\alpha _{m}}=0. \end{aligned}$$

(4.1)

It follows that the sequence $\{\frac{\theta _m}{\alpha _{m}}\Vert x_m-x_{m-1}\Vert \}$ is bounded. Hence, there exists $N_1>0$ such that $\frac{\theta _m}{\alpha _{m}}\Vert x_m-x_{m-1}\Vert \le N_2,$ for all $m\in {\mathbb {N}}.$ Then using Algorithm 3.2, we have

$$\begin{aligned} \Vert w_m -p\Vert&= \Vert x_m+\theta _{m}(x_{m}-x_{m-1})-p\Vert \nonumber \\&\le \Vert x_m -p\Vert +\theta _{n}\Vert x_{n}-x_{n-1}\Vert \nonumber \\&= \Vert x_m -p\Vert +\alpha _{n}\frac{\theta _{m}}{\alpha _{m}}\Vert x_{m}-x_{m-1}\Vert \nonumber \\&\le \Vert x_m -p\Vert +\alpha _{m}N_1. \end{aligned}$$

(4.2)

Similarly, we get

$$\begin{aligned} \Vert s_m -p\Vert&= \Vert x_m+\delta _{m}(x_{m}-x_{m-1})-p\Vert \nonumber \\&\le \Vert x_m -p\Vert +\delta _{m}\Vert x_{m}-x_{m-1}\Vert \nonumber \\&= \Vert x_m -p\Vert +\alpha _{m}\frac{\delta _{m}}{\alpha _{m}}\Vert x_{m}-x_{m-1}\Vert \nonumber \\&\le \Vert x_m -p\Vert +\alpha _{m}N_2. \end{aligned}$$

(4.3)

Furthermore, using Algorithm 3.2 and the step size, we have

$$\begin{aligned} \Vert v_m-p\Vert ^2&=\Vert w_m+\gamma _mT^*(z_m -Tw_m) -p \Vert \nonumber \\&=\Vert w_m -p\Vert ^2 + \gamma _m^2\Vert T^*(z_n-Tw_m)\Vert ^2 + 2\gamma _m\langle w_m -p, T^*(z_m-Tw_m)\rangle \nonumber \\&=\Vert w_m -p\Vert ^2 + \gamma _m^2\Vert T^*(z_m-Tw_m)\Vert ^2 + 2\gamma _m\langle Tw_m-Tp, z_m-Tw_m\rangle \nonumber \\&=\Vert w_m -p\Vert ^2 + \gamma _m^2\Vert T^*(z_m-Tw_m)\Vert ^2 + \gamma _m\Vert z_m-Tp\Vert ^2 -\gamma _m\Vert Tw_m -Tp\Vert ^2\nonumber \\&\quad -\gamma _m\Vert z_m -Tw_m\Vert ^2\nonumber \\&\le \Vert w_m- p\Vert ^2 + \gamma _n^2\Vert T^*(z_m-Tw_m)\Vert ^2 + \gamma _m\Vert Tw_m-Tp\Vert ^2 -\gamma _m\Vert Tw_m -Tp\Vert ^2\nonumber \\&\quad -\gamma _m\Vert z_m -Tw_m\Vert ^2\nonumber \\&\le \Vert w_m- p\Vert ^2 + \gamma _m^2\Vert T^*(z_m-Tw_m)\Vert ^2 -\gamma _m(\gamma _m+\epsilon )\Vert T^*(z_m-Tw_m)\Vert ^2\nonumber \\&= \Vert w_m- p\Vert ^2 -\gamma _m\epsilon \Vert T^*(z_m-Tw_m)\Vert ^2\nonumber \\&\le \Vert w_m -p\Vert ^2, \end{aligned}$$

(4.4)

which implies that

$$\begin{aligned} \Vert v_m-p\Vert&\le \Vert w_m -p\Vert . \end{aligned}$$

(4.5)

Also, using same approach as in (3.17), we have

$$\begin{aligned} \Vert q_m-p\Vert ^2&\le \Vert v_m-p\Vert ^2-\left( 1-\phi \frac{\tau _{m}}{\tau _{m+1}}\right) [\Vert v_m -u_m\Vert ^2 + \Vert q_m-u_m\Vert ^2]\nonumber \\&\le \Vert v_m-p\Vert ^2. \end{aligned}$$

(4.6)

which implies that

$$\begin{aligned} \Vert q_m-p\Vert&\le \Vert v_m -p\Vert \le \Vert w_m-p\Vert . \end{aligned}$$

(4.7)

Finally, using Algorithm 3.2, (4.2), (4.3), and (4.7), we get

$$\begin{aligned} \Vert x_{m+1}-p\Vert&=\Vert \alpha _mh(s_m) + \eta _{m}q_m + \beta _{m}Sq_m -p\Vert \nonumber \\&=\Vert \alpha _m(h(s_m)-p) + \eta _{m}(q_m-p) + \beta _{m}(Sq_m -p)\Vert \nonumber \\&\le \alpha _m\Vert h(s_m)- h(p)\Vert + \alpha _{m}\Vert h(p)-p\Vert + \eta _{m}\Vert q_m-p\Vert + \beta _{m}\Vert Sq_m -p\Vert \nonumber \\&\le \alpha _m\rho \Vert s_m-p\Vert + \alpha _{m}\Vert h(p)-p\Vert + \eta _{m}\Vert q_m-p\Vert + \beta _{m}\Vert q_m -p\Vert \nonumber \\&\le \alpha _m\rho \Vert s_m-p\Vert + \alpha _{m}\Vert h(p)-p\Vert +(1-\alpha _{m})\Vert q_m-p\Vert \nonumber \\&\le \alpha _m\rho \Vert x_m-p\Vert +\alpha _{m}N_2+ \alpha _{m}\Vert h(p)-p\Vert +(1-\alpha _{m})\Vert x_m-p\Vert + \alpha _{m}N_1\nonumber \\&= (1-\alpha _m(1-\rho ))\Vert x_m-p\Vert +\alpha _{m}N_3+ \alpha _{m}\Vert h(p)-p\Vert \nonumber \\&\le (1-\alpha _m(1-\rho )) \Vert x_m-p\Vert + \alpha _{m}(1-\rho )\frac{N_3 + \Vert h(p)-p\Vert }{(1-\rho )}\nonumber \\&\le \max \bigg \{\Vert x_m -p\Vert , \frac{N_3 + \Vert h(p)-p\Vert }{1- \rho }\bigg \}\nonumber \\&\le \nonumber \\&\vdots \nonumber \\&\le \max \bigg \{\Vert x_0 -p\Vert , \frac{N_3 + \Vert h(p)-p\Vert }{1-\rho }\bigg \}, \end{aligned}$$

(4.8)

where $N_3=N_1+N_2.$ Thus, $\{x_m\}$ is bounded. $\square$

Theorem 4.2

Let $\{x_m\}$ be the sequence generated by Algorithm 3.2. Then, under the Assumption 3.1, $\{x_m\}$ converges strongly to $p \in \Omega ,$ where $p=P_{\Omega }\circ h(p).$

Proof

Let $p\in \Omega ,$ and using (3.9), we have

$$\begin{aligned} \Vert w_m -p\Vert ^2&= \Vert x_m + \theta _m(x_m - x_{m-1}) -p\Vert ^2\nonumber \\&= \Vert (x_m-p) + \theta _m(x_m - x_{m-1})\Vert ^2\nonumber \\&= \Vert x_m-p\Vert ^2 + 2\theta _m\langle x_m -p, x_m -x_{m-1} \rangle + \theta _m^2 \Vert x_m -x_{m-1}\Vert ^2\nonumber \\&\le \Vert x_m-p\Vert ^2 + 2\theta _m\Vert x_{m}-x_{m-1}\Vert \Vert x_m -p\Vert + \theta _m^2\Vert x_m - x_{m-1}\Vert ^2\nonumber \\&\le \Vert x_m-p\Vert ^2 + \theta _m\Vert x_m -x_{m-1}\Vert [2\Vert x_m -p\Vert + \theta _m\Vert x_m - x_{m-1}\Vert ]\nonumber \\&= \Vert x_m-p\Vert ^2 + \theta _m\Vert x_m -x_{m-1}\Vert [2\Vert x_m -p\Vert + \alpha _{m}\frac{\theta _m}{\alpha _{m}}\Vert x_m - x_{m-1}\Vert ]\nonumber \\&\le \Vert x_m-p\Vert ^2 + \theta _m\Vert x_m -x_{m-1}\Vert [2\Vert x_m -p\Vert + \alpha _{m}N_1]\nonumber \\&\le \Vert x_m-p\Vert ^2 + \theta _m\Vert x_m -x_{m-1}\Vert N_3, \end{aligned}$$

(4.9)

for some $N_3=\sup _{m\in {\mathbb {N}}}\{ 2\Vert x_m -p\Vert , \alpha _{m}N_1\}>0.$

Similarly, we get

$$\begin{aligned} \Vert s_m-p\Vert ^2&\le \Vert x_m -p\Vert ^2 + \delta _m\Vert x_m -x_{m-1}\Vert N_4. \end{aligned}$$

(4.10)

In addition, we obtain

$$\begin{aligned} \Vert x_{m+1}-p\Vert ^2&=\Vert \alpha _mh(s_m) + \eta _{m}q_m + \beta _{m}Sq_m -p\Vert ^2\nonumber \\&=\Vert \alpha _m(h(s_m)-h(p)) + \eta _{m}(q_m-p) + \beta _{m}(Sq_m -p) + \alpha _{m}(h(p)-p)\Vert ^2\nonumber \\&\le \Vert \alpha _m(h(s_m)-h(p)) + \eta _{m}(q_m-p) + \beta _{m}(Sq_m -p)\Vert ^2\nonumber \\&\quad +2 \alpha _{m}\langle h(p)-p, x_{n+1}-p\rangle \nonumber \\&\le \alpha _m\Vert h(s_m)-h(p)\Vert ^2 + \eta _{m}\Vert q_m-p\Vert ^2 + \beta _{m}\Vert Sq_m -p\Vert ^2\nonumber \\&\quad +2 \alpha _{m}\langle h(p)-p, x_{m+1}-p\rangle \nonumber \\&\le \alpha _m\rho ^2\Vert s_m-p\Vert ^2 + \eta _{m}\Vert q_m-p\Vert ^2 + \beta _{m}\Vert q_m -p\Vert ^2 \nonumber \\&\quad +2 \alpha _{m}\langle h(p)-p, x_{m+1}-p\rangle \nonumber \\&\le \alpha _m\rho \rho \Vert s_m-p\Vert ^2 + (1-\alpha _{m})\Vert q_m -p\Vert ^2\nonumber \\&\quad +2 \alpha _{m}\langle h(p)-p, x_{m+1}-p\rangle \nonumber \\&\le \alpha _m\rho \Vert s_m-p\Vert ^2 + (1-\alpha _{m})\Vert v_m -p\Vert ^2 +2 \alpha _{m}\langle h(p)-p, x_{m+1}-p\rangle \nonumber \\&\le \alpha _m\rho \Vert s_m-p\Vert ^2 + (1-\alpha _{m})\Vert w_m -p\Vert ^2 +2 \alpha _{m}\langle h(p)-p, x_{m+1}-p\rangle \nonumber \\&\le \alpha _m\rho \Vert x_m -p\Vert ^2 + \delta _m\Vert x_m-x_{m-1}\Vert N_4 \nonumber \\&\quad + (1-\alpha _{m})[\Vert x_m -p\Vert ^2 + \theta _m\Vert x_m -x_{m-1}\Vert N_3]\nonumber \\&\quad +2 \alpha _{m}\langle h(p)-p, x_{m+1}-p\rangle \nonumber \\&=(1-\alpha _{m}(1-\rho ))\Vert x_m -p\Vert ^2 \nonumber \\&\quad + \alpha _{m}(1-\rho )\bigg [\frac{\delta _m\Vert x_m -x_{m-1}\Vert N_4}{\alpha _{m}(1-\rho )} + \frac{\theta _m\Vert x_m -x_{m-1}N_3\Vert }{\alpha _{m}(1-\rho )}\nonumber \\&\quad + \frac{2}{(1-\rho )}\langle h(p)-p, x_{m+1}-p\rangle \bigg ]\nonumber \\&=(1-\alpha _{m}(1-\rho ))\Vert x_m -p\Vert ^2 + \alpha _{m}(1-\rho )\Psi _m, \end{aligned}$$

(4.11)

where $\Psi _m= \bigg [\frac{\delta _m\Vert x_m -x_{m-1}\Vert N_3}{\alpha _{m}(1-\rho )} + \frac{\theta _m\Vert x_m -x_{m-1}\Vert N_4}{\alpha _{m}(1-\rho )} + \frac{2}{(1-\rho )}\langle h(p)-p, x_{m+1}-p\rangle \bigg ].$ According to Lemma 2.4, to conclude our proof, it is sufficient to establish that $\limsup _{k\rightarrow \infty } \Psi _{m_k} \le 0$ for every subsequence $\{\Vert x_{m_k}-p\Vert \}$ of $\{\Vert x_{m}-p\Vert \}$ satisfying the condition:

$$\begin{aligned} \liminf _{k\rightarrow \infty }\{\Vert x_{m_{k}+1}-p\Vert - \Vert x_{m_k}-p\Vert \}\ge 0. \end{aligned}$$

(4.12)

From (4.11), (4.6), (4.10) and (4.9), we get

$$\begin{aligned} \Vert x_{{m_k}+1}-p\Vert ^2&\le \alpha _{m_k}\Vert s_{m_k}-p\Vert ^2 + (1-\alpha _{m_k})\Vert q_{m_k} -p\Vert ^2 +2 \alpha _{m_k}\langle h(p)-p, x_{{m_k}+1}-p\rangle \nonumber \\&\le \alpha _{m_k} [ \Vert x_{m_k}-p\Vert ^2 + \delta _{m_k}\Vert x_{m_k}-x_{{m_k}-1}\Vert N_4]+ (1-\alpha _{m_k})(\Vert v_{m_k}-p\Vert ^2\nonumber \\&\quad -\left( 1-\phi \frac{\tau _{m_k}}{\tau _{{m_k}+1}}\right) [\Vert v_{m_k} -u_{m_k}\Vert ^2 + \Vert q_{m_k}-u_{m_k}\Vert ^2])\nonumber \\&\quad +2 \alpha _{m_k}\langle h(p)-p, x_{m+1}-p\rangle \nonumber \\&\le \alpha _{m_k} [ \Vert x_{m_k}-p\Vert ^2 + \delta _{m_k}\Vert x_{m_k}-x_{{m_k}-1}\Vert N_4]+ (1-\alpha _{m_k})(\Vert w_{m_k}-p\Vert ^2\nonumber \\&\quad -\left( 1-\phi \frac{\tau _{m}}{\tau _{{m_k}+1}}\right) [\Vert v_{m_k}-u_{m_k}\Vert ^2 + \Vert q_{m_k}-u_{m_k}\Vert ^2])\nonumber \\&\quad +2 \alpha _{m_k}\langle h(p)-p, x_{m+1}-p\rangle \nonumber \\&\le \alpha _{m_k}\Vert x_{m_k}-p\Vert ^2 + \delta _{m_k}\Vert x_{m_k}-x_{{m_k}-1}\Vert N_4 \nonumber \\&\quad + (1-\alpha _{m_k})[\Vert x_{m_k}-p\Vert ^2 + \theta _{m_k}\Vert x_{m_k} -x_{m-1}\Vert N_5]\nonumber \\&\quad -(1-\alpha _{m_k})\left( 1-\phi \frac{\tau _{m_k}}{\tau _{{m_k}+1}}\right) [\Vert v_{m_k} -u_{m_k}\Vert ^2 + \Vert q_{m_k}-u_{m_k}\Vert ^2]\nonumber \\&\quad +2 \alpha _{m_k}\langle h(p)-p, x_{{m_k}+1}-p\rangle \nonumber \\&\le \Vert x_{m_k}-p\Vert ^2 + \delta _{m_k}\Vert x_{m_k} -x_{{m_k}-1}\Vert N_4 + \theta _{m_k}\Vert x_{m_k} -x_{m-1}\Vert N_5\nonumber \\&\quad -(1-\alpha _{m_k})\left( 1-\phi \frac{\tau _{m_k}}{\tau _{{m_k}+1}}\right) [\Vert v_{m_k}-u_{m_k}\Vert ^2 + \Vert q_{m_k}-u_{m_k}\Vert ^2] \nonumber \\&\quad +2 \alpha _{m_k}\langle h(p)-p, x_{{m_k}+1}-p\rangle , \end{aligned}$$

(4.13)

which implies that

$$\begin{aligned}&\limsup _{k\rightarrow \infty }\bigg ( (1-\alpha _{m_k})\bigg (1-\phi \frac{\tau _{n_k}}{\tau _{m_k+1}}\bigg )[\Vert v_{m_k} -u_{m_k}\Vert ^2 + \Vert q_{m_k}-u_{m_k}\Vert ^2] \bigg )\\&\quad \le \limsup _{k\rightarrow \infty }\bigg [\Vert x_{m_k}-p\Vert ^2 + \alpha _{m_k}\frac{\theta _{m_k}}{\alpha _{m_k}}\Vert x_{m_k} -x_{m_k-1}\Vert N_5 + \alpha _{m_k}\frac{\delta _{m_k}}{\alpha _{m_k}}\Vert x_{m_k} -x_{m_k-1}\Vert N_4\\&\qquad +2 \alpha _{m_k}\langle h(p)-p, x_{m_k+1}-p\rangle -\Vert x_{m_k+1}-p\Vert ^2\bigg ] \\&\quad \le -\liminf _{k\rightarrow \infty }[\Vert x_{{m_k}+1}-p\Vert ^2-\Vert x_{m_k}-p\Vert ^2 ]\le 0. \end{aligned}$$

Thus, we have

$$\begin{aligned} \lim \limits _{k\rightarrow \infty }\Vert v_{m_k} -u_{m_k}\Vert = 0= \lim \limits _{k\rightarrow \infty }\Vert q_{m_k}-u_{m_k}\Vert . \end{aligned}$$

(4.14)

From (4.14), we have

$$\begin{aligned} \Vert q_{m_k}-v_{m_k}\Vert \le \Vert q_{m_k} -u_{m_k}\Vert + \Vert u_{m_k} -v_{m_k}\Vert \rightarrow 0~\text {as}~n\rightarrow \infty . \end{aligned}$$

(4.15)

Furthermore, using (4.11), (4.4), (4.10) and (4.9), we get

$$\begin{aligned} \Vert x_{{m_k}+1}-p\Vert ^2&\le \alpha _{m_k}\Vert s_{m_k}-p\Vert ^2 + (1-\alpha _{m_k})\Vert v_{m_k}-p\Vert ^2\nonumber \\&\quad +2 \alpha _{m_k}\langle h(p)-p, x_{{m_k}+1}-p\rangle \nonumber \\&\le \alpha _{m_k}\Vert s_{m_k}-p\Vert ^2 + (1-\alpha _{m_k})[\Vert w_{m_k}- p\Vert ^2 \nonumber \\&\quad -\gamma _{m_k}\epsilon \Vert T^*(z_{m_k}-Tw_{m_k})\Vert ^2] +2 \alpha _{m_k}\langle h(p)-p, x_{{m_k}+1}-p\rangle \nonumber \\&\le \alpha _{m_k}[\Vert x_{m_k} -p\Vert ^2 + \delta _{m_k}\Vert x_{m_k} -x_{{m_k}-1}\Vert N_4]\nonumber \\&\quad + (1-\alpha _{m_k})[\Vert x_{m_k} -p\Vert ^2 + \theta _{m_k}\Vert x_{m_k} -x_{m-1}\Vert N_3 ]\nonumber \\&\quad -(1-\alpha _{m_k})\gamma _{m_k}\epsilon \Vert T^*(z_{m_k}-Tw_{m_k})\Vert ^2 +2 \alpha _{m_k}\langle h(p)-p, x_{{m_k}+1}-p\rangle \nonumber \\&\le \Vert x_{m_k} -p\Vert ^2 + \delta _{m_k}\Vert x_{m_k} -x_{{m_k}-1}\Vert N_4 + \theta _{m_k}\Vert x_{m_k} -x_{{m_k}-1}\Vert N_3 \nonumber \\&\quad -(1-\alpha _{m_k})\epsilon ^2\Vert T^*(z_{m_k}-Tw_{m_k})\Vert ^2 +2 \alpha _{m_k}\langle h(p)-p, x_{{m_k}+1}-p\rangle , \end{aligned}$$

(4.16)

which implies that

$$\begin{aligned}&\limsup _{k\rightarrow \infty }\bigg ( (1-\alpha _{m_k})\epsilon ^2\Vert T^*(z_{m_k}-Tw_{m_k})\Vert ^2 \bigg )\\&\quad \le \limsup _{k\rightarrow \infty }\bigg [\Vert x_{m_k}-p\Vert ^2 + \alpha _{m_k}\frac{\theta _{m_k}}{\alpha _{m_k}}\Vert x_{m_k} -x_{m_k-1}\Vert N_5 + \alpha _{m_k}\frac{\delta _{m_k}}{\alpha _{m_k}}\Vert x_{m_k} -x_{m_k-1}\Vert N_4\\&\qquad +2 \alpha _{m_k}\langle h(p)-p, x_{{m_k}+1}-p\rangle -\Vert x_{m_k+1}-p\Vert ^2\bigg ] \\&\quad \le -\liminf _{k\rightarrow \infty }[\Vert x_{{m_k}+1}-p\Vert ^2-\Vert x_{m_k}-p\Vert ^2 ]\le 0. \end{aligned}$$

Thus, we have

$$\begin{aligned} \lim \limits _{k\rightarrow \infty }\Vert T^*(z_{m_k}-Tw_{m_k})\Vert =0. \end{aligned}$$

(4.17)

Thus, using (4.17), (3.19) and the boundedness of $\{w_n\}$, we obtain

$$\begin{aligned} \lim _{k\rightarrow \infty }\Vert Tw_{m_k}-z_{m_k}\Vert =0. \end{aligned}$$

(4.18)

Now, observe that

$$\begin{aligned} \Vert z_{m_k}-Tp\Vert ^2&=\Vert Tw_{m_k}-Tp\Vert ^2 + \Vert Tw_{m_k} -z_{m_k}\Vert ^2-2\langle Tw_{m_k}-Tp, Tw_{m_k} -z_{m_k}\rangle \nonumber \\&\ge \Vert Tw_{m_k}-Tp\Vert ^2 + \Vert Tw_{m_k} -z_{m_k}\Vert ^2-2 \Vert Tw_{m_k}-Tp\Vert \Vert Tw_{m_k} -z_{m_k}\Vert , \end{aligned}$$

(4.19)

which implies that

$$\begin{aligned} -\Vert z_{m_k}-Tp\Vert ^2&\le -\Vert Tw_{m_k}-Tp\Vert ^2 - \Vert Tw_{m_k} -z_{m_k}\Vert ^2+2 \Vert Tw_{m_k}-Tp\Vert \Vert Tw_{m_k} -z_{m_k}\Vert \end{aligned}$$

(4.20)

Adding (4.20) and (3.17), we get

$$\begin{aligned}&\left( 1-\frac{\nu \lambda _{m_k}}{\lambda _{{m_k}+1}}\right) [\Vert Tw_{m_k}-y_{m_k}\Vert ^2+\Vert z_{m_k}-y_{m_k}\Vert ^2]\nonumber \\&\quad \le 2 \Vert Tw_{m_k}-Tp\Vert \Vert Tw_{m_k} -z_{m_k}\Vert - \Vert Tw_{m_k} -z_{m_k}\Vert ^2. \end{aligned}$$

(4.21)

Thus, using (4.18), we get

$$\begin{aligned}&\lim _{k\rightarrow \infty }\Vert Tw_{m_k}-y_{m_k}\Vert =0=\lim _{k\rightarrow \infty }\Vert z_{m_k}-y_{m_k}\Vert . \end{aligned}$$

(4.22)

$$\begin{aligned}&\Vert Tw_{m_k}-z_{m_k}\Vert \le \Vert Tw_{m_k} -y_{m_k}\Vert + \Vert y_{m_k} -z_{m_k}\Vert \rightarrow 0~\text {as}~k\rightarrow \infty . \end{aligned}$$

(4.23)

In addition, using Algorithm (3.2) and (2.5), we have

$$\begin{aligned} \Vert x_{m_k+1}-p\Vert ^2&= \Vert \alpha _{m_k} (h(s_{m_k})-p) + \eta _{{m_k}}(q_{m_k}-p) + \beta _{m_k} (Sq_{m_k}-p)\Vert ^2\nonumber \\&\le \alpha _{m_k} \Vert h(s_{m_k})-p\Vert ^2 \nonumber \\&\quad + \eta _{m_k}\Vert q_{m_k}-p\Vert ^2 + \beta _{m_k} \Vert Sq_{m_k}-p\Vert ^2 - \beta _{m_k}\eta _{m_k}\Vert q_{m_k} -Sq_{m_k}\Vert ^2\nonumber \\&\le \alpha _{m_k} [ \Vert h(s_{m_k})-h(p)\Vert + \Vert h(p)-p\Vert ]^2 \nonumber \\&\quad + (1-\alpha _{m_k})\Vert q_{m_k}-p\Vert ^2 - \beta _{m_k}\eta _{m_k}\Vert q_{m_k} -Sq_{m_k}\Vert ^2\nonumber \\&\le \alpha _{m_k} [ \rho \Vert s_{m_k}-p\Vert + \Vert h(p)-p\Vert ]^2 \nonumber \\&\quad + (1-\alpha _{m_k})\Vert q_{m_k}-p\Vert ^2 - \beta _{m_k}\eta _{m_k}\Vert q_{m_k} -Sq_{m_k}\Vert ^2\nonumber \\&\le \alpha _{m_k} [ \rho ^2\Vert s_{m_k}-p\Vert ^2+ \Vert h(p)-p\Vert ^2 \nonumber \\&\quad + 2\rho \Vert h(p)-p\Vert \Vert s_{m_k}-p\Vert ]^2 + (1-\alpha _{m_k})\Vert w_{m_k}-p\Vert ^2 \nonumber \\&\quad - \beta _{m_k}\eta _{m_k}\Vert q_{m_k} -Sq_{m_k}\Vert ^2\nonumber \\&\le \alpha _{m_k} \Vert s_{m_k}-p\Vert + \alpha _{m_k}\Vert h(p)-p\Vert ^2\nonumber \\&\quad + 2\rho \alpha _{m_k}\Vert h(p)-p\Vert \Vert s_m-p\Vert + (1-\alpha _{m_k})\Vert w_{m_k}-p\Vert ^2\nonumber \\&\quad - \beta _{m_k}\eta _{m_k}\Vert q_{m_k} -Sq_{m_k}\Vert ^2\nonumber \\&\le \alpha _{m_k} [\Vert x_{m_k} -p\Vert ^2 + \delta _{m_k}\Vert x_{m_k}-x_{{m_k}-1}\Vert N_4]\nonumber \\&\quad + \alpha _{m_k}\Vert h(p)-p\Vert ^2 + 2\rho \alpha _{m_k}\Vert h(p)-p\Vert \Vert s_{m_k}-p\Vert \nonumber \\&\quad + (1-\alpha _{m_k})[\Vert x_{m_k} -p\Vert ^2 + \theta _{m_k}\Vert x_{m_k}-x_{{m_k}-1}\Vert N_4] - \beta _{m_k}\eta _{m_k}\Vert q_{m_k} -Sq_{m_k}\Vert ^2\nonumber \\&\le \Vert x_{m_k} -p\Vert ^2 + \alpha _{m_k}\delta _{m_k}\Vert x_{m_k}-x_{{m_k}-1}\Vert N_3+ \alpha _{m_k}\Vert h(p)-p\Vert ^2\nonumber \\&\quad + 2\rho \alpha _{m_k}\Vert f(p)-p\Vert \Vert s_{m_k}-p\Vert \nonumber \\&\quad + \theta _{m_k}\Vert x_{m_k}-x_{{m_k}-1}\Vert N_4 - \beta _{m_k}\eta _{m_k}\Vert q_{m_k} -Sq_{m_k}\Vert ^2 \end{aligned}$$

(4.24)

which implies

$$\begin{aligned}&\limsup _{k\rightarrow \infty }\bigg ( \beta _{m_k}\eta _{m_k}\Vert q_{m_k} -Sq_{m_k}\Vert ^2 \bigg )\\&\quad \le \limsup _{k\rightarrow \infty }\bigg [\Vert x_{m_k}-p\Vert ^2 + \alpha _{m_k}\frac{\theta _{m_k}}{\alpha _{m_k}}\Vert x_{m_k} -x_{m_k-1}\Vert N_4\\&\qquad + \alpha _{m_k}\delta _{m_k}\Vert x_{m_k} -x_{m_k-1}\Vert N_3+ \alpha _{m_k}\Vert h(p)-p\Vert ^2\\&\qquad + 2\rho \alpha _{m_k}\Vert h(p)-p\Vert \Vert z_{m_k}-p\Vert -\Vert x_{m_k+1}-p\Vert ^2\bigg ] \\&\quad \le -\liminf _{k\rightarrow \infty }[\Vert x_{{m_k}+1}-p\Vert ^2-\Vert x_{m_k}-p\Vert ^2 ]\le 0. \end{aligned}$$

Thus, we have

$$\begin{aligned} \lim \limits _{k\rightarrow \infty }\Vert q_{m_k} -Sq_{m_k}\Vert =0. \end{aligned}$$

(4.25)

Using, Algorithm 3.2, we have

$$\begin{aligned} \lim \limits _{k\rightarrow \infty }\Vert s_{m_k} -x_{m_k}\Vert \le \lim \limits _{k\rightarrow \infty } \alpha _{m_k}\cdot \frac{\delta _{m_k}}{\alpha _{m_k}}\Vert x_{m_k} -x_{m_k-1}\Vert =0, \end{aligned}$$

(4.26)

and

$$\begin{aligned}&\lim \limits _{k\rightarrow \infty }\Vert w_{m_k} -x_{m_k}\Vert \le \lim \limits _{k\rightarrow \infty } \alpha _{m_k}\cdot \frac{\theta _{m_k}}{\alpha _{m_k}}\Vert x_{m_k} -x_{m_k-1}\Vert =0. \end{aligned}$$

(4.27)

$$\begin{aligned}&\Vert v_{m_k}-x_{m_k}\Vert = \Vert w_{m_k}-x_{m_k}\Vert + \gamma _{m_k} \Vert T^*(z_{m_k} -Tw_{m_k})\Vert \rightarrow 0~\text {as}~k \rightarrow \infty . \end{aligned}$$

(4.28)

Using (4.26), (4.27), (4.25) and (4.14), we get

$$\begin{aligned}&\lim \limits _{k\rightarrow \infty }\Vert w_{m_k} -v_{m_k}\Vert \le \lim \limits _{k\rightarrow \infty }\Vert w_{m_k} -x_{m_k}\Vert + \lim \limits _{k\rightarrow \infty } \Vert x_{m_k} -v_{n_k}\Vert =0 \end{aligned}$$

(4.29)

$$\begin{aligned}&\lim \limits _{k\rightarrow \infty }\Vert q_{m_k} -x_{m_k}\Vert \le \lim \limits _{k\rightarrow \infty } \Vert q_{m_k} -v_{m_k}\Vert + \lim \limits _{k\rightarrow \infty }\Vert v_{m_k} -x_{m_k}\Vert =0, \end{aligned}$$

(4.30)

$$\begin{aligned}&\lim \limits _{k\rightarrow \infty }\Vert u_{m_k} -x_{m_k}\Vert \le \lim \limits _{k\rightarrow \infty } \Vert u_{m_k} -v_{m_k}\Vert + \lim \limits _{k\rightarrow \infty }\Vert v_{m_k} -x_{m_k}\Vert =0 \end{aligned}$$

(4.31)

$$\begin{aligned}&\lim \limits _{k\rightarrow \infty }\Vert x_{m_k+1}- q_{m_k}\Vert \le \lim \limits _{k\rightarrow \infty }\alpha _{m_k} \Vert h(z_{m_k}) - q_{m_k}\Vert \nonumber \\&\quad +\lim \limits _{k\rightarrow \infty } \eta _{m_k}\Vert q_{m_k}-q_{m_k}\Vert + \lim \limits _{k\rightarrow \infty }\beta _{m_k} \Vert Sq_{m_k}-q_{m_k}\Vert =0 \end{aligned}$$

(4.32)

$$\begin{aligned}&\lim \limits _{k\rightarrow \infty }\Vert x_{m_k+1}-x_{m_k}\Vert \le \lim \limits _{k\rightarrow \infty }\Vert x_{m_k+1}-q_{m_k} \Vert + \lim \limits _{k\rightarrow \infty }\Vert q_{m_k}-x_{m_k}\Vert =0. \end{aligned}$$

(4.33)

Now, since $\{x_{m_k}\}$ is bounded, then, there exists a subsequence $\{x_{m_{k_j}}\}$ of $\{x_{n_k}\}$ such that $\{x_{m_{k_j}}\}$ converges weakly to $x^*\in H.$ In addition, using (4.30) and the boundedness of $\{q_{m_k}\}$, there exists a subsequence $\{q_{m_{k_j}}\}$ of $\{q_{m_k}\}$ such that $\{q_{m_{k_j}}\}$ converges weakly to $x^*\in H_1$ and since S is demiclosed with (4.25), we have that $x^*\in F(S).$ We now establish that $x^*\in Sol(g,C).$ Also, (4.14), we obtain that $\{u_{m_{k_j}}\}$ converges weakly to $x^*.$ Next, we claim that $x^*\in SOL(g,C).$ To see this, from the definition of $u_{m_k}$ and Lemma 2.3, we get

$$\begin{aligned} 0\in \partial \bigg (\tau _{m_k}g(v_{m_k}, u) + \frac{1}{2}\Vert v_{m_k} -u\Vert ^2\bigg )(u_{m_k}) + N_C(u_{m_k}). \end{aligned}$$

Thus, there exists $\mu _{m_k}\in N_{C}(u_{m_k})$ and $\psi _{m_k}\in \partial g(v_{m_k}, \cdot )(u_{m_k})$ such that

$$\begin{aligned} \tau _{m}\psi _{m_k} +u_{m_k}-v_{m_k}+\mu _{m_k}=0. \end{aligned}$$

(4.34)

Since $\mu _{m_k}\in N_{C}(u_{m_k})$, it follows that $\langle \mu _{m_k}, u-u_{m_k}\rangle \le 0,$ for all $u\in C.$ From (4.34), we have

$$\begin{aligned} \tau _{m_k}\langle \mu _{n_k}, u-u_{m_k}\rangle \ge \langle v_{m_k}-u_{m_k}, u-u_{m_k}\rangle ~~\forall ~u\in C. \end{aligned}$$

(4.35)

Also, since $\psi _{m_k}\in \partial g(v_{m_k}, \cdot )(u_{m_k}),$ we have

$$\begin{aligned} g(v_{m_k}, y)- g(v_{m_k},u_{m_k})\ge \langle \mu _{m_k}, u-u_{m_k}\rangle ~~\forall ~u\in C. \end{aligned}$$

(4.36)

Thus,

$$\begin{aligned} \tau _{m_k}\bigg [g(v_{m_k}, u)-g(v_{m_k},u_{m_k})\bigg ]\ge \langle v_{m_k}-u_{m_k}, u-u_{m_k}\rangle ~~\forall ~u\in C. \end{aligned}$$

(4.37)

Taking limit as $k\rightarrow \infty ,$ and using conditions and (4.14), we have

$$\begin{aligned} g((x^*,u))\ge 0~~\forall ~u\in C. \end{aligned}$$

Thus, $x^* \in SOL(g,C).$ We now establish that $Tx^*\in SOL(f,Q).$ To see this observe that from the definition of $y_{m_k}$ and Lemma 2.3, we get

$$\begin{aligned} 0\in \partial \bigg (\lambda _{m_k}f(Tw_{m_k}, y) + \frac{1}{2}\Vert Tw_{m_k} -y\Vert ^2\bigg )(y_{m_k}) + N_C(y_{m_k}). \end{aligned}$$

Thus, there exists $\rho _{m_k}\in N_{Q}(y_{m_k})$ and $\chi _{m_k}\in \partial f(Tw_{m_k}, \cdot )(y_{m_k})$ such that

$$\begin{aligned} \lambda _{n}\chi _{m_k} +y_{m_k}-Tw_{m_k}+\rho _{m_k}=0. \end{aligned}$$

(4.38)

Since $\rho _{m_k}\in N_{Q}(y_{n_k})$, it follows that $\langle \rho _{m_k}, y-y_{m_k}\rangle \le 0,$ for all $y\in Q.$ From (4.38), we have

$$\begin{aligned} \lambda _{m_k}\langle \rho _{m_k}, y-y_{m_k}\rangle \ge \langle Tw_{m_k}-y_{m_k}, y-y_{m_k}\rangle ~~\forall ~y\in C. \end{aligned}$$

(4.39)

Also, since $\chi _{m_k}\in \partial f(Tw_{m_k}, \cdot )(y_{m_k}),$ we have

$$\begin{aligned} f(Tw_{m_k}, y)- f(Tw_{m_k},y_{m_k})\ge \langle \rho _{m_k}, y-y_{m_k}\rangle ~~\forall ~y\in Q. \end{aligned}$$

(4.40)

Thus,

$$\begin{aligned} \lambda _{m_k}\bigg [f(Tw_{m_k}, y)-f(Tw_{m_k},y_{m_k})\bigg ]\ge \langle Tw_{m_k}-y_{m_k}, y-y_{m_k}\rangle ~~\forall ~y\in Q. \end{aligned}$$

(4.41)

From (4.27), we obtain that $w_{m_k}$ converges to $x^{*}$ and since T is a bounded linear operator, we obtain that $\{Tw_{m_k}\}$ converges $Tx^{*}$. Hence, using (4.22), we have that $\{y_{m_k}\}$ also converges to $Tx^{*}.$ Thus, taking limit as $k\rightarrow \infty ,$ in (4.41) and using the fact that $\lim _{k\rightarrow \infty }\lambda _{m_k}>0$, we have

$$\begin{aligned} f((Tx^*,y))\ge 0~~\forall ~y\in Q. \end{aligned}$$

Thus, $Tx^* \in SOL(f,Q).$ Hence, we have that $x^*\in \Omega .$ Furthermore, we get

$$\begin{aligned} \limsup _{k\rightarrow \infty }\langle h(p)-p, x_{m_k}-p\rangle&=\limsup _{j\rightarrow \infty }\langle h(p)-p, x_{m_{k_j}}-p\rangle \nonumber \\&= \langle h(p)-p, x^*-p \rangle \le 0, \end{aligned}$$

(4.42)

using (4.33) and (4.42), we have

$$\begin{aligned} \limsup _{k\rightarrow \infty }\langle h(p)-p, x_{m_k+1}-p\rangle&= \limsup _{k\rightarrow \infty }\langle h(p)-p, x_{m_k+1}-x_{n_k}\rangle \nonumber \\&\quad + \limsup _{k\rightarrow \infty }\langle h(p)-p, x_{m_k}-p\rangle \nonumber \\&= \langle h(p)-p, x^*-p \rangle \le 0. \end{aligned}$$

(4.43)

which implies that

$$\begin{aligned} \limsup _{k\rightarrow \infty }\langle h(p)-p, x_{m_{k}+1}-p\rangle \le 0. \end{aligned}$$

(4.44)

Using our Condition B (1) and the above inequality, we have that $\limsup _{k\rightarrow \infty }\Psi _{m_k} = \limsup _{k\rightarrow \infty } \bigg [\frac{\delta _{m_k}\Vert x_{m_k}-x_{{m_k}-1}\Vert N_3}{\alpha _{m_k}(1-\rho )} + \frac{\theta _{m_k}\Vert x_{m_k} -x_{m-1}\Vert N_4}{\alpha _{m_k}(1-\rho )} + \frac{2}{(1-\rho )}\langle h(p)-p, x_{{m_k}+1}-p\rangle \bigg ]\le 0.$ Thus, by Lemma 2.4, we have $\lim \limits _{m\rightarrow \infty }\Vert x_m-p\Vert =0.$ Thus, $\{x_m\}$ converges strongly to $p\in \Omega .$ $\square$

5 Application to split variational inequality problem

In this section, we will apply our results to split variational inequality problem (SVIP). Let H be a real Hilbert space and C be a nonempty closed convex subset H. The classical variational inequality problem (VIP) for an operator $B:C\rightarrow C$ is formulated as follows: find $w^*\in C$ such that

$$\begin{aligned} \langle Bw^*,y-w^*\rangle \ge 0,\forall y\in H. \end{aligned}$$

(5.1)

The solution set of the VIP (5.1) is denoted by VI(C, B). Let $H_1$ and $H_2$ be two real Hilbert spaces with nonempty closed convex subset C and Q, respectively. Let $B:C\rightarrow C$ and $D:Q\rightarrow Q$ be $L^*_1$ and $L^*_2$–Lipschitz continuous on C and D, respectively. Let $T:H_1\rightarrow H_2$ be a bounded linear operator with its adjoint $T^*$. The split variational inequality problem is generated as: Find $w^*\in C$ such that

$$\begin{aligned} \langle Bw^*,y-w^*\rangle \ge 0,\forall y\in C, \end{aligned}$$

(5.2)

and ${z}=T{w^*}\in Q$ solves

$$\begin{aligned} \langle Dz,r-z\rangle \ge 0,\forall r\in Q. \end{aligned}$$

(5.3)

We denote the solution set of SVIP (5.2)–(5.3) by $\Gamma ^*=\{w^*\in VI(C,B):Tw^*\in VI(Q,D)\}$. Now, we consider the following conditions for solving the SVIP (5.2)–(5.3):

($A_1$):

$B:C\rightarrow C$ and $D:Q\rightarrow Q$ are pseudomonotone operators, i.e.

$$\begin{aligned} \langle Bw,y-w\rangle \ge 0\implies \langle By,w-y\rangle \le 0,\forall w,y\in C. \end{aligned}$$

($A_2$):

$B:C\rightarrow C$ and $D:Q\rightarrow Q$ are $L^*_1$ and $L^*_2$–Lipschitz continuous operators, i.e. there exist $L^*_1>0$ such that

$$\begin{aligned} \Vert Bw-By\Vert \le L^*_1\Vert w-y\Vert ,\forall w,y\in C. \end{aligned}$$

($A_3$):

$B:C\rightarrow C$ and $D:Q\rightarrow Q$ are sequentially weakly continuous operators.

Set $f(Tw,y)=\langle DTw,y-Tw\rangle$, $\forall y\in Q$ and $g(v,u)=\langle Bv,u-v\rangle$, $\forall u,v\in C$, then the (SEP) becomes the (SVIP) with $L^*_2=2L_1=2L_2$ and $L^*_1==2L_3=2L_3$. Moreover, we have

$$\begin{aligned} y_m=\text {argmin}_{y\in Q}\{\lambda _mf(Tw_m,y)+\frac{1}{2}\Vert y-Tw_m\Vert ^2\}=P_Q(Tw_m-\lambda _{m}DTw_m) \end{aligned}$$

and

$$\begin{aligned} u_m=\text {argmin}_{u\in C}\{\tau _mg(v_m,u)+\frac{1}{2}\Vert u-v_m\Vert ^2\}=P_C(v_m-\tau _{m}Bv_m), \end{aligned}$$

where $P_C$ and $P_Q$ are called the metric projections of $H_1$ and $H_2$ onto C and Q. Hence, we obtain the following result:

Corollary 5.1

Let $H_1$ and $H_2$ be two real Hilbert spaces with nonempty closed convex subset C and Q, respectively. Let $T:H_1\rightarrow H_2$ be a bounded linear operator with its adjoint $T^*$. Assume that condition B and assumption ($A_1$)–($A_3$) hold. Let $h:H_1\rightarrow H_1$ be a contraction mapping with contraction constant $\rho \in [0,1)$ and $S:H_1\rightarrow H_1$ be a quasinonexpasive mapping such that the solution set $\Omega ^*=\Gamma ^*\cap F(S)\ne \emptyset$. Then, the sequence generated by Algorithm 5.2 strongly converges to an element $\Omega ^*$.

6 Numerical experiments

In this section, we present some numerical examples to validate our mains results. We compare the proposed Algorithm 3.2 (briefly, Alg. 3.2) with Algorithm (1.9) (briefly, KD Alg. 1.8), Algorithm 1 (briefly, SPK Alg. 1) of Suantai et al. [26] and Algorithm 2 (briefly, SPK Alg. 2) of Suantai et al. [26]. We perform all numerical simulations using MATLAB R2020b and carried out on PC Desktop Intel® Core$^{\textrm{TM}}$ i7-3540 M CPU @ 3.00GHz $\times$ 4 memory 400.00GB.

Example 6.1

Let $H_1={\mathbb {R}}^k$ and $H_2={\mathbb {R}}^m$ be two real Hilbert spaces with the Euclidean norm. We consider the bifunctions ${\tilde{f}}$ and ${\tilde{g}}$ which are generated from Nash-Cournot oligopolistic equilibrium models of electricity markets [10, 21],

$$\begin{aligned} {\tilde{f}}(x,y)&=\langle U_1x+V_1y,y-x\rangle ,\,\,\forall x,y\in {\mathbb {R}}^k, \end{aligned}$$

(6.1)

$$\begin{aligned} {\tilde{g}}(x,y)&=\langle U_2u+V_2v,v-u\rangle \,\,\forall u,v\in {\mathbb {R}}^m, \end{aligned}$$

(6.2)

where $U_1,V_1\in {\mathbb {R}}^{k\times k}$ and $U_2,V_2\in {\mathbb {R}}^{m\times m}$ are matrices such that $V_1,V_2$ are symmetric positive semidefinite and $V_1-U_1$, $V_2-U_2$ are negative semidefinite. Note that ${\tilde{f}}(x,y)= {\tilde{g}}(x,y)=(x-y)^T(V_1-U_1)(x-y),\forall x,y\in {\mathbb {R}}^k$. Furthermore, from the property of $V_1-U_1$, we have that the operator ${\tilde{f}}$ is monotone. Also, we have that ${\tilde{g}}$ is a monotone operator.

Now, we consider the two bifunctions f and g which are defined by

$$\begin{aligned} {f}(x,y)= {\left\{ \begin{array}{ll} {\tilde{f}}(x,y),\,\,\text {if}\,\,(x,y)\in Q\times Q\\ 0, \hspace{1cm}\text {otherwise}, \end{array}\right. } \end{aligned}$$

(6.3)

and

$$\begin{aligned} {g}(u,v)= {\left\{ \begin{array}{ll} {\tilde{g}}(u,v),\,\,\text {if}\,\,(x,y)\in C\times C\\ 0, \hspace{1cm}\text {otherwise}, \end{array}\right. } \end{aligned}$$

(6.4)

where $C=\prod _{i=1}^{k}[-5,5]$ and $Q=\prod _{i=1}^{k}[-20,20]$ are the constrained boxes. We observe that f and g are Lipschitz-type continuous with constants $L_1=L_2=\frac{1}{2}\Vert U_1-Q_1\Vert$ and $L_3=L_4=\frac{1}{2}\Vert U_2-Q_2\Vert$, respectively (see [28]). Take $c_1=\max \{L_1,L_3\}$, and $c_2=\max \{L_2,L_4\}$. Then, the bifunctions f and g are Lipschitz-type continuous with constants $c_1$ and $c_2$.

In this numerical test, the matrices $U_1,V_1,U_2$ and $V_2$ are generated randomly in the interval [-5,5] such that they fulfill the required conditions above and the linear operator $T:{\mathbb {R}}^k\rightarrow {\mathbb {R}}^k$ is a $k\times k$ matrix such that its entries is randomly generated in the interval [-2,2], and $h:C\rightarrow C,$ and $S: C\rightarrow C$ be a $k\times k$ matrix such that $\Vert h\Vert <1$ and $S=I$ (identity mapping). Clearly, the solution set $\Omega =\{0\}$. We consider the following control parameters for KD Alg. 1.8, SPK Alg. 1 and SPK Alg. 2: $\eta =\frac{1}{2\Vert T\Vert ^2}$, $\alpha =0.5$, $\lambda _{m}=\mu _m=\frac{1}{4\max \{c_1,c_2\}}$, $\epsilon _m=\frac{1}{(m+1)^2}$, $\gamma _m=\frac{1}{m+1}$, and $\beta _{m}=0.5(1-\gamma _m)$. Now, for Alg. 3.2, we consider the following control parameters: $\epsilon _m=\delta _m=\frac{1}{(m+1)^2}$, $\alpha _m=\frac{1}{m+1}$, $\eta _{m}=\beta _{m}=0.5(1-\alpha _{m})$, $\lambda _{1}=2.5$ $\tau _{1}=1.7$, $\psi =0.5$, $\theta =0.4$, $\phi =0.6$ and $\nu =0.8$. The starting point $x_0=x_1\in {\mathbb {R}}^k$ are generated randomly in [-5,5] and the stopping criteria for the concerned algorithm is given as $\Vert x_{m+1}-x_m\Vert \le 10^{-9}$. The parameters k and m are picked as follows: $(k=40,m=20)$, $(k=80,m=40)$, $(k=120,m=60).$

Fig. 1

Example 6.1, Case 1 (top left); Case 2 (top right); Case 3 (bottom )

Example 6.2

Let $H_2=(\ell _2({\mathbb {R}}),\Vert \cdot \Vert _{\ell _2})=H_1$, where $\ell _2({\mathbb {R}})=\{x=(x_1,x_2,x_3,\cdots ), u_i\in {\mathbb {R}}:\sum _{j=1}^{\infty }|x_j|<\infty \}$ and $\Vert x\Vert _{\ell _2}=(\sum _{j=1}^{\infty }|x_j|^2)^\frac{1}{2}$, $\forall u\in \ell _2({\mathbb {R}})$ (Fig. 1). We now define the operator $B:\ell _2({\mathbb {R}})\rightarrow \ell _2({\mathbb {R}})$ by

$$\begin{aligned} Tx=\left( 0,x_1,\frac{x_2}{2},\frac{x_3}{3},\cdots \right) ,\,\,\forall x\in \ell _2({\mathbb {R}}). \end{aligned}$$

Then, T is a bounded and linear operator on $\ell _2({\mathbb {R}})$ with adjoint

$$\begin{aligned} T^*v=\left( v_2,\frac{v_3}{2},\frac{v_4}{3},\cdots \right) ,\,\,\forall v\in \ell _2({\mathbb {R}}). \end{aligned}$$

To show this, let $x=(x_1,x_2,x_3,\cdots )$ and $v=(v_1,v_2,v_3,\cdots )$ be arbitrary points in $\ell _2({\mathbb {R}})$ and $\beta _1$, $\beta _2$ be arbitrary scalar in ${\mathbb {R}}$. Then,

$$\begin{aligned} B(\beta _1x+\beta _2 v)=&\left( 0,\beta _1x_1+\beta _2 v_1,\frac{\beta _1x_2+\beta _2 v_2}{2},\frac{\beta _1x_3+\beta _2 v_3}{3},\cdots \right) \\ =&\beta _1B(x)+\beta _2B(v). \end{aligned}$$

Thus, this shows that T is linear. In a addition, $\Vert Tx\Vert _{\ell _2}\Vert \le \Vert x\Vert _{\ell _2},\,\forall x\in \ell _2({\mathbb {R}})$. Hence, the operator T is also bounded. Showing that $T^*$ is the adjoint of T follows immediately from definition.

Fig. 2

Example 6.2, Case 1 (top left); Case 2 (top right); Case 3 (bottom )

Let $C=Q=\{x\in \ell _2({\mathbb {R}}):\Vert x-p\Vert _{\ell _2}\le q\}$, where $p=\left( 1,\frac{1}{2},\frac{1}{3},\cdots \right)$, $q=4$ for C and $p=\left( \frac{1}{2},\frac{1}{4},\frac{1}{8}\cdots \right)$, for $q=1$ for Q. Then, obviously C and Q are nonempty, convex and closed subsets of $\ell _2({\mathbb {R}})$. We now define the metric projections $P_C$ and $P_Q$ as follows:

$$\begin{aligned} P_C(x)=P_Q(x)=&{\left\{ \begin{array}{ll}x,\,\,\,\hspace{2cm}\text {if}\,\,x\in \Vert x-p\Vert _{\ell _2}\le q, \\ \frac{x-p}{\Vert x-p\Vert _{\ell _2}}q+p,\hspace{.5cm}\text {otherwise}. \end{array}\right. } \end{aligned}$$

Next, let the operators $B,D:\ell _2({\mathbb {R}})\rightarrow \ell _2({\mathbb {R}})$ be defined by

$$\begin{aligned} B(x_1,x_2,x_3,\cdots )=(x_1e^{-{x^2_1}},0,0,\cdots ), \,\,D(x_1,x_2,x_3,\cdots )=(6x_1e^{-{x^2_1}},0,0,\cdots ). \end{aligned}$$

It is not hard to verify that B and D are pseudomonotone operators. Furthermore, we define the mapping $S:\ell _2({\mathbb {R}})\rightarrow \ell _2({\mathbb {R}})$ by

$$\begin{aligned} Sx=(0,x_1,x_2,\cdots ),\,\,\forall x\in \ell _2({\mathbb {R}}). \end{aligned}$$

We define $h:\ell _2({\mathbb {R}})\rightarrow \ell _2({\mathbb {R}})$ by

$$\begin{aligned} hx=\left( 0,\frac{x_1}{2}, \frac{x_2}{2},\frac{x_3}{2},\cdots \right) ,\,\,\forall x\in \ell _2({\mathbb {R}}). \end{aligned}$$

Throughout the numerical experiments, we use

$$\begin{aligned} TOL_m=\frac{1}{2}(\Vert x_m-P_{Q_m}(u_m-\tau _mBx_k)\Vert ^2+\Vert Tx_m-P_C(Tx_m-\lambda _mDTx_m)\Vert ^2)<\upsilon , \end{aligned}$$

for stopping criterion, where $\upsilon =10^{-6}$ and we consider the following cases (Fig. 2):

Case I: $u_0=\left( \frac{1}{5},\frac{1}{6},\frac{1}{7},\cdots \right)$ and $u_1=\left( 1,\frac{1}{3},\frac{1}{4},\cdots \right)$.

Case II: $u_0=\left( \frac{1}{6},\frac{1}{8},\frac{1}{11},\cdots \right)$ and $u_1=\left( 1,\frac{1}{9},\frac{1}{11},\cdots \right)$.

Case III: $u_0=\left( 1,\frac{1}{8},\frac{1}{10},\cdots \right)$ and $u_=\left( \frac{1}{5},\frac{1}{7},\frac{1}{8}\cdots \right)$.

For this numerical experiment, we compare our Algorithm (3.2) (shortly, Alg.3.2) with Algorithm (4.8) of Tiang and Jiang [27] (briefly, TT Alg 4.8), Algorithm 1 of Huy et al. [15] (briefly, HHT Alg 1) and Algorithm 2 of Ogwo et al. [20] (briefly, OIM Alg. 2). We choose the following control parameters for various algorithms:

• (1) Alg 3.2: $\epsilon _m=\delta _m=\frac{1}{(m+1)^2}$, $\alpha _m=\frac{1}{m+1}$, $\eta _{m}=\beta _{m}=0.5(1-\alpha _{m})$, $\lambda _{1}=2.5$ $\tau _{1}=1.7$, $\psi =0.5$, $\theta =0.4$, $\phi =0.6$ and $\nu =0.8$.

• (2) TT Alg 4.8: $\alpha _{m}=\frac{1}{m+1}$ and $\mu =0.66$.

• (3) HHT Alg 1: $\alpha _{m}=\frac{1}{m+1}$, $\lambda _0=2.5$, $\mu _0=1.9$, $\mu =0.66$ and $\lambda =0.72$.

• (3) OIM Alg. 2: $\delta _{m}=\frac{1}{m+1}$, $\tau _m=\frac{100}{(k+1)^2}$, $\theta _m=\frac{1}{2}-\alpha _{m}$, $a_i=0.84$ $(i=1,2)$, $\gamma _i=1.84$ $(i=1,2)$, $\alpha =3$ and $\eta =0.96$.

7 Conclusion

In this work, we have introduced an efficient method for approximating the solution of a split equilibrium problem and a fixed point problem in the framework of Hilbert spaces. The strong convergence result of the proposed method is achieved under some weak conditions on the control parameters. Furthermore, we apply our results to solve split variational inequality problem and fixed point problem. In addition, we presented some numerical experiments to show the efficiency and the applicability of our method in comparison with other existing method. It is easy to see that from the numerical experiments that our proposed method is better than some existing methods in the literature.