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\title[A modified inertial shrinking projection algorithm]{A modified inertial shrinking projection algorithm with adaptive step size for solving split generalized equilibrium, monotone inclusion and fixed point problems}
\author{A- O. E. Owolabi}
\address{School of Mathematics, Statistics and Computer Science, \\  University of KwaZulu-Natal,\\
Durban, South Africa}
\email{218086824@stu.ukzn.ac.za}
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\author{T. O. Alakoya}
\address{School of Mathematics, Statistics and Computer Science, \\  University of KwaZulu-Natal,\\
Durban, South Africa}
\email{alakoyat1@ukzn.ac.za, timimaths@gmail.com}

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\author{O. T. Mewomo}
\address{School of Mathematics, Statistics and Computer Science, \\  University of KwaZulu-Natal,\\
Durban, South Africa}
\email{mewomoo@ukzn.ac.za}
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\subjclass{65K15, 47J25, 65J15}
\keywords{Split generalized equilibrium problem, Monotone inclusion problem, Inertial method, Fixed point problem, Strict Pseudo-contractions, Multivalued mappings.}
\begin{abstract}
In this paper, we study the common solution problem of split generalized equilibrium problem, monotone inclusion problem and common fixed point problem for a countable family of strict pseudo-contractive multivalued mappings. We propose a modified shrinking projection algorithm of inertial form with self-adaptive step sizes for finding a common solution of the aforementioned problem. The self-adaptive step size eliminates the difficulty of computing the operator norm while the  inertial term accelerates the rate of convergence of the  proposed algorithm. Moreover, unlike several of the existing results in the literature, the monotone inclusion problem considered is a more general problem involving the sum of Lipschitz continuous monotone operators and maximal monotone operators, and knowledge of the Lipschitz constant is not required to implement our algorithm. Under some mild conditions, we establish strong convergence result for the proposed method. Finally, we present some applications and numerical experiments to illustrate the usefulness and applicability of our algorithm as well as comparing it with some related methods. Our results improve and extend  corresponding results in the literature.

\end{abstract}
\maketitle

\section{Introduction}
	\noindent Let $H$ be a real Hilbert space with induced norm $\|\cdot\|$ and inner product $\langle \cdot, \cdot \rangle$. Let $C$ be a nonempty closed convex subset of a real Hilbert space and let $F : C \times C \rightarrow \mathbb{R}$  be A bifunction. The \textit{equilibrium problem} (shortly, (EP)) in the sense of Blum and Oettli \cite{BlOe} is to find $\hat{x}\in C$ such that
	\begin{equation}\label{c1x}
		F(\hat{x},y)\geq 0,\quad \forall~ y\in C.
	\end{equation}
	\noindent The set of all solutions of EP \eqref{c1x} is denoted by $EP(F).$ The EP attracts considerable research efforts and serves as a unifying framework for studying many well-known
	problems, such as the Nonlinear Complementarity Problems (NCPs), Optimization Problems (OPs), Variational Inequality Problems (VIPs), Saddle Point Problems (SPPs), the Fixed Point Problem (FPP), the Nash equilibria and many others, and has many applications in physics and economics, (see, for example \cite{alakoya,CeYa,CeGiRe,ogwo,ogwo1,uzor} and the references therein).
	
	\noindent On the other hand, the {\it generalized equilibrium problem} (GEP) is defined as finding a point $x \in C$ such that
	\begin{equation}\label{c2x}
		F(x, y) + \phi(x, y) \geq 0, \forall y \in C,
	\end{equation} 	
	
	
	\noindent where $F, \phi : C \times C \rightarrow \mathbb{R}$ are bifunctions. We denote  the solution set of GEP \eqref{c2x} by $GE{P(F, \phi)}$. If $\phi = 0$, then the GEP \eqref{c2x} reduces to the equilibrium problem \eqref{c1x}.
	
	\noindent Let $C \subseteq H_1$ and $Q \subseteq H_2$ where $H_1$ and $H_2$ are real Hilbert spaces. Let $F_1, \phi_1 : C \times C \rightarrow \mathbb{R}$ and
	$F_2, \phi_2 : Q \times Q \rightarrow \mathbb{R}$ be nonlinear bifunctions, and $A : H_1 \rightarrow H_2$ be a bounded linear operator. The {\it split generalized equilibrium problem} (SGEP) introduced by Kazmi and Rizvi \cite{kaz} is defined as follows: Find $\bar{x} \in C$ such that
	\begin{equation}\label{c3x}
		F_1(\bar{x}, x) + \phi_1(\bar{x}, x) \geq 0, \forall x \in C,
	\end{equation}
	and such that
	\begin{equation}\label{c4x}
		\bar{y} = A\bar{x} \in Q~~\text{solves}~~F_2(\bar{y}, y) + \phi_2(\bar{y}, y) \geq 0, ~~\forall y \in Q.
	\end{equation}	
	\noindent The solution set of the split generalized equilibrium problem is denoted by
	\begin{equation}\label{c*}
		SGEP(F_1, \phi_1, F_2, \phi_2) = \{\bar{x} \in C : \bar{x} \in GEP(F_1, \phi_1)~~\text{and}~~A\bar{x} \in GEP(F_2, \phi_2)\}.
	\end{equation}	
	\noindent If $\phi_1 = 0$ and $\phi_2 = 0$, we obtain a special case of the split generalized equilibrium problem \eqref{c3x}-\eqref{c4x} called the {\it split equilibrium problem} (SEP) which is defined as follows:
	\begin{equation}\label{c5x}
		F_1(\bar{x}, x) \geq 0, \forall x \in C,
	\end{equation}
	and such that
	\begin{equation}\label{c6x}
		\bar{y} = A\bar{x} \in Q~~\text{solves}~~F_2(\bar{y}, y)  \geq 0, ~~\forall y \in Q.
	\end{equation}	 	
	\noindent We denote the solution set of the SEP \eqref{c5x}-\eqref{c6x}  by $\Omega := \{\bar{x} \in EP(F_1) : A\bar{x} \in EP(F_2)\}$. The split generalized equilibrium problem has been studied by numerous authors and several iterative algorithms have been proposed by many authors for solving the problem (see, \cite{phu, sitt}).
	
	\noindent Another important problem that we consider is the \textit{monotone inclusion problem} (MIP), which is defined as finding a point $z\in H$ such that
	\begin{equation}\label{abx}
		0 \in (B + D)z,
	\end{equation}
	where $B : H\to H$ is a nonlinear operator and $D : H\to 2^H$ is a set-valued operator. We denote the set of solutions of \eqref{abx} by  $(B+D)^{-1}(0).$ The MIP \eqref{abx} and related optimization problems have been studied by several authors with various iterative algorithms proposed for approximating their solutions in Hilbert spaces and  Banach spaces (see, for instance \cite{alakoya2,mou, Nak,tse, yuy,uzor1}). One of the most efficient methods for solving the MIP is the forward-backward splitting method (see \cite{atto,bot, cho, eck1, eck2, lor}).
	
	\noindent Martinez \cite{mart1} first introduced the Proximal Point Algorithm (PPA) for finding the zero point of a maximal monotone operator $B$. The sequence generated by  PPA is defined as follows:
	\begin{equation*}
		x_{n+1} = J_{r_n}^Dx_n,
	\end{equation*}
	where $0 < r_n < \infty$, $J_{r_n}^D = (I + r_nD)^{-1}$ is the resolvent operator of $D$ and $I$ is the identity mapping. This algorithm was eventually modified by Rockafellar \cite{rock} to the following PPA with errors:
	\begin{equation*}
		x_{n+1} = J_{r_n}^Dx_n + e_n,
	\end{equation*}
	where $\{e_n\}$ is an error sequence. It was proved that if $e_n\to 0$ such that $\sum\limits_{n=1}^{\infty}\|e_n\| < +\infty$, and the solution set $D^{-1}(0) \neq \emptyset$ and $\liminf\limits_{n\to\infty}r_n > 0$, then the sequence $\{x_n\}$ converges weakly to a zero point of $D.$
	
	\noindent Also, Moudafi and Th\'era \cite{mou} introduced the following iterative algorithm for solving MIP \eqref{abx}:
	\begin{equation}
		\begin{cases}
			x_n = J^D_{r}v_n,\\
			v_{n+1} = tv_n + (1 - t)x_n - \mu(1 - t)Bx_n,
		\end{cases}
	\end{equation}
	where $t \in (0,1),$ $r>0$, $B$ is Lipschitz continuous and strongly monotone and $D$ is maximal monotone. They proved that the sequence $\{x_n\}$ generated by the iterative algorithm converges weakly to an element in $(B + D)^{-1}(0)$.
	
	\noindent Alvarez and Attouch \cite{alv} proposed the following  modified PPA of inertial form:
	\begin{equation}\label{eq1.8}
		\begin{cases}
			y_n = x_n + \mu_n(x_n - x_{n-1}),\\
			x_{n+1} = J_{\lambda_n}^Dy_n, \qquad n\geq 1,
		\end{cases}
	\end{equation}
	where $\{\mu_n\}\subset [0,1)$, $\{\lambda_n\}$ is non-decreasing and
	\begin{equation}\label{eq1.9}
		\sum_{n=1}^{\infty}\mu_n\|x_n - x_{n-1}\|^2 < \infty,\quad \forall \mu_n < \frac{1}{3}.
	\end{equation}
	It was proved that Algorithm \eqref{eq1.8} converges weakly to a zero of $D$.
	
	\noindent Recently,  Moudafi and Oliny \cite{mou1} introduced the following inertial PPA for approximating the zero point problem of the sum of two monotone operators:
	\begin{equation}\label{eq1.10}
		\begin{cases}
			y_n = x_n + \mu_n(x_n - x_{n-1}),\\
			x_{n+1} = J_{\lambda_n}^D(y_n - \lambda_nBx_n), n\geq 1,
		\end{cases}
	\end{equation}
	where $D : H\to 2^H$ is maximal monotone and $B$ is Lipschitz continuous. They proved that the sequence generated by  Algorithm \eqref{eq1.10} converges weakly if $\lambda_n < \frac{2}{L}$, where $L$ is the Lipschitz constant of $B$.
	
	\noindent Moreover, the following inertial forward-backward algorithm was introduced by Lorenz and Pock \cite{lor}:
	\begin{equation}\label{eq1.11}
		\begin{cases}
			y_n = x_n + \mu_n(x_n - x_{n-1}),\\
			x_{n+1} = J_{\lambda_n}^D(y_n - \lambda_nBy_n), n\geq 1,
		\end{cases}
	\end{equation}
	where $\{\lambda_n\}$ is a positive real sequence. Algorithm \eqref{eq1.11} differs from Algorithm \eqref{eq1.10} since the operator $B$ is evaluated as the inertial extrapolate $y_n.$ The proposed algorithm was also proved to converge weakly to a solution of the MIP \eqref{abx}.
	

	\noindent In 2016, Deepho \cite{dph} introduced the general Ces\'aro mean iterative method for approximating a common solution of split generalized equilibrium, fixed point of nonexpansive mappings $T_j$ and variational inequality problems:
	
	\begin{equation}\label{algdep}
		\begin{cases}
			z_n = T_{r_n}^{(F_1, \phi_1)}(x_n + \gamma A^*(T_{r_n}^{(F_2, \phi_2)} - I)Ax_n),\\
			u_n = P_C(z_n - \lambda_nGz_n),\\
			x_{n+1} = \alpha_n\eta f(x_n) + \beta x_n + ((1 - \beta_n)I - \alpha_nK)\frac{1}{n+1}\sum_{j = 0}^{n} T_ju_n, ~~\forall n \geq 0,
		\end{cases}
	\end{equation}
	where $\{\alpha_n\}, \{\beta_n\} \subset (0, 1)$, $\{\lambda_n\} \in [a, b] \subset (0, 2\beta)$ and $\{r_n\} \subset (0, \alpha)$ and $\gamma \in \bigl(0, \frac{1}{L}\bigr)$, $L$ is the spectral radius of the operator $A^*A$ and $A^*$ is the adjoint of $A$. Under the following conditions:
	
	\begin{enumerate}
		\item[(C1)] $\lim_{n \to \infty}\alpha_n = 0,$ $\sum_{n = 0}^{\infty}\alpha_n = \infty;$\\
		\item[(C2)] $ 0 \leq \liminf_{n\to \infty}\beta_n \leq \limsup_{n \to \infty}\beta_n < 1;$\\
		\item[(C3)] $\lim_{n \to \infty}|\lambda_{n + 1} - \lambda_n| = 0;$\\
		\item[(C4)] $\liminf_{n\to \infty}r_n > 0, \lim_{n \to \infty}|r_{n+1} - r_n| = 0.$
	\end{enumerate}
	\noindent the authors proved that the sequence $\{x_n\}$ converges strongly to an element $q$ in the solution set $\Omega,$ where $q = P_{\Omega}(I - K + \gamma f)(q)$ is the unique solution of the variational inequality problem
	
	\begin{equation*}
		\langle (K - \gamma f)q, x - q \rangle \geq 0,~~\forall x \in \Omega.
	\end{equation*}
	
	\noindent Also, in 2017, Sitthithakerngkiet \cite{sitt} proposed and studied the following iterative method for approximating a common solution of split generalized equilibrium, variational inequality for an inverse-strongly monotone mapping and fixed point problems of nonexpansive mappings in Hilbert spaces:
	
	\begin{equation}\label{algsitth}
		\begin{cases}
			z_n = T_{r_n}^{(F_1, \phi_1)}(x_n + \gamma A^*(T_{r_n}^{(F_2, \phi_2)} - I)Ax_n),\\
			x_{n+1} = \alpha_nf(x_n) + \beta_nx_n + \xi_nT[\sigma_nv + (1 - \sigma_n)P_C(z_n - \lambda_nGz_n)],
		\end{cases}
	\end{equation}
	where $v\in C$ is a fixed point, $r_n \in (0, \infty), \mu \in \bigl(0, \frac{1}{L}\bigr)$, $L$ is the spectral radius of the operator $A^*A$, $A^*$ is the adjoint of $A$, sequences $\{\alpha_n\}, \{\beta_n\}, \{\xi_n\}$ and $\{\sigma_n\}$ are in $(0, 1)$ and satisfy $\alpha_n + \beta_n + \xi_n = 1, \lambda_n \in [a, b]$ for some $a, b$ with $0 < a < b < 2\beta_n$ and $\{\gamma_n\} \subset [c, 1]$ for some $c \in (0, 1)$. Assume that the following conditions are satisfied:
	
	\begin{enumerate}
		\item[(C1)] $\lim_{n \to \infty}\alpha_n = 0,$ $\sum_{n = 0}^{\infty}\alpha_n = \infty;$\\
		\item[(C2)] $\lim_{n\to \infty}\sigma_n = 0;$\\
		\item[(C3)] $ 0 \leq \liminf_{n\to \infty}\beta_n \leq \limsup_{n \to \infty}\beta_n < 1;$\\
		\item[(C4)] $\lim_{n \to \infty}|\lambda_{n + 1} - \lambda_n| = 0;$\\
		\item[(C5)] $\liminf_{n\to \infty}r_n > 0, \lim_{n \to \infty}|r_{n+1} - r_n| = 0,$
	\end{enumerate}
	\noindent the authors proved that  the sequence $\{x_n\}$ converges strongly to $z \in \Omega,$ where $z = P_{\Omega}f(z).$
	
	
	
	
	\noindent Recently, Phuengrattana and Lerkchaiyaphum \cite{phu} introduced the following shrinking projection method for solving SGEP and FPP for a countable family of nonexpansive  multivalued mappings: For $x_1\in C$ and $C_1=C,$ then
	\begin{equation} \label{algphu}
		\begin{cases}
			z_n= T^{(F_1, \phi_1)}_{r_n} (I - \gamma A^*(I - T^{(F_2, \phi_2)}_{r_n}) A)x_n, \\
			y_n = \delta_{n, 0}x_n + \sum_{j = 1}^n\delta_{n, j}u_{n, j},\quad\quad u_{n,j} \in P_j z_n,\\
			C_{n+1} = \{ p \in C_n : \|y_n - p \|^2 \le \| x_n - p\|^2 \},\\
			x_{n+1} = P_{C_{n+1}} x_1,\quad\quad n\in \mathbb{N}.
		\end{cases}
	\end{equation}
	They proved that if
	\begin{enumerate}
		\item[(i)] $\lim \inf _{n \rightarrow \infty}$ $r_n > 0$,
		\item[(ii)] The limits $\lim_{n\rightarrow \infty}\delta_{n,j} \in (0,1)$ exist for all $j \ge 0$,
	\end{enumerate}
	then the sequence $\{x_n\}$ generated by $\eqref{algphu}$ converges strongly to $P_{\Gamma}x_1$, where $\Gamma = \bigcap_{j=1}^ \infty$ $F(P_j)\cap SGEP(F_1,\phi_1, F_2, \phi_2) \ne \emptyset,$ $F(P_j)$ is the set of fixed points of $P_j$ and $P_j$ is a countable family of nonexpansive multivalued mappings.
	
	\noindent In 2021, Olona et al. \cite{olona} proposed an inertial shrinking projection defined as follows for split generalized equilibrium and fixed point problems for a countable family of nonexpansive multivalued mappings : for $x_0, x_1 \in C$ with $C_1 = C$, then
	
	\begin{equation} \label{xii}
		\begin{cases}
			w_n = x_n + \theta_n(x_n - x_{n-1}),\\
			u_n = T^{(F_1, \phi_1)}_{r_n}(I - \gamma_n A^*(I - T^{(F_2, \phi_2)}_{r_n})A)w_n,\\
			z_n = \delta_{n,0}u_n + \sum_{i=1}^{n} \delta_{n,j}y_{n,j},~~ y_{n,j} \in P_ju_n,\\
			C_{n+1} =\{ p \in C_n : \|z_n - p\|^2 \le \|x_n - p\|^2 \\
- 2\theta_n\langle x_n - p, x_{n-1}-x_n \rangle + \theta_n^2\|x_{n-1} - x_n\|^2\},\\
			x_{n+1} = P_{C_{n+1}} x_1,\quad n \in \mathbb{N},
		\end{cases}
	\end{equation}
	
	\begin{equation*}
		\gamma_n =
		\begin{cases}
			\frac{\tau_n \|(I- T_{r_n}^{(F_2, \phi_2)}) Aw_n \|^2}{\| A^* (I - T^{(F_2, \phi_2)}_{r_n}) Aw_n\|^2}\quad\quad \text{if}~~~ Aw_n \ne\	 T_{r_n}^{(F_2, \phi_2)} Aw_n,\\
			\gamma\hspace{100pt}	\text{otherwise ($\gamma$ being any nonnegative real number)},
		\end{cases}
	\end{equation*}
	\noindent where $A : H_1 \to H_2$ is a bounded linear operator, $0<a\leq\tau_n\leq b<1, \{\theta_n\} \subset \mathbb{R},$ $\{\delta_{n,j}\} \subset (0,1),$ such that $\sum_{j=0}^{n}\delta_{n,j} = 1,$ and $\{r_n\} \subset (0,\infty).$ $\{P_j\}$ is a countable family of nonexpansive multivalued  mappings,  $F_1,\phi_1:C\times C\rightarrow\mathbb{R},~ F_2,\phi_2:Q\times Q\rightarrow\mathbb{R}$ are bifunctions. Under some appropriate conditions, it was proved that the sequence $\{x_n\}$ converges strongly to $P_\Omega x_1,$ where $\Omega = \bigcap_{j=1}^{\infty} F(P_j) \cap SGEP (F_1,\phi_1, F_2, \phi_2) \neq \emptyset$.
	
	
	\noindent Motivated by the above results and the current research interest in this direction, in this paper, we propose a new iterative algorithm of inertial type with self-adaptive step size for approximating the common solution of SGEP \eqref{c3x}-\eqref{c4x}, MIP \eqref{abx} and FPP of strictly pseudo-contractive multivalued mappings. We prove that the sequence generated by our algorithm converges strongly to a solution of the investigated problem. Finally, we present some applications and numerical examples to illustrate the usefulness and efficiency of the proposed method in comparison with some related methods. Our proposed method uses self-adaptive step size and employs inertial technique to accelerate the rate of convergence of the proposed method. The implementation of our proposed algorithm does not require a prior knowledge of the norm of the bounded linear operator.
	
	\noindent  Subsequent sections of this paper are organised as follows: In Section \ref{Sec2}, we recall some basic definitions and lemmas that are relevant in establishing our main results. In Section \ref{Sec3}, we present our proposed algorithm and highlight some of its features. In Section \ref{Sec4}, we prove some lemmas that are useful in establishing the strong convergence of our proposed algorithm and also prove the strong convergence theorem for the algorithm. In Section \ref{Sec5}, we apply our result to study some optimization problems while in Section \ref{Sec6}, we present some numerical experiments to illustrate the performance of our method and compare it with some related methods in the literature. Finally, in Section \ref{Sec7} we give a concluding remark.
	
	
	\section{Preliminaries}\label{Sec2}
	\noindent Let $C$ be a nonempty, closed and convex subset of a real Hilbert space $H$ with inner product $\langle \cdot, \cdot \rangle$ and norm $\|\cdot\|$. We denote $x_n \rightarrow x$ to mean that sequence $\{x_n\}$ converges strongly to $x$ and $x_n \rightharpoonup x$ to indicate that the sequence $\{x_n\}$ converges weakly to $x.$ We write $w_\omega(x_n)$ to denote set of weak limits of $\{x_n\},$ that is,
	$$\omega_w(x_n):= \{x\in H: x_{n_j}\rightharpoonup x~ \text{for some subsequence}~ \{x_{n_j}\}~ \text{of} ~\{x_{n}\}\}.$$ The nearest point projection of $H$ onto $C$ denoted by $P_C$ is defined for each $x\in H,$ as the unique element $P_Cx\in C$ such that
	\begin{equation}
		\|x - P_Cx\| \leq \|x - y\|,~~\forall~y \in C.
	\end{equation}
	\noindent It is well known that $P_C$ is nonexpansive and has the following  characteristics (see \cite{alakoya1,Godwin}:
	
	\begin{equation}
		\|P_Cx - P_Cy\|^2 \leq \langle x - y, P_Cx - P_Cy \rangle,~~\forall~x, y \in H_1,
	\end{equation}
	
	\begin{equation}
		\langle x - P_Cx, y - P_Cx \rangle \leq 0,	
	\end{equation}
	
	\begin{equation}
		\|x - y\|^2 \leq \|x - P_Cx\|^2 + \|y - P_Cx\|^2,~~\forall x \in H, y \in C,
	\end{equation}
	
	\begin{equation}
		\|(x - y) - (P_Cx - P_Cy)\|^2 \geq \|x - y\|^2 - \|P_Cx - P_Cy\|^2,~~x, y \in H.	
	\end{equation}
	
	\noindent A mapping $B : C \rightarrow H$ is said to be monotone if
	\begin{equation}
		\langle Bu - Bv, u - v \rangle \geq 0,~~\forall u, v \in C.
	\end{equation}
	\noindent Moreover, if $B$ satisfies
	\begin{equation}
		\langle Bu - Bv, u - v \rangle \geq \alpha\|Bu - Bv\|^2,~~\forall u, v \in C,
	\end{equation}
	for some positive real number $\alpha$. Then, $B$ is called an $\alpha$-inverse-strongly monotone  mapping. It is clear that every inverse-strongly monotone mapping is monotone.   	
	\begin{lemma}\cite{olona, marn}\label{lem3f}
		Let $H$ be a real Hilbert space, $\lambda \in \mathbb{R}$, then $\forall x, y \in H,$ we have
		\begin{enumerate}
			\item[(i)] $\|x + y\|^2 = \|x\|^2 + 2\langle x, y \rangle + \|y\|^2$;
			\item[(ii)] $\|x - y\|^2 = \|x\|^2 - 2\langle x, y \rangle + \|y\|^2$;
			\item[(iii)] $\|x + y\|^2 \leq \|x\|^2 + 2\langle y, x + y \rangle$;
			\item[(iv)] $\|\lambda x + (1 - \lambda)y\|^2 = \lambda\|x\|^2 + (1 - \lambda)\|y\|^2 - \lambda(1 - \lambda)\|x - y\|^2$.
		\end{enumerate}
	\end{lemma}
	
	\begin{lemma}\label{lem3a1}\cite{Nak}
		Let $C$ be a nonempty closed convex subset of a real Hilbert space $H$, and let
		$P_C : H \rightarrow C$ be the metric projection. Then
		
		\begin{equation*}
			\|y - P_Cx\|^2 + \|x - P_Cx\|^2 \leq \|x - y\|^2,~~\forall x \in H, y \in C.
		\end{equation*}
		
	\end{lemma}
	
	\begin{lemma}\label{lem3b}\cite{ZeSh}
		Let $x_i \in H, (1 \leq i \leq m), \sum_{i=1}^{m}\alpha_i = 1$, where $\{\alpha_i\} \subseteq (0, 1)$. Then
		\begin{equation*}
			\bigg\|\sum_{i=1}^{m}\alpha_ix_i\bigg\|^2 = \sum_{i=1}^{m}\alpha_i\|x_i\|^2 - \sum_{1\le i<j\le m}\alpha_i\alpha_j\|x_i - x_j\|^2.
		\end{equation*}
	\end{lemma}
	
	
	
	\begin{lemma}\cite{KiXu}\label{Lem2.4}
		Let $C$ be a nonempty, closed and convex subset of a real Hilbert space $H.$ Given $x,y,z\in H$ and $a\in\mathbb(R),$ the set $D = \{v\in C: \|y-v\|^2\le \|x-v\|^2 + \langle z,v \rangle + a \}$ is convex and closed.
		
		
	\end{lemma}
	
	
	
	
	\begin{assumption}\label{Ass 1}
		Let $C$ be a nonempty closed convex subset of a Hilbert space $H.$ Let $F_1 : C \times C \rightarrow \mathbb{R}$ and $\phi_1 : C \times C \rightarrow \mathbb{R}$ be two bifunctions that satisfy the following conditions:
		\begin{enumerate}
			\item [(A1)] $F_1(x,x) = 0$ for all $x \in C,$
			\item [(A2)] $F$ is monotone, that is, $F_1(x,y) + F_1(y,x) \le 0$ for all $x,y \in C,$
			\item[(A3)] $F$  is upper hemicontinuous, that is, for all $ x,y,z \in C$,\\
 $\lim_{t \downarrow 0} F\big(tz + (1-t)x,y \big)\le F(x,y),$
			\item [(A4)] for each $x \in C, y \mapsto F_1(x,y)$ is convex and lower semicontinuous,
			\item [(A5)] $\phi_1(x, x) \geq 0$, for all $x \in C,$
			\item[(A6)] for each $y \in C, x \mapsto \phi_1(x, y)$ is upper semicontinuous,
			\item[(A7)] for each $x \in C, y \mapsto \phi_1(x, y)$ is convex and lower semicontinuous,
		\end{enumerate}
		\noindent and assume that for fixed $r > 0$ and $z \in C$, there exists a nonempty compact convex subset $K$ of $H_1$ and $x \in C \cap K$ such that
		\begin{equation*}
			F_1(y, x) + \phi_1(y, x) + \frac{1}{r}\langle y - x, x - z \rangle < 0, ~~~\forall y \in C~~\backslash K.
		\end{equation*}
	\end{assumption}
	
	\begin{lemma}\label{lem3c}\cite{ma}
		Let $C$ be a nonempty closed convex subset of a Hilbert space $H$. Let $F : C \times C \rightarrow \mathbb{R}$ and $\phi_1 : C \times C \rightarrow \mathbb{R}$ be two bifunctions that satisfy Assumption \ref{Ass 1}. Assume that $\phi$ is monotone. For $r > 0$ and and $x \in H$. Define mapping $T_r^{(F, \phi)} : H \rightarrow C$ as follows:
		\begin{equation*}
			T_r^{(F, \phi)}(x) = \bigl\{z \in C: F(z, y) + \phi(z, y) + \frac{1}{r}\langle y - z, z - x \rangle \geq 0, ~~~\forall y \in C\bigr\}
		\end{equation*}
		\noindent for all $x \in H_1$. Then
		\begin{enumerate}
			\item[(1)] for each $x \in H_1$, $T_r^{(F, \phi)} \neq \emptyset,$
			\item[(2)] $T_r^{(F, \phi)}$ is single-valued,
			\item[(3)] $T_r^{(F, \phi)}$ is firmly nonexpansive, that is, for any $x, y \in H_1$,
			\begin{equation*}
				\|T_r^{(F, \phi)}x - T_r^{(F, \phi)}y\|^2 \leq \langle T_r^{(F, \phi)}x - T_r^{(F, \phi)}y, x - y\rangle,
			\end{equation*}
			\item[(4)] $F(T_r^{(F, \phi)}) = GEP(F, \phi),$
			\item[(5)] GEP$(F, \phi)$ is closed and convex.
			
		\end{enumerate}
		
	\end{lemma}
	
	
	\begin{lemma}\label{lem3g}\cite{suant}
		Let $X$ be a Banach space space satisfying Opial's condition and let $\{x_n\}$ be a sequence in $X$. Let $u, v \in X$ be such that
		\begin{equation*}
			\lim_{n\to \infty}\|x_n - u\|~~\text{and}~~\lim_{n\to \infty}\|x_n - v\|~~\text{exist}.
		\end{equation*}
		If $\{x_{n_k}\}$ and $\{x_{m_k}\}$ are subsequences of $\{x_n\}$ which converge weakly to $u$ and $v$, respectively, then $u = v$.
	\end{lemma}
	
	\begin{lemma}\label{lem3h}\cite{brez}
		Let $B : H \to 2^H$ be a maximal monotone mapping and $A : H \to H$ be a Lipschitz continuous and monotone mapping. Then, the mapping $A + B$ is a maximal monotone mapping.
	\end{lemma}
	
	\begin{lemma}\cite{tse1}
		Let $B : H \to 2^H$ be a maximal monotone operator and $A : H \to H$ be a mapping on $H$. Define $T_{\lambda} := (I + \lambda B)^{-1}(I - \lambda A)$, $\lambda > 0$. Then, we have the following
		\begin{equation}
			Fix(T_{\lambda}) = (A + B)^{-1}(0),~~\forall \lambda > 0.
		\end{equation}
		
	\end{lemma}
	
	
	
	\noindent Let $D$ be a nonempty subset of $H$. $D$ is said to be proximal if there exists  $y \in D$ such that
	\begin{equation*}
		\|x - y\| = d(x, D),~~x \in H.
	\end{equation*}
	\noindent Let $CC(C), CB(C)$ and $P(C)$ be the family of nonempty closed convex subset of $H$, nonempty closed bounded subsets of $H$ and nonempty proximal bounded subsets of $H$ respectively. The Hausdorff metric on $CB(C)$ is defined as follows:
	\begin{equation*}
		H(A, B) : = \max\biggl\{\sup_{x\in A} d(x, B), \sup_{y\in B} d(y, A) ,~~~\forall A, B \in CB(C)\biggr\}.
	\end{equation*}
	Let $S:C\rightarrow 2^C$ be a multivalued mapping.
	An element $x \in H$ is said to be a fixed point of $S$ if $x \in Sx.$ We say that $S$ satisfies the endpoint condition if $Sp=\{p\}$ for all $p\in F(S).$ For multivalued mappings $S_i: H\rightarrow 2^{H}~(i\in\mathbb{N})$ with $\cap_{i=1}^\infty F(S_i)\neq\emptyset,$ we say $S_i$ satisfies the common endpoint condition if $S_i(p)=\{p\}$ for all $i\in\mathbb{N},~ p\in\cap_{i=1}^\infty F(S_i).$
	
	
	\begin{definition}
		\noindent Let $A : H \to H$ be a nonlinear operator. Then $A$ is called
		
		\begin{enumerate}
		\item[(i)] Lipschitz continuous if for all $L > 0$
			\begin{equation*}
				\|Ax - Ay\| \leq L\|x - y\|,~~ \forall x, y \in H;
			\end{equation*}
			if $0 \leq L < 1$, then $A$ is a contraction mapping,	
		\item[(ii)] $\beta-$strongly monotone if for all $\beta > 0$
			\begin{equation*}
				\langle Ax - Ay, x - y \rangle \geq \beta\|x - y\|^2,~~ \forall x, y \in H.
			\end{equation*}
		\end{enumerate}
	\end{definition}
	
	\begin{definition}
		Let $S : C \to CB(C)$ be a multivalued mapping. $S$ is said to be
		
		\begin{enumerate}
			\item[(i)] nonexpansive if
			\begin{equation*}
				H(Sx, Sy) \leq \|x - y\|,~~ \forall x, y \in C,
			\end{equation*}
			\item[(ii)] quasi-nonexpansive if $F(S) \neq \emptyset$ such that
			\begin{equation*}
				H(Sx, Sp) \leq \|x - p\|,~~ \forall x \in C, p \in F(S),
			\end{equation*}
			\item[(iii)] $k$- strictly pseudo-contractive if there exists a constant $k \in [0, 1)$ such that
			\begin{equation}\label{st}
				(H(Sx, Sy))^2 \leq \|x - y\|^2 + k\|(x - u) - (y - v)\|^2,~~\forall u\in Sx, v \in Sy 	
			\end{equation}
			\noindent	If $k = 1$ in \eqref{st}, then the mapping $S$ is said to be pseudo-contractive.
		\end{enumerate}
		\noindent Clearly, the class of $k$-strict pseudo-contractive mappings properly contains the class of nonexpansive mappings. That is, $S$ is nonexpansive if and only if $S$ is $0$-strict pseudo-contractive. It is known that if $S$ is a $k$-strict pseudo-contraction and $F(S)\neq\emptyset,$ then $F(S)$ is a closed convex subset of $H$ (see \cite{Zho}). Strict pseudo-contractions have many applications, due to their ties with inverse strongly monotone operators. It is known that, if $B$ is a strongly monotone operator, then $S = I - B$ is a strict pseudo-contraction, and so we can recast a problem of zeros for $B$ as a fixed point problem for $S,$ and vice versa (see e.g. \cite{ChYa, ShZe}).
		
		\noindent Let $S : H \to CB(H)$ be a multivalued mapping. The multivalued mapping $I-S$ is said to be demiclosed at zero if for any sequence $\{x_n\}\subset H$ which converges weakly to $p$ and the sequence $\{\|x_n-u_n\|\}$ converges strongly to 0, where $u_n\in Sx_n,$ then $p\in F(S).$
	\end{definition}
	\section{Proposed Method}\label{Sec3}
	\noindent In this section, we present our proposed algorithm.
	
	\noindent Let $C$ and $Q$ be nonempty closed convex subsets of real Hilbert spaces $H_1$ and $H_2$, respectively. Let $A : H_1 \rightarrow H_2$ be a bounded linear operator, and let $\{S_i\}_{i=1}^m$ be a countable family of $k_i$-strictly pseudo-contractive multivalued mappings of $C$ into $CB(C)$ such that $I-S_i$ is demiclosed at zero for each $i=1,2,\ldots,m,$ $S_ip = \{p\}$ for each $p \in \cap_{i=1}^mF(S_i)$ and $k=\max\{k_i\}.$ Let $F_1, \phi_1 : C \times C \rightarrow \mathbb{R}, F_2, \phi_2 : Q \times Q \rightarrow $ be bifunctions satisfying Assumptions \ref{Ass 1}. Let $\phi_1, \phi_2$ be monotone, $\phi_1$ be upper hemicontinuous, and $F_2$ and $\phi_2$ be upper semicontinuous in the first argument. Let $B : H_1 \rightarrow H_1$ be $L$-Lipschitz continuous and monotone and $D : H_1 \rightarrow 2^{H_1}$ be a maximal monotone operator such that $\Gamma = SGEP(F_1, \phi_1, F_2, \phi_2)\cap  \cap_{i=1}^mF(S_i) \cap (B + D)^{-1}(0) \neq \emptyset.$ We establish the convergence of our algorithm under the following conditions on the control parameters:
	
	\begin{enumerate}
		\item[(C1)] $0 < a \leq \tau_n \leq b < 2, \{r_n\} \subset (0, \infty), \liminf_{n\to \infty}r_n > 0,$
		\item[(C2)] $\liminf_n\alpha_{n, i}(\alpha_{n, 0} - k)>0$ and $\lim_{n\to \infty}\alpha_{n, i} \in (0, 1)$ exists for all $i \geq 0$.
	\end{enumerate}
	Now, we present our proposed algorithm as follows:
	
	
	\hrule\hrule
	\begin{algorithm}\label{alg3a}\noindent
		\hrule
		
		\noindent \textbf{Initialization:} Select $x_0, x_1 \in H_1$, $s_1>0, \mu \in (0, 1),\theta_n \in [-\theta, \theta]$ for some $\theta>0$ and $C_1=C.$
		
		\noindent \textbf{Iterative Step:} Given the current iterate $x_n$, calculate the next iterate as follows:
		
		\noindent \textbf{Step 1 :} Compute
		\begin{equation*}
			w_n = x_n + \theta_n(x_n - x_{n - 1}).
		\end{equation*}
		\noindent \textbf{Step 2 :} Compute
		\begin{equation*}
			z_n = T_{r_n}^{(F_1, \phi_1)}(I - \gamma_nA^*(I - T_{r_n}^{(F_2, \phi_2)})A)w_n.
		\end{equation*}
		\noindent \textbf{Step 3 :} Compute
		\begin{equation*}
			y_n = \alpha_{n, 0}z_n + \sum_{i = 1}^m\alpha_{n, i}u_{n, i},\quad u_{n, i} \in S_iz_n.
		\end{equation*}
		\noindent \textbf{Step 4 :} Compute
		\begin{equation*}
			\begin{cases}
				v_n = (I + s_nD)^{-1}(I - s_nB)y_n=J_{s_n}^D(I - s_nB)y_n \\
				t_n = v_n - s_n(Bv_n - By_n)\\
				C_{n+1} = \{p \in C_n : \|t_n - p\|^2 \leq \|w_n - p\|^2 - \bigg(1 - \mu^2\frac{s_n^2}{s_{n+1}^2}\bigg)\|y_n - v_n\|^2\}\\
				x_{n+1} = P_{C_{n+1}}x_0,
			\end{cases}
		\end{equation*}
		\noindent \textbf{Step 5 :} Compute
		\begin{equation}\label{ck}
			s_{n+1} = \begin{cases}
				\min\biggl\{\frac{\mu\|y_n - v_n\|}{\|By_n - Bv_n\|} , s_n\biggr\} & \text{if}~~By_n - Bv_n \neq 0.\\
				s_n   & \text{otherwise},
			\end{cases}
		\end{equation}
		\noindent Set $n := n + 1$ and return to {\bf Step 1},\\
		
		\noindent where
		\begin{equation*}
			\gamma_n =
			\begin{cases}
				\tau_n\frac{||(I - T_{r_n}^{(F_2, \phi_2)})Aw_n||^2}{||A^*(I - T_{r_n}^{(F_2, \phi_2)})Aw_n||^2} &  \text{If}~~ Aw_n \neq T^{(F_2, \phi_2)}_{r_n}Aw_n\\
				
				\gamma  &  \text{otherwise}~~(\gamma~~\text{being any non-negative real number}).
			\end{cases}
		\end{equation*}
		
	\end{algorithm}
	\hrule\hrule
	
	\begin{remark}
		\noindent We observe that
		\begin{enumerate}
		\item[(i)] The implementation of our proposed algorithm does not require prior knowledge of the operator norm.  Hence, this makes our method easily implementable.
		\item[(ii)] We employ the inertial technique to accelerate the rate of convergence.
		\item[(iii)] The underlying single-valued operator $B : H_1 \rightarrow H_1$ for most of the results on monotone inclusion problem in the literature are either strongly monotone or inverse strongly monotone while the single-valued operator in our proposed algorithm is only required to be monotone and Lipschitz continuous. Moreover, knowledge of the Lipschitz constant of the operator is
		not required to implement our proposed algorithm. Thus, our method is more applicable than several
		of the existing methods in the literature.
		
		\item[(iv)] Our result extends and improves on the results of Deepho et al. \cite{dph}, Sitthithakerngkiet et al. \cite{sitt}, Phuengrattana and Lerkchaiyaphum \cite{phu}, Olona et al. \cite{olona} and several other results in the current literature in this direction.
		\end{enumerate}
		
	\end{remark}
	
	
	\section{Convergence Analysis}\label{Sec4}
	\noindent In this section, we analyze the convergence of our proposed algorithm.	
	
	
	\begin{lemma}\label{Lem4.1}
		Let $\{s_n\}$ be a sequence generated by \eqref{ck}. Then,  $\{s_n\}$ is a nonincreasing sequence and
		
		\begin{equation}
			\lim_{n \to \infty}s_n = s \geq \min\biggl\{s_1, \frac{\mu}{L}\biggr\}.
		\end{equation}
	\end{lemma}
	\begin{proof}
		\noindent\\
		\noindent From \eqref{ck}, it is clear that $\{s_n\}$ is a nonincreasing sequence. Moreover, observe that if $By_n - Bv_n \neq 0,$ then
		
		\begin{equation}
			\frac{\mu\|y_n - v_n\|}{\|By_n - Bv_n\|} \geq \frac{\mu}{L}.
		\end{equation}
		Hence, the sequence $\{s_n\}$ has the lower bound $\min\biggl\{s_1, \frac{\mu}{L}\biggr\}$.
	\end{proof}
	
	\begin{lemma}\cite{tse1}\label{Lem4.2}
		Let $\{x_n\}$ be a sequence generated by Algorithm \ref{alg3a}. Then the following inequality holds for all $p\in\Gamma:$
	
		\begin{equation}\label{c1}
			\|t_n - p\|^2 \leq \|y_n - p\|^2 - \bigg(1 - \mu^2\frac{s_n^2}{s_{n+1}^2}\bigg)\|y_n - v_n\|^2, ~~p \in \Gamma,
		\end{equation}
		and
		\begin{equation}\label{c^2}
			\|t_n - v_n\| \leq \mu\frac{s_n}{s_{n+1}}\|y_n - v_n\|.
		\end{equation} 	

	\end{lemma}

	
\begin{proof}
		\noindent By the definition of $s_n$, we have
		\begin{equation}\label{csd1}
			\|By_n - Bv_n\| \leq \frac{\mu}{s_{n+1}}\|y_n - v_n\|\quad\forall~n\in\mathbb{N}.
		\end{equation}
		\noindent Clearly, if $By_n = Bv_n$, then \eqref{csd1} holds. Otherwise , we have
		
		\begin{equation*}
			s_{n+1} = \min\biggl\{\frac{\mu\|y_n - v_n\|}{\|By_n - Bv_n\|}, s_n\biggr\} \leq \frac{\mu\|y_n - v_n\|}{\|By_n - Bv_n\|}.
		\end{equation*}
		\noindent This implies that
		\begin{equation*}
			\|By_n - Bv_n\| \leq \frac{\mu}{s_{n+1}}\|y_n - v_n\|.
		\end{equation*}
		\noindent Thus, \eqref{csd1} holds when $By_n = Bv_n$ and $By_n\ne Bv_n.$ Let $p \in \Gamma,$ then by Lemma \ref{lem3f}, we have
		\begin{align}
			\|t_n - p\|^2 & = \|v_n - s_n(Bv_n - By_n) - p\|^2\nonumber\\
			& = \|v_n - p\|^2 + s^2_n\|Bv_n - By_n\|^2 - 2s_n\langle v_n - p, Bv_n - By_n \rangle\nonumber\\
			& = \|y_n - p\|^2 + \|y_n - v_n\|^2 + 2\langle v_n - y_n, y_n - p \rangle \nonumber\\
	        &+ s^2_n\|Bv_n - By_n\|^2 - 2s_n\langle v_n - p, Bv_n - By_n \rangle\nonumber\\
			& = \|y_n - p\|^2 + \|y_n - v_n\|^2 - 2\langle v_n - y_n, v_n - y_n \rangle + 2\langle v_n - y_n, v_n - p \rangle \nonumber\\
	       &+ s^2_n\|Bv_n - By_n\|^2 -2s_n\langle v_n - p, Bv_n - By_n\rangle\nonumber\\
			& = \|y_n - p\|^2 - \|y_n - v_n\|^2 + 2\langle v_n - y_n, y_n - p \rangle \nonumber\\
	       &+ s^2_n\|Bv_n - By_n\|^2 - 2s_n\langle v_n - p, Bv_n - By_n \rangle\nonumber\\
			& = \|y_n - p\|^2 - \|y_n - v_n\|^2 - 2\langle y_n - v_n - s_n(By_n - Bv_n), v_n - p\rangle \nonumber\\
	       &+ s^2_n\|Bv_n - By_n\|^2\label{cg}.
		\end{align}
		\noindent By applying \eqref{csd1} in \eqref{cg}, we obtain
		\begin{equation}
			\|t_n - p\|^2  \leq \|y_n - p\|^2 - \bigg(1 - \mu^2\frac{s^2_n}{s^2_{n+1}}\bigg)\|y_n - v_n\|^2 - 2\langle y_n - v_n - s_n(By_n - Bv_n), v_n - p \rangle \label{chl}.
		\end{equation}
		\noindent We now prove that $\langle y_n - v_n - s_n(By_n - Bv_n), v_n - p \rangle \geq 0.$
		\noindent Since $v_n = (I + s_nD)^{-1}(I - s_nB)y_n$, then we have $(I - s_nB)y_n \in (I + s_nD)v_n$. Recall that $D$ is maximal monotone. Then there exists $u_n \in Dy_n$ such that
		\begin{equation*}
			(I - s_nB)y_n = v_n + s_nu_n,
		\end{equation*}
		\noindent  from which we obtain
		\begin{equation}\label{dj}
			u_n = \frac{1}{s_n}(y_n - v_n - s_nBy_n).
		\end{equation}
		\noindent Moreover, we have $0 \in (B + D)p$ and $Bv_n + u_n \in (B + D)v_n.$ Since $B + D$ is maximal monotone, we get
		\begin{equation}\label{dxt}
			\langle Bv_n + u_n, v_n - p\rangle \geq 0.
		\end{equation}
		By substituting \eqref{dj} into \eqref{dxt}, we obtain
		\begin{equation*}
			\frac{1}{s_n}\langle y_n - v_n - s_nBy_n + s_nBv_n, v_n - p \rangle \geq 0.
		\end{equation*}
		This implies that
		\begin{equation}\label{dzf}
			\langle y_n - v_n - s_n(By_n- Bv_n), v_n - p \rangle \geq 0.
		\end{equation}
		\noindent By applying \eqref{dzf} in \eqref{chl}, we have
		\begin{equation}\label{lgi}
			\|t_n - p\|^2  \leq \|y_n - p\|^2 - \bigg(1 - \mu^2\frac{s^2_n}{s^2_{n+1}}\bigg)\|y_n - v_n\|^2.
		\end{equation}


\noindent On the other hand, one can see that \eqref{c^2} follows from \eqref{csd1}.	


\end{proof}
	
	
	
	\begin{remark}\label{Rem4.3}
	\noindent By Lemma \ref{Lem4.1} and $\mu\in (0,1),$ there exists $n_0\in\mathbb{N}$ such that $1 - \mu^2\frac{s_n^2}{s_{n+1}^2}>\epsilon>0$ for all $n\ge n_0.$ Consequently, it follows from \eqref{c1} that for all $p\in \Gamma$ and $n\ge n_0$
	
	\begin{equation*}
		\|t_n - p\|^2 \leq \|y_n - p\|^2 - \epsilon\|y_n - v_n\|^2.
	\end{equation*}
\end{remark}	
	

	
	
	

\begin{theorem}\label{the3}
		\noindent Let $C$ and $Q$ be nonempty closed convex subsets of real Hilbert spaces $H_1$ and $H_2$, respectively. Let $A : H_1 \rightarrow H_2$ be a bounded linear operator, and let $\{S_i\}$ be a countable family of $k_i$-strictly pseudo-contractive multivalued mappings of $C$ into $CB(C).$ Let $F_1, \phi_1 : C \times C \rightarrow \mathbb{R}, F_2, \phi_2 : Q \times Q \rightarrow \mathbb{R} $ be bifunctions satisfying Assumptions \ref{Ass 1}. Suppose $\phi_1, \phi_2$ are monotone, $\phi_1$ is upper hemicontinuous, and $F_2$ and $\phi_2$ are upper semicontinuous in the first argument. Let $B : H_1 \rightarrow H_1$ be an $L-$Lipschitz continuous monotone mapping  and $D : H_1 \rightarrow 2^{H_1}$ be a maximal monotone operator such that $\Gamma = SGEP(F_1, \phi_1, F_2, \phi_2) \cap \bigcap_{i=1}^mF(S_i) \cap \Omega \neq \emptyset$, where $\Omega = (B + D)^{-1}(0)$ and $S_ip = \{p\}$ for each $p \in \cap_{i=1}^mF(S_i)$. Let $\{x_n\}$ be a sequence generated by Algorithm \ref{alg3a} such that conditions (C1) and (C2) hold. Then, the sequence $\{x_n\}$ converges strongly to $q = P_{\Gamma}x_0.$
	\end{theorem}	
	
	
	\begin{proof}
		\noindent We divide the proof of the strong convergence Theorem \ref{the3} into various steps as follows:	 	
		
		\noindent \underline{Step 1:} We show that sequence $\{x_n\}$ generated by Algorithm \ref{alg3a} is bounded and well defined.
		
		\noindent Let $p \in \Gamma,$ then we have $p = T_{r_n}^{(F_1, \phi_1)}p$ and $Ap = T_{r_n}^{(F_1, \phi_1)}Ap, S_ip = p,$ for all $i = 1, 2, ...m.$
		
		\noindent Since $T_{r_n}^{(F_1, \phi_1)}$ is nonexpansive, then by Lemma \ref{lem3f} we have
		\begin{align}
			\|z_n - p\|^2 & = \|T_{r_n}^{(F_1, \phi_1)}(w_n - \gamma_nA^*(I - T_{r_n}^{(F_2, \phi_2)})Aw_n) - p\|^2\nonumber\\
			& \leq \|w_n - \gamma_nA^*(I - T_{r_n}^{(F_2, \phi_2)})Aw_n - p\|^2\nonumber\\
			&= \|w_n - p\|^2 + \gamma_n^2\|A^*(I - T_{r_n}^{(F_2, \phi_2)})Aw_n\|^2 \nonumber\\
	       &- 2\gamma_n\langle w_n - p, A^*(I - T_{r_n}^{(F_2, \phi_2)})Aw_n\rangle. \label{c5}
		\end{align}
		By the firmly nonexpansivity of $I - T_{r_n}^{(F_2, \phi_2)}$, we get
		\begin{align}
			\langle w_n - p, A^*(I - T_{r_n}^{(F_2, \phi_2)})Aw_n \rangle & = \langle Aw_n - Ap, (I - T_{r_n}^{(F_2, \phi_2)})Aw_n \rangle\nonumber\\
			& = \langle Aw_n - Ap, (I - T_{r_n}^{(F_2, \phi_2)})Aw_n \nonumber\\
	       &- (I - T_{r_n}^{(F_2, \phi_2)})Ap \rangle\nonumber\\
			& \geq \|(I - T_{r_n}^{(F_2, \phi_2)})Aw_n\|^2.\label{c6}
		\end{align}
		By substituting \eqref{c6} in \eqref{c5} and applying the condition on $\tau_n,$ we have
		\begin{align}
			\|z_n - p\|^2 & \le \|w_n - p\|^2 + \gamma_n^2\|A^*(I - T_{r_n}^{(F_2, \phi_2)})Aw_n\|^2 \nonumber\\
	        &- 2\gamma_n\|(I - T_{r_n}^{(F_2, \phi_2)})Aw_n\|^2\nonumber\\
			& = \|w_n - p\|^2 - \gamma_n\bigl[2\|(I - T_{r_n}^{(F_2, \phi_2)})Aw_n\|^2 \nonumber\\
	       &- \gamma_n\|A^*(I - T_{r_n}^{(F_2, \phi_2)})Aw_n\|^2\bigr]\nonumber\\
			& = \|w_n - p\|^2 - \gamma_n(2 - \tau_n)\|(I - T_{r_n}^{(F_2, \phi_2)})Aw_n\|^2\label{cx}\\
			& \leq \|w_n - p\|^2.\label{c7}
		\end{align}
		
		\noindent By Lemma \ref{lem3b} and applying the fact that $S_i,~i=1,2,\ldots,m$ is strictly pseudo-contractive together with condition (C2), we get
		\begin{align}
			\|y_n - p\|^2 & = \|\alpha_{n, 0}z_n + \sum_{i=1}^{m}\alpha_{n, i}u_{n, i} - p\|^2\nonumber\\
			& = \alpha_{n, 0}\|z_n - p\|^2 + \sum_{i=1}^{m}\alpha_{n, i}\|u_{n, i} - p\|^2 \nonumber\\
	       &- \sum_{i=1}^{m}\alpha_{n, 0}\alpha_{n, i}\|u_{n, i} - z_n\|^2 - \sum_{1 \leq i < j\le m}\alpha_{n,i}\alpha_{n,j}\|u_{n, i} - u_{n, j}\|^2\nonumber\\
			& \leq \alpha_{n, 0}\|z_n - p\|^2 + \sum_{i=1}^{m} \alpha_{n,i}\big(H(S_iz_n, S_ip)\big)^2 \nonumber\\
            &- \sum_{i=1}^{m}\alpha_{n, 0}\alpha_{n, i}\|u_{n, i} - z_n\|^2 - \sum_{1 \leq i < j\le m}\alpha_{n, i}\alpha_{n, j}\|u_{n, i} - u_{n, j}\|^2\nonumber\\
			& \leq \alpha_{n, 0}\|z_n - p\|^2 + \sum_{i=1}^{m}\alpha_{n, i}\bigl(\|z_n - p\|^2 + k_i\|u_{n, i} - z_n\|^2\bigr) \nonumber\\
	       &- \sum_{i=1}^{m}\alpha_{n, 0}\alpha_{n, i}\|u_{n, i} - z_n\|^2\nonumber\\
			&~~ - \sum_{1 \leq i < j\le m}\alpha_{n, i}\alpha_{n, j}\|u_{n, i} - u_{n, j}\|^2\nonumber\\
			& \leq \|z_n - p\|^2 - \sum_{i=1}^{m}\alpha_{n, i}(\alpha_{n, 0} - k_i)\|u_{n, i} - z_n\|^2 \label{c2}\\
			& \leq \|z_n - p\|^2, \label{c3}
		\end{align}
		
		\noindent which implies that
		\begin{equation}\label{c4}
			\|y_n - p\| \leq \|z_n - p\|.
		\end{equation}
		
		\noindent By applying \eqref{c3} and \eqref{c7} into \eqref{lgi}, we get
		
		\begin{equation}\label{lmg}
			\|t_n - p\|^2  \leq \|w_n - p\|^2 - \bigg(1 - \mu^2\frac{s^2_n}{s^2_{n+1}}\bigg)\|y_n - v_n\|^2, \forall p \in \Gamma.
		\end{equation}
		
		\noindent By Lemma \ref{Lem2.4}, we have that $C_{n+1}$ is closed and convex. Furthermore, from \eqref{lmg} it follows that $p \in C_{n+1}.$ Hence, we have $\Gamma \subset C_{n+1} \subset C_n$ for all $n$ and thus $x_{n+1} = P_{C_{n+1}}x_0$ is well defined. Therefore, $\{x_n\}$ is well defined.
		
		\noindent We now show that $\{x_n\}$ is bounded.
		\noindent It is known that $\Gamma$ is a nonempty closed convex subset of $H_1$, then there exists a unique $q \in \Gamma$ such that $q = P_{\Gamma}x_0$. From $x_n = P_{C_n}x_0$ and $x_{n+1} \in C_{n+1}$ for all $n \in \mathbb{N},$ we obtain
		\begin{equation*}
			\|x_n - x_0\| \leq \|x_{n+1} - x_0\|,~~\forall~~n \in \mathbb{N}.
		\end{equation*}
		On the other hand, since $\Gamma \subset C_n,$ we get
		\begin{equation*}
			\|x_n - x_0\| \leq \|q - x_0\|,~~\forall~n \in \mathbb{N}.
		\end{equation*}
		This implies that $\{\|x_n - x_0\|\}$ is bounded. Hence, $\{x_n\}$ is bounded. Consequently $\{w_n\}, \{t_n\}, \{z_n\}$ and $\{y_n\}$ are bounded. Thus, $\lim_{n\to \infty}\|x_n - x_0\|$ exists.
		
		
		\noindent \underline{Step 2:} We claim that $\lim_{n\to \infty}x_n = q$, for some $q \in C.$
		
		\noindent It is clear from the definition of $C_n$ that $x_m = P_{C_m}x_0 \in C_m \subset C_n$, $m > n \geq 1.$ By Lemma \ref{lem3a1}, we obtain
		\begin{equation}\label{c11}
			\|x_m - x_n\|^2 \leq \|x_m - x_0\|^2 - \|x_n - x_0\|^2.
		\end{equation}
		Since $\lim_{n\to \infty}\|x_n - x_0\|$ exists, then it follows from \eqref{c11} that $\|x_m - x_n\| \rightarrow 0$ as $n \rightarrow \infty.$ Thus, $\{x_n\}$ is a Cauchy sequence. Since $H_1$ is complete and $C$ is closed, there exists $q\in C$ such that $x_n \rightarrow q $ as $n \to \infty.$
		
		
		\noindent \underline{Step 3:} We now show that $q\in\Gamma.$
		
		\noindent From \eqref{c11}, we obtain
		\begin{equation}\label{c12}
			\lim_{n\to \infty}\|x_{n+1} - x_n\| = 0.
		\end{equation}
		
		\noindent From the definition of $w_n$ and by applying  \eqref{c12}, we get
		\begin{align}\label{c15}
			\|w_n - x_n\| = |\theta_n|\|x_n - x_{n-1}\|\le |\theta|\|x_n - x_{n-1}\|\to 0,\quad n\to\infty .
		\end{align}
		
		\noindent From \eqref{c12} and \eqref{c15}, we obtain
		
		\begin{align}\label{Eq4.15}
			\|w_n - x_{n+1}\| \to 0,\quad n\to\infty .
		\end{align}
		
		\noindent We known that $x_{n+1} \in C_{n+1}.$ Then, from the definition of $C_{n+1}$ we obtain
		
		\begin{equation*}
			\|t_n - x_{n+1}\|^2 \leq \|w_n - x_{n+1}\|^2.
		\end{equation*}
		
		\noindent Combining this with \eqref{Eq4.15}
		gives
		
		\begin{equation}\label{c13}
			\lim_{n\to \infty}\|t_n - x_{n+1}\| = 0.
		\end{equation}
		
		\noindent From \eqref{c12} and \eqref{c13}, we obtain
		
		\begin{equation}\label{c14}
			\lim_{n\to \infty}\|t_n - x_n\| = 0.
		\end{equation}
		
		
		
		\noindent From \eqref{c15} and \eqref{c14}, we obtain
		\begin{equation}\label{c16}
			\lim_{n\to \infty}\|t_n - w_n\| = 0.
		\end{equation}
		
		
		\noindent By applying \eqref{c3} and \eqref{c7} into Remark \ref{Rem4.3}, we have

\begin{equation*}
\|t_n - p\|^2 \leq \|w_n - p\|^2 - \epsilon\|y_n - v_n\|^2.
\end{equation*}

\noindent From which we get	
		\begin{align*}
			\epsilon\|y_n - v_n\|^2	&\leq \|w_n - p\|^2 - \|t_n - p\|^2\\
			&\leq \|w_n - t_n\|(\|w_n - p\| + \|t_n - p\|),
		\end{align*}
		
		
		\noindent which together with \eqref{c16} implies that
		
		
		\begin{align}\label{Eq4.19}
			\|y_n - v_n\|\to 0,\quad n\to\infty.
		\end{align}
		
		\noindent Applying Lemma \ref{Lem4.1} together with \eqref{Eq4.19} to \eqref{c^2}, we have
		
		
		
		\begin{align}\label{Eq4.20}
			\|t_n - v_n\|\to 0,\quad n\to\infty.
		\end{align}
		
		\noindent From \eqref{c16}-\eqref{Eq4.20}, we obtain
		
		
		\begin{align}\label{Eq4.21}
			\|y_n - w_n\|\to 0,\quad n\to\infty.
		\end{align}
		
		
		\noindent From \eqref{c7} and \eqref{c2}, we obtain
		
		\begin{equation*}
			\|y_n - p\|^2\leq \|w_n - p\|^2 - \sum_{i=1}^{m}\alpha_{n, i}(\alpha_{n, 0} - k_i)\|u_{n, i} - z_n\|^2.
		\end{equation*}
		
		\noindent From this we have
		
		\begin{align*}
			\alpha_{n, i}(\alpha_{n, 0} - k_i)\|u_{n, i} - z_n\|^2 &\le\sum_{i=1}^{m}\alpha_{n, i}(\alpha_{n, 0} - k_i)\|u_{n, i} - z_n\|^2\\
			&\le\|w_n - p\|^2-\|y_n - p\|^2\\
			&\le (\|w_n - y_n\|)(\|w_n - p\|+ \|y_n - p\|).
		\end{align*}
		
		\noindent By applying Condition (C2) and \eqref{Eq4.21}, we get
		
		
		\begin{align}\label{Eq4.26}
			\|u_{n, i} - z_n\|\to 0,\quad n\to\infty.
		\end{align}
		
		
		\noindent From the definition of $y_n$ and by applying \eqref{Eq4.26}, we get
		
		
		
		\begin{equation}\label{Eq4.23}
			\|y_n - z_n\| \le \alpha_{n, 0}\|z_n-z_n\| + \sum_{i=1}^{m}\alpha_{n, i}\|u_{n, i} - z_n\|\to 0,\quad n\to\infty.
		\end{equation}
		
		
		
		
		
		
		
		\noindent Also, by applying \eqref{c15}, \eqref{Eq4.21} and \eqref{Eq4.23}, we obtain
		\begin{equation}\label{cxh}
			\lim_{n \to \infty}\|w_n - z_n\|= 0;\quad
			\lim_{n \to \infty}\|z_n - x_n\| =0 .
		\end{equation}
		
		\noindent From \eqref{cx}, we have
		\begin{equation*}
			\|z_n - p\|^2 \leq \|w_n - p\|^2 - \gamma_n(2 - \gamma_n)\|(I - T_{r_n}^{(F_2, \phi_2)})Aw_n\|^2,
		\end{equation*}
		\noindent which implies that
		
		\begin{align*}
			\gamma_n(2 - \gamma_n)\|(I - T_{r_n}^{(F_2, \phi_2)})Aw_n\|^2 & \leq \|w_n - p\|^2 - \|z_n - p\|^2\\
			& \leq \|w_n - z_n\|(\|w_n - p\| + \|z_n - p\|).
		\end{align*}
		Using the definition of $\gamma_n$, the condition on $\tau_n$ and applying \eqref{cxh}, it follows that
		\begin{equation*}
			\frac{\tau_n(2 - \tau_n)\|(I - T_{r_n}^{(F_2, \phi_2)})Aw_n\|^4}{\|A^*(I - T_{r_n}^{(F_2, \phi_2)Aw_n})Aw_n\|^2} \rightarrow 0\quad\text{as}~~n \rightarrow \infty.
		\end{equation*}
		From which we get
		\begin{equation*}
			\frac{\|(I - T_{r_n}^{(F_2, \phi_2)})Aw_n\|^2}{\|A^*(I - T_{r_n}^{(F_2, \phi_2)})Aw_n\|} \rightarrow 0\quad\text{as}~~n \rightarrow \infty.
		\end{equation*}
		
		
		\noindent Since $\|A^*(I - T_{r_n}^{(F_2, \phi_2)})Aw_n\|$ is bounded, then it follows that
		
		\begin{equation}\label{Eq4.24}
			\|(I - T_{r_n}^{(F_2, \phi_2)})Aw_n\| \rightarrow 0,\quad~~n \rightarrow \infty.
		\end{equation}
		
		\noindent Consequently, we have
		\begin{align}
			\|A^*(I - T_{r_n}^{(F_2, \phi_2)})Aw_n)\| & \leq \|A^*\|\|(I - T_{r_n}^{(F_2, \phi_2)})Aw_n)\|\nonumber\\
			&= \|A\|\|(I - T_{r_n}^{(F_2, \phi_2)})Aw_n)\|\rightarrow 0~~\text{as}~~n \rightarrow \infty.\label{c17}
		\end{align}
		
		\noindent Since $t_n = v_n - s_n(Bv_n - By_n)$ and $B$ is Lipschitz continuous, then by applying \eqref{Eq4.19} we have
		\begin{align*}
			\|t_n - v_n\|  = \|v_n - s_n(Bv_n - By_n) - v_n\| = s_n\|By_n - Bv_n\|\to 0,\quad n\to\infty.
		\end{align*}
		
		
		\noindent Since $\{x_n\}$ is bounded, then $w_\omega(x_n)$ is nonempty. Let $q\in w_\omega(x_n)$ be an arbitrary element. Then there exists a subsequence $\{x_{n_k}\}$ of $\{x_n\}$ such that $x_{n_k}\rightharpoonup q$ as $k\to\infty.$ Let $z\in w_\omega(x_n)$ and $\{x_{n_j}\}\subset \{x_n\}$ be such that $x_{n_j}\rightharpoonup z$ as $j\to\infty.$ From \eqref{cxh}, we get $z_{n_k}\rightharpoonup q$ and $z_{n_j}\rightharpoonup z.$
		Since $I-S_i$ is demiclosed at zero for each $i=1,2,\ldots,m,$ then it follows from \eqref{Eq4.26} that $q,z\in F(S_i)$ for all $i=1,2,\ldots,m,$ which implies that $q,z\in\cap_{i=1}^m F(S_i).$
		
		
		
		\noindent Next, let $(g, h) \in \text{Graph}(B + D),$ that is $h - Bg\in Dg.$ Since $v_{n_k} = (I + s_{n_k}D)^{-1}(I - s_{n_k}B)y_{n_k},$ we have
		
		\begin{equation*}
			(I - s_{n_k}B)y_{n_k} \in (I + s_{n_k}D)v_{n_k},
		\end{equation*}
		which implies that
		\begin{equation*}
			\frac{1}{s_{n_k}}(y_{n_k} - v_{n_k} - s_{n_k}By_{n_k}) \in Dv_{n_k}.
		\end{equation*}
		Since $D$ is maximal monotone, we get
		\begin{equation*}
			\biggl\langle g - v_{n_k}, h - Bg - \frac{1}{s_{n_k}}(y_{n_k} - v_{n_k} - s_{n_k}By_{n_k}) \biggr\rangle \geq 0.
		\end{equation*}
		
		
		\noindent From this we obtain
		
		\begin{align*}
			\langle g - v_{n_k}, h \rangle & \geq \biggl\langle g - v_{n_k}, Bg + \frac{1}{s_{n_k}}(y_{n_k} - v_{n_k} - s_{n_k}By_{n_k}) \biggr\rangle\\
			& = \langle g - v_{n_k}, Bg - By_{n_k} \rangle + \biggl\langle g - v_{n_k}, \frac{1}{s_{n_k}}(y_{n_k} - v_{n_k}) \biggr\rangle\\
			& = \langle g - v_{n_k}, Bg - Bv_{n_k} \rangle + \langle g - v_{n_k}, Bv_{n_k} - By_{n_k} \rangle \nonumber\\
	       &+ \biggl\langle g - v_{n_k}, \frac{1}{s_{n_k}}(y_{n_k} - v_{n_k}) \biggr\rangle\\
			& \geq \langle g - v_{n_k}, Bv_{n_k} - By_{n_k} \rangle + \biggl\langle g - v_{n_k}, \frac{1}{s_{n_k}}(y_{n_k} - v_{n_k}) \biggr\rangle.
		\end{align*}
		Since $B$ is Lipschitz continuous and $\lim_{n \to \infty}\|v_n - y_n\| = 0$, we have $\lim_{n \to \infty}\|Bv_{n_k} - By_{n_k}\| = 0.$ Applying this together with $\lim_{n\to \infty}s_n = s \geq \min\biggl\{s_1, \frac{\mu}{L}\biggr\}$, we get
		
		\begin{equation}\label{Eq4.27}
			\langle g - q, h \rangle = \lim_{k \to \infty}\langle g - v_{n_k}, h \rangle \geq 0.
		\end{equation}
		
		
		\noindent Following similar argument, we obtain
		
		\begin{equation}\label{Eq4.28}
			\langle g - z, h \rangle = \lim_{j \to \infty}\langle g - v_{n_j}, h \rangle \geq 0.
		\end{equation}
		
		
		\noindent By the maximal monotonicity of $(B + D)$, it follows from \eqref{Eq4.27} and \eqref{Eq4.28} that
		$q,z \in (B + D)^{-1}(0).$
		
		
		\noindent Next, since $z_{n_k} = T^{(F_1, \phi_1)}_{r_{n_k}}(I - \gamma_{n_k}A^*(I - T_{r_{n_k}}^{F_2, \phi_2})A)w_{n_k},$ then by applying Lemma \ref{lem3c}, we get
		\begin{align*}
			F_1(z_{n_k}, y) + \phi_1(z_{n_k}, y) & \nonumber\\
	& + \frac{1}{r_{n_k}}\langle y - z_{n_k}, z_{n_k} - w_{n_k} - \gamma_{n_k}A^*(I - T_{r_{n_k}}^{(F_2, \phi_2)})Aw_{n_k} \rangle \nonumber\\
	& \geq 0,~~\forall y \in C,
		\end{align*}
		which implies that
		\begin{align*}
			F_1(z_{n_k}, y) + \phi_1(z_{n_k}, y) &\\
&+ \frac{1}{r_{n_k}}\langle y - z_{n_k}, z_{n_k} - w_{n_k}\rangle &\nonumber\\
	 &-\frac{1}{r_{n_k}}\langle y - z_{n_k}, \gamma_{n_k}A^*(I - T_{r_{n_k}}^{(F_2, \phi_2)})Aw_{n_k} \rangle \nonumber\\
	&\geq 0,~~\forall y \in C.
		\end{align*}
		From the monotonicity of $F_1$ and $\phi_1$, it follows that
		\begin{align*}
			\frac{1}{r_{n_k}}\langle y - z_{n_k}, z_{n_k} - w_{n_k}\rangle & \nonumber\\
	&-\frac{1}{r_{n_k}}\langle y - z_{n_k}, \gamma_{n_k}A^*(I - T_{r_{n_k}}^{(F_2, \phi_2)})Aw_{n_k} \rangle\nonumber\\
	& \geq F_1(y, z_{n_k}) + \phi_1(y, z_{n_k}),~~\forall y \in C.
		\end{align*}
		\noindent By \eqref{cxh} and $x_{n_k} \rightharpoonup q$, we obtain $z_{n_k} \rightharpoonup q$. Applying condition $(C1)$, \eqref{cxh}, \eqref{c17} and Assumption \ref{Ass 1} (A1)-(A7), we obtain
		\begin{equation*}
			0 \geq F_1(y, q) + \phi_1(y, q),~~\forall y \in C.
		\end{equation*}
		Suppose $y_t = ty + (1 - t)q, \forall t \in (0, 1]$ and $y \in C.$ Then, $y_t \in C$ and $F_1(y_t, q) + \phi_1(y_t, q) \leq 0.$ Therefore, by Assumption \ref{Ass 1} (A1)-(A7), we get
		\begin{align*}
			0 & \leq F_1(y_t, y_t) + \phi_1(y_t, y_t)\\
			& \leq t\bigl(F_1(y_t, y) + \phi_1(y_t, y)\bigr) + (1 - t)\bigl(F_1(y_t, q) + \phi_1(y_t, q)\bigr)\\
			& \leq t\bigl(F_1(y_t, y) + \phi_1(y_t, y)\bigr).
		\end{align*}
		Thus, we have
		\begin{equation*}
			F_1(y_t, y) + \phi_1(y_t, y) \geq 0,~~\forall y \in C.
		\end{equation*}
		Letting $t \to 0,$ and applying condition (A3) together   with the upper hemicontinuity of $\phi_1,$ we have
		\begin{equation}\label{Eq4.29}
			F_1(q, y) + \phi_1(q, y) \geq 0,~~\forall y \in C.
		\end{equation}
		
		\noindent By similar argument, we have
		
		
		\begin{equation}\label{Eq4.30}
			F_1(z, y) + \phi_1(z, y) \geq 0,~~\forall y \in C.
		\end{equation}
		
		\noindent It follows from \eqref{Eq4.29} and \eqref{Eq4.30} that
		$q,z \in GEP(F_1, \phi_1)$.
		
		
		
		\noindent Next, we show that $Aq,Az \in GEP(F_2, \phi_2)$. Since $A$ is a bounded linear operator, then by \eqref{c15} we have $Aw_{n_k} \rightharpoonup Aq.$ Hence, from \eqref{Eq4.24}, we obtain
		\begin{equation}\label{c32}
			T_{r_{n_k}}^{(F_2, \phi_2)}Aw_{n_k} \rightharpoonup Aq,\quad k\to\infty.
		\end{equation}
		By the definition of $T_{r_{n_k}}^{(F_2, \phi_2)}Aw_{n_k}$, we have
		\begin{align*}
			F_2(T_{r_{n_k}}^{(F_2, \phi_2)}Aw_{n_k}, y) & \nonumber\\
	&+ \phi_2(T_{r_{n_k}}^{(F_2, \phi_2)}Aw_{n_k}, y) \nonumber\\
	&+ \frac{1}{r_{n_k}}\langle y - T_{r_{n_k}}^{(F_2, \phi_2)}Aw_{n_k}, T_{r_{n_k}}^{(F_2, \phi_2)}Aw_{n_k} - Aw_{n_k} \rangle \nonumber\\
	&\geq 0,~~\forall y \in Q.
		\end{align*}
		Since $F_2$ and $\phi_2$ are upper semicontinuous in the first argument, then by \eqref{Eq4.24}, \eqref{c32} and $\liminf_{k\to \infty}r_{n_k} > 0,$ we have
		
		\begin{equation}\label{Eq4.32}
			F_2(Aq, y) + \phi_2(Aq, y) \geq 0,~~\forall y \in Q.
		\end{equation}
		
		
		\noindent Following similar argument, we have
		
		\begin{equation}\label{Eq4.33}
			F_2(Az, y) + \phi_2(Az, y) \geq 0,~~\forall y \in Q.
		\end{equation}
		
		\noindent From \eqref{Eq4.32} and \eqref{Eq4.33}, it follows that $Aq,Az \in GEP(F_2, \phi_2).$ Therefore $q,z \in SGEP(F_1, \phi_1, F_2, \phi_2).$   By Invoking Lemma \ref{lem3g}, we get $q=z.$ Hence, we have that $q\in\Gamma.$
		
		
		
		\noindent \underline{Step 4.} Lastly, we show that $q = P_{\Gamma}x_0.$
		
		\noindent\\
		\noindent Since $x_n = P_{C_n}x_0$ and $\Gamma \subset C_n,$  we have $\langle x_0 - x_n, x_n - p \rangle \geq 0$ for all $p \in \Gamma$. By taking limit as $n\to\infty,$ we have $\langle x_0 - q, q - p \rangle \geq 0$ for all $p \in \Gamma$. This shows that $q = P_{\Gamma}x_0$.
		
		\noindent Therefore, we can conclude by the steps above that $\{x_n\}$ converges strongly to $q = P_{\Gamma}x_0$. This completes the proof.
	\end{proof}
	
	\noindent If $\phi_1 = \phi_2 = 0$ in \eqref{c3x}-\eqref{c4x}, then the split generalized equilibrium problem reduces to split equilibrium problem. Hence from Theorem \ref{alg3a} , we obtain the following consequent result.
	
	\begin{corollary}\label{corl1x}
		\noindent Let $C$ and $Q$ be nonempty closed convex subsets of real Hilbert spaces $H_1$ and $H_2$, respectively. Let $A : H_1 \rightarrow H_2$ be a bounded linear operator, and let $\{S_i\}_{i=1}^m$ be a countable family of $k_i$-strictly pseudo-contractive multivalued mappings of $C$ into $CB(C)$ such that $I-S_i$ is demiclosed at zero for each $i=1,2,\ldots,m,$  $S_ip = \{p\}$ for each $p \in \cap_{i=1}^mF(S_i)$ and $k=\max\{k_i\}.$ Let $F_1: C \times C \rightarrow \mathbb{R}, F_2: Q \times Q \rightarrow \mathbb{R}$ be bifunctions satisfying Assumptions \ref{Ass 1} such that $F_2$ is upper semicontinuous in the first argument. Let $B : H_1 \rightarrow H_1$ be $L$-Lipschitz continuous and monotone and $D : H_1 \rightarrow 2^{H_1}$ be a maximal monotone operator such that $\Gamma = SGEP(F_1, F_2)  \cap_{i=1}^mF(S_i) \bigcap (B + D)^{-1}(0) \neq \emptyset.$ Let $\{x_n\}$ be a sequence generated as follows:
		
\hrule		
		\begin{algorithm}\label{alg4a}\noindent
			\hrule \hrule
			
			\noindent \textbf{Initialization:} Select $x_0, x_1 \in H_1$, $\mu \in (0, 1), \theta_n \in [-\theta, \theta]$ for some $\theta>0$ and $C_1=C.$
			
			\noindent \textbf{Iterative Step:} Given the current iterate $x_n$, calculate the next iterate as follows:
			
			\noindent \textbf{Step 1 :} Compute
			\begin{equation*}
				w_n = x_n + \theta_n(x_n - x_{n - 1}).
			\end{equation*}
			\noindent \textbf{Step 2 :} Compute
			\begin{equation*}
				z_n = T_{r_n}^{F_1}(I - \gamma_nA^*(I - T_{r_n}^{F_2})A)w_n.
			\end{equation*}
			\noindent \textbf{Step 3 :} Compute
			\begin{equation*}
				y_n = \alpha_{n, 0}z_n + \sum_{i = 1}^{m}\alpha_{n, i}u_{n, i},\quad u_{n, i} \in S_iz_n.
			\end{equation*}
			\noindent \textbf{Step 4 :} Compute
			\begin{equation*}
				\begin{cases}
					v_n = (I + s_nD)^{-1}(I - s_nB)y_n \\
					t_n = v_n - s_n(Bv_n - By_n)\\
					C_{n+1} = \{p \in C_n : \|t_n - p\|^2 \leq \|x_n - p\|^2 - 2\theta_n\langle
					x_n - p, x_{n-1} - x_n\rangle \\
                    + \theta_n^2\|x_{n-1} - x_n\|^2\}\\
					x_{n+1} = P_{C_{n+1}}x_0,
				\end{cases}
			\end{equation*}
			\noindent \textbf{Step 5 :} Compute
			\begin{equation}
				s_{n+1} = \begin{cases}
					\min\biggl\{\frac{\mu\|y_n - v_n\|}{\|By_n - Bv_n\|} , s_n\biggr\} & \text{if}~~By_n - Bv_n \neq 0.\\
					s_n   & \text{otherwise},
				\end{cases}
			\end{equation}
			\noindent Set $n := n + 1$ and return to {\bf Step 1}.
			\noindent where
			\begin{equation*}
				\gamma_n =
				\begin{cases}
					\tau_n\frac{||(I - T_{r_n}^{(F_2})Aw_n||^2}{||A^*(I - T_{r_n}^{(F_2})Aw_n||^2} &  \text{If~~} Aw_n \neq T^{(F_2}_{r_n}Aw_n\\
					
					\gamma  &  \text{otherwise}~~(\gamma~~\text{being any non-negative real number.})
				\end{cases}
			\end{equation*}
			
		\end{algorithm}
		\hrule
		
		
		\noindent Suppose other conditions of Theorem \ref{alg3a} hold. Then, the sequence $\{x_n\}$ converges strongly to $q = P_{\Gamma}x_0.$
	\end{corollary}
	
	
	
	
	
	
	
	\section{Applications}\label{Sec5}
	
	
	\subsection{Split Minimization Problem}
	\noindent \\ Let $H_1,~ H_2$ be two real Hilbert spaces, and let $C\subset H_1$ and $Q\subset H_2$ be nonempty, closed, and convex subsets. Let $f:C\to\mathbb{R}, ~~ ~~ g:Q\to\mathbb{R}$ be two operators and $A:H_1\to H_2$ be a bounded linear operator. The split minimization problem (SMP) is formulated as finding
	\begin{equation}\label{Eq5.1B}
		x^*\in C \quad \text{such that}\quad f(x^*)\le f(x), ~~ ~ \forall x\in C,
	\end{equation}
	
	and
	
	\begin{equation}\label{Eq5.2A}
		y^*=Ax^* \quad \text{such that}\quad g(y^*)\le g(y), ~~ ~ y\in Q.
	\end{equation}
	\noindent Let $\Omega$ denote the set of solution of SMP \eqref{Eq5.1B}-\eqref{Eq5.2A}, and we assume $\Omega \ne \emptyset.$ Let $\phi_1=\phi_2=0,$ and
	$$F_1(x,y):= f(y)-f(x)~~ ~~\quad \text{for all}~~ ~ x,y\in C;$$
	\noindent and
	$$F_2(u,v):= g(v)-g(u)\quad \text{for all}~~ ~ u,v\in Q.$$
	
	\noindent Suppose $f$ and $g$ are convex and lower semi-continuous on $C$ and $Q,$ respectively. Then, $F_1, F_2, \phi_1$ and $\phi_2$ satisfy all the conditions of Assumption \ref{Ass 1}. Consequently, from Theorem \ref{alg3a} we obtain a strong convergence theorem for approximating a common solution of split minimization problem, monotone variational inclusion problem and fixed point problem for a countable family of strict pseudo-contractive multivalued mappings in Hilbert spaces.
	
	
	
	\subsection{Split Variational Inequality Problem}
	\noindent \\ Let $C$ be a nonempty closed convex subset of a real Hilbert space $H$, and $f:H\to H$ be a single-valued mapping. The variational inequality problem (VIP) introduced independently by Fichera \cite{FicG} and Stampacchia \cite{StampG} is formulated as follows:
	\begin{align} \label{Eq5.2}
		\text{find}~~ x^* \in C ~~ \text{such that}~~ \langle y-x^*, fx^* \rangle \ge 0, \quad \quad \forall~ y\in C.
	\end{align}
	\noindent The VIP can be modelled to solve several optimization problems and has vast applications  in different fields, such as in physics, engineering,  economics, etc, (see \cite{alakoya2, BlOe, CeGiRe, dph, ogwo, ogwo3, sitt}).
	
	\noindent The split variational inequality problem (SVIP), which was first introduced by Censor et al. \cite{CeGiRe} is defined as finding a point:\
	\begin{align}
		& x^* \in C ~~ \text{such that}~~ \langle x-x^*, f(x^*) \rangle \ge 0 \quad \quad \forall~ x\in C,\label{Eq5.3}\\ \text{and}\quad&\nonumber\\
		& y^* =Ax^*\in Q ~~ \text{solves}~~ \langle y-y^*, g(y^*) \rangle \ge 0 \quad \quad \forall~ y\in Q,\label{Eq5.4}
	\end{align}
	\noindent where $C$ and $Q$ are nonempty, closed, convex subsets of real Hilbert spaces $H_1$ and $H_2,$  respectively, $f:H_1\to H_1$ and $g:H_2\to H_2$ are monotone mappings, and $A:H_1\to H_2$ is a bounded linear operator, see \cite{LoKaF}. Let $\Omega \ne \emptyset$ denote the set of solution of SVIP \eqref{Eq5.3}-\eqref{Eq5.4}. By setting $\phi_1=\phi_2=0,$ and
	$$F_1(x,y):= \langle y-x, f(x) \rangle \quad \text{for all}~~ ~ x,y\in C;$$
	\noindent\hspace{110pt} and
	$$F_2(u,v):= \langle v-u, g(u) \rangle \quad \text{for all}~~ ~ u,v\in Q.$$
	\noindent Then, $F_1, F_2, \phi_1$ and $\phi_2$ satisfy all the conditions of Assumption \ref{Ass 1}. Hence, from Theorem \ref{alg3a}, we obtain a strong convergence theorem for approximating a common solution of split variational inequality problem, monotone variational inclusion problem and fixed point problem for a countable family of strict pseudo-contractive multivalued mappings in Hilbert spaces.
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	\section{Numerical Examples}\label{Sec6}
	
	\noindent In this section, we present a numerical experiments to illustrate the
	performance of our Algorithm \ref{alg3a} as well as comparing it with Algorithm \eqref{algdep}, Algorithm \eqref{algsitth}, Algorithm \eqref{algphu} and Algorithm \eqref{xii} in the literature.
	
	
	
	\noindent In our computation,  we choose $\alpha_{n,0}=\frac{n}{2n+1}, \alpha_{n,i}=\frac{n+1}{5(2n+1)},~i=1,2,\ldots,5, \tau_n=1.5, \theta_n=1.9, r_n=2.0, s_0=0.1$ and $\mu=0.7$ in our Algorithm \ref{alg3a}. $Gx=\frac{1}{3}x,fx=\frac{2}{3}x,Kx=\frac{2}{5}x, \lambda_n=\frac{2n}{5n+1}, \alpha_n=\frac{2}{2n+3},\beta_n =\frac{n+1}{2n+3}, \eta=\frac{2}{5},\gamma=0.2, T_jx=\frac{2}{(3+j)}x$ in Algorithm \eqref{algdep},$ \beta_n=\xi_n=\frac{1}{2}(1-\alpha_n),\sigma_n=\frac{2}{2n+1}$ in Algorithm \eqref{algsitth} while in
	Algorithm \eqref{algphu} and Algorithm \eqref{xii}. Let the sequences $\{\delta_{n,j}\}$ be defined as follows for each $j\in\mathbb{N}\cup\{0\}$ and $n\in\mathbb{N}:$
	\begin{equation}\label{Eq5.1A}
		\delta_{n,j} =\begin{cases}
			\frac{1}{b^{j+1}}(\frac{n}{n+1}),\hspace{67pt} n> j,\\
			1-\frac{n}{n+1}(\sum_{k=1}^{n}\frac{1}{b^k}),\hspace{30pt} n =j,\\
			0, \hspace{105pt} n<j,
		\end{cases}
	\end{equation}
	\noindent where $b>1.$
	
	
	
	
	\begin{example}\label{EX1}
		Let $H_1=H_2=\mathbb{R}$ and $C=Q=[0,10].$ Let $A:H_1\to H_2$ be defined by $Ax=\frac{x}{5}$ for all $x\in H_1.$ Then, we have that $A^*y=\frac{y}{5}$ for all $y\in H_2.$ For $x\in C,j\in\mathbb{N}~~$and $ i=1,2,\ldots,5,$ let $P_j,S_i:C\to CB(C)$ be multivalued mappings defined as follows:
		\begin{equation}
			P_j(x) = \bigg[0, \frac{x}{10j}\bigg],\quad
			S_i(x) = \bigg[0, \frac{x}{10i}\bigg] .
		\end{equation}	
		\noindent One can easily verify that $P_j$ and $S_i$ are nonexpansive and strictly pseudo-contractive, respectively.  Define mappings $B:H_1\to H_1$ by $Bx=2x,$ $D:H_1\to H_1$ by $Dx=3x,$ and	
		let the bifunctions $F_1,\phi_1:C\times C\to\mathbb{R}$ be defined by $F_1(x,y)=y^2+3xy-4x^2$ and $\phi_1(x,y)=y^2-x^2$ for $x,y\in C,$ and $F_2,\phi_2:Q\times Q\to\mathbb{R}$ by $F_2(w,v)=2v^2+wv-3w^2$ and $\phi_2(w,v)=w-v$ for $w,v\in Q.$ It is easy to verify that all the conditions of Theorem \ref{the3} are satisfied. Next, we compute $T_r^{(F_1,\phi_1)}(x).$ We find $u\in C$ such that for all $z\in C$
		\begin{align*}
			0&\leq F_1(u,z) + \phi_1(u,z) + \frac{1}{r}\langle z - u, u - x \rangle \\
			&= 2z^2 + 3uz - 5u^2 + \frac{1}{r}\langle z - u, u - x \rangle\\
			&\Leftrightarrow\\
			0&\leq 2rz^2 + 3ruz - 5ru^2 + (z-u)(u-x)\\
			&= 2rz^2 + 3ruz - 5ru^2 + uz - xz - u^2 + ux\\
			&= 2rz^2 + (3ru+u-x)z + (-5ru^2 - u^2 + ux).
		\end{align*}
		\noindent Suppose $h(z) = 2rz^2 + (3ru+u-x)z + (-5ru^2 - u^2 + ux).$ Then, $h(z)$ is a quadratic function of $z$ with coefficients $a=2r, b=3ru+u-x,$ and $c=-5ru^2 - u^2 + ux.$ We determine the discriminant $\triangle$ of $h(z)$ as follows:
		\begin{align}
			\triangle &= (3ru+u-x)^2 -4(2r)(-5ru^2 - u^2 + ux)\nonumber\\
			&= 49r^2u^2 + 14ru^2 - 14rux + u^2 - 2ux + x^2\nonumber\\
			&= ((7r+1)u-x)^2.\label{cdx}
		\end{align}
		\noindent By Lemma \ref{lem3c},  $T_r^{(F_1,\phi_1)}$ is single-valued. Thus, it follows that $h(z)$ has at most one solution in $\mathbb{R}.$ Hence, from \eqref{cdx}, we have that $u= \frac{y}{7r+1}.$ This implies that $T_r^{(F_1,\phi_1)}(y)=\frac{y}{7r+1}.$ Similarly, we compute $T^{(F_2, \phi_2)}_r (y).$ Find $w\in Q$ such that for all $d\in Q$
		\begin{equation*}
			T^{(F_2, \phi_2)}_r (y) = \left\{ w \in Q : F_2 (w,d) + \phi_2(w,d) + \frac{1}{r} \langle d - w, w - y \rangle\ge 0,\quad \forall~ d \in Q \right\}.
		\end{equation*}
		By following similar procedure as above, we obtain $w=\frac{y+r}{5r+1}.$ This implies that $T^{(F_2, \phi_2)}_r (y)=\frac{y+r}{5r+1}.$
		
		\noindent In this example, we set the parameter $b$ on $\{\delta_{n,i}\}$ in \eqref{Eq5.1A} to be $b=40, v=3.5$ and we choose different initial values as follows:\\
		\noindent {\it Case I:} $x_0 = 7, x_1 = 3;$\\
		\noindent {\it Case II:} $x_0 = 6 , x_1 = 2;$\\
		\noindent {\it Case III:} $x_0 = 8, x_1 = 4;$\\
		\noindent {\it Case IV:} $x_0 = 9, x_1 = 5.$\\\\
		\noindent We compare the performance of our Algorithm \ref{alg3a} with Algorithms \eqref{algdep}, \eqref{algsitth},  \eqref{algphu} and \eqref{xii}. The stopping criterion used for our computation is $|x_{n+1}-x_{n}|< 10^{-4}$. We plot the graphs of errors against the number of iterations in each case.  The numerical results are reported in Figure \ref{fig1} and Table \ref{tab1}.
	
\begin{table}[h]
		\caption{\bf Numerical results for Example \ref{EX1}}
		\label{tab1}
		\begin{tabular}{ |p{1.3cm} |p{2.5cm}| p{1.2cm}| p{1.2cm}| p{1.2cm}| p{1.2cm}|  p{1.2cm}|}
			\hline
			\noindent & \noindent & Alg. \eqref{algdep} & Alg. \eqref{algsitth} & Alg. \eqref{algphu} & Alg. \eqref{xii} & Alg. \ref{alg3a}\\
			\hline
			Case I  & No. of Iter.  & 9 & 20 & 4 & 9 & 2\\
			&   CPU time (sec)  & 0.0057 & 0.0078 & 1.6693 & 0.3383 & 0.0032\\
			\hline
			Case II & No. of Iter. & 8 & 20 & 4 & 8 & 2\\
			&   CPU time (sec)  & 0.0051 & 0.0059 & 1.6884 & 0.3124 & 0.0039\\
			\hline
			Case III  & No. of Iter. & 9 & 20 & 4 & 9 & 2\\
			&   CPU time (sec)  & 0.0053 & 0.0057 & 1.6625 & 0.3566 & 0.0041\\
			\hline
			Case IV & No. of Iter. & 9 & 20 & 4 & 9 & 2 \\
			&   CPU time (sec)  & 0.0054 & 0.0067 & 1.6623 & 0.3449 & 0.0039\\
			\hline

			
			
		\end{tabular}
	\end{table}
	
	
	\begin{figure}
		\begin{center}
			\includegraphics[height=4cm]{Case1.pdf}	
			\includegraphics[height=4cm]{Case2.pdf}\\
			\includegraphics[height=4cm]{Case3.pdf}	
			\includegraphics[height=4cm]{Case4.pdf}\\
		\end{center}
		\caption{ Top left: Case I ; Top right: Case II; Bottom left:  Case III ; Bottom right: Case IV.}\label{fig1}
	\end{figure}
	
		

	\end{example}
	
	
	\begin{example}\label{EX2}
		Let $H_1=H_2 = L_2([0,1])$  with the inner product defined as
		\begin{equation*}
			\langle x, y \rangle = \int_0^1 x(t)y(t)dt, \quad \forall x,y\in L_2([0,1]).
		\end{equation*}
		\noindent Let
		$$C:=\{x\in H_1:\langle a,x \rangle \ge d\},$$
		\noindent where $a=2t^2$ and $d= 0.$ Here, we have
		$$P_C(x) = x + \frac{d-\langle a,x \rangle}{||a||^2}a.$$
		\noindent Also, let
		$$Q:=\{x\in H_2:\langle c,x \rangle\leq e\},$$
		\noindent where $c=\frac{t}{3},$  $e=1$ and we have
		$$P_Q(x) = x + \max \bigg\{0, \frac{e-\langle c,x \rangle}{||c||^2}c\bigg\}.$$
		\noindent Let $F_1:C\times C\rightarrow\mathbb{R}$ and $F_2:Q\times Q \rightarrow\mathbb{R}$ be defined as  $F_1(x,y) =\langle L_1x, y-x \rangle$ and $F_2(x,y) =\langle L_2x, y-x \rangle,$ where $L_1x(t) =\frac{x(t)}{3}$ and $L_2x(t) =\frac{x(t)}{4}.$ It can easily be verified that $F_1$ and $F_2$ satisfy conditions (A1)-(A4). Also, let $\phi_1=\phi_2=0.$ Furthermore, define $B:H_1\to H_1$ by $Bx=3x,$ $D:H_1\to H_1$ by $Dx=7x,$ and	let $A:L_2([0,1])\rightarrow L_2([0,1])$ be defined by $Ax(t)=\frac{x(t)}{3}$ and $A^*y(t)=\frac{y(t)}{3}.$  Then, $A$ is a bounded linear operator. We consider the case for which the multivalued mappings $\{S_j\}$ and $\{S_i\}$ are single-valued.	Let $S_j,S_i: L^2([0,1])\rightarrow L^2([0,1])$ be defined by
		$$
		(S_jx)(t) = \int_0^1 t^jx(s)ds\quad\quad\text{and}\quad\quad (S_ix)(t) = \int_0^1 t^ix(s)ds\;\;\;\text{for all}\,\, t\in [0,1].
		$$
		Note that $S_i$ and $S_j$ are nonexpansive for each $i,j.$ Select $r_n=\frac{2n}{2n+1}, \theta_n=0.8, \tau_n=0.7.$ It can easily be checked that all the conditions of Theorem \ref{the3} are satisfied. Now, we compute $T_r^{(F_1,\phi_1)}(x).$ We find $z\in C$ such that for all $y\in C$
		\begin{align}\label{Eq5.1}
			F_1(z&,y) + \phi_1(z,y) + \frac{1}{r}\langle y - z, z - x \rangle \geq 0\nonumber\\
			\Leftrightarrow & \langle \frac{z}{2}, y-z \rangle + \frac{1}{r}\langle y - z, z - x \rangle \geq 0\nonumber\\
			\Leftrightarrow & \frac{z}{3}(y-z) + \frac{1}{r}(y - z)(z - x)\geq 0\nonumber\\
			\Leftrightarrow & (y - z)[rz + 3(z-x)]\geq 0\nonumber\\
			\Leftrightarrow & (y - z)[(r+3)z -3x]\geq 0.
		\end{align}
		By Lemma \ref{lem3c}, we obtain
		\begin{equation*}
			T^{(F_1, \phi_1)}_r (x) = \left\{z \in C : F_1(z,y) + \phi_1(z,y) + \frac{1}{r} \langle y - z, z - x \rangle \ge 0, ~~ \forall~ y \in C \right\}, 
		\end{equation*}
($\forall~~x \in H_1$), is single-valued. Thus, from (\ref{Eq5.1}) we obtain $z=\frac{3x}{r+3}.$ This implies that $T^{(F_1, \phi_1)}_r (x) = \frac{3x}{r+3}.$  Similarly, we compute $T^{(F_2, \phi_2)}_r (v).$ We find $w\in Q$ such that for all $d\in Q$
		\begin{equation*}
			T^{(F_2, \phi_2)}_s (v) = \left\{ w \in Q : F_2 (w,d) + \phi_2(w,d) + \frac{1}{s} \langle d - w, w - v \rangle\ge 0,\quad \forall~ d \in Q \right\}.
		\end{equation*}
		By using similar approach as above, we obtain $w=\frac{4v}{s+4}.$ This implies that $T^{(F_2, \phi_2)}_s (v)=\frac{4v}{s+4}.$
		
	\noindent Here, we set the parameter $b$ on $\{\delta_{n,i}\}$ in \eqref{Eq5.1A} to be $b=3, v=t^2$ and we choose different initial values as follows:\\
		\noindent {\it Case I:} $x_0 = t^4, x_1 = t^2 + t^4 + t^6 + 3;$\\
		\noindent {\it Case II:} $x_0 = t^5,  x_1 = t^2 + t^5 + 2;$\\
		\noindent {\it Case III:} $x_0 = t^4, x_1 = t^3 + t^5 + t^7 + 2;$\\
		\noindent {\it Case IV:} $x_0 = t^5, x_1 = t + t^2 + 1.$\\\\
		\noindent We compare the performance of our Algorithm \ref{alg3a} with Algorithms \eqref{algdep}, \eqref{algsitth},  \eqref{algphu} and \eqref{xii}. The stopping criterion used for our computation is $||x_{n+1}-x_{n}||< 10^{-4}$. We plot the graphs of errors against the number of iterations in each case.  The numerical results are reported in Figure \ref{fig2} and Table \ref{tab2}.
	

\begin{table}[h]
		\caption{\bf Numerical results for Example \ref{EX2}}
		\label{tab2}
		\begin{tabular}{ |p{1.3cm} |p{2.5cm}| p{1.2cm}| p{1.2cm}| p{1.2cm}| p{1.2cm}|  p{1.2cm}|}
			\hline
			\noindent & \noindent & Alg. \eqref{algdep} & Alg. \eqref{algsitth} & Alg. \eqref{algphu} & App. \eqref{xii} & Alg. \ref{alg3a}\\
			\hline
			Case I & No. of Iter.  & 10 & 14 & 10 & 6 & 6\\
			&   CPU time (sec)  & 0.7297 & 0.7237 & 1.2541 & 0.2548 & 0.3256\\
			\hline
			Case II & No. of Iter. & 9 & 14 & 9 & 6 & 6\\
			&   CPU time (sec)  & 0.6743 & 0.7004 & 1.1791 & 0.2628 & 0.3091\\
			\hline
			Case III & No. of Iter. & 9 & 14 & 9 & 6 & 6\\
			&   CPU time (sec)  & 0.6507 & 0.6825 & 1.1474 & 0.2599 & 0.3087\\
			\hline
			Case IV & No. of Iter. & 9 & 13 & 8 & 6 & 6 \\
			&   CPU time (sec)  & 0.6353 & 0.6458 & 1.1130 & 0.2631 & 0.3166\\
			\hline

			
			
		\end{tabular}
	\end{table}
	
	
	\begin{figure}
		\begin{center}
			\includegraphics[height=4cm]{CaseA.pdf}	
			\includegraphics[height=4cm]{CaseB.pdf}\\
			\includegraphics[height=4cm]{CaseC.pdf}	
			\includegraphics[height=4cm]{CaseD.pdf}\\
		\end{center}
		\caption{ Top left: Case I; Top right: Case II; Bottom left:  Case III; Bottom right: Case IV.}\label{fig2}
	\end{figure}
	
		
	
	\end{example}	
	
	
	
	
	
	
	
	\section{Conclusion}\label{Sec7}
	\noindent In this article, we proposed a new modified inertial shrinking projection algorithm for finding common solution of split generalized equilibrium problem, monotone inclusion problem and fixed point problems for a countable family of strictly pseudo-contractive multivalued mappings. We established strong convergence result for the proposed method. We applied our results to study related optimization problems and presented  some numerical examples to demonstrate the efficiency of our proposed method in comparison with other existing methods. Our results extend and improve several existing results in this direction in the current literature.
	
\noindent {\bf Acknowlegment:} The first author acknowledges with thanks the bursary and financial support from Department of Science and Innovation and National Research Foundation, Republic of South Africa Center of Excellence in Mathematical and Statistical Sciences (DSI-NRF COE-MaSS) Doctoral Bursary. Opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the CoE-MaSS.

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