Multi-step iterative algorithms with regularization for triple hierarchical variational inequalities with constraints of mixed equilibria, variational inclusions, and convex minimization

Abstract

In this paper, we introduce and analyze a relaxed iterative algorithm by virtue of Korpelevich’s extragradient method, the viscosity approximation method, the hybrid steepest-descent method, the regularization method, and the averaged mapping approach to the gradient-projection algorithm. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to a common element of the fixed point set of infinitely many nonexpansive mappings, the solution set of finitely many generalized mixed equilibrium problems (GMEPs), the solution set of finitely many variational inclusions and the set of minimizers of convex minimization problem (CMP), which is just a unique solution of a triple hierarchical variational inequality (THVI) in a real Hilbert space. In addition, we also consider the application of the proposed algorithm to solving a hierarchical fixed point problem with constraints of finitely many GMEPs, finitely many variational inclusions, and CMP. The results obtained in this paper improve and extend the corresponding results announced by many others.

MSC:49J30, 47H09, 47J20, 49M05.

1 Introduction

Let C be a nonempty closed convex subset of a real Hilbert space H and $P_C$ be the metric projection of H onto C. Let $S:C\to H$ be a nonlinear mapping on C. We denote by $Fix(S)$ the set of fixed points of S and by R the set of all real numbers. A mapping $S:C\to H$ is called L-Lipschitz continuous if there exists a constant $L\ge 0$ such that

$$\parallel Sx-Sy\parallel \le L\parallel x-y\parallel ,\mathrm{\forall }x,y\in C.$$

In particular, if $L=1$ then S is called a nonexpansive mapping; if $L\in [0,1)$ then S is called a contraction.

Let $A:C\to H$ be a nonlinear mapping on C. We consider the following variational inequality problem (VIP): find a point $x\in C$ such that

$$〈Ax,y-x〉\ge 0,\mathrm{\forall }y\in C.$$

(1.1)

The solution set of VIP (1.1) is denoted by $VI(C,A)$.

The VIP (1.1) was first discussed by Lions [1]. There are many applications of VIP (1.1) in various fields; see, e.g., [2–5]. It is well known that, if A is a strongly monotone and Lipschitz continuous mapping on C, then VIP (1.1) has a unique solution. In 1976, Korpelevich [6] proposed an iterative algorithm for solving the VIP (1.1) in Euclidean space ${\mathbf{R}}^n$:

$$\{\begin{array}{c}{y}_n=P_C(x_n-\tau A{x}_n),\hfill \\ x_{n+1}=P_C(x_n-\tau A{y}_n),\mathrm{\forall }n\ge 0,\hfill \end{array}$$

with $\tau >0$ a given number, which is known as the extragradient method. The literature on the VIP is vast and Korpelevich’s extragradient method has received great attention from many authors, who improved it in various ways; see, e.g., [7–20] and references therein, to name but a few.

Let $\phi :C\to \mathbf{R}$ be a real-valued function, $A:H\to H$ be a nonlinear mapping and $\Theta :C\times C\to \mathbf{R}$ be a bifunction. In 2008, Peng and Yao [8] introduced the generalized mixed equilibrium problem (GMEP) of finding $x\in C$ such that

$$\Theta (x,y)+\phi (y)-\phi (x)+〈Ax,y-x〉\ge 0,\mathrm{\forall }y\in C.$$

(1.2)

We denote the set of solutions of GMEP (1.2) by $GMEP(\Theta ,\phi ,A)$. The GMEP (1.2) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problems in noncooperative games and others. The GMEP is further considered and studied; see, e.g., [10, 16, 18, 19, 21–23]. In particular, if $\phi =0$, then GMEP (1.2) reduces to the generalized equilibrium problem (GEP) which is to find $x\in C$ such that

$$\Theta (x,y)+〈Ax,y-x〉\ge 0,\mathrm{\forall }y\in C.$$

It was introduced and studied by Takahashi and Takahashi [24]. The set of solutions of GEP is denoted by $GEP(\Theta ,A)$.

If $A=0$, then GMEP (1.2) reduces to the mixed equilibrium problem (MEP) which is to find $x\in C$ such that

$$\Theta (x,y)+\phi (y)-\phi (x)\ge 0,\mathrm{\forall }y\in C.$$

It was considered and studied in [25]. The set of solutions of MEP is denoted by $MEP(\Theta ,\phi )$.

If $\phi =0$, $A=0$, then GMEP (1.2) reduces to the equilibrium problem (EP) which is to find $x\in C$ such that

$$\Theta (x,y)\ge 0,\mathrm{\forall }y\in C.$$

It was considered and studied in [26, 27]. The set of solutions of EP is denoted by $EP(\Theta )$. It is worth to mention that the EP is an unified model of several problems, namely, variational inequality problems, optimization problems, saddle point problems, complementarity problems, fixed point problems, Nash equilibrium problems, etc.

It was assumed in [8] that $\Theta :C\times C\to \mathbf{R}$ is a bifunction satisfying conditions (A1)-(A4) and $\phi :C\to \mathbf{R}$ is a lower semicontinuous and convex function with restriction (B1) or (B2), where

• (A1) $T(x,x)=0$ for all $x?C$;

• (A2) T is monotone, i.e., $T(x,y)+T(y,x)=0$ for any $x,y?C$;

• (A3) T is upper-hemicontinuous, i.e., for each $x,y,z?C$,

  $$\underset{t?0^{+}}{lim?sup}T(tz+(1-t)x,y)=T(x,y);$$

• (A4) $T(x,·)$ is convex and lower semicontinuous for each $x?C$;

• (B1) for each $x?H$ and $r>0$, there exists a bounded subset $D_x?C$ and $y_x?C$ such that, for any $z?C\backslash D_x$,

  $$T(z,y_x)+f(y_x)-f(z)+\frac{1}{r}{y}_x-z,z-x><0;$$

• (B2) C is a bounded set.

Given a positive number $r>0$. Let $T_r^{(\Theta ,\phi )}:H\to C$ be the solution set of the auxiliary mixed equilibrium problem, that is, for each $x\in H$,

$$T_r^{(\Theta ,\phi )}(x):=\{y\in C:\Theta (y,z)+\phi (z)-\phi (y)+\frac{1}{r}〈y-x,z-y〉\ge 0,\mathrm{\forall }z\in C\}.$$

On the other hand, let B be a single-valued mapping of C into H and R be a multivalued mapping with $D(R)=C$. Consider the following variational inclusion: find a point $x\in C$ such that

$$0\in Bx+Rx.$$

(1.3)

We denote by $\mathrm{I}(B,R)$ the solution set of the variational inclusion (1.3). In particular, if $B=R=0$, then $\mathrm{I}(B,R)=C$. If $B=0$, then problem (1.3) becomes the inclusion problem introduced by Rockafellar [28]. It is well known that problem (1.3) provides a convenient framework for the unified study of optimal solutions in many optimization related areas including mathematical programming, complementarity problems, variational inequalities, optimal control, mathematical economics, equilibria, and game theory, etc. Let a set-valued mapping $R:D(R)\subset H\to 2^H$ be maximal monotone. We define the resolvent operator $J_{R,\lambda }:H\to \overline{D(R)}$ associated with R and λ as follows:

$$J_{R,\lambda }={(I+\lambda R)}^{-1},\mathrm{\forall }x\in H,$$

where λ is a positive number.

In 1998, Huang [9] studied problem (1.3) in the case where R is maximal monotone and B is strongly monotone and Lipschitz continuous with $D(R)=C=H$. Subsequently, Zeng et al. [29] further studied problem (1.3) in the case which is more general than Huang’s one [9]. Moreover, the authors of [29] obtained the same strong convergence conclusion as in Huang’s result [9]. In addition, the authors also gave the geometric convergence rate estimate for approximate solutions. Also, various types of iterative algorithms for solving variational inclusions have been further studied and developed; for more details, refer to [11, 12, 30, 31] and the references therein.

Let $f:C\to \mathbf{R}$ be a convex and continuously Fréchet differentiable functional. Consider the convex minimization problem (CMP) of minimizing f over the constraint set C,

$$\text{minimize }\{f(x):x\in C\}.$$

(1.4)

It and its special cases were considered and studied in [13, 14, 32–34]. We denote by Γ the set of minimizers of CMP (1.4). The gradient-projection algorithm (GPA) generates a sequence $\{x_n\}$ determined by the gradient ∇f and the metric projection $P_C$:

$$x_{n+1}:=P_C(x_n-\lambda \mathrm{\nabla }f(x_n)),\mathrm{\forall }n\ge 0,$$

(1.5)

or, more generally,

$$x_{n+1}:=P_C(x_n-{\lambda }_n\mathrm{\nabla }f(x_n)),\mathrm{\forall }n\ge 0,$$

(1.6)

where, in both (1.5) and (1.6), the initial guess $x_0$ is taken from C arbitrarily, the parameters λ or ${\lambda }_n$ are positive real numbers. The convergence of algorithms (1.5) and (1.6) depends on the behavior of the gradient ∇f. As a matter of fact, it is well known that, if ∇f is α-strongly monotone and L-Lipschitz continuous, then, for $0<\lambda <\frac{2\alpha }{L^2}$, the operator $P_C(I-\lambda \mathrm{\nabla }f)$ is a contraction; hence, the sequence $\{x_n\}$ defined by the GPA (1.5) converges in norm to the unique solution of CMP (1.4). More generally, if $\{{\lambda }_n\}$ is chosen to satisfy the property

$$0<\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}{\lambda }_n\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\lambda }_n<\frac{2\alpha }{L^2},$$

then the sequence $\{x_n\}$ defined by the GPA (1.6) converges in norm to the unique minimizer of CMP (1.4). If the gradient ∇f is only assumed to be Lipschitz continuous, then $\{x_n\}$ can only be weakly convergent if H is infinite-dimensional (a counterexample is given in Section 5 of Xu [33]). Recently, Xu [33] used averaged mappings to study the convergence analysis of the GPA, which is hence an operator-oriented approach.

Very recently, Ceng and Al-Homidan [23] introduced and analyzed the following iterative algorithm by the hybrid steepest-descent viscosity method and derived its strong convergence under appropriate conditions.

Theorem CA (see [[23], Theorem 21])

Let C be a nonempty closed convex subset of a real Hilbert space H. Let $f:C\to \mathbf{R}$ be a convex functional with L-Lipschitz continuous gradient ∇f. Let M, N be two integers. Let ${\Theta }_k$ be a bifunction from $C\times C$ to R satisfying (A1)-(A4) and ${\phi }_k:C\to \mathbf{R}\cup \{+\mathrm{\infty }\}$ be a proper lower semicontinuous and convex function, where $k\in \{1,2,\dots ,M\}$. Let $B_k:H\to H$ and $A_i:C\to H$ be ${\mu }_k$-inverse strongly monotone and ${\eta }_i$-inverse strongly monotone, respectively, where $k\in \{1,2,\dots ,M\}$, $i\in \{1,2,\dots ,N\}$. Let $F:H\to H$ be a κ-Lipschitzian and η-strongly monotone operator with positive constants $\kappa ,\eta >0$. Let $V:H\to H$ be an l-Lipschitzian mapping with constant $l\ge 0$. Let $0<\mu <\frac{2\eta }{{\kappa }^2}$ and $0\le \gamma l<\tau$, where $\tau =1-\sqrt{1-\mu (2\eta -\mu {\kappa }^2)}$. Assume that $\Omega :={\bigcap }_{k=1}^{M}GMEP({\Theta }_k,{\phi }_k,B_k)\cap {\bigcap }_{i=1}^{N}VI(C,A_i)\cap \Gamma \ne \mathrm{\varnothing }$ and that either (B1) or (B2) holds. For arbitrarily given $x_1\in H$, let $\{x_n\}$ be a sequence generated by

$$\{\begin{array}{c}{u}_n=T_{r_{M,n}}^{({\Theta }_M,{\phi }_M)}(I-r_{M,n}{B}_M)T_{r_{M-1,n}}^{({\Theta }_{M-1},{\phi }_{M-1})}(I-r_{M-1,n}{B}_{M-1})\cdots T_{r_{1,n}}^{({\Theta }_1,{\phi }_1)}(I-r_{1,n}{B}_1)x_n,\hfill \\ v_n=P_C(I-{\lambda }_{N,n}{A}_N)P_C(I-{\lambda }_{N-1,n}{A}_{N-1})\cdots P_C(I-{\lambda }_{2,n}{A}_2)P_C(I-{\lambda }_{1,n}{A}_1)u_n,\hfill \\ x_{n+1}=s_n\gamma V{x}_n+{\beta }_n{x}_n+((1-{\beta }_n)I-s_n\mu F)T_n{v}_n,\mathrm{\forall }n\ge 1,\hfill \end{array}$$

where $P_C(I-{\lambda }_n\mathrm{\nabla }f)=s_{n}I+(1-s_n)T_n$ (here $T_n$ is nonexpansive, $s_n=\frac{2-{\lambda }_{n}L}{4}\in (0,\frac{1}{2})$ for each ${\lambda }_n\in (0,\frac{2}{L})$). Assume that the following conditions hold:

• (i) $s_n\in (0,\frac{1}{2})$ for each ${\lambda }_n\in (0,\frac{2}{L})$, and ${lim}_{n\to \mathrm{\infty }}{s}_n=0$ ($\iff {lim}_{n\to \mathrm{\infty }}{\lambda }_n=\frac{2}{L}$);

• (ii) $\{{\beta }_n\}\subset (0,1)$ and $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_n\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\beta }_n<1$;

• (iii) $\{{\lambda }_{i,n}\}\subset [a_i,b_i]\subset (0,2{\eta }_i)$ and ${lim}_{n\to \mathrm{\infty }}|{\lambda }_{i,n+1}-{\lambda }_{i,n}|=0$ for all $i\in \{1,2,\dots ,N\}$;

• (iv) $\{r_{k,n}\}\subset [e_k,f_k]\subset (0,2{\mu }_k)$ and ${lim}_{n\to \mathrm{\infty }}|r_{k,n+1}-r_{k,n}|=0$ for all $k\in \{1,2,\dots ,M\}$.

Then $\{x_n\}$ converges strongly as ${\lambda }_n\to \frac{2}{L}$ ($\iff s_n\to 0$) to a point $x^{\ast }\in \Omega$, which is a unique solution in Ω to the VIP:

$$〈(\mu F-\gamma V)x^{\ast },p-x^{\ast }〉\ge 0,\mathrm{\forall }p\in \Omega .$$

Equivalently, $x^{\ast }=P_{\Omega }(I-(\mu F-\gamma V))x^{\ast }$.

In 2009, Yao et al. [35] considered the following hierarchical fixed point problem (HFPP): find hierarchically a fixed point of a nonexpansive mapping T with respect to another nonexpansive mapping S, namely; find $\tilde{x}\in Fix(T)$ such that

$$〈\tilde{x}-S\tilde{x},\tilde{x}-x〉\le 0,\mathrm{\forall }x\in Fix(T).$$

(1.7)

The solution set of HFPP (1.7) is denoted by Λ. It is not hard to check that solving HFPP (1.7) is equivalent to the fixed point problem of the composite mapping $P_{Fix(T)}S$, i.e., find $\tilde{x}\in C$ such that $\tilde{x}=P_{Fix(T)}S\tilde{x}$. The authors of [35] introduced and analyzed the following iterative algorithm for solving HFPP (1.7):

$$\{\begin{array}{c}{y}_n={\beta }_{n}S{x}_n+(1-{\beta }_n)x_n,\hfill \\ x_{n+1}={\alpha }_{n}V{x}_n+(1-{\alpha }_n)T{y}_n,\mathrm{\forall }n\ge 0.\hfill \end{array}$$

(1.8)

Theorem YLM (see [[35], Theorem 3.2])

Let C be a nonempty closed convex subset of a real Hilbert space H. Let S and T be two nonexpansive mappings of C into itself. Let $V:C\to C$ be a fixed contraction with $\alpha \in (0,1)$. Let $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ be two sequences in $(0,1)$. For any given $x_0\in C$, let $\{x_n\}$ be the sequence generated by (1.8). Assume that the sequence $\{x_n\}$ is bounded and that

• (i) ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$;

• (ii) ${lim}_{n\to \mathrm{\infty }}\frac{1}{{\alpha }_n}|\frac{1}{{\beta }_n}-\frac{1}{{\beta }_{n-1}}|=0$, ${lim}_{n\to \mathrm{\infty }}\frac{1}{{\beta }_n}|1-\frac{{\alpha }_{n-1}}{{\alpha }_n}|=0$;

• (iii) ${lim}_{n\to \mathrm{\infty }}{\beta }_n=0$, ${lim}_{n\to \mathrm{\infty }}\frac{{\alpha }_n}{{\beta }_n}=0$ and ${lim}_{n\to \mathrm{\infty }}\frac{{\beta }_n^2}{{\alpha }_n}=0$;

• (iv) $Fix(T)\cap intC\ne \mathrm{\varnothing }$;

• (v) there exists a constant $k>0$ such that $\parallel x-Tx\parallel \ge kDist(x,Fix(T))$ for each $x\in C$, where $Dist(x,Fix(T))={inf}_{y\in Fix(T)}\parallel x-y\parallel$. Then $\{x_n\}$ converges strongly to $\tilde{x}=P_{\Lambda }V\tilde{x}$ which solves the VIP: $〈\tilde{x}-S\tilde{x},\tilde{x}-x〉\le 0$, $\mathrm{\forall }x\in Fix(T)$.

Very recently, Iiduka [36, 37] considered a variational inequality with a variational inequality constraint over the set of fixed points of a nonexpansive mapping. Since this problem has a triple structure in contrast with bilevel programming problems or hierarchical constrained optimization problems or hierarchical fixed point problem, it is referred to as a triple hierarchical constrained optimization problem (THCOP). He presented some examples of THCOP and developed iterative algorithms to find the solution of such a problem. The convergence analysis of the proposed algorithms is also studied in [36, 37]. Since the original problem is a variational inequality, in this paper, we call it a triple hierarchical variational inequality (THVI). Subsequently, Ceng et al. [38] introduced and considered the following triple hierarchical variational inequality (THVI):

Problem I Let $S,T:C\to C$ be two nonexpansive mappings with $Fix(T)\ne \mathrm{\varnothing }$, $V:C\to H$ be a ρ-contractive mapping with constant $\rho \in [0,1)$ and $F:C\to H$ be a κ-Lipschitzian and η-strongly monotone mapping with constants $\kappa ,\eta >0$. Let $0<\mu <2\eta /{\kappa }^2$ and $0<\gamma \le \tau$ where $\tau =1-\sqrt{1-\mu (2\eta -\mu {\kappa }^2)}$. Consider the following THVI: find $x^{\ast }\in \Xi$ such that

$$〈(\mu F-\gamma V)x^{\ast },x-x^{\ast }〉\ge 0,\mathrm{\forall }x\in \Xi ,$$

in which Ξ denotes the solution set of the following hierarchical variational inequality (HVI): find $z^{\ast }\in Fix(T)$ such that

$$〈(\mu F-\gamma S)z^{\ast },z-z^{\ast }〉\ge 0,\mathrm{\forall }z\in Fix(T),$$

where the solution set Ξ is assumed to be nonempty.

The authors of [38] proposed both implicit and explicit iterative methods and studied the convergence analysis of the sequences generated by the proposed methods. In this paper, we introduce and study the following triple hierarchical variational inequality (THVI) with constraints of mixed equilibria, variational inequalities, and convex minimization problem.

Problem II Let M, N be two integers. Let $f:C\to \mathbf{R}$ be a convex functional with L-Lipschitz continuous gradient ∇f. Let ${\Theta }_k$ be a bifunction from $C\times C$ to R satisfying (A1)-(A4) and ${\phi }_k:C\to \mathbf{R}\cup \{+\mathrm{\infty }\}$ be a proper lower semicontinuous and convex function, where $k\in \{1,2,\dots ,M\}$. Let $R_i:C\to 2^H$ be a maximal monotone mapping and let $A_k:H\to H$ and $B_i:C\to H$ be ${\mu }_k$-inverse strongly monotone and ${\eta }_i$-inverse strongly monotone, respectively, where $k\in \{1,2,\dots ,M\}$, $i\in \{1,2,\dots ,N\}$. Let $S:H\to H$ be a nonexpansive mapping and ${\{T_n\}}_{n=1}^{\mathrm{\infty }}$ be a sequence of nonexpansive mappings on H. Let $F:H\to H$ be a κ-Lipschitzian and η-strongly monotone operator with positive constants $\kappa ,\eta >0$. Let $V:H\to H$ be an l-Lipschitzian mapping with constant $l\ge 0$. Let $0<\mu <\frac{2\eta }{{\kappa }^2}$, $0<\gamma \le \tau$, and $0\le \gamma l<\tau$, where $\tau =1-\sqrt{1-\mu (2\eta -\mu {\kappa }^2)}$. Consider the following triple hierarchical variational inequality (THVI): find $x^{\ast }\in \Xi$ such that

$$〈(\mu F-\gamma V)x^{\ast },x-x^{\ast }〉\ge 0,\mathrm{\forall }x\in \Xi ,$$

(1.9)

where Ξ denotes the solution set of the following hierarchical variational inequality (HVI): find $z^{\ast }\in \Omega :={\bigcap }_{n=1}^{\mathrm{\infty }}Fix(T_n)\cap {\bigcap }_{k=1}^{M}GMEP({\Theta }_k,{\phi }_k,B_k)\cap {\bigcap }_{i=1}^N\mathrm{I}(B_i,R_i)\cap \Gamma$ such that

$$〈(\mu F-\gamma S)z^{\ast },z-z^{\ast }〉\ge 0,\mathrm{\forall }z\in \Omega ,$$

(1.10)

where the solution set Ξ is assumed to be nonempty.

Motivated and inspired by the above facts, we introduce and analyze a relaxed iterative algorithm by virtue of Korpelevich’s extragradient method, the viscosity approximation method, the hybrid steepest-descent method, the regularization method, and the averaged mapping approach to the GPA. It is proven that, under appropriate assumptions, the proposed algorithm converges strongly to a common element $x^{\ast }\in \Omega :={\bigcap }_{n=1}^{\mathrm{\infty }}Fix(T_n)\cap {\bigcap }_{k=1}^{M}GMEP({\Theta }_k,{\phi }_k,A_k)\cap {\bigcap }_{i=1}^N\mathrm{I}(B_i,R_i)\cap \Gamma$ of the fixed point set of infinitely many nonexpansive mappings ${\{T_n\}}_{n=1}^{\mathrm{\infty }}$, the solution set of finitely many GMEPs, the solution set of finitely many variational inclusions and the set of minimizers of CMP (1.4), which is just a unique solution of the THVI (1.9). In addition, we also consider the application of the proposed algorithm to solving a hierarchical fixed point problem with constraints of finitely many GMEPs, finitely many variational inclusions and CMP (1.4). That is, under very mild conditions, it is proven that the proposed algorithm converges strongly to a unique solution $x^{\ast }\in \Omega$ of the VIP: $〈(\gamma V-\mu F)x^{\ast },x-x^{\ast }〉\le 0$, $\mathrm{\forall }x\in \Omega$; equivalently, $P_{\Omega }(I-(\mu F-\gamma V))x^{\ast }=x^{\ast }$. The results obtained in this paper improve and extend the corresponding results announced by many others.

2 Preliminaries

Throughout this paper, we assume that H is a real Hilbert space whose inner product and norm are denoted by $〈\cdot ,\cdot 〉$ and $\parallel \cdot \parallel$, respectively. Let C be a nonempty closed convex subset of H. We write $x_n\rightharpoonup x$ to indicate that the sequence $\{x_n\}$ converges weakly to x and $x_n\to x$ to indicate that the sequence $\{x_n\}$ converges strongly to x. Moreover, we use ${\omega }_w(x_n)$ to denote the weak ω-limit set of the sequence $\{x_n\}$, i.e.,

$${\omega }_w(x_n):=\{x\in H:x_{n_i}\rightharpoonup x\text{ for some subsequence }\{x_{n_i}\}\text{ of }\{x_n\}\}.$$

Recall that a mapping $A:C\to H$ is called

• (i) monotone if

  $$〈Ax-Ay,x-y〉\ge 0,\mathrm{\forall }x,y\in C;$$

• (ii) η-strongly monotone if there exists a constant $\eta >0$ such that

  $$〈Ax-Ay,x-y〉\ge \eta {\parallel x-y\parallel }^2,\mathrm{\forall }x,y\in C;$$

• (iii) α-inverse strongly monotone if there exists a constant $\alpha >0$ such that

  $$〈Ax-Ay,x-y〉\ge \alpha {\parallel Ax-Ay\parallel }^2,\mathrm{\forall }x,y\in C.$$

It is obvious that if A is α-inverse strongly monotone, then A is monotone and $\frac{1}{\alpha }$-Lipschitz continuous. Moreover, we also have, for all $u,v\in C$ and $\lambda >0$,

$$\begin{array}{rl}{\parallel (I-\lambda A)u-(I-\lambda A)v\parallel }^2& ={\parallel (u-v)-\lambda (Au-Av)\parallel }^2\\ ={\parallel u-v\parallel }^2-2\lambda 〈Au-Av,u-v〉+{\lambda }^2{\parallel Au-Av\parallel }^2\\ \le {\parallel u-v\parallel }^2+\lambda (\lambda -2\alpha ){\parallel Au-Av\parallel }^2.\end{array}$$

(2.1)

So, if $\lambda \le 2\alpha$, then $I-\lambda A$ is a nonexpansive mapping from C to H.

The metric (or nearest point) projection from H onto C is the mapping $P_C:H\to C$ which assigns to each point $x\in H$ the unique point $P_{C}x\in C$ satisfying the property

$$\parallel x-P_{C}x\parallel =\underset{y\in C}{inf}\parallel x-y\parallel =:d(x,C).$$

Some important properties of projections are gathered in the following proposition.

Proposition 2.1 For given $x\in H$ and $z\in C$:

• (i) $z=P_{C}x\iff 〈x-z,y-z〉\le 0$, $\mathrm{\forall }y\in C$;

• (ii) $z=P_{C}x\iff {\parallel x-z\parallel }^2\le {\parallel x-y\parallel }^2-{\parallel y-z\parallel }^2$, $\mathrm{\forall }y\in C$;

• (iii) $〈P_{C}x-P_{C}y,x-y〉\ge {\parallel P_{C}x-P_{C}y\parallel }^2$, $\mathrm{\forall }y\in H$.

Consequently, $P_C$ is nonexpansive and monotone.

Definition 2.1 A mapping $T:H\to H$ is said to be:

• (a) nonexpansive if

  $$\parallel Tx-Ty\parallel \le \parallel x-y\parallel ,\mathrm{\forall }x,y\in H;$$

• (b) firmly nonexpansive if $2T-I$ is nonexpansive, or equivalently, if T is 1-inverse strongly monotone (1-ism),

  $$〈x-y,Tx-Ty〉\ge {\parallel Tx-Ty\parallel }^2,\mathrm{\forall }x,y\in H;$$

alternatively, T is firmly nonexpansive if and only if T can be expressed as

$$T=\frac{1}{2}(I+S),$$

where $S:H\to H$ is nonexpansive; projections are firmly nonexpansive.

It can easily be seen that if T is nonexpansive, then $I-T$ is monotone. It is also easy to see that a projection $P_C$ is 1-ism. Inverse strongly monotone (also referred to as co-coercive) operators have been applied widely in solving practical problems in various fields.

Definition 2.2 A mapping $T:H\to H$ is said to be an averaged mapping if it can be written as the average of the identity I and a nonexpansive mapping, that is,

$$T\equiv (1-\alpha )I+\alpha S,$$

where $\alpha \in (0,1)$ and $S:H\to H$ is nonexpansive. More precisely, when the last equality holds, we say that T is α-averaged. Thus firmly nonexpansive mappings (in particular, projections) are $\frac{1}{2}$-averaged mappings.

Proposition 2.2 (see [39])

Let $T:H\to H$ be a given mapping.

• (i) T is nonexpansive if and only if the complement $I-T$ is $\frac{1}{2}$-ism.

• (ii) If T is ν-ism, then for $\gamma >0$, γT is $\frac{\nu }{\gamma }$-ism.

• (iii) T is averaged if and only if the complement $I-T$ is ν-ism for some $\nu >1/2$. Indeed, for $\alpha \in (0,1)$, T is α-averaged if and only if $I-T$ is $\frac{1}{2\alpha }$-ism.

Proposition 2.3 (see [39, 40])

Let $S,T,V:H\to H$ be given operators.

• (i) If $T=(1-\alpha )S+\alpha V$ for some $\alpha \in (0,1)$ and if S is averaged and V is nonexpansive, then T is averaged.

• (ii) T is firmly nonexpansive if and only if the complement $I-T$ is firmly nonexpansive.

• (iii) If $T=(1-\alpha )S+\alpha V$ for some $\alpha \in (0,1)$ and if S is firmly nonexpansive and V is nonexpansive, then T is averaged.

• (iv) The composite of finitely many averaged mappings is averaged. That is, if each of the mappings ${\{T_i\}}_{i=1}^N$ is averaged, then so is the composite $T_1\cdot \cdot \cdot T_N$. In particular, if $T_1$ is ${\alpha }_1$-averaged and $T_2$ is ${\alpha }_2$-averaged, where ${\alpha }_1,{\alpha }_2\in (0,1)$, then the composite $T_1{T}_2$ is α-averaged, where $\alpha ={\alpha }_1+{\alpha }_2-{\alpha }_1{\alpha }_2$.

• (v) If the mappings ${\{T_i\}}_{i=1}^N$ are averaged and have a common fixed point, then

  $$\bigcap _{i=1}^{N}Fix(T_i)=Fix(T_1\cdot \cdot \cdot T_N).$$

The notation $Fix(T)$ denotes the set of all fixed points of the mapping T, that is, $Fix(T)=\{x\in H:Tx=x\}$.

Next we list some elementary conclusions for the MEP.

Proposition 2.4 (see [25])

Assume that $\Theta :C\times C\to \mathbf{R}$ satisfies (A1)-(A4) and let $\phi :C\to \mathbf{R}$ be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For $r>0$ and $x\in H$, define a mapping $T_r^{(\Theta ,\phi )}:H\to C$ as follows:

$$T_r^{(\Theta ,\phi )}(x)=\{z\in C:\Theta (z,y)+\phi (y)-\phi (z)+\frac{1}{r}〈y-z,z-x〉\ge 0,\mathrm{\forall }y\in C\}$$

for all $x\in H$. Then the following hold:

• (i) for each $x\in H$, $T_r^{(\Theta ,\phi )}(x)$ is nonempty and single-valued;

• (ii) $T_r^{(\Theta ,\phi )}$ is firmly nonexpansive, that is, for any $x,y\in H$,

  $${\parallel T_r^{(\Theta ,\phi )}x-T_r^{(\Theta ,\phi )}y\parallel }^2\le 〈T_r^{(\Theta ,\phi )}x-T_r^{(\Theta ,\phi )}y,x-y〉;$$

• (iii) $Fix(T_r^{(\Theta ,\phi )})=MEP(\Theta ,\phi )$;

• (iv) $MEP(\Theta ,\phi )$ is closed and convex;

• (v) ${\parallel T_s^{(\Theta ,\phi )}x-T_t^{(\Theta ,\phi )}x\parallel }^2\le \frac{s-t}{s}〈T_s^{(\Theta ,\phi )}x-T_t^{(\Theta ,\phi )}x,T_s^{(\Theta ,\phi )}x-x〉$ for all $s,t>0$ and $x\in H$.

We need some facts and tools in a real Hilbert space H, which are listed as lemmas below.

Lemma 2.1 Let X be a real inner product space. Then we have the following inequality:

$${\parallel x+y\parallel }^2\le {\parallel x\parallel }^2+2〈y,x+y〉,\mathrm{\forall }x,y\in X.$$

Lemma 2.2 Let $A:C\to H$ be a monotone mapping. In the context of the variational inequality problem the characterization of the projection (see Proposition  2.1(i)) implies

$$u\in VI(C,A)\iff u=P_C(u-\lambda Au),\lambda >0.$$

Lemma 2.3 (see [[41], Demiclosedness principle])

Let C be a nonempty closed convex subset of a real Hilbert space H. Let T be a nonexpansive self-mapping on C with $Fix(T)\ne \mathrm{\varnothing }$. Then $I-T$ is demiclosed. That is, whenever $\{x_n\}$ is a sequence in C weakly converging to some $x\in C$ and the sequence $\{(I-T)x_n\}$ strongly converges to some y, it follows that $(I-T)x=y$. Here I is the identity operator of H.

Let ${\{T_n\}}_{n=1}^{\mathrm{\infty }}$ be an infinite family of nonexpansive self-mappings on C and ${\{{\lambda }_n\}}_{n=1}^{\mathrm{\infty }}$ be a sequence of nonnegative numbers in $[0,1]$. For any $n\ge 0$, define a mapping $W_n$ on C as follows:

$$\{\begin{array}{c}{U}_{n,n+1}=I,\hfill \\ U_{n,n}={\lambda }_n{T}_n{U}_{n,n+1}+(1-{\lambda }_n)I,\hfill \\ U_{n,n-1}={\lambda }_{n-1}{T}_{n-1}{U}_{n,n}+(1-{\lambda }_{n-1})I,\hfill \\ \cdots \hfill \\ U_{n,k}={\lambda }_k{T}_k{U}_{n,k+1}+(1-{\lambda }_k)I,\hfill \\ U_{n,k-1}={\lambda }_{k-1}{T}_{k-1}{U}_{n,k}+(1-{\lambda }_{k-1})I,\hfill \\ \cdots \hfill \\ U_{n,2}={\lambda }_2{T}_2{U}_{n,3}+(1-{\lambda }_2)I,\hfill \\ W_n=U_{n,1}={\lambda }_1{T}_1{U}_{n,2}+(1-{\lambda }_1)I.\hfill \end{array}$$

(2.2)

Such a mapping $W_n$ is called the W-mapping generated by $T_n,T_{n-1},\dots ,T_1$ and ${\lambda }_n,{\lambda }_{n-1},\dots ,{\lambda }_1$.

Lemma 2.4 (see [[42], Lemma 3.2])

Let C be a nonempty closed convex subset of a real Hilbert space H. Let ${\{T_n\}}_{n=1}^{\mathrm{\infty }}$ be a sequence of nonexpansive self-mappings on C such that ${\bigcap }_{n=1}^{\mathrm{\infty }}Fix(T_n)\ne \mathrm{\varnothing }$ and let ${\{{\lambda }_n\}}_{n=1}^{\mathrm{\infty }}$ be a sequence in $(0,b]$ for some $b\in (0,1)$. Then, for every $x\in C$ and $k\ge 1$ the limit ${lim}_{n\to \mathrm{\infty }}{U}_{n,k}x$ exists where $U_{n,k}$ is defined as in (2.2).

Remark 2.1 (see [[43], Remark 3.1])

It can be found from Lemma 2.4 that if D is a nonempty bounded subset of C, then for $ϵ>0$ there exists $n_0\ge k$ such that, for all $n>n_0$,

$$\underset{x\in D}{sup}\parallel U_{n,k}x-U_{k}x\parallel \le ϵ.$$

Remark 2.2 (see [[43], Remark 3.2])

Utilizing Lemma 2.4, we define a mapping $W:C\to C$ as follows:

$$Wx=\underset{n\to \mathrm{\infty }}{lim}{W}_{n}x=\underset{n\to \mathrm{\infty }}{lim}{U}_{n,1}x,\mathrm{\forall }x\in C.$$

Such a W is called the W-mapping generated by $T_1,T_2,\dots$ and ${\lambda }_1,{\lambda }_2,\dots$ . Since $W_n$ is nonexpansive, $W:C\to C$ is also nonexpansive. If $\{x_n\}$ is a bounded sequence in C, then we put $D=\{x_n:n\ge 1\}$. Hence, it is clear from Remark 2.1 that, for an arbitrary $ϵ>0$, there exists $N_0\ge 1$ such that, for all $n>N_0$,

$$\parallel W_n{x}_n-W{x}_n\parallel =\parallel U_{n,1}{x}_n-U_1{x}_n\parallel \le \underset{x\in D}{sup}\parallel U_{n,1}x-U_{1}x\parallel \le ϵ.$$

This implies that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel W_n{x}_n-W{x}_n\parallel =0.$$

Lemma 2.5 (see [[42], Lemma 3.3])

Let C be a nonempty closed convex subset of a real Hilbert space H. Let ${\{T_n\}}_{n=1}^{\mathrm{\infty }}$ be a sequence of nonexpansive self-mappings on C such that ${\bigcap }_{n=1}^{\mathrm{\infty }}Fix(T_n)\ne \mathrm{\varnothing }$, and let ${\{{\lambda }_n\}}_{n=1}^{\mathrm{\infty }}$ be a sequence in $(0,b]$ for some $b\in (0,1)$. Then $Fix(W)={\bigcap }_{n=1}^{\mathrm{\infty }}Fix(T_n)$.

The following lemma can easily be proven, and therefore we omit the proof.

Lemma 2.6 Let $V:H\to H$ be an l-Lipschitzian mapping with constant $l\ge 0$, and $F:H\to H$ be a κ-Lipschitzian and η-strongly monotone operator with positive constants $\kappa ,\eta >0$. Then for $0\le \gamma l<\mu \eta$,

$$〈(\mu F-\gamma V)x-(\mu F-\gamma V)y,x-y〉\ge (\mu \eta -\gamma l){\parallel x-y\parallel }^2,\mathrm{\forall }x,y\in H.$$

That is, $\mu F-\gamma V$ is strongly monotone with constant $\mu \eta -\gamma l>0$.

Let C be a nonempty closed convex subset of a real Hilbert space H. We introduce some notations. Let λ be a number in $(0,1]$ and let $\mu >0$. Associated with a nonexpansive mapping $T:C\to H$, we define the mapping $T^{\lambda }:C\to H$ by

$$T^{\lambda }x:=Tx-\lambda \mu F(Tx),\mathrm{\forall }x\in C,$$

where $F:H\to H$ is an operator such that, for some positive constants $\kappa ,\eta >0$, F is κ-Lipschitzian and η-strongly monotone on H; that is, F satisfies the conditions:

$$\parallel Fx-Fy\parallel \le \kappa \parallel x-y\parallel \text{and}〈Fx-Fy,x-y〉\ge \eta {\parallel x-y\parallel }^2$$

for all $x,y\in H$.

Lemma 2.7 (see [[44], Lemma 3.1])

$T^{\lambda }$ is a contraction provided $0<\mu <\frac{2\eta }{{\kappa }^2}$; that is,

$$\parallel T^{\lambda }x-T^{\lambda }y\parallel \le (1-\lambda \tau )\parallel x-y\parallel ,\mathrm{\forall }x,y\in C,$$

where $\tau =1-\sqrt{1-\mu (2\eta -\mu {\kappa }^2)}\in (0,1]$.

Lemma 2.8 (see [45])

Let $\{a_n\}$ be a sequence of nonnegative real numbers satisfying the property

$$a_{n+1}\le (1-s_n)a_n+s_n{t}_n+{ϵ}_n,\mathrm{\forall }n\ge 0,$$

where $\{s_n\}\subset [0,1]$ and $\{t_n\}$ are such that

• (i) ${\sum }_{n=1}^{\mathrm{\infty }}{s}_n=\mathrm{\infty }$;

• (ii) either ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{t}_n\le 0$ or ${\sum }_{n=1}^{\mathrm{\infty }}{s}_n|t_n|<\mathrm{\infty }$;

• (iii) ${\sum }_{n=1}^{\mathrm{\infty }}{ϵ}_n<\mathrm{\infty }$ where ${ϵ}_n\ge 0$, $\mathrm{\forall }n\ge 0$.

Then ${lim}_{n\to \mathrm{\infty }}{a}_n=0$.

Lemma 2.9 (see [41])

Let H be a real Hilbert space. Then the following hold:

• (a) ${\parallel x-y\parallel }^2={\parallel x\parallel }^2-{\parallel y\parallel }^2-2〈x-y,y〉$ for all $x,y\in H$;

• (b) ${\parallel \lambda x+\mu y\parallel }^2=\lambda {\parallel x\parallel }^2+\mu {\parallel y\parallel }^2-\lambda \mu {\parallel x-y\parallel }^2$ for all $x,y\in H$ and $\lambda ,\mu \in [0,1]$ with $\lambda +\mu =1$;

• (c) if $\{x_n\}$ is a sequence in H such that $x_n\rightharpoonup x$, it follows that

  $$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\parallel x_n-y\parallel }^2=\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\parallel x_n-x\parallel }^2+{\parallel x-y\parallel }^2,\mathrm{\forall }y\in H.$$

Finally, recall that a set-valued mapping $T:D(T)\subset H\to 2^H$ is called monotone if for all $x,y\in D(T)$, $f\in Tx$ and $g\in Ty$ imply

$$〈f-g,x-y〉\ge 0.$$

A set-valued mapping T is called maximal monotone if T is monotone and $(I+\lambda T)D(T)=H$ for each $\lambda >0$, where I is the identity mapping of H. We denote by $G(T)$ the graph of T. It is well known that a monotone mapping T is maximal if and only if, for $(x,f)\in H\times H$, $〈f-g,x-y〉\ge 0$ for every $(y,g)\in G(T)$ implies $f\in Tx$. Next we provide an example to illustrate the concept of maximal monotone mapping.

Let $A:C\to H$ be a monotone, k-Lipschitz continuous mapping and let $N_{C}v$ be the normal cone to C at $v\in C$, i.e.,

$$N_{C}v=\{u\in H:〈v-p,u〉\ge 0,\mathrm{\forall }p\in C\}.$$

Define

$$\tilde{T}v=\{\begin{array}{cc}Av+N_{C}v,\hfill & \text{if }v\in C,\hfill \\ \mathrm{\varnothing },\hfill & \text{if }v\notin C.\hfill \end{array}$$

Then $\tilde{T}$ is maximal monotone (see [28]) such that

$$0\in \tilde{T}v⟺v\in VI(C,A).$$

(2.3)

Let $R:D(R)\subset H\to 2^H$ be a maximal monotone mapping. Let $\lambda ,\mu >0$ be two positive numbers.

Lemma 2.10 (see [46])

We have the resolvent identity

$$J_{R,\lambda }x=J_{R,\mu }(\frac{\mu }{\lambda }x+(1-\frac{\mu }{\lambda })J_{R,\lambda }x),\mathrm{\forall }x\in H.$$

Remark 2.3 For $\lambda ,\mu >0$, we have the following relation:

$$\parallel J_{R,\lambda }x-J_{R,\mu }y\parallel \le \parallel x-y\parallel +|\lambda -\mu |(\frac{1}{\lambda }\parallel J_{R,\lambda }x-y\parallel +\frac{1}{\mu }\parallel x-J_{R,\mu }y\parallel ),\mathrm{\forall }x,y\in H.$$

(2.4)

Indeed, whenever $\lambda \ge \mu$, utilizing Lemma 2.10 we deduce that

$$\begin{array}{rl}\parallel J_{R,\lambda }x-J_{R,\mu }y\parallel & =\parallel J_{R,\mu }(\frac{\mu }{\lambda }x+(1-\frac{\mu }{\lambda })J_{R,\lambda }x)-J_{R,\mu }y\parallel \\ \le \parallel \frac{\mu }{\lambda }x+(1-\frac{\mu }{\lambda })J_{R,\lambda }x-y\parallel \\ \le \frac{\mu }{\lambda }\parallel x-y\parallel +(1-\frac{\mu }{\lambda })\parallel J_{R,\lambda }x-y\parallel \\ \le \parallel x-y\parallel +\frac{|\lambda -\mu |}{\lambda }\parallel J_{R,\lambda }x-y\parallel .\end{array}$$

Similarly, whenever $\lambda <\mu$, we get

$$\parallel J_{R,\lambda }x-J_{R,\mu }y\parallel \le \parallel x-y\parallel +\frac{|\lambda -\mu |}{\mu }\parallel x-J_{R,\mu }y\parallel .$$

Combining the above two cases we conclude that (2.4) holds.

In terms of Huang [9] (see also [29]), we have the following property for the resolvent operator $J_{R,\lambda }:H\to \overline{D(R)}$.

Lemma 2.11 $J_{R,\lambda }$ is single-valued and firmly nonexpansive, i.e.,

$$〈J_{R,\lambda }x-J_{R,\lambda }y,x-y〉\ge {\parallel J_{R,\lambda }x-J_{R,\lambda }y\parallel }^2,\mathrm{\forall }x,y\in H.$$

Consequently, $J_{R,\lambda }$ is nonexpansive and monotone.

Lemma 2.12 (see [12])

Let R be a maximal monotone mapping with $D(R)=C$. Then for any given $\lambda >0$, $u\in C$ is a solution of problem (1.6) if and only if $u\in C$ satisfies

$$u=J_{R,\lambda }(u-\lambda Bu).$$

Lemma 2.13 (see [29])

Let R be a maximal monotone mapping with $D(R)=C$ and let $B:C\to H$ be a strongly monotone, continuous, and single-valued mapping. Then for each $z\in H$, the equation $z\in (B+\lambda R)x$ has a unique solution $x_{\lambda }$ for $\lambda >0$.

Lemma 2.14 (see [12])

Let R be a maximal monotone mapping with $D(R)=C$ and $B:C\to H$ be a monotone, continuous and single-valued mapping. Then $(I+\lambda (R+B))C=H$ for each $\lambda >0$. In this case, $R+B$ is maximal monotone.

3 Strong convergence theorems for the THVI and HFPP

In this section, we will introduce and analyze a relaxed iterative algorithm for finding a solution of the THVI (1.9) with constraints of several problems: finitely many GMEPs, finitely many variational inclusions, and CMP (1.4) in a real Hilbert space. This algorithm is based on Korpelevich’s extragradient method, the viscosity approximation method, the hybrid steepest-descent method, the regularization method, and the averaged mapping approach to the GPA. We prove the strong convergence of the proposed algorithm to a unique solution of THVI (1.9) under suitable conditions. In addition, we also consider the application of the proposed algorithm to solving a hierarchical fixed point problem with the same constraints.

Let $f:C\to \mathbf{R}$ be a convex functional with L-Lipschitz continuous gradient ∇f. It is worth emphasizing that the regularization, in particular, the traditional Tikhonov regularization, is usually used to solve ill-posed optimization problems. Consider the regularized minimization problem

$$\underset{x\in C}{min}{f}_{\alpha }(x):=f(x)+\frac{\alpha }{2}{\parallel x\parallel }^2,$$

where $\alpha >0$ is the regularization parameter.

The advantage of a regularization method is its possible strong convergence to the minimum-norm solution of the optimization problem under investigation. The disadvantage is, however, its implicity, and hence explicit iterative methods seem more attractive, with which Xu was also concerned in [33, 47]. Very recently, some approximation methods are proposed in [13, 14, 32, 48] to solve the vector optimization problem and split feasibility problem by virtue of the regularization method.

We are now in a position to state and prove the first main result in this paper.

Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let M, N be two integers. Let $f:C\to \mathbf{R}$ be a convex functional with L-Lipschitz continuous gradient ∇f. Let ${\Theta }_k$ be a bifunction from $C\times C$ to R satisfying (A1)-(A4) and ${\phi }_k:C\to \mathbf{R}\cup \{+\mathrm{\infty }\}$ be a proper lower semicontinuous and convex function, where $k\in \{1,2,\dots ,M\}$. Let $R_i:C\to 2^H$ be a maximal monotone mapping and let $A_k:H\to H$ and $B_i:C\to H$ be ${\mu }_k$-inverse strongly monotone and ${\eta }_i$-inverse strongly monotone, respectively, where $k\in \{1,2,\dots ,M\}$, $i\in \{1,2,\dots ,N\}$. Let $S:H\to H$ be a nonexpansive mapping, ${\{T_n\}}_{n=1}^{\mathrm{\infty }}$ be a sequence of nonexpansive mappings on H and ${\{{\lambda }_n\}}_{n=1}^{\mathrm{\infty }}$ be a sequence in $(0,b]$ for some $b\in (0,1)$. Let $F:H\to H$ be a κ-Lipschitzian and η-strongly monotone operator with positive constants $\kappa ,\eta >0$. Let $V:H\to H$ be an l-Lipschitzian mapping with constant $l\ge 0$. Let $0<\lambda <\frac{2}{L}$, $0<\mu <\frac{2\eta }{{\kappa }^2}$, $0<\gamma \le \tau$ and $0\le \gamma l<\tau$, where $\tau =1-\sqrt{1-\mu (2\eta -\mu {\kappa }^2)}$. Assume that either (B1) or (B2) holds. Let $\{{\beta }_n\}$ and $\{{\theta }_n\}$ be sequences in $(0,1)$ and $\{{\alpha }_n\}$ be a sequence in $(0,\mathrm{\infty })$ with ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n<\mathrm{\infty }$. For arbitrarily given $x_1\in H$, let $\{x_n\}$ be a sequence generated by

$$\{\begin{array}{c}{u}_n=T_{r_{M,n}}^{({\Theta }_M,{\phi }_M)}(I-r_{M,n}{A}_M)T_{r_{M-1,n}}^{({\Theta }_{M-1},{\phi }_{M-1})}(I-r_{M-1,n}{A}_{M-1})\cdots \hfill \\ \phantom{u_n=}\times T_{r_{1,n}}^{({\Theta }_1,{\phi }_1)}(I-r_{1,n}{A}_1)x_n,\hfill \\ v_n=J_{R_N,{\lambda }_{N,n}}(I-{\lambda }_{N,n}{B}_N)J_{R_{N-1},{\lambda }_{N-1,n}}(I-{\lambda }_{N-1,n}{B}_{N-1})\cdots J_{R_1,{\lambda }_{1,n}}(I-{\lambda }_{1,n}{B}_1)u_n,\hfill \\ y_n={\theta }_n\gamma S{x}_n+(I-{\theta }_n\mu F)P_C(v_n-\lambda \mathrm{\nabla }{f}_{{\alpha }_n}(v_n)),\hfill \\ x_{n+1}={\beta }_n\gamma V{x}_n+(I-{\beta }_n\mu F)W_n{P}_C(y_n-\lambda \mathrm{\nabla }{f}_{{\alpha }_n}(y_n)),\mathrm{\forall }n\ge 1,\hfill \end{array}$$

(3.1)

where $W_n$ is the W-mapping defined by (2.2). Suppose that the following conditions are satisfied:

• (H1) ${?}_{n=1}^{\mathrm{8}}{ß}_n=\mathrm{8}$ and ${lim}_{n?\mathrm{8}}\frac{1}{{ß}_n}|1-\frac{{?}_{n-1}}{{?}_n}|=0$;

• (H2) ${lim}_{n?\mathrm{8}}\frac{1}{{ß}_n}|\frac{1}{{?}_n}-\frac{1}{{?}_{n-1}}|=0$ and ${lim}_{n?\mathrm{8}}\frac{1}{{?}_n}|1-\frac{{ß}_{n-1}}{{ß}_n}|=0$;

• (H3) ${lim}_{n?\mathrm{8}}{?}_n=0$, ${lim}_{n?\mathrm{8}}\frac{a_n}{{?}_n}=0$ and ${lim}_{n?\mathrm{8}}\frac{{?}_n}{{ß}_n}=0$;

• (H4) ${lim}_{n?\mathrm{8}}\frac{|a_n-a_{n-1}|}{{ß}_n{?}_n}=0$ and ${lim}_{n?\mathrm{8}}\frac{b^n}{{ß}_n{?}_n}=0$;

• (H5) $\{{?}_{i,n}\}?[a_i,b_i]?(0,2{?}_i)$ and ${lim}_{n?\mathrm{8}}\frac{|{?}_{i,n}-{?}_{i,n-1}|}{{ß}_n{?}_n}=0$ for all $i?\{1,2,\dots ,N\}$;

• (H6) $\{r_{k,n}\}?[e_k,f_k]?(0,2{\mu }_k)$ and ${lim}_{n?\mathrm{8}}\frac{|r_{k,n}-r_{k,n-1}|}{{ß}_n{?}_n}=0$ for all $k?\{1,2,\dots ,M\}$.

Then we have the following:

• (i) ${lim}_{n\to \mathrm{\infty }}\frac{\parallel x_{n+1}-x_n\parallel }{{\theta }_n}=0$;

• (ii) ${\omega }_w(x_n)\subset \Omega$;

• (iii) ${\omega }_w(x_n)\subset \Xi$ provided $\parallel x_n-y_n\parallel =o({\theta }_n)$ additionally.

Proof First of all, let us show that $P_C(I-\lambda \mathrm{\nabla }{f}_{\alpha })$ is ξ-averaged for each $\lambda \in (0,\frac{2}{\alpha +L})$, where

$$\xi =\frac{2+\lambda (\alpha +L)}{4}\in (0,1).$$

Indeed, note that the Lipschitzian property of ∇f implies that ∇f is $\frac{1}{L}$-ism [49] (see also [33]), that is,

$$〈\mathrm{\nabla }f(x)-\mathrm{\nabla }f(y),x-y〉\ge \frac{1}{L}{\parallel \mathrm{\nabla }f(x)-\mathrm{\nabla }f(y)\parallel }^2.$$

Observe that

$$\begin{array}{r}(\alpha +L)〈\mathrm{\nabla }{f}_{\alpha }(x)-\mathrm{\nabla }{f}_{\alpha }(y),x-y〉\\ =(\alpha +L)[\alpha {\parallel x-y\parallel }^2+〈\mathrm{\nabla }f(x)-\mathrm{\nabla }f(y),x-y〉]\\ ={\alpha }^2{\parallel x-y\parallel }^2+\alpha 〈\mathrm{\nabla }f(x)-\mathrm{\nabla }f(y),x-y〉+\alpha L{\parallel x-y\parallel }^2\\ +L〈\mathrm{\nabla }f(x)-\mathrm{\nabla }f(y),x-y〉\\ \ge {\alpha }^2{\parallel x-y\parallel }^2+2\alpha 〈\mathrm{\nabla }f(x)-\mathrm{\nabla }f(y),x-y〉+{\parallel \mathrm{\nabla }f(x)-\mathrm{\nabla }f(y)\parallel }^2\\ ={\parallel \alpha (x-y)+\mathrm{\nabla }f(x)-\mathrm{\nabla }f(y)\parallel }^2\\ ={\parallel \mathrm{\nabla }{f}_{\alpha }(x)-\mathrm{\nabla }{f}_{\alpha }(y)\parallel }^2.\end{array}$$

Hence, it follows that $\mathrm{\nabla }{f}_{\alpha }=\alpha I+\mathrm{\nabla }f$ is $\frac{1}{\alpha +L}$-ism. Thus, $\lambda \mathrm{\nabla }{f}_{\alpha }$ is $\frac{1}{\lambda (\alpha +L)}$-ism according to Proposition 2.2(ii). By Proposition 2.2(iii) the complement $I-\lambda \mathrm{\nabla }{f}_{\alpha }$ is $\frac{\lambda (\alpha +L)}{2}$-averaged. Therefore, noting that $P_C$ is $\frac{1}{2}$-averaged and utilizing Proposition 2.3(iv), we know that, for each $\lambda \in (0,\frac{2}{\alpha +L})$, $P_C(I-\lambda \mathrm{\nabla }{f}_{\alpha })$ is ξ-averaged with

$$\xi =\frac{1}{2}+\frac{\lambda (\alpha +L)}{2}-\frac{1}{2}\cdot \frac{\lambda (\alpha +L)}{2}=\frac{2+\lambda (\alpha +L)}{4}\in (0,1).$$

This shows that $P_C(I-\lambda \mathrm{\nabla }{f}_{\alpha })$ is nonexpansive. Furthermore, for $\lambda \in (0,\frac{2}{L})$, utilizing the fact that ${lim}_{n\to \mathrm{\infty }}\frac{2}{{\alpha }_n+L}=\frac{2}{L}$, we may assume that

$$0<\lambda <\frac{2}{{\alpha }_n+L},\mathrm{\forall }n\ge 1.$$

Consequently, it follows that, for each integer $n\ge 0$, $P_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})$ is ${\xi }_n$-averaged with

$${\xi }_n=\frac{1}{2}+\frac{\lambda ({\alpha }_n+L)}{2}-\frac{1}{2}\cdot \frac{\lambda ({\alpha }_n+L)}{2}=\frac{2+\lambda ({\alpha }_n+L)}{4}\in (0,1).$$

This immediately implies that $P_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})$ is nonexpansive for all $n\ge 1$. Put

$${\Delta }_n^k=T_{r_{k,n}}^{({\Theta }_k,{\phi }_k)}(I-r_{k,n}{A}_k)T_{r_{k-1,n}}^{({\Theta }_{k-1},{\phi }_{k-1})}(I-r_{k-1,n}{A}_{k-1})\cdots T_{r_{1,n}}^{({\Theta }_1,{\phi }_1)}(I-r_{1,n}{A}_1)x_n$$

for all $k\in \{1,2,\dots ,M\}$ and $n\ge 1$,

$${\Lambda }_n^i=J_{R_i,{\lambda }_{i,n}}(I-{\lambda }_{i,n}{B}_i)J_{R_{i-1},{\lambda }_{i-1,n}}(I-{\lambda }_{i-1,n}{B}_{i-1})\cdots J_{R_1,{\lambda }_{1,n}}(I-{\lambda }_{1,n}{B}_1)$$

for all $i\in \{1,2,\dots ,N\}$, ${\Delta }_n^0=I$ and ${\Lambda }_n^0=I$, where I is the identity mapping on H. Then we have $u_n={\Delta }_n^M{x}_n$ and $v_n={\Lambda }_n^N{u}_n$.

We divide the rest of the proof into several steps.

Step 1. We prove that $\{x_n\}$ is bounded.

Indeed, taking into account the assumption $\Xi \ne \mathrm{\varnothing }$ in Problem II, we know that $\Omega \ne \mathrm{\varnothing }$. Take $p\in \Omega$ arbitrarily. Then from (2.1) and Proposition 2.4(ii) we have

$$\begin{array}{rl}\parallel u_n-p\parallel & =\parallel T_{r_{M,n}}^{({\Theta }_M,{\phi }_M)}(I-r_{M,n}{B}_M){\Delta }_n^{M-1}{x}_n-T_{r_{M,n}}^{({\Theta }_M,{\phi }_M)}(I-r_{M,n}{B}_M){\Delta }_n^{M-1}p\parallel \\ \le \parallel (I-r_{M,n}{B}_M){\Delta }_n^{M-1}{x}_n-(I-r_{M,n}{B}_M){\Delta }_n^{M-1}p\parallel \\ \le \parallel {\Delta }_n^{M-1}{x}_n-{\Delta }_n^{M-1}p\parallel \\ \cdots \\ \le \parallel {\Delta }_n^0{x}_n-{\Delta }_n^{0}p\parallel \\ =\parallel x_n-p\parallel .\end{array}$$

(3.2)

Similarly, we have

$$\begin{array}{rl}\parallel v_n-p\parallel & =\parallel J_{R_N,{\lambda }_{N,n}}(I-{\lambda }_{N,n}{A}_N){\Lambda }_n^{N-1}{u}_n-J_{R_N,{\lambda }_{N,n}}(I-{\lambda }_{N,n}{A}_N){\Lambda }_n^{N-1}p\parallel \\ \le \parallel (I-{\lambda }_{N,n}{A}_N){\Lambda }_n^{N-1}{u}_n-(I-{\lambda }_{N,n}{A}_N){\Lambda }_n^{N-1}p\parallel \\ \le \parallel {\Lambda }_n^{N-1}{u}_n-{\Lambda }_n^{N-1}p\parallel \\ \cdots \\ \le \parallel {\Lambda }_n^0{u}_n-{\Lambda }_n^{0}p\parallel \\ =\parallel u_n-p\parallel .\end{array}$$

(3.3)

Combining (3.2) and (3.3), we have

$$\parallel v_n-p\parallel \le \parallel x_n-p\parallel .$$

(3.4)

Utilizing Lemma 2.7, from (3.1), and (3.4) we obtain

$$\begin{array}{r}\parallel y_n-p\parallel \\ =\parallel {\theta }_n\gamma (S{x}_n-Sp)+(I-{\theta }_n\mu F)P_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})v_n-(I-{\theta }_n\mu F)P_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})p\\ +(I-{\theta }_n\mu F)P_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})p-(I-{\theta }_n\mu F)P_C(I-\lambda \mathrm{\nabla }f)p+{\theta }_n(\gamma S-\mu F)p\parallel \\ \le {\theta }_n\gamma \parallel S{x}_n-Sp\parallel +\parallel (I-{\theta }_n\mu F)P_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})v_n-(I-{\theta }_n\mu F)P_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})p\parallel \\ +\parallel (I-{\theta }_n\mu F)P_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})p-(I-{\theta }_n\mu F)P_C(I-\lambda \mathrm{\nabla }f)p\parallel +{\theta }_n\parallel (\gamma S-\mu F)p\parallel \\ \le {\theta }_n\gamma \parallel x_n-p\parallel +(1-{\theta }_n\tau )\parallel P_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})v_n-P_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})p\parallel \\ +(1-{\theta }_n\tau )\parallel P_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})p-P_C(I-\lambda \mathrm{\nabla }f)p\parallel +{\theta }_n\parallel (\gamma S-\mu F)p\parallel \\ \le {\theta }_n\gamma \parallel x_n-p\parallel +(1-{\theta }_n\tau )\parallel v_n-p\parallel +(1-{\theta }_n\tau )\parallel (I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})p-(I-\lambda \mathrm{\nabla }f)p\parallel \\ +{\theta }_n\parallel (\gamma S-\mu F)p\parallel \\ ={\theta }_n\gamma \parallel x_n-p\parallel +(1-{\theta }_n\tau )\parallel v_n-p\parallel +(1-{\theta }_n\tau )\lambda {\alpha }_n\parallel p\parallel +{\theta }_n\parallel (\gamma S-\mu F)p\parallel \\ \le {\theta }_n\gamma \parallel x_n-p\parallel +(1-{\theta }_n\tau )\parallel x_n-p\parallel +\lambda {\alpha }_n\parallel p\parallel +{\theta }_n\parallel (\gamma S-\mu F)p\parallel \\ =(1-{\theta }_n(\tau -\gamma ))\parallel x_n-p\parallel +{\theta }_n\parallel (\gamma S-\mu F)p\parallel +\lambda {\alpha }_n\parallel p\parallel \\ =(1-{\theta }_n(\tau -\gamma ){\theta }_n)\parallel x_n-p\parallel +{\theta }_n(\tau -\gamma )\frac{\parallel (\gamma S-\mu F)p\parallel }{\tau -\gamma }+\lambda {\alpha }_n\parallel p\parallel \\ \le max\{\parallel x_n-p\parallel ,\frac{\parallel (\gamma S-\mu F)p\parallel }{\tau -\gamma }\}+\lambda {\alpha }_n\parallel p\parallel ,\end{array}$$

and hence

$$\begin{array}{r}\parallel x_{n+1}-p\parallel \\ =\parallel {\beta }_n\gamma (V{x}_n-Vp)+(I-{\beta }_n\mu F)W_n{P}_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})y_n\\ -(I-{\beta }_n\mu F)W_n{P}_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})p+(I-{\beta }_n\mu F)W_n{P}_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})p\\ -(I-{\beta }_n\mu F)W_n{P}_C(I-\lambda \mathrm{\nabla }f)p+{\beta }_n(\gamma V-\mu F)p\parallel \\ \le {\beta }_n\gamma \parallel V{x}_n-Vp\parallel \\ +\parallel (I-{\beta }_n\mu F)W_n{P}_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})y_n-(I-{\beta }_n\mu F)W_n{P}_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})p\parallel \\ +\parallel (I-{\beta }_n\mu F)W_n{P}_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})p-(I-{\beta }_n\mu F)W_n{P}_C(I-\lambda \mathrm{\nabla }f)p\parallel \\ +{\beta }_n\parallel (\gamma V-\mu F)p\parallel \\ \le {\beta }_n\gamma l\parallel x_n-p\parallel +(1-{\beta }_n\tau )\parallel P_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})y_n-P_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})p\parallel \\ +(1-{\beta }_n\tau )\parallel P_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})p-P_C(I-\lambda \mathrm{\nabla }f)p\parallel +{\beta }_n\parallel (\gamma V-\mu F)p\parallel \\ \le {\beta }_n\gamma l\parallel x_n-p\parallel +(1-{\beta }_n\tau )\parallel y_n-p\parallel +(1-{\beta }_n\tau )\parallel (I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})p-(I-\lambda \mathrm{\nabla }f)p\parallel \\ +{\beta }_n\parallel (\gamma V-\mu F)p\parallel \\ ={\beta }_n\gamma l\parallel x_n-p\parallel +(1-{\beta }_n\tau )\parallel y_n-p\parallel +(1-{\beta }_n\tau )\lambda {\alpha }_n\parallel p\parallel +{\beta }_n\parallel (\gamma V-\mu F)p\parallel \\ \le {\beta }_n\gamma l\parallel x_n-p\parallel +(1-{\beta }_n\tau )[max\{\parallel x_n-p\parallel ,\frac{\parallel (\gamma S-\mu F)p\parallel }{\tau -\gamma }\}+\lambda {\alpha }_n\parallel p\parallel ]\\ +\lambda {\alpha }_n\parallel p\parallel +{\beta }_n\parallel (\gamma V-\mu F)p\parallel \\ \le (1-{\beta }_n(\tau -\gamma l))max\{\parallel x_n-p\parallel ,\frac{\parallel (\gamma S-\mu F)p\parallel }{\tau -\gamma }\}+2\lambda {\alpha }_n\parallel p\parallel +{\beta }_n\parallel (\gamma V-\mu F)p\parallel \\ =(1-{\beta }_n(\tau -\gamma l))max\{\parallel x_n-p\parallel ,\frac{\parallel (\gamma S-\mu F)p\parallel }{\tau -\gamma }\}\\ +{\beta }_n(\tau -\gamma l)\frac{\parallel (\gamma V-\mu F)p\parallel }{\tau -\gamma l}+2\lambda {\alpha }_n\parallel p\parallel \\ \le max\{\parallel x_n-p\parallel ,\frac{\parallel (\gamma S-\mu F)p\parallel }{\tau -\gamma },\frac{\parallel (\gamma V-\mu F)p\parallel }{\tau -\gamma l}\}+2\lambda {\alpha }_n\parallel p\parallel .\end{array}$$

Let us show that, for all $n\ge 1$,

$$\parallel x_{n+1}-p\parallel \le max\{\parallel x_1-p\parallel ,\frac{\parallel (\gamma S-\mu F)p\parallel }{\tau -\gamma },\frac{\parallel (\gamma V-\mu F)p\parallel }{\tau -\gamma l}\}+2\sum _{i=1}^n\lambda {\alpha }_i\parallel p\parallel .$$

(3.5)

Indeed, for $n=1$, it is clear that (3.5) holds. Assume that (3.5) holds for some $n\ge 1$. Observe that

$$\begin{array}{r}\parallel x_{n+2}-p\parallel \\ \le max\{\parallel x_{n+1}-p\parallel ,\frac{\parallel (\gamma S-\mu F)p\parallel }{\tau -\gamma },\frac{\parallel (\gamma V-\mu F)p\parallel }{\tau -\gamma l}\}+2\lambda {\alpha }_{n+1}\parallel p\parallel \\ \le max\{max\{\parallel x_1-p\parallel ,\frac{\parallel (\gamma S-\mu F)p\parallel }{\tau -\gamma },\frac{\parallel (\gamma V-\mu F)p\parallel }{\tau -\gamma l}\}\\ +2\sum _{i=1}^n\lambda {\alpha }_i\parallel p\parallel ,\frac{\parallel (\gamma S-\mu F)p\parallel }{\tau -\gamma },\frac{\parallel (\gamma V-\mu F)p\parallel }{\tau -\gamma l}\}+2\lambda {\alpha }_{n+1}\parallel p\parallel \\ \le max\{\parallel x_1-p\parallel ,\frac{\parallel (\gamma S-\mu F)p\parallel }{\tau -\gamma },\frac{\parallel (\gamma V-\mu F)p\parallel }{\tau -\gamma l}\}+2\sum _{i=1}^n\lambda {\alpha }_i\parallel p\parallel +2\lambda {\alpha }_{n+1}\parallel p\parallel \\ =max\{\parallel x_1-p\parallel ,\frac{\parallel (\gamma S-\mu F)p\parallel }{\tau -\gamma },\frac{\parallel (\gamma V-\mu F)p\parallel }{\tau -\gamma l}\}+2\sum _{i=1}^{n+1}\lambda {\alpha }_i\parallel p\parallel .\end{array}$$

By induction, (3.5) holds for all $n\ge 1$. Taking into account ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n<\mathrm{\infty }$, we know that $\{x_n\}$ is bounded and so are the sequences $\{u_n\}$, $\{v_n\}$, $\{y_n\}$.

Step 2. We prove that ${lim}_{n\to \mathrm{\infty }}\frac{\parallel x_{n+1}-x_n\parallel }{{\theta }_n}=0$.

Indeed, for simplicity, put ${\tilde{v}}_n=P_C(v_n-\lambda \mathrm{\nabla }{f}_{{\alpha }_n}(v_n))$ and ${\tilde{y}}_n=P_C(y_n-\lambda \mathrm{\nabla }{f}_{{\alpha }_n}(y_n))$. Then $y_n={\theta }_n\gamma S{x}_n+(I-{\theta }_n\mu F){\tilde{v}}_n$ and $x_{n+1}={\beta }_n\gamma V{x}_n+(I-{\beta }_n\mu F)W_n{\tilde{y}}_n$ for every $n\ge 1$. We observe that

$$\begin{array}{rl}\parallel {\tilde{v}}_n-{\tilde{v}}_{n-1}\parallel \le & \parallel P_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})v_n-P_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})v_{n-1}\parallel \\ +\parallel P_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})v_{n-1}-P_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_{n-1}})v_{n-1}\parallel \\ \le & \parallel v_n-v_{n-1}\parallel +\parallel P_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})v_{n-1}-P_C(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_{n-1}})v_{n-1}\parallel \\ \le & \parallel v_n-v_{n-1}\parallel +\parallel (I-\lambda \mathrm{\nabla }{f}_{{\alpha }_n})v_{n-1}-(I-\lambda \mathrm{\nabla }{f}_{{\alpha }_{n-1}})v_{n-1}\parallel \\ =& \parallel v_n-v_{n-1}\parallel +\parallel \lambda \mathrm{\nabla }{f}_{{\alpha }_n}(v_{n-1})-\lambda \mathrm{\nabla }{f}_{{\alpha }_{n-1}}(v_{n-1})\parallel \\ =& \parallel v_n-v_{n-1}\parallel +\lambda |{\alpha }_n-{\alpha }_{n-1}|\parallel v_{n-1}\parallel .\end{array}$$

(3.6)

Similarly, we get

$$\parallel {\tilde{y}}_n-{\tilde{y}}_{n-1}\parallel \le \parallel y_n-y_{n-1}\parallel +\lambda |{\alpha }_n-{\alpha }_{n-1}|\parallel y_{n-1}\parallel .$$

(3.7)

Also, it is easy to see from (3.1) that

$$\{\begin{array}{c}{y}_n={\theta }_n\gamma S{x}_n+(I-{\theta }_n\mu F){\tilde{v}}_n,\hfill \\ y_{n-1}={\theta }_{n-1}\gamma S{x}_{n-1}+(I-{\theta }_{n-1}\mu F){\tilde{v}}_{n-1}\hfill \end{array}$$

and

$$\{\begin{array}{c}{x}_{n+1}={\beta }_n\gamma V{x}_n+(I-{\beta }_n\mu F)W_n{\tilde{y}}_n,\hfill \\ x_n={\beta }_{n-1}\gamma V{x}_{n-1}+(I-{\beta }_{n-1}\mu F)W_{n-1}{\tilde{y}}_{n-1}.\hfill \end{array}$$

Hence we obtain

$$\begin{array}{rl}{y}_n-y_{n-1}=& {\theta }_n(\gamma S{x}_n-\gamma S{x}_{n-1})+({\theta }_n-{\theta }_{n-1})(\gamma S{x}_{n-1}-\mu F{\tilde{v}}_{n-1})\\ +(I-{\theta }_n\mu F){\tilde{v}}_n-(I-{\theta }_n\mu F){\tilde{v}}_{n-1}\end{array}$$

and

$$\begin{array}{rl}{x}_{n+1}-x_n=& {\beta }_n(\gamma V{x}_n-\gamma V{x}_{n-1})+({\beta }_n-{\beta }_{n-1})(\gamma V{x}_{n-1}-\mu F{W}_{n-1}{\tilde{y}}_{n-1})\\ +(I-{\beta }_n\mu F)W_n{\tilde{y}}_n-(I-{\beta }_n\mu F)W_{n-1}{\tilde{y}}_{n-1}.\end{array}$$

Utilizing Lemma 2.7, we deduce from (3.6) and (3.7) that

$$\begin{array}{rl}\parallel y_n-y_{n-1}\parallel \le & {\theta }_n\parallel \gamma S{x}_n-\gamma S{x}_{n-1}\parallel +|{\theta }_n-{\theta }_{n-1}|\parallel \gamma S{x}_{n-1}-\mu F{\tilde{v}}_{n-1}\parallel \\ +\parallel (I-{\theta }_n\mu F){\tilde{v}}_n-(I-{\theta }_n\mu F){\tilde{v}}_{n-1}\parallel \\ \le & {\theta }_n\gamma \parallel x_n-x_{n-1}\parallel +|{\theta }_n-{\theta }_{n-1}|\parallel \gamma S{x}_{n-1}-\mu F{\tilde{v}}_{n-1}\parallel \\ +(1-{\theta }_n\tau )\parallel {\tilde{v}}_n-{\tilde{v}}_{n-1}\parallel \\ \le & {\theta }_n\gamma \parallel x_n-x_{n-1}\parallel +|{\theta }_n-{\theta }_{n-1}|\parallel \gamma S{x}_{n-1}-\mu F{\tilde{v}}_{n-1}\parallel \\ +(1-{\theta }_n\tau )(\parallel v_n-v_{n-1}\parallel +\lambda |{\alpha }_n-{\alpha }_{n-1}|\parallel v_{n-1}\parallel )\end{array}$$

(3.8)

and

$$\begin{array}{rl}\parallel x_{n+1}-x_n\parallel \le & {\beta }_n\parallel \gamma V{x}_n-\gamma V{x}_{n-1}\parallel +|{\beta }_n-{\beta }_{n-1}|\parallel \gamma V{x}_{n-1}-\mu F{W}_{n-1}{\tilde{y}}_{n-1}\parallel \\ +\parallel (I-{\beta }_n\mu F)W_n{\tilde{y}}_n-(I-{\beta }_n\mu F)W_{n-1}{\tilde{y}}_{n-1}\parallel \\ \le & {\beta }_n\gamma l\parallel x_n-x_{n-1}\parallel +|{\beta }_n-{\beta }_{n-1}|\parallel \gamma V{x}_{n-1}-\mu F{W}_{n-1}{\tilde{y}}_{n-1}\parallel \\ +(1-{\beta }_n\tau )\parallel W_n{\tilde{y}}_n-W_{n-1}{\tilde{y}}_{n-1}\parallel \\ \le & {\beta }_n\gamma l\parallel x_n-x_{n-1}\parallel +|{\beta }_n-{\beta }_{n-1}|\parallel \gamma V{x}_{n-1}-\mu F{W}_{n-1}{\tilde{y}}_{n-1}\parallel \\ +(1-{\beta }_n\tau )(\parallel W_n{\tilde{y}}_n-W_n{\tilde{y}}_{n-1}\parallel +\parallel W_n{\tilde{y}}_{n-1}-W_{n-1}{\tilde{y}}_{n-1}\parallel )\\ \le & {\beta }_n\gamma l\parallel x_n-x_{n-1}\parallel +|{\beta }_n-{\beta }_{n-1}|\parallel \gamma V{x}_{n-1}-\mu F{W}_{n-1}{\tilde{y}}_{n-1}\parallel \\ +(1-{\beta }_n\tau )(\parallel {\tilde{y}}_n-{\tilde{y}}_{n-1}\parallel +\parallel W_n{\tilde{y}}_{n-1}-W_{n-1}{\tilde{y}}_{n-1}\parallel )\\ \le & {\beta }_n\gamma l\parallel x_n-x_{n-1}\parallel +|{\beta }_n-{\beta }_{n-1}|\parallel \gamma V{x}_{n-1}-\mu F{W}_{n-1}{\tilde{y}}_{n-1}\parallel \\ +(1-{\beta }_n\tau )(\parallel y_n-y_{n-1}\parallel +\lambda |{\alpha }_n-{\alpha }_{n-1}|\parallel y_{n-1}\parallel \\ +\parallel W_n{\tilde{y}}_{n-1}-W_{n-1}{\tilde{y}}_{n-1}\parallel ),\end{array}$$

(3.9)

where $\tau =1-\sqrt{1-\mu (2\eta -\mu {\kappa }^2)}$.

Utilizing (2.1) and (2.4), we obtain

$$\begin{array}{rcl}\parallel v_{n+1}-v_n\parallel & =& \parallel {\Lambda }_{n+1}^N{u}_{n+1}-{\Lambda }_n^N{u}_n\parallel \\ =& \parallel J_{R_N,{\lambda }_{N,n+1}}(I-{\lambda }_{N,n+1}{B}_N){\Lambda }_{n+1}^{N-1}{u}_{n+1}-J_{R_N,{\lambda }_{N,n}}(I-{\lambda }_{N,n}{B}_N){\Lambda }_n^{N-1}{u}_n\parallel \\ \le & \parallel J_{R_N,{\lambda }_{N,n+1}}(I-{\lambda }_{N,n+1}{B}_N){\Lambda }_{n+1}^{N-1}{u}_{n+1}-J_{R_N,{\lambda }_{N,n+1}}(I-{\lambda }_{N,n}{B}_N){\Lambda }_{n+1}^{N-1}{u}_{n+1}\parallel \\ +\parallel J_{R_N,{\lambda }_{N,n+1}}(I-{\lambda }_{N,n}{B}_N){\Lambda }_{n+1}^{N-1}{u}_{n+1}-J_{R_N,{\lambda }_{N,n}}(I-{\lambda }_{N,n}{B}_N){\Lambda }_n^{N-1}{u}_n\parallel \\ \le & \parallel (I-{\lambda }_{N,n+1}{B}_N){\Lambda }_{n+1}^{N-1}{u}_{n+1}-(I-{\lambda }_{N,n}{B}_N){\Lambda }_{n+1}^{N-1}{u}_{n+1}\parallel \\ +\parallel (I-{\lambda }_{N,n}{B}_N){\Lambda }_{n+1}^{N-1}{u}_{n+1}-(I-{\lambda }_{N,n}{B}_N){\Lambda }_n^{N-1}{u}_n\parallel \\ +|{\lambda }_{N,n+1}-{\lambda }_{N,n}|\\ \times (\frac{1}{{\lambda }_{N,n+1}}\parallel J_{R_N,{\lambda }_{N,n+1}}(I-{\lambda }_{N,n}{B}_N){\Lambda }_{n+1}^{N-1}{u}_{n+1}-(I-{\lambda }_{N,n}{B}_N){\Lambda }_n^{N-1}{u}_n\parallel \\ +\frac{1}{{\lambda }_{N,n}}\parallel (I-{\lambda }_{N,n}{B}_N){\Lambda }_{n+1}^{N-1}{u}_{n+1}-J_{R_N,{\lambda }_{N,n}}(I-{\lambda }_{N,n}{B}_N){\Lambda }_n^{N-1}{u}_n\parallel )\\ \le & |{\lambda }_{N,n+1}-{\lambda }_{N,n}|(\parallel B_N{\Lambda }_{n+1}^{N-1}{u}_{n+1}\parallel +\tilde{M})+\parallel {\Lambda }_{n+1}^{N-1}{u}_{n+1}-{\Lambda }_n^{N-1}{u}_n\parallel \\ \le & |{\lambda }_{N,n+1}-{\lambda }_{N,n}|(\parallel B_N{\Lambda }_{n+1}^{N-1}{u}_{n+1}\parallel +\tilde{M})\\ +|{\lambda }_{N-1,n+1}-{\lambda }_{N-1,n}|(\parallel B_{N-1}{\Lambda }_{n+1}^{N-2}{u}_{n+1}\parallel +\tilde{M})+\parallel {\Lambda }_{n+1}^{N-2}{u}_{n+1}-{\Lambda }_n^{N-2}{u}_n\parallel \\ \cdots \\ \le & |{\lambda }_{N,n+1}-{\lambda }_{N,n}|(\parallel B_N{\Lambda }_{n+1}^{N-1}{u}_{n+1}\parallel +\tilde{M})\\ +|{\lambda }_{N-1,n+1}-{\lambda }_{N-1,n}|(\parallel B_{N-1}{\Lambda }_{n+1}^{N-2}{u}_{n+1}\parallel +\tilde{M})+\cdots \\ +|{\lambda }_{1,n+1}-{\lambda }_{1,n}|(\parallel B_1{\Lambda }_{n+1}^0{u}_{n+1}\parallel +\tilde{M})+\parallel {\Lambda }_{n+1}^0{u}_{n+1}-{\Lambda }_n^0{u}_n\parallel \\ \le & {\tilde{M}}_0\sum _{i=1}^N|{\lambda }_{i,n+1}-{\lambda }_{i,n}|+\parallel u_{n+1}-u_n\parallel ,\end{array}$$

(3.10)

where

$$\begin{array}{r}\underset{n\ge 1}{sup}\{\frac{1}{{\lambda }_{N,n+1}}\parallel J_{R_N,{\lambda }_{N,n+1}}(I-{\lambda }_{N,n}{B}_N){\Lambda }_{n+1}^{N-1}{u}_{n+1}-(I-{\lambda }_{N,n}{B}_N){\Lambda }_n^{N-1}{u}_n\parallel \\ +\frac{1}{{\lambda }_{N,n}}\parallel (I-{\lambda }_{N,n}{B}_N){\Lambda }_{n+1}^{N-1}{u}_{n+1}-J_{R_N,{\lambda }_{N,n}}(I-{\lambda }_{N,n}{B}_N){\Lambda }_n^{N-1}{u}_n\parallel \}\\ \le \tilde{M},\end{array}$$

for some $\tilde{M}>0$ and ${sup}_{n\ge 1}\{{\sum }_{i=1}^N\parallel B_i{\Lambda }_{n+1}^{i-1}{u}_{n+1}\parallel +\tilde{M}\}\le {\tilde{M}}_0$ for some ${\tilde{M}}_0>0$.

Utilizing Proposition 2.4(ii), (v) we deduce that

$$\begin{array}{r}\parallel u_{n+1}-u_n\parallel \\ =\parallel {\Delta }_{n+1}^M{x}_{n+1}-{\Delta }_n^M{x}_n\parallel \\ =\parallel T_{r_{M,n+1}}^{({\Theta }_M,{\phi }_M)}(I-r_{M,n+1}{A}_M){\Delta }_{n+1}^{M-1}{x}_{n+1}-T_{r_{M,n}}^{({\Theta }_M,{\phi }_M)}(I-r_{M,n}{A}_M){\Delta }_n^{M-1}{x}_n\parallel \\ \le \parallel T_{r_{M,n+1}}^{({\Theta }_M,{\phi }_M)}(I-r_{M,n+1}{A}_M){\Delta }_{n+1}^{M-1}{x}_{n+1}-T_{r_{M,n}}^{({\Theta }_M,{\phi }_M)}(I-r_{M,n}{A}_M){\Delta }_{n+1}^{M-1}{x}_{n+1}\parallel \\ +\parallel T_{r_{M,n}}^{({\Theta }_M,{\phi }_M)}(I-r_{M,n}{A}_M){\Delta }_{n+1}^{M-1}{x}_{n+1}-T_{r_{M,n}}^{({\Theta }_M,{\phi }_M)}(I-r_{M,n}{A}_M){\Delta }_n^{M-1}{x}_n\parallel \\ \le \parallel T_{r_{M,n+1}}^{({\Theta }_M,{\phi }_M)}(I-r_{M,n+1}{A}_M){\Delta }_{n+1}^{M-1}{x}_{n+1}-T_{r_{M,n}}^{({\Theta }_M,{\phi }_M)}(I-r_{M,n+1}{A}_M){\Delta }_{n+1}^{M-1}{x}_{n+1}\parallel \\ +\parallel T_{r_{M,n}}^{({\Theta }_M,{\phi }_M)}(I-r_{M,n+1}{A}_M){\Delta }_{n+1}^{M-1}{x}_{n+1}-T_{r_{M,n}}^{({\Theta }_M,{\phi }_M)}(I-r_{M,n}{A}_M){\Delta }_{n+1}^{M-1}{x}_{n+1}\parallel \\ +\parallel (I-r_{M,n}{A}_M){\Delta }_{n+1}^{M-1}{x}_{n+1}-(I-r_{M,n}{A}_M){\Delta }_n^{M-1}{x}_n\parallel \\ \le \frac{|r_{M,n+1}-r_{M,n}|}{r_{M,n+1}}\parallel T_{r_{M,n+1}}^{({\Theta }_M,{\phi }_M)}(I-r_{M,n+1}{A}_M){\Delta }_{n+1}^{M-1}{x}_{n+1}-(I-r_{M,n+1}{A}_M){\Delta }_{n+1}^{M-1}{x}_{n+1}\parallel \\ +|r_{M,n+1}-r_{M,n}|\parallel A_M{\Delta }_{n+1}^{M-1}{x}_{n+1}\parallel +\parallel {\Delta }_{n+1}^{M-1}{x}_{n+1}-{\Delta }_n^{M-1}{x}_n\parallel \\ =|r_{M,n+1}-r_{M,n}|[\parallel A_M{\Delta }_{n+1}^{M-1}{x}_{n+1}\parallel \\ +\frac{1}{r_{M,n+1}}\parallel T_{r_{M,n+1}}^{({\Theta }_M,{\phi }_M)}(I-r_{M,n+1}{A}_M){\Delta }_{n+1}^{M-1}{x}_{n+1}-(I-r_{M,n+1}{A}_M){\Delta }_{n+1}^{M-1}{x}_{n+1}\parallel ]\\ +\parallel {\Delta }_{n+1}^{M-1}{x}_{n+1}-{\Delta }_n^{M-1}{x}_n\parallel \\ \cdots \\ \le |r_{M,n+1}-r_{M,n}|[\parallel A_M{\Delta }_{n+1}^{M-1}{x}_{n+1}\parallel \\ +\frac{1}{r_{M,n+1}}\parallel T_{r_{M,n+1}}^{({\Theta }_M,{\phi }_M)}(I-r_{M,n+1}{A}_M){\Delta }_{n+1}^{M-1}{x}_{n+1}-(I-r_{M,n+1}{A}_M){\Delta }_{n+1}^{M-1}{x}_{n+1}\parallel ]+\cdots \\ +|r_{1,n+1}-r_{1,n}|[\parallel A_1{\Delta }_{n+1}^0{x}_{n+1}\parallel \\ +\frac{1}{r_{1,n+1}}\parallel T_{r_{1,n+1}}^{({\Theta }_1,{\phi }_1)}(I-r_{1,n+1}{A}_1){\Delta }_{n+1}^0{x}_{n+1}-(I-r_{1,n+1}{A}_1){\Delta }_{n+1}^0{x}_{n+1}\parallel ]\\ +\parallel {\Delta }_{n+1}^0{x}_{n+1}-{\Delta }_n^0{x}_n\parallel \\ \le {\tilde{M}}_1\sum _{k=1}^M|r_{k,n+1}-r_{k,n}|+\parallel x_{n+1}-x_n\parallel ,\end{array}$$

(3.11)

where ${\tilde{M}}_1>0$ is a constant such that, for each $n\ge 1$,

$$\begin{array}{r}\sum _{k=1}^M[\parallel A_k{\Delta }_{n+1}^{k-1}{x}_{n+1}\parallel +\frac{1}{r_{k,n+1}}\parallel T_{r_{k,n+1}}^{({\Theta }_k,{\phi }_k)}(I-r_{k,n+1}{A}_k){\Delta }_{n+1}^{k-1}{x}_{n+1}-(I-r_{k,n+1}{A}_k){\Delta }_{n+1}^{k-1}{x}_{n+1}\parallel ]\\ \le {\tilde{M}}_1.\end{array}$$

In the meantime, from (2.2), since $W_n$, $T_n$, and $U_{n,i}$ are all nonexpansive, we have

$$\begin{array}{rl}\parallel W_{n+1}{\tilde{y}}_n-W_n{\tilde{y}}_n\parallel & =\parallel {\lambda }_1{T}_1{U}_{n+1,2}{\tilde{y}}_n-{\lambda }_1{T}_1{U}_{n,2}{\tilde{y}}_n\parallel \\ \le {\lambda }_1\parallel U_{n+1,2}{\tilde{y}}_n-U_{n,2}{\tilde{y}}_n\parallel \\ ={\lambda }_1\parallel {\lambda }_2{T}_2{U}_{n+1,3}{\tilde{y}}_n-{\lambda }_2{T}_2{U}_{n,3}{\tilde{y}}_n\parallel \\ \le {\lambda }_1{\lambda }_2\parallel U_{n+1,3}{\tilde{y}}_n-U_{n,3}{\tilde{y}}_n\parallel \\ \cdots \\ \le {\lambda }_1{\lambda }_2\cdots {\lambda }_n\parallel U_{n+1,n+1}{\tilde{y}}_n-U_{n,n+1}{\tilde{y}}_n\parallel \\ \le {\tilde{M}}_2\prod _{i=1}^n{\lambda }_i,\end{array}$$

(3.12)

where ${\tilde{M}}_2$ is a constant such that $\parallel U_{n+1,n+1}{\tilde{y}}_n\parallel +\parallel U_{n,n+1}{\tilde{y}}_n\parallel \le {\tilde{M}}_2$ for each $n\ge 1$. So, from (3.8)-(3.12), and $\{{\lambda }_n\}\subset (0,b]\subset (0,1)$ it follows that

$$\begin{array}{r}\parallel y_n-y_{n-1}\parallel \\ \le {\theta }_n\gamma \parallel x_n-x_{n-1}\parallel +|{\theta }_n-{\theta }_{n-1}|\parallel \gamma S{x}_{n-1}-\mu F{\tilde{v}}_{n-1}\parallel \\ +(1-{\theta }_n\tau )(\parallel v_n-v_{n-1}\parallel +\lambda |{\alpha }_n-{\alpha }_{n-1}|\parallel v_{n-1}\parallel )\\ \le {\theta }_n\gamma \parallel x_n-x_{n-1}\parallel +|{\theta }_n-{\theta }_{n-1}|\parallel \gamma S{x}_{n-1}-\mu F{\tilde{v}}_{n-1}\parallel \\ +(1-{\theta }_n\tau )({\tilde{M}}_0\sum _{i=1}^N|{\lambda }_{i,n}-{\lambda }_{i,n-1}|+\parallel u_n-u_{n-1}\parallel +\lambda |{\alpha }_n-{\alpha }_{n-1}|\parallel v_{n-1}\parallel )\\ \le {\theta }_n\gamma \parallel x_n-x_{n-1}\parallel +|{\theta }_n-{\theta }_{n-1}|\parallel \gamma S{x}_{n-1}-\mu F{\tilde{v}}_{n-1}\parallel \\ +(1-{\theta }_n\tau )({\tilde{M}}_0\sum _{i=1}^N|{\lambda }_{i,n}-{\lambda }_{i,n-1}|+{\tilde{M}}_1\sum _{k=1}^M|r_{k,n}-r_{k,n-1}|\\ +\parallel x_n-x_{n-1}\parallel +\lambda |{\alpha }_n-{\alpha }_{n-1}|\parallel v_{n-1}\parallel )\\ \le (1-{\theta }_n(\tau -\gamma ))\parallel x_n-x_{n-1}\parallel +|{\theta }_n-{\theta }_{n-1}|\parallel \gamma S{x}_{n-1}-\mu F{\tilde{v}}_{n-1}\parallel \\ +{\tilde{M}}_0\sum _{i=1}^N|{\lambda }_{i,n}-{\lambda }_{i,n-1}|+{\tilde{M}}_1\sum _{k=1}^M|r_{k,n}-r_{k,n-1}|+\lambda |{\alpha }_n-{\alpha }_{n-1}|\parallel v_{n-1}\parallel \\ \le \parallel x_n-x_{n-1}\parallel +|{\theta }_n-{\theta }_{n-1}|\parallel \gamma S{x}_{n-1}-\mu F{\tilde{v}}_{n-1}\parallel \\ +{\tilde{M}}_0\sum _{i=1}^N|{\lambda }_{i,n}-{\lambda }_{i,n-1}|+{\tilde{M}}_1\sum _{k=1}^M|r_{k,n}-r_{k,n-1}|+\lambda |{\alpha }_n-{\alpha }_{n-1}|\parallel v_{n-1}\parallel ,\end{array}$$

and hence

$$\begin{array}{r}\parallel x_{n+1}-x_n\parallel \\ \le {\beta }_n\gamma l\parallel x_n-x_{n-1}\parallel +|{\beta }_n-{\beta }_{n-1}|\parallel \gamma V{x}_{n-1}-\mu F{W}_{n-1}{\tilde{y}}_{n-1}\parallel \\ +(1-{\beta }_n\tau )(\parallel y_n-y_{n-1}\parallel +\lambda |{\alpha }_n-{\alpha }_{n-1}|\parallel y_{n-1}\parallel +\parallel W_n{\tilde{y}}_{n-1}-W_{n-1}{\tilde{y}}_{n-1}\parallel )\\ \le {\beta }_n\gamma l\parallel x_n-x_{n-1}\parallel +|{\beta }_n-{\beta }_{n-1}|\parallel \gamma V{x}_{n-1}-\mu F{W}_{n-1}{\tilde{y}}_{n-1}\parallel \\ +(1-{\beta }_n\tau )\{\parallel x_n-x_{n-1}\parallel +|{\theta }_n-{\theta }_{n-1}\end{array}$$