Variational inequality problems over split fixed point sets of strict pseudo-nonspreading mappings and quasi-nonexpansive mappings with applications

Abstract

In this paper, we first establish a strong convergence theorem for a variational inequality problem over split fixed point sets of a finite family of strict pseudo-nonspreading mappings and a countable family of quasi-nonexpansive mappings. As applications, we establish a strong convergence theorem of split fixed point sets of a finite family of strict pseudo-nonspreading mappings and a countable family of strict pseudo-nonspreading mappings without semicompact assumption on the strict pseudo-nonspreading mappings. We also study the variational inequality problems over split common solutions of fixed points for a finite family of strict pseudo-nonspreading mappings, fixed points of a countable family of pseudo-contractive mappings (or strict pseudo-nonspreading mappings) and solutions of a countable family of nonlinear operators. We study fixed points of a countable family of pseudo-contractive mappings with hemicontinuity assumption, neither Lipschitz continuity nor closedness assumption is needed.

1 Introduction

The split feasibility problem (SFP) in finite dimensional Hilbert spaces was first introduced by Censor and Elfving [1] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction. Since then, the split feasibility problem (SFP) has received much attention due to its applications in signal processing, image reconstruction, with particular progress in intensity-modulated radiation therapy, approximation theory, control theory, biomedical engineering, communications, and geophysics. For example, one can see [2–5].

Let C be a nonempty closed convex subset of a real Hilbert space $H_1$ with the inner product $〈\cdot ,\cdot 〉$ and the norm $\parallel \cdot \parallel$.

Let f be a contraction on C, let $T:C\to C$ be a nonexpansive mapping and $\{{\alpha }_n\}$ be a sequence in $[0,1]$. In 2004, Xu [6] proved that under some condition on $\{{\alpha }_n\}$, the sequence $\{x_n\}$ generated by

$$x_{n+1}={\alpha }_{n}f{x}_n+(1-{\alpha }_n)T{x}_n$$

strongly converges to $x^{\ast }$ in $Fix(T)$ which is the unique solution of the variational inequality

$$〈(I-f)x^{\ast },x-x^{\ast }〉\ge 0$$

for all $x\in Fix(T)$.

Xu [7] studied the following minimization problem over the set of fixed point set of a nonexpansive operator T on a real Hilbert space $H_1$:

$$\underset{x\in Fix(T)}{min}\frac{1}{2}〈Bx,x〉-〈a,x〉,$$

where a is a given point in $H_1$ and B is a strongly positive bounded linear operator on $H_1$. In [7], Xu proved that the sequence $\{x_n\}$ defined by the following iterative method:

$$x_{n+1}=(I-{\alpha }_{n}B)T{x}_n+{\alpha }_{n}a$$

converges strongly to the unique solution of the minimization problem of a quadratic function.

Let $H_1$, $H_2$ be two real Hilbert spaces, $A:H_1\to H_2$ be a bounded linear operator, ${\{S_i\}}_{i=1}^{\mathrm{\infty }}:H_1\to H_1$ be an infinite family of $k_i$-strictly pseudo-nonspreading mappings and ${\{T_i\}}_{i=1}^N:H_2\to H_2$ be a finite family of ${\rho }_i$-strict pseudo-nonspreading mappings, $C={\bigcap }_{i=1}^{\mathrm{\infty }}Fix(S_i)$ and $Q={\bigcap }_{i=1}^{N}Fix(T_i)$, Chang et al. [8] studied the following iterative sequence.

Let $\{x_n\}\subset H_1$ be defined by

$$\{\begin{array}{l}{x}_1\in C\text{chosen arbitrarily},\\ y_n=[I-\gamma A^{\ast }(I-T_{n(modN)})A]x_n,\\ x_{n+1}=a_{0,n}{y}_n+{\sum }_{i=1}^{\mathrm{\infty }}{a}_{i,n}{S}_{i,\beta }{y}_n\text{for all }n\ge 1,\end{array}$$

where $S_{i,\beta }=\beta I+(1-\beta )S_i$, $\beta \in (0,1)$ is a constant, I is an identity function on $H_1$.

Chang et al. [8] proved a weak convergence theorem to the solution of the following problem: Find $\overline{x}\in C$, $A\overline{x}\in Q$.

In addition, if there exists some positive integer m such that $S_m$ is semicompact, then $\{x_n\}$, $\{y_n\}$ converge strongly to a point $\overline{x}\in \mathrm{\Gamma }=\{x\in C,Ax\in Q\}$. In 2013, Tang [9] studied common solutions of fixed points of continuous pseudo-contractive mappings and zero points of the sum of monotone mappings. In 2014, Zegeye and Shahzad [10] and Yao et al. [11] studied common solutions of fixed points of Lipschitz continuous pseudo-contractive mappings and zero points of the sum of monotone mappings. Wangkeree and Nammanee [12] studied common solutions of fixed points of continuous pseudo-contractive mappings and solutions of a continuous monotone variational inequality.

In 2013, Cheng et al. [13] constructed a three-step iteration and obtained a convergence theorem for a countable family of uniformly Lipschitz and uniformly closed pseudo-contractive mappings. Deng [14] constructed another iteration and obtained a convergence theorem for a countable family of closed and Lipschitz pseudo-contractive mappings. Both the results of Cheng et al. [13] and Deng [14] used too strong conditions to study common fixed points of a countable family of pseudo-contractive mappings. The fixed point theorems of pseudo-contractive mappings in the literature assumed the Lipschitz continuous condition; see, for example, [15, 16].

Motivated and inspired by the above results, in this paper, we first establish a strong convergence theorem for a variational inequality problem over split fixed point sets of a finite family of strict pseudo-nonspreading mappings and a countable family of quasi-nonexpansive mappings. As applications, we establish a strong convergence theorem of split fixed point sets of a finite family of strict pseudo-nonspreading mappings and a countable family of strict pseudo-nonspreading mappings without semicompact assumption on the strict pseudo-nonspreading mappings. We also study the variational inequality problems over split common solutions of a fixed point for a finite family of strict pseudo-nonspreading mappings, fixed points of a countable family of pseudo-contractive mappings (or strict pseudo-nonspreading mappings) and solutions of a countable family of nonlinear operators. We study a fixed point of a countable family of pseudo-contractive mappings with hemicontinuity assumption, neither Lipschitz continuity nor closedness assumptions is needed.

2 Preliminaries

Throughout this paper, let ℕ be the set of positive integers, and let ℝ be the set of real numbers, $H_1$ be a (real) Hilbert space. Let $〈\cdot ,\cdot 〉$ and $\parallel \cdot \parallel$ denote the inner product and the norm of $H_1$, respectively. Let C be a nonempty closed convex subset of $H_1$. We denote the strong convergence and the weak convergence of $\{x_n\}$ to $x\in H_1$ by $x_n\to x$ and $x_n\rightharpoonup x$, respectively.

Let $T:C\to H_1$ be a mapping, and let $Fix(T):=\{x\in C:Tx=x\}$ denote the set of fixed points of T. A mapping $T:H_1\to H_1$ is called

• (i) pseudo-contractive if for each $x,y\in H_1$, we have $〈Tx-Ty,x-y〉\le {\parallel x-y\parallel }^2$.

  Note that the above inequality can be equivalently written as

  $${\parallel Tx-Ty\parallel }^2\le {\parallel x-y\parallel }^2+{\parallel (I-T)x-(I-T)y\parallel }^2\text{for all }x,y\in H_1;$$

• (ii) a k-strictly pseudo-contractive mapping if there exists a constant $k\in [0,1)$ such that $〈x-y,Tx-Ty〉\le {\parallel x-y\parallel }^2-k{\parallel (I-T)x-(I-T)y\parallel }^2$ for all $x,y\in H_1$;

• (iii) nonexpansive if $\parallel Tx-Ty\parallel \le \parallel x-y\parallel$ for all $x,y\in H_1$;

• (iv) firmly nonexpansive if ${\parallel Tx-Ty\parallel }^2\le {\parallel x-y\parallel }^2-{\parallel (I-T)x-(I-T)y\parallel }^2$ for every $x,y\in C$;

• (v) quasi-nonexpansive if $Fix(T)\ne \mathrm{\varnothing }$ and $\parallel Tx-p\parallel \le \parallel x-p\parallel$ for all $x\in H_1$, $p\in Fix(T)$;

• (vi) nonspreading [17] if $2{\parallel Tx-Ty\parallel }^2\le {\parallel Tx-y\parallel }^2+{\parallel Ty-x\parallel }^2$ for all $x,y\in H_1$, that is, ${\parallel Tx-Ty\parallel }^2\le {\parallel x-y\parallel }^2+2〈x-Tx,y-Ty〉$ for all $x,y\in H_1$;

• (vii) α-strictly pseudo-nonspreading [18] if there exists $\alpha \in [0,1)$ such that

  $${\parallel Tx-Ty\parallel }^2\le {\parallel x-y\parallel }^2+\alpha {\parallel x-Tx-(y-Ty)\parallel }^2+2〈x-Tx,y-Ty〉$$

for all $x,y\in H_1$;

• (viii) strongly monotone if there exists $\overline{\gamma }>0$ such that $〈x-y,Tx-Ty〉\ge \overline{\gamma }{\parallel x-y\parallel }^2$ for all $x,y\in H_1$;

• (ix) Lipschitz continuous if there exists $L>0$ such that $\parallel Tx-Ty\parallel \le L\parallel x-y\parallel$ for all $x,y\in H_1$;

• (x) hemicontinuous ([19], p.204) if, for all $x,y\in H_1$, the mapping $g:[0,1]\to H_1$ defined by $g(t)=T(tx+(1-t)y)$ is continuous, where $H_1$ has a weak topology;

• (xi) α-inverse-strongly monotone if $〈x-y,Vx-Vy〉\ge \alpha {\parallel Tx-Ty\parallel }^2$ for all $x,y\in H_1$ and $\alpha >0$;

• (xi) monotone if $〈x-y,Tx-Ty〉\ge 0$ for all $x,y\in H_1$.

We also know that (i) if V is an α-inverse-strongly monotone mapping and $0<\lambda \le 2\alpha$, then $I-\lambda V:C\to H_1$ is a nonexpansive mapping; (ii) if V is a monotone mapping, then $T=I-V:C\to H_1$ is a pseudo-contractive mapping.

Let $T:C\to H_1$ be a mapping. Then $p\in C$ is called an asymptotic fixed point of T [20] if there exists $\{x_n\}\subseteq C$ such that $x_n\rightharpoonup p$, and ${lim}_{n\to \mathrm{\infty }}\parallel x_n-T{x}_n\parallel =0$. We denote by $Fix(\hat{T})$ the set of asymptotic fixed points of T. A mapping $T:C\to H$ is said to be demiclosed if it satisfies $Fix(T)=Fix(\hat{T})$.

Let $B:H_1⊸H_1$ be a multivalued mapping. The effective domain of B is denoted by $D(B)$, that is, $D(B)=\{x\in H_1:Bx\ne \mathrm{\varnothing }\}$.

Then $B:H_1⊸H_1$ is called

• (i) a monotone operator on $H_1$ if $〈x-y,u-v〉\ge 0$ for all $x,y\in D(B)$, $u\in Bx$, and $v\in By$;

• (ii) a maximal monotone operator on $H_1$ if B is a monotone operator on $H_1$ and its graph is not properly contained in the graph of any other monotone operator on $H_1$.

For a maximal monotone operator B on $H_1$ and $r>0$, we may define a single-valued operator $J_r={(I+rB)}^{-1}:H_1\to D(B)$, which is called the resolvent of B for r. Let $B^{-1}0=\{x\in H_1:0\in Bx\}$.

The following lemmas are needed in this paper.

A mapping $T:H_1\to H_1$ is said to be averaged if $T=(1-\alpha )I+\alpha S$, where $\alpha \in (0,1)$ and $S:H_1\to H_1$ is nonexpansive. In this case, we also say that T is α-averaged. A firmly nonexpansive mapping is $\frac{1}{2}$-averaged.

Lemma 2.1 [5, 21]

Let $T:C\to C$ be a mapping. Then the following are satisfied:

• (i) T is nonexpansive if and only if the complement $(I-T)$ is $1/2$-ism.

• (ii) If S is υ-ism, then, for $\gamma >0$, γS is $\upsilon /\gamma$-ism.

• (iii) S is averaged if and only if the complement $I-S$ is υ-ism for some $\upsilon >1/2$.

• (iv) If S and T are both averaged, then the product (composite) ST is averaged.

• (v) If the mappings ${\{T_i\}}_{i=1}^n$ are averaged and have a common fixed point, then ${\bigcap }_{i=1}^{n}Fix(T_i)=Fix(T_1\cdots T_n)$.

Lemma 2.2 [22]

Let $V:H_1\to H_1$ be a $\overline{\gamma }$-strongly monotone and L-Lipschitz continuous operator with $\overline{\gamma }>0$ and $L>0$. Let $\theta \in H_1$ and $V_1:H_1\to H_1$ such that $V_{1}x=Vx-\theta$. Then $V_1$ is a $\overline{\gamma }$-strongly monotone and L-Lipschitz continuous mapping. Furthermore, there exists a unique fixed point $z_0$ in C satisfying $z_0=P_C(z_0-V{z}_0+\theta )$. This point $z_0\in C$ is also a unique solution of the hierarchical variational inequality

$$〈V{z}_0-\theta ,q-z_0〉\ge 0,\mathrm{\forall }q\in C.$$

Lemma 2.3 [19]

Let $B:H_1⊸H_1$ be maximal monotone.

• (i) For each $\beta >0$, $J_{\beta }^B$ is single-valued and firmly nonexpansive.

• (ii) $\mathcal{D}(J_{\beta }^B)=H_1$ and $Fix(J_{\beta }^B)=\{x\in \mathcal{D}(B):0\in Bx\}$.

Lemma 2.4 [23]

Let $\{a_n\}$ be a sequence of real numbers such that there exists a subsequence $\{n_i\}$ of $\{n\}$ such that $a_{n_i}<a_{n_i+1}$ for all $i\in \mathbb{N}$. Then there exists a nondecreasing sequence $\{m_k\}\subseteq \mathbb{N}$ such that $m_k\to \mathrm{\infty }$ and the following properties are satisfied by all (sufficiently large) numbers $k\in \mathbb{N}$:

$$a_{m_k}\le a_{m_k+1}\mathit{\text{and}}{a}_k\le a_{m_k+1}.$$

In fact, $m_k=max\{j\le k:a_j<a_{j+1}\}$.

Lemma 2.5 [24]

Let ${\{a_n\}}_{n\in \mathbb{N}}$ be a sequence of nonnegative real numbers, $\{{\alpha }_n\}$ be a sequence of real numbers in $[0,1]$ with ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$, $\{u_n\}$ be a sequence of nonnegative real numbers with ${\sum }_{n=1}^{\mathrm{\infty }}{u}_n<\mathrm{\infty }$, $\{t_n\}$ be a sequence of real numbers with $lim\hspace{0.17em}sup{t}_n\le 0$. Suppose that $a_{n+1}\le (1-{\alpha }_n)a_n+{\alpha }_n{t}_n+u_n$ for each $n\in \mathbb{N}$. Then ${lim}_{n\to \mathrm{\infty }}{a}_n=0$.

The equilibrium problem is to find $z\in C$ such that

$$(\mathrm{EP})g(z,y)\ge 0\text{for each }y\in C,$$

where $g:C\times C\to \mathbb{R}$ is a bifunction.

This problem includes fixed point problems, optimization problems, variational inequality problems, Nash equilibrium problems, minimax inequalities, and saddle point problems as special cases. (For example, one can see [25] and related literature.) The solution set of equilibrium problem (EP) is denoted by $EP(g)$.

For solving the equilibrium problem, let us assume that the bifunction $g:C\times C\to \mathbb{R}$ satisfies the following conditions:

(A1) $g(x,x)=0$ for each $x\in C$.

(A2) g is monotone, i.e., $g(x,y)+g(y,x)\le 0$ for any $x,y\in C$.

(A3) For each $x,y,z\in C$, ${lim\hspace{0.17em}sup}_{t\downarrow 0}g(tz+(1-t)x,y)\le g(x,y)$.

(A4) For each $x\in C$, the scalar function $y\to g(x,y)$ is convex and lower semicontinuous.

We have the following result from Blum and Oettli [25].

Theorem 2.1 [25]

Let $g:C\times C\to \mathbb{R}$ be a bifunction which satisfies conditions (A1)-(A4). Then, for each $r>0$ and each $x\in H_1$, there exists $z\in C$ such that

$$g(z,y)+\frac{1}{r}〈y-z,z-x〉\ge 0$$

for all $y\in C$.

In 2005, Combettes and Hirstoaga [26] established the following important properties of a resolvent operator.

Theorem 2.2 [26]

Let $g:C\times C\to \mathbb{R}$ be a function satisfying conditions (A1)-(A4). For $r>0$, define $T_r^g:H_1\to C$ by

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

for all $x\in H_1$. Then the following hold:

• (i) $T_r^g$ is single-valued.

• (ii) $T_r^g$ is firmly nonexpansive, that is, ${\parallel T_r^{g}x-T_r^{g}y\parallel }^2\le 〈x-y,T_r^{g}x-T_r^{g}y〉$ for all $x,y\in H$.

• (iii) $\{x\in H_1:T_r^{g}x=x\}=\{x\in C:g(x,y)\ge 0,\mathrm{\forall }y\in C\}$.

• (iv) $\{x\in C:g(x,y)\ge 0,\mathrm{\forall }y\in C\}$ is a closed and convex subset of C.

We call such $T_r^g$ the resolvent of g for $r>0$.

Lemma 2.6 [27]

Let E be a uniformly convex Banach space, $r>0$ be a constant. Then there exists a continuous, strictly increasing and convex function $g:[0,2r)\to [0,\mathrm{\infty })$ with $g(0)=0$ such that

$${\parallel \sum _{k=1}^{\mathrm{\infty }}{\alpha }_k{x}_k\parallel }^2\le \sum _{k=1}^{\mathrm{\infty }}{\alpha }_k{\parallel x_k\parallel }^2-{\alpha }_i{\alpha }_{j}g(\parallel x_i-x_j\parallel )$$

for all $i,j\in \mathbb{N}$, $x_k\in B_r:=\{z\in E:\parallel z\parallel \le r\}$, ${\alpha }_k\in \mathbb{N}$ with ${\sum }_{k=1}^{\mathrm{\infty }}{\alpha }_k=1$.

Takahashi et al. [28] showed the following result.

Lemma 2.7 [28]

Let $g:C\times C\to \mathbb{R}$ be a bifunction satisfying the conditions (A1)-(A4). Define $A_g$ as follows:

$$(\mathrm{L}4.1)A_{g}x=\{\begin{array}{ll}\{z\in H_1:g(x,y)\ge 〈y-x,z〉,\mathrm{\forall }y\in C\},& \mathrm{\forall }x\in C;\\ \mathrm{\varnothing },& \mathrm{\forall }x\notin C.\end{array}$$

Then $EP(g)=A_g^{-1}0$ and $A_g$ is a maximal monotone operator with the domain of $A_g\subset C$. Furthermore, for any $x\in H_1$ and $r>0$, the resolvent $T_r^g$ of g coincides with the resolvent of $A_g$, i.e., $T_r^{g}x={(I+r{A}_g)}^{-1}x$.

Lemma 2.8 [18]

Let H be a real Hilbert space, C be a nonempty and closed convex subset of H, and $T:C\to C$ be a k-strictly pseudo-nonspreading mapping. Then the following hold:

• (i) If $Fix(T)\ne \mathrm{\varnothing }$, then $Fix(T)$ is closed and convex.

• (ii) T is demiclosed.

Lemma 2.9 [8]

Let H be a real Hilbert space, C be a nonempty and closed convex subset of H, and $T:C\to C$ be a k-strictly pseudo-nonspreading mapping and $Fix(T)\ne \mathrm{\varnothing }$. Let $T_{\gamma }=\gamma I+(1-\gamma )T$, $k\le \gamma <1$. Then $T_{\gamma }$ is a quasi-nonexpansive mapping.

3 Convergence theorems of hierarchical problems

Let C and Q be nonempty closed convex subsets of real Hilbert spaces $H_1$ and $H_2$, respectively. For each $i\in \mathbb{N}$, let $G_i$ be a maximal monotone mapping on $H_1$ such that the domains of $G_i$ are included in C, and let $J_{{\lambda }_i}^{G_i}={(I+{\lambda }_i{G}_i)}^{-1}$ for each ${\lambda }_i>0$. For $i\in \mathbb{N}$, ${\kappa }_i>0$, let $B_i$ be a ${\kappa }_i$-inverse-strongly monotone mapping of C into $H_1$. Let $A:H_1\to H_2$ be a bounded linear operator and $A^{\ast }$ be the adjoint of A, and let R be the spectral radius of the operator $A^{\ast }A$. Let $\{{\theta }_n\}\subset H_1$ be a sequence, and let V be a $\overline{\gamma }$-strongly monotone and L-Lipschitz continuous operator with $\overline{\gamma }>0$ and $L>0$. Let ${\{F_i\}}_{i=1}^{\mathrm{\infty }}$ and ${\{W_i\}}_{i=1}^{\mathrm{\infty }}$ be countable families of quasi-nonexpansive mappings from C into itself with demiclosed property, and let ${\{T_i\}}_{i=1}^N:H_2\to H_2$ be a finite family of ${\rho }_i$-strictly pseudo-nonspreading mappings with $\rho =max\{{\rho }_i:i=1,2,\dots ,N\}\in (0,1)$. Let ${\{S_i\}}_{i=1}^{\mathrm{\infty }}$ be a countable family of ${\delta }_i$-strictly pseudo-nonspreading mappings from C into itself with $\delta ={sup}_{i\ge 1}{\delta }_i$, and let ${\mathrm{\Psi }}_i:C\to C$ be a countable family of pseudo-contractive mappings. Throughout this paper, we use these notations and assumptions unless specified otherwise.

In this paper, we first study the variational inequality problem over split common solutions for a fixed point of a finite family of strict pseudo-nonspreading mappings, and a solution of a countable family of quasi-nonexpansive mappings.

Theorem 3.1 Suppose that ${\mathrm{\Pi }}_1:=\{x\in C:x\in {\bigcap }_{i=1}^{\mathrm{\infty }}Fix(F_i)\mathit{\text{ and }}Ax\in {\bigcap }_{i=1}^{N}Fix(T_i)\}\ne \mathrm{\varnothing }$. Take $\mu \in \mathbb{R}$ as follows:

$$0<\mu <\frac{2\overline{\gamma }}{L^2}.$$

Let $\{x_n\}\subset H$ be defined by

$$(3.1)\{\begin{array}{l}{x}_1\in C\mathit{\text{chosen arbitrarily}},\\ y_n=[I-b{A}^{\ast }(I-T_{n(modN)})A]x_n,\\ s_n=a_{n,0}{y}_n+{\sum }_{i=1}^{\mathrm{\infty }}{a}_{n,i}{F}_i{y}_n,\\ x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)({\beta }_n{\theta }_n+(1-{\beta }_{n}V)s_n)\end{array}$$

for each $n\in \mathbb{N}$, $\{a_{n,i}:n,i\in \mathbb{N}\}\subset [0,1]$ and $\{{\alpha }_n,{\beta }_n\}\subset (0,1)$. Assume the following:

• (i) $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\alpha }_n<1$;

• (ii) ${lim}_{n\to \mathrm{\infty }}{\beta }_n=0$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\beta }_n=\mathrm{\infty }$;

• (iii) $a_{n,0}+{\sum }_{i=1}^{\mathrm{\infty }}{a}_{n,i}=1$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{a}_{n,0}{a}_{n,i}>0$, $\mathrm{\forall }i\in \mathbb{N}$ and $0<b<\frac{1-\rho }{{\parallel A\parallel }^2}$;

• (iv) ${lim}_{n\to \mathrm{\infty }}{\theta }_n=\theta$ for some $\theta \in H$.

Then ${lim}_{n\to \mathrm{\infty }}{x}_n=\overline{x}$, where $\overline{x}=P_{{\mathrm{\Pi }}_1}(\overline{x}-V\overline{x}+\theta )$. This point $\overline{x}$ is also a unique solution of the following hierarchical variational inequality:

$$〈V\overline{x}-\theta ,q-\overline{x}〉\ge 0,\mathrm{\forall }q\in {\mathrm{\Pi }}_1.$$

Proof Take any $\overline{x}\in {\mathrm{\Pi }}_1$ and let $\overline{x}$ be fixed. Then $\overline{x}\in {\bigcap }_{i=1}^{\mathrm{\infty }}{F}_i\overline{x}$ and $A\overline{x}\in {\bigcap }_{i=1}^N{T}_i$. For each $n\in \mathbb{N}$, by Lemma 2.6, we have

$$\begin{array}{rcl}{\parallel s_n-\overline{x}\parallel }^2& =& {\parallel a_{n,0}{y}_n+\sum _{i=1}^{\mathrm{\infty }}{a}_{n,i}{F}_i{y}_n-\overline{x}\parallel }^2\\ \le & a_{n,0}{\parallel y_n-\overline{x}\parallel }^2+\sum _{i=1}^{\mathrm{\infty }}{a}_{n,i}{\parallel F_i{y}_n-\overline{x}\parallel }^2-a_{n,0}{a}_{n,i}g(\parallel y_n-F_i{y}_n\parallel )\\ \le & a_{n,0}{\parallel y_n-\overline{x}\parallel }^2+\sum _{i=1}^{\mathrm{\infty }}{a}_{n,i}{\parallel y_n-\overline{x}\parallel }^2-a_{n,0}{a}_{n,i}g(\parallel y_n-F_i{y}_n\parallel )\\ \le & {\parallel y_n-\overline{x}\parallel }^2-a_{n,0}{a}_{n,i}g(\parallel y_n-F_i{y}_n\parallel )\\ \le & {\parallel y_n-\overline{x}\parallel }^2.\end{array}$$

(1)

In Theorem 3.1 in [8], Chang et al. showed that

$$\begin{array}{rcl}{\parallel y_n-\overline{x}\parallel }^2& =& {\parallel [I-b{A}^{\ast }(I-T_{n(modN)})A]x_n-\overline{x}\parallel }^2\\ \le & {\parallel x_n-\overline{x}\parallel }^2-b(1-\rho -b{\parallel A\parallel }^2){\parallel (I-T_{n(modN)})A{x}_n\parallel }^2\\ \le & {\parallel x_n-\overline{x}\parallel }^2.\end{array}$$

(2)

Let $z_n={\beta }_n{\theta }_n+(I-{\beta }_{n}V)s_n$ and $\tau =\overline{\gamma }-\frac{L^2\mu }{2}$, following the same argument as in the proof of Theorem 3.1 in [22], we have

$$\parallel x_{n+1}-\overline{x}\parallel \le max\{\parallel x_n-\overline{x}\parallel ,M\},$$

where $M=max\{\frac{\parallel {\theta }_n-V\overline{x}\parallel }{\tau }:n\in \mathbb{N}\}$. By mathematical induction, we know

$$\parallel x_n-\overline{x}\parallel \le max\{\parallel x_1-\overline{x}\parallel ,M\}.$$

This implies that the sequence $\{x_n\}$ is bounded. Furthermore, $\{z_n\}$, $\{s_n\}$ and $\{F_i{x}_n\}$ are bounded for all $i\in \mathbb{N}$.

Following the same argument as in the proof of Theorem 3.1 in [22] again, we have

$$\begin{array}{c}{\parallel x_{n+1}-\overline{x}\parallel }^2-{\parallel x_n-\overline{x}\parallel }^2+(1-{\alpha }_n){\alpha }_n{\parallel s_n-x_n\parallel }^2\hfill \\ \le 2(1-{\alpha }_n){\beta }_n〈{\theta }_n,x_n-\overline{x}〉-2(1-{\alpha }_n){\beta }_n〈V{s}_n,x_n-\overline{x}〉\hfill \\ +{(1-{\alpha }_n)}^2[{\beta }_n^2{\parallel {\theta }_n-V{s}_n\parallel }^2+2{\beta }_n\parallel {\theta }_n-V{s}_n\parallel \parallel s_n-x_n\parallel ].\hfill \end{array}$$

(3)

We will divide the proof into two cases as follows.

Case 1: There exists a natural number N such that $\parallel x_{n+1}-\overline{x}\parallel \le \parallel x_n-\overline{x}\parallel$ for each $n\ge N$. So, ${lim}_{n\to \mathrm{\infty }}\parallel x_n-\overline{x}\parallel$ exists. Hence, it follows from (3), (i) and (ii) that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel s_n-x_n\parallel =0.$$

(4)

Following the same argument as in the proof of Theorem 3.1 in [22] again, we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_{n+1}-x_n\parallel =0,$$

(5)

$$\underset{n\to \mathrm{\infty }}{lim}\parallel z_n-s_n\parallel =0$$

(6)

and

$$\underset{n\to \mathrm{\infty }}{lim}\parallel z_n-x_n\parallel =0.$$

(7)

By (1) and (2), we have

$${\parallel s_n-\overline{x}\parallel }^2\le {\parallel x_n-\overline{x}\parallel }^2-b(1-\rho -b{\parallel A\parallel }^2){\parallel (I-T_{n(modN)})A{x}_n\parallel }^2-a_{n,0}{a}_{n,i}g(\parallel y_n-F_i{y}_n\parallel )$$

for all $i\in \mathbb{N}$.

Therefore,

$$\begin{array}{c}b(1-\rho -b{\parallel A\parallel }^2){\parallel (I-T_{n(modN)})A{x}_n\parallel }^2+a_{n,0}{a}_{n,i}g(\parallel y_n-F_i{y}_n\parallel )\hfill \\ \le {\parallel x_n-\overline{x}\parallel }^2-{\parallel s_n-\overline{x}\parallel }^2\hfill \\ \le \parallel x_n-s_n\parallel (\parallel s_n-\overline{x}\parallel +\parallel x_n-\overline{x}\parallel )\hfill \end{array}$$

(8)

for all $i\in \mathbb{N}$.

Thus, by (4), (8), and condition (iii), we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel (I-T_{n(modN)})A{x}_n\parallel =0\text{and}\underset{n\to \mathrm{\infty }}{lim}g(\parallel y_n-F_i{y}_n\parallel )=0$$

(9)

for all $i\in \mathbb{N}$.

From the properties of g, we conclude that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel y_n-F_i{y}_n\parallel =0$$

(10)

for all $i\in \mathbb{N}$.

By Lemma 2.8 and the assumption, we have that ${\mathrm{\Pi }}_1:=\{x\in C:x\in {\bigcap }_{i=1}^{\mathrm{\infty }}Fix(F_i)\text{ and }Ax\in {\bigcap }_{i=1}^{N}Fix(T_i)\}$ is a nonempty closed convex subset of H. Hence, by Lemma 2.2, we can take ${\overline{x}}_0\in {\mathrm{\Pi }}_1$ such that

$${\overline{x}}_0=P_{{\mathrm{\Pi }}_1}({\overline{x}}_0-V{\overline{x}}_0+\theta ).$$

This point ${\overline{x}}_0$ is also a unique solution of the hierarchical variational inequality

$$〈V{\overline{x}}_0-\theta ,q-{\overline{x}}_0〉\ge 0,\mathrm{\forall }q\in {\mathrm{\Pi }}_1.$$

(11)

We show that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈V{\overline{x}}_0-\theta ,z_n-{\overline{x}}_0〉\ge 0.$$

Without loss of generality, there exists a subsequence $\{z_{n_k}\}$ of $\{z_n\}$ such that $z_{n_k}\rightharpoonup w$ for some $w\in H$ and

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈V{\overline{x}}_0-\theta ,z_n-{\overline{x}}_0〉=\underset{k\to \mathrm{\infty }}{lim}〈V{\overline{x}}_0-\theta ,z_{n_k}-\overline{x_0}〉.$$

(12)

By (7), we have that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-z_n\parallel =0\text{and}{x}_{n_k}\rightharpoonup w.$$

(13)

Since A is a bounded linear operator, this implies that $A{x}_{n_k}\rightharpoonup Aw$. Also, by equation (10), we have

$$\underset{n_k\to \mathrm{\infty }}{lim}\parallel (I-T_{n_k(modN)})A{x}_{n_k}\parallel =0.$$

(14)

Hence there exist a positive integer $j\in \{1,2,\dots ,N\}$ and a subsequence $\{n_{k_i}\}$ of $\{n_k\}$ with $n_{k_i}(modN)=j$ such that

$$\underset{n_{k_i}\to \mathrm{\infty }}{lim}\parallel (I-T_j)A{x}_{n_{k_i}}\parallel =0.$$

(15)

Since $A{x}_{n_{k_i}}\rightharpoonup Aw$, it follows from Lemma 2.8 that $I-T_j$ is demiclosed at 0. This implies that $Aw\in Fix(T_j)$.

By equations (13) and (14), we have

$$y_{n_k}=x_{n_k}-\gamma A^{\ast }(I-T_j)A{x}_{n_{k_i}}\rightharpoonup w.$$

(16)

For each $i\in \mathbb{N}$, by $F_i$ is demiclosed, equations (10) and (16), we have $w\in Fix(F_i)$. Hence, $w\in {\mathrm{\Pi }}_1$. Thus, from (11) and (12) we have

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈V{\overline{x}}_0-\theta ,z_n-{\overline{x}}_0〉=\underset{k\to \mathrm{\infty }}{lim}〈V{\overline{x}}_0-\theta ,z_{n_k}-{\overline{x}}_0〉=〈V{\overline{x}}_0-\theta ,w-{\overline{x}}_0〉\ge 0.$$

(17)

Following the same argument as in the proof of Theorem 3.1 in [22] again, we have that

$$\begin{array}{rcl}{\parallel x_{n+1}-{\overline{x}}_0\parallel }^2& \le & [1-2(1-{\alpha }_n){\beta }_n\tau ]{\parallel x_n-{\overline{x}}_0\parallel }^2+2(1-{\alpha }_n){\beta }_n\tau (\frac{{\beta }_n\tau {\parallel x_n-{\overline{x}}_0\parallel }^2}{2}\\ +\frac{〈{\theta }_n-\theta ,z_n-{\overline{x}}_0〉}{\tau }+\frac{〈\theta -V{\overline{x}}_0,z_n-{\overline{x}}_0〉}{\tau }).\end{array}$$

(18)

By (17), (18), assumptions, and Lemma 2.5, we know that ${lim}_{n\to \mathrm{\infty }}{x}_n={\overline{x}}_0$, where

$${\overline{x}}_0=P_{{\mathrm{\Pi }}_1}({\overline{x}}_0-V{\overline{x}}_0+\theta ).$$

Case 2: Suppose that there exists $\{n_i\}$ of $\{n\}$ such that $\parallel x_{n_i}-\overline{x}\parallel \le \parallel x_{n_{i+1}}-\overline{x}\parallel$ for all $i\in \mathbb{N}$. By Lemma 2.4, there exists a nondecreasing sequence $\{m_j\}$ in ℕ such that $m_j\to \mathrm{\infty }$ and

$$\parallel x_{m_j}-\overline{x}\parallel \le \parallel x_{m_j+1}-\overline{x}\parallel \text{and}\parallel x_j-\overline{x}\parallel \le \parallel x_{m_j+1}-\overline{x}\parallel .$$

(19)

Following the same argument as in the proof of Theorem 3.1 in [22] again, we have

$$\underset{j\to \mathrm{\infty }}{lim}\parallel x_{m_{j+1}}-{\overline{x}}_0\parallel =0.$$

(20)

By equations (19) and (20),

$$\underset{j\to \mathrm{\infty }}{lim}\parallel x_j-{\overline{x}}_0\parallel \le \underset{j\to \mathrm{\infty }}{lim}\parallel x_{m_j+1}-{\overline{x}}_0\parallel =0.$$

Thus, the proof is completed. □

Remark 3.1 Theorem 3.1 improves Theorem 3.1 in [29], Theorem 3.3 in [30] and Theorem 3.1 in [31] in the sense that our convergence is for the common fixed point of a countable family of quasi-nonexpansive mappings and for the split feasibility problem.

Applying Theorem 3.1, we study a variational inequality problem over split common solutions for a fixed point of a finite family of strict pseudo-nonspreading mappings and a common fixed point of countable families of mappings.

Theorem 3.2 Suppose that ${\mathrm{\Pi }}_2:=\{x\in C:x\in {\bigcap }_{i=1}^{\mathrm{\infty }}Fix(S_i)\cap {\bigcap }_{i=1}^{\mathrm{\infty }}Fix(W_i)\mathit{\text{ and }}Ax\in {\bigcap }_{i=1}^{N}Fix(T_i)\}\ne \mathrm{\varnothing }$. Take $\mu \in \mathbb{R}$ as follows:

$$0<\mu <\frac{2\overline{\gamma }}{L^2}.$$

Let $\{x_n\}\subset H$ be defined by

$$(3.2)\{\begin{array}{l}{x}_1\in C\mathit{\text{chosen arbitrarily}},\\ y_n=[I-b{A}^{\ast }(I-T_{n(modN)})A]x_n,\\ s_n=a_{n,0}{y}_n+{\sum }_{i=1}^{\mathrm{\infty }}{b}_{n,i}{W}_i{y}_n+{\sum }_{i=1}^{\mathrm{\infty }}{a}_{n,i}{S}_{i,\gamma }{y}_n,\\ x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)({\beta }_n{\theta }_n+(1-{\beta }_{n}V)s_n),\end{array}$$

where $S_{i,\gamma }=\gamma I+(1-\gamma )S_i$, $i\ge 1$, $\gamma \in [\delta ,1)$ for each $n\in \mathbb{N}$, $\{{\alpha }_n,{\beta }_n\}\subset (0,1)$, $\{a_{n,i}:n,i\in \mathbb{N}\}\subset [0,1]$ and $\{b_{n,i}:n,i\in \mathbb{N}\}\subset [0,1]$. Assume the following:

• (i) $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\alpha }_n<1$;

• (ii) ${lim}_{n\to \mathrm{\infty }}{\beta }_n=0$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\beta }_n=\mathrm{\infty }$;

• (iii) $a_{n,0}+{\sum }_{i=1}^{\mathrm{\infty }}{b}_{n,i}+{\sum }_{i=1}^{\mathrm{\infty }}{a}_{n,i}=1$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{a}_{n,0}{a}_{n,i}>0$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{a}_{n,0}{b}_{n,i}>0$, $\mathrm{\forall }i\in \mathbb{N}$ and $0<b<\frac{1-\rho }{{\parallel A\parallel }^2}$;

• (iv) ${lim}_{n\to \mathrm{\infty }}{\theta }_n=\theta$ for some $\theta \in H$.

Then ${lim}_{n\to \mathrm{\infty }}{x}_n=\overline{x}$, where $\overline{x}=P_{{\mathrm{\Pi }}_2}(\overline{x}-V\overline{x}+\theta )$. This point $\overline{x}$ is also a unique solution of the following hierarchical variational inequality:

$$〈V\overline{x}-\theta ,q-\overline{x}〉\ge 0,\mathrm{\forall }q\in {\mathrm{\Pi }}_2.$$

Proof By ${\mathrm{\Pi }}_2\ne \mathrm{\varnothing }$, there exists $w\in C$ such that $w\in {\bigcap }_{i=1}^{\mathrm{\infty }}Fix(S_i){\bigcap }_{i=1}^{\mathrm{\infty }}Fix(W_i)={\bigcap }_{i=1}^{\mathrm{\infty }}Fix(S_{i,\gamma }){\bigcap }_{i=1}^{\mathrm{\infty }}Fix(W_i)$. For each $i\in \mathbb{N}$, $S_i$ is a ${\delta }_i$-strictly pseudo-nonspreading mapping and $\gamma \in [\delta ,1)$, by Lemma 2.9, we have $S_{i,\gamma }$ is quasi-nonexpansive. For each $i\in \mathbb{N}$, by Lemma 2.8 and $\parallel x-S_{i,\gamma }x\parallel =(1-\gamma )\parallel x-S_{i}x\parallel$, we see that $S_{i,\gamma }$ is demiclosed.

For each $i\in \mathbb{N}$, let $F_{2i}=S_{i,\gamma }$ and $F_{2i-1}=W_i$ in Theorem 3.1, then $F_i$ is quasi-nonexpansive with demiclosed property and ${\mathrm{\Pi }}_2={\mathrm{\Pi }}_1$. It follows from Theorem 3.1 that ${lim}_{n\to \mathrm{\infty }}{x}_n=\overline{x}$, where $\overline{x}=P_{{\mathrm{\Pi }}_2}(\overline{x}-V\overline{x}+\theta )$. This point $\overline{x}$ is also a unique solution of the following hierarchical variational inequality:

$$〈V\overline{x}-\theta ,q-\overline{x}〉\ge 0,\mathrm{\forall }q\in {\mathrm{\Pi }}_2.$$

 □

Applying Theorem 3.1, we study split common solutions for common fixed points of a finite family of strict pseudo-nonspreading mappings, and common fixed points of a countable family of strict pseudo-nonspreading mappings. Our result improves Theorem 3.1 in [8].

Theorem 3.3 Suppose that ${\mathrm{\Pi }}_3:=\{x\in C:x\in {\bigcap }_{i=1}^{\mathrm{\infty }}Fix(S_i)\mathit{\text{ and }}Ax\in {\bigcap }_{i=1}^{N}Fix(T_i)\}\ne \mathrm{\varnothing }$. Take $\mu \in \mathbb{R}$ as follows:

$$0<\mu <\frac{2\overline{\gamma }}{L^2}.$$

Let $\{x_n\}\subset H$ be defined by

$$(3.3)\{\begin{array}{l}{x}_1\in C\mathit{\text{chosen arbitrarily}},\\ y_n=[I-b{A}^{\ast }(I-T_{n(modN)})A]x_n,\\ s_n=s_n=c_{n,0}{y}_n+{\sum }_{i=1}^{\mathrm{\infty }}{c}_{n,i}{S}_{i,\gamma }{y}_n,\\ x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)({\beta }_n{\theta }_n+(1-{\beta }_n)s_n),\end{array}$$

where $S_{i,\gamma }=\gamma I+(1-\gamma )S_i$, $i\ge 1$, $\gamma \in [\delta ,1)$ for each $n\in \mathbb{N}$, $\{{\alpha }_n,{\beta }_n\}\subset (0,1)$ and $\{c_{n,i}:n,i\in \mathbb{N}\}\subset [0,1]$. Assume the following:

• (i) $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\alpha }_n<1$;

• (ii) ${lim}_{n\to \mathrm{\infty }}{\beta }_n=0$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\beta }_n=\mathrm{\infty }$;

• (iii) $c_{n,0}+{\sum }_{i=1}^{\mathrm{\infty }}{c}_{n,i}=1$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{c}_{n,0}{c}_{n,i}>0$, $\mathrm{\forall }i\in \mathbb{N}$ and $0<b<\frac{1-\rho }{{\parallel A\parallel }^2}$;

• (iv) ${lim}_{n\to \mathrm{\infty }}{\theta }_n=\theta$ for some $\theta \in H$.

Then ${lim}_{n\to \mathrm{\infty }}{x}_n=\overline{x}$, where $\overline{x}=P_{{\mathrm{\Pi }}_3}(\theta )$.

Proof By ${\mathrm{\Pi }}_3\ne \mathrm{\varnothing }$, there exists $w\in C$ such that $w\in {\bigcap }_{i=1}^{\mathrm{\infty }}Fix(S_i)={\bigcap }_{i=1}^{\mathrm{\infty }}Fix(S_{i,\gamma })$. For each $i\in \mathbb{N}$, $S_i$ is a ${\delta }_i$-strictly pseudo-nonspreading mapping and $\gamma \in [\delta ,1)$, by Lemma 2.9, we have $S_{i,\gamma }$ is quasi-nonexpansive and $Fix(S_i)=Fix(S_{i,\gamma })$. For each $i\in \mathbb{N}$, by Lemma 2.8 and $\parallel x-S_{i,\gamma }x\parallel =(1-\gamma )\parallel x-S_{i}x\parallel$, we see that $S_{i,\gamma }$ is demiclosed.

For each $i\in \mathbb{N}$, let $F_i=S_{i,\gamma }$ and $V(x)=x$ for all $x\in C$ in Theorem 3.1, then $F_i$ is quasi-nonexpansive with demiclosed property and ${\mathrm{\Pi }}_3={\mathrm{\Pi }}_1$. It follows from Theorem 3.1 that ${lim}_{n\to \mathrm{\infty }}{x}_n=\overline{x}$, where $\overline{x}=P_{{\mathrm{\Pi }}_3}(\theta )$. □

Remark 3.2 Theorem 3.3 improves Theorem 3.1 in [8]. Indeed, Theorem 3.1 in [8] establishes a weak convergence. In Theorem 3.1 in [8], if there exists some positive integer m such that $S_m$ is semicompact, then $\{x_n\}$ converges strongly to $\overline{x}\in {\mathrm{\Pi }}_3$. However, Theorem 3.3 is a strong convergence theorem without semicompact assumption.

Let $\gamma =\delta ={\delta }_i=0$ for all $i\in \mathbb{N}$ and $\rho ={\rho }_j=0$ for all $j=1,2,\dots ,N$ in Theorem 3.2, we have the following theorem.

Theorem 3.4 Suppose that for each $i\in \mathbb{N},S_i$ is a nonspreading mapping, and let ${\{T_i\}}_{i=1}^N:H_2\to H_2$ be a finite family of nonspreading mappings. Let ${\mathrm{\Pi }}_4:=\{x\in C:x\in {\bigcap }_{i=1}^{\mathrm{\infty }}Fix(S_i)\cap {\bigcap }_{i=1}^{\mathrm{\infty }}Fix(W_i)\mathit{\text{ and }}Ax\in {\bigcap }_{i=1}^{N}Fix(T_i)\}\ne \mathrm{\varnothing }$. Take $\mu \in \mathbb{R}$ as follows:

$$0<\mu <\frac{2\overline{\gamma }}{L^2}.$$

Let $\{x_n\}\subset H$ be defined by

$$(3.4)\{\begin{array}{l}{x}_1\in C\mathit{\text{chosen arbitrarily}},\\ y_n=[I-b{A}^{\ast }(I-T_{n(modN)})A]x_n,\\ s_n=a_{n,0}{y}_n+{\sum }_{i=1}^{\mathrm{\infty }}{b}_{n,i}{W}_i{y}_n+{\sum }_{i=1}^{\mathrm{\infty }}{a}_{n,i}{S}_i{y}_n,\\ x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)({\beta }_n{\theta }_n+(1-{\beta }_{n}V)s_n)\end{array}$$

for each $n\in \mathbb{N}$, $\{{\alpha }_n,{\beta }_n\}\subset (0,1)$, $\{a_{n,i}:n,i\in \mathbb{N}\}\subset [0,1]$ and $\{b_{n,i}:n,i\in \mathbb{N}\}\subset [0,1]$. Assume the following:

• (i) $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\alpha }_n<1$;

• (ii) ${lim}_{n\to \mathrm{\infty }}{\beta }_n=0$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\beta }_n=\mathrm{\infty }$;

• (iii) $a_{n,0}+{\sum }_{i=1}^{\mathrm{\infty }}{b}_{n,i}+{\sum }_{i=1}^{\mathrm{\infty }}{a}_{n,i}=1$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{a}_{n,0}{a}_{n,i}>0$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{a}_{n,0}{b}_{n,i}>0$, $\mathrm{\forall }i\in \mathbb{N}$ and $0<b<\frac{1-\rho }{{\parallel A\parallel }^2}$;

• (iv) ${lim}_{n\to \mathrm{\infty }}{\theta }_n=\theta$ for some $\theta \in H$.

Then ${lim}_{n\to \mathrm{\infty }}{x}_n=\overline{x}$, where $\overline{x}=P_{{\mathrm{\Pi }}_4}(\overline{x}-V\overline{x}+\theta )$. This point $\overline{x}$ is also a unique solution of the following hierarchical variational inequality:

$$〈V\overline{x}-\theta ,q-\overline{x}〉\ge 0,\mathrm{\forall }q\in {\mathrm{\Pi }}_4.$$

Proof For each $i\in \mathbb{N}$, $S_i$ is a nonspreading mapping, hence $S_i$ is a quasi-nonexpansive mapping and a ${\delta }_i$-strictly pseudo-nonspreading mapping with ${\delta }_i=0$. For each $i=1,2,\dots ,N$, $T_i$ is a nonspreading mapping, hence $T_i$ is a ${\rho }_i$-strictly pseudo-nonspreading mapping with ${\rho }_i=0$.

Let $\gamma =\delta ={\delta }_i=0$ for all $i\in \mathbb{N}$ and $\rho ={\rho }_j=0$ for all $j=1,2,\dots ,N$ in Theorem 3.2. It follows from Theorem 3.2 and ${\mathrm{\Pi }}_4={\mathrm{\Pi }}_2$ that ${lim}_{n\to \mathrm{\infty }}{x}_n=\overline{x}$, where $\overline{x}=P_{{\mathrm{\Pi }}_4}(\overline{x}-V\overline{x}+\theta )$. This point $\overline{x}$ is also a unique solution of the following hierarchical variational inequality:

$$〈V\overline{x}-\theta ,q-\overline{x}〉\ge 0,\mathrm{\forall }q\in {\mathrm{\Pi }}_4.$$

 □

• (I) We study split common fixed points for a finite family of ${\rho }_i$-strictly pseudo-nonspreading mappings, zeros for a countable family of mappings which are the sum of two monotone mappings and common fixed points for a countable family of quasi-nonexpansive mappings:

  $$\{\begin{array}{l}\text{Find }\overline{x}\in C\text{ such that}\\ \overline{x}\in {\bigcap }_{i=1}^{\mathrm{\infty }}{(G_i+B_i)}^{-1}0\cap {\bigcap }_{i=1}^{\mathrm{\infty }}Fix(W_i)\end{array}$$

and

$$\{\begin{array}{l}\overline{u}=A\overline{x}\in H_2\text{ such that}\\ \overline{u}\in {\bigcap }_{i=1}^{N}Fix(T_i).\end{array}$$

Theorem 3.5 Suppose that ${\mathrm{\Pi }}_5:=\{x\in C:x\in {\bigcap }_{i=1}^{\mathrm{\infty }}{(B_i+G_i)}^{-1}0{\bigcap }_{i=1}^{\mathrm{\infty }}Fix(W_i)\mathit{\text{ and }}Ax\in {\bigcap }_{i=1}^{N}Fix(T_i)\}\ne \mathrm{\varnothing }$. Take $\mu \in \mathbb{R}$ as follows:

$$0<\mu <\frac{2\overline{\gamma }}{L^2}.$$

Let $\{x_n\}\subset H$ be defined by

$$(3.5)\{\begin{array}{l}{x}_1\in C\mathit{\text{chosen arbitrarily}},\\ y_n=[I-b{A}^{\ast }(I-T_{n(modN)})A]x_n,\\ s_n=a_{n,0}{y}_n+{\sum }_{i=1}^{\mathrm{\infty }}{b}_{n,i}{W}_i{y}_n+{\sum }_{i=1}^{\mathrm{\infty }}{a}_{n,i}{J}_{r_i}^{G_i}(I-r_i{B}_i)y_n,\\ x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)({\beta }_n{\theta }_n+(1-{\beta }_{n}V)s_n)\end{array}$$

for each $n\in \mathbb{N}$, $\{{\alpha }_n,{\beta }_n\}\subset (0,1)$, $\{a_{n,i}:n,i\in \mathbb{N}\}\subset [0,1]$, $\{b_{n,i}:n,i\in \mathbb{N}\}\subset [0,1]$ and $\{r_i\}\subset (0,\mathrm{\infty })$. Assume the following:

• (i) $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\alpha }_n<1$;

• (ii) ${lim}_{n\to \mathrm{\infty }}{\beta }_n=0$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\beta }_n=\mathrm{\infty }$;

• (iii) $a_{n,0}+{\sum }_{i=1}^{\mathrm{\infty }}{b}_{n,i}+{\sum }_{i=1}^{\mathrm{\infty }}{a}_{n,i}=1$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{a}_{n,0}{a}_{n,i}>0$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{a}_{n,0}{b}_{n,i}>0$, $\mathrm{\forall }i\in \mathbb{N}$ and $0<b<\frac{1-\rho }{{\parallel A\parallel }^2}$;

• (iv) $0<r_i<2{\kappa }_i$ for all $i\in \mathbb{N}$ and ${lim}_{n\to \mathrm{\infty }}{\theta }_n=\theta$ for some $\theta \in H$.

Then ${lim}_{n\to \mathrm{\infty }}{x}_n=\overline{x}$, where $\overline{x}=P_{{\mathrm{\Pi }}_5}(\overline{x}-V\overline{x}+\theta )$. This point $\overline{x}$ is also a unique solution of the following hierarchical variational inequality:

$$〈V\overline{x}-\theta ,q-\overline{x}〉\ge 0,\mathrm{\forall }q\in {\mathrm{\Pi }}_5.$$

Proof By Theorem 4.1 in [32] and Lemma 2.1(iv), (v), we have that

$$J_{r_i}^{G_{2,i}}(I-r_i{B}_{2,i})\text{ is averaged}$$

for each $i\in \mathbb{N}$.

For each $i\in \mathbb{N}$, let $F_{2i}=W_i$ and let $F_{2i-1}=J_{r_i}^{G_i}(I-r_i{B}_i)$. Then $F_i$ is a quasi-nonexpansive mapping and

$$\bigcap _{i=1}^{\mathrm{\infty }}Fix(F_i)=\bigcap _{i=1}^{\mathrm{\infty }}Fix(J_{r_i}^{G_i}(I-r_i{B}_i))\cap \bigcap _{i=1}^{\mathrm{\infty }}Fix(W_i).$$

Thus, ${\mathrm{\Pi }}_5={\mathrm{\Pi }}_1$. By Theorem 3.1, we have that ${lim}_{n\to \mathrm{\infty }}{x}_n=\overline{x}$, where $\overline{x}=P_{{\mathrm{\Pi }}_5}(\overline{x}-V\overline{x}+\theta )$. This point $\overline{x}$ is also a unique solution of the following hierarchical variational inequality:

$$〈V\overline{x}-\theta ,q-\overline{x}〉\ge 0,\mathrm{\forall }q\in {\mathrm{\Pi }}_5,$$

and the proof is completed. □

Remark 3.3 Theorem 3.5 improves Theorem 10 in [33] in the sense that our convergence is for the fixed point of a countable family of quasi-nonexpansive mappings and for the split feasibility problem.

• (II) We study split common fixed points for a finite family of ${\rho }_i$-strictly pseudo-nonspreading mappings, zeros for a countable family of mappings which are the sum of two monotone mappings and common fixed points of a countable family of ${\delta }_i$-strictly pseudo-nonspreading mappings:

  $$\{\begin{array}{l}\text{Find }\overline{x}\in C\text{ such that}\\ \overline{x}\in {\bigcap }_{i=1}^{\mathrm{\infty }}{(G_i+B_i)}^{-1}0\cap {\bigcap }_{i=1}^{\mathrm{\infty }}Fix(S_i)\end{array}$$

and

$$\{\begin{array}{l}\overline{u}=A\overline{x}\in H_2\text{ such that}\\ \overline{u}\in {\bigcap }_{i=1}^{N}Fix(T_i).\end{array}$$

Theorem 3.6 Suppose that ${\mathrm{\Pi }}_6:=\{x\in C:x\in {\bigcap }_{i=1}^{\mathrm{\infty }}Fix(J_{r_i}^{G_i}(I-r_i{B}_i)){\bigcap }_{i=1}^{\mathrm{\infty }}Fix(S_i)\mathit{\text{ and }}Ax\in {\bigcap }_{i=1}^{N}Fix(T_i)\}\ne \mathrm{\varnothing }$. Take $\mu \in \mathbb{R}$ as follows:

$$0<\mu <\frac{2\overline{\gamma }}{L^2}.$$

Let $\{x_n\}\subset H$ be defined by

$$(3.6)\{\begin{array}{l}{x}_1\in C\mathit{\text{chosen arbitrarily}},\\ y_n=[I-b{A}^{\ast }(I-T_{n(modN)})A]x_n,\\ s_n=a_{n,0}{y}_n+{\sum }_{i=1}^{\mathrm{\infty }}{b}_{n,i}{S}_{i,\gamma }{y}_n+{\sum }_{i=1}^{\mathrm{\infty }}{a}_{n,i}{J}_{r_i}^{G_i}(I-r_i{B}_i)y_n,\\ x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)({\beta }_n{\theta }_n+(1-{\beta }_{n}V)s_n)\end{array}$$

for each $n\in \mathbb{N}$, $\{{\alpha }_n,{\beta }_n\}\subset (0,1)$, $\{a_{n,i}:n,i\in \mathbb{N}\}\subset [0,1]$, $\{b_{n,i}:n,i\in \mathbb{N}\}\subset [0,1]$ and $\{r_i\}\subset (0,\mathrm{\infty })$. Assume the following:

• (i) $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\alpha }_n<1$;

• (ii) ${lim}_{n\to \mathrm{\infty }}{\beta }_n=0$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\beta }_n=\mathrm{\infty }$;

• (iii) $a_{n,0}+{\sum }_{i=1}^{\mathrm{\infty }}{b}_{n,i}+{\sum }_{i=1}^{\mathrm{\infty }}{a}_{n,i}=1$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{a}_{n,0}{a}_{n,i}>0$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{a}_{n,0}{b}_{n,i}>0$, $\mathrm{\forall }i\in \mathbb{N}$ and $0<b<\frac{1-\rho }{{\parallel A\parallel }^2}$;

• (iv) $0<r_i<2{\kappa }_i$ for all $i\in \mathbb{N}$ and ${lim}_{n\to \mathrm{\infty }}{\theta }_n=\theta$ for some $\theta \in H$.

Then ${lim}_{n\to \mathrm{\infty }}{x}_n=\overline{x}$, where $\overline{x}=P_{{\mathrm{\Pi }}_4}(\overline{x}-V\overline{x}+\theta )$. This point $\overline{x}$ is also a unique solution of the following hierarchical variational inequality:

$$〈V\overline{x}-\theta ,q-\overline{x}〉\ge 0,\mathrm{\forall }q\in {\mathrm{\Pi }}_6.$$

Proof For each $i\in \mathbb{N}$, let $S_{i,\gamma }=\gamma S_i+(1-\gamma )I$. Then $S_{i,\gamma }$ is a quasi-nonexpansive mapping and $Fix{S}_i=Fix{S}_{i,\gamma }$ for each $i\in \mathbb{N}$. Then Theorem 3.6 follows immediately from Theorem 3.5. □

• (III) We study split common fixed points for a finite family of ${\rho }_i$-strictly pseudo-nonspreading mappings, zeros for a countable family of monotone mappings and common fixed points for a countable family of ${\delta }_i$-strictly pseudo-nonspreading mappings:

  $$\{\begin{array}{l}\text{Find }\overline{x}\in C\text{ such that}\\ \overline{x}\in {\bigcap }_{i=1}^{\mathrm{\infty }}{B}_i^{-1}0\cap {\bigcap }_{i=1}^{\mathrm{\infty }}Fix(S_i)\end{array}$$

and

$$\{\begin{array}{l}\overline{u}=A\overline{x}\in H_2,\overline{u_i}=A_i\overline{x}\text{ such that}\\ \overline{u}\in {\bigcap }_{i=1}^{N}Fix(T_i).\end{array}$$

Theorem 3.7 Suppose that ${\mathrm{\Pi }}_7:=\{x\in C:x\in {\bigcap }_{i=1}^{\mathrm{\infty }}{B}_i^{-1}0{\bigcap }_{i=1}^{\mathrm{\infty }}Fix(S_i)\mathit{\text{ and }}Ax\in {\bigcap }_{i=1}^{N}Fix(T_i)\}\ne \mathrm{\varnothing }$. Take $\mu \in \mathbb{R}$ as follows:

$$0<\mu <\frac{2\overline{\gamma }}{L^2}.$$

Let $\{x_n\}\subset H$ be defined by

$$(3.7)\{\begin{array}{l}{x}_1\in C\mathit{\text{chosen arbitrarily}},\\ y_n=[I-b{A}^{\ast }(I-T_{n(modN)})A]x_n,\\ s_n=a_{n,0}{y}_n+{\sum }_{i=1}^{\mathrm{\infty }}{b}_{n,i}{S}_{i,\gamma }{y}_n+{\sum }_{i=1}^{\mathrm{\infty }}{a}_{n,i}(I-r_i{B}_i)y_n,\\ x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)({\beta }_n{\theta }_n+(1-{\beta }_{n}V)s_n),\end{array}$$

where $S_{i,\gamma }=\gamma I+(1-\gamma )S_i$, $i\ge 1$, $\gamma \in [\delta ,1)$ for each $n\in \mathbb{N}$, $\{{\alpha }_n,{\beta }_n\}\subset (0,1)$, $\{a_{n,i}:n,i\in \mathbb{N}\}\subset [0,1]$, $\{b_{n,i}:n,i\in \mathbb{N}\}\subset [0,1]$ and $\{r_i\}\subset (0,\mathrm{\infty })$. Assume the following:

• (i) $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\alpha }_n<1$;

• (ii) ${lim}_{n\to \mathrm{\infty }}{\beta }_n=0$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\beta }_n=\mathrm{\infty }$;

• (iii) $a_{n,0}+{\sum }_{i=1}^{\mathrm{\infty }}{b}_{n,i}+{\sum }_{i=1}^{\mathrm{\infty }}{a}_{n,i}=1$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{a}_{n,0}{a}_{n,i}>0$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{a}_{n,0}{b}_{n,i}>0$, $\mathrm{\forall }i\in \mathbb{N}$ and $0<b<\frac{1-\rho }{{\parallel A\parallel }^2}$;

• (iv) $0<r_i<2{\kappa }_i$ for all $i\in \mathbb{N}$ and ${lim}_{n\to \mathrm{\infty }}{\theta }_n=\theta$ for some $\theta \in H$.

Then ${lim}_{n\to \mathrm{\infty }}{x}_n=\overline{x}$, where $\overline{x}=P_{{\mathrm{\Pi }}_4}(\overline{x}-V\overline{x}+\theta )$. This point $\overline{x}$ is also a unique solution of the following hierarchical variational inequality:

$$〈V\overline{x}-\theta ,q-\overline{x}〉\ge 0,\mathrm{\forall }q\in {\mathrm{\Pi }}_7.$$

Proof For each $i\in \mathbb{N}$, $S_i$ is a ${\delta }_i$-strictly pseudo-nonspreading mapping and $\gamma \in [\delta ,1)$, by Lemma 2.9, we have $S_{i,\gamma }$ is quasi-nonexpansive. By Lemma 2.1, we see that $(I-r_i{B}_i)$ is averaged for each $i\in \mathbb{N}$.

For each $i\in \mathbb{N}$, let $F_{2i}=S_{i,\gamma }$ and let $F_{2i-1}=(I-r_i{B}_i)$ in Theorem 3.1. Then $F_i$ is a quasi-nonexpansive mapping and

$$\bigcap _{i=1}^{\mathrm{\infty }}Fix(F_i)=\bigcap _{i=1}^{\mathrm{\infty }}Fix((I-r_i{B}_i))\cap \bigcap _{i=1}^{\mathrm{\infty }}Fix(S_{i,\gamma }).$$

Thus, ${\mathrm{\Pi }}_7={\mathrm{\Pi }}_1$. Then Theorem 3.7 follows immediately from Theorem 3.1. □

• (IV) Let $f_i:C\times C\to \mathbb{R}$ be a bifunction satisfying conditions (A1)-(A4). We study split common solutions for a countable family of equilibrium problems, common fixed points for a countable family of quasi-nonexpansive mappings and common fixed points of a finite family of ${\rho }_i$-strictly pseudo-nonspreading mappings.

That is,

$$\{\begin{array}{l}\text{Find }\overline{x}\in C\text{ such that}\\ \overline{x}\in {\bigcap }_{i=1}^{\mathrm{\infty }}EP(f_i)\cap {\bigcap }_{i=1}^{\mathrm{\infty }}Fix(W_i)\end{array}$$

for all $i\in \mathbb{N}$, and

$$\{\begin{array}{l}\overline{u}=A\overline{x}\in H_2\text{ such that}\\ \overline{u}\in {\bigcap }_{i=1}^{N}Fix(T_i).\end{array}$$

Theorem 3.8 For each $i\in \mathbb{N}$, let $A_{f_i}$ be as in Lemma  2.7. Suppose that ${\mathrm{\Pi }}_8:=\{x\in C:x\in {\bigcap }_{i=1}^{\mathrm{\infty }}EP(f_i)\cap {\bigcap }_{i=1}^{\mathrm{\infty }}Fix(W_i)\mathit{\text{ and }}Ax\in {\bigcap }_{i=1}^{N}Fix(T_i)\}\ne \mathrm{\varnothing }$. Take $\mu \in \mathbb{R}$ as follows:

$$0<\mu <\frac{2\overline{\gamma }}{L^2}.$$

Let $\{x_n\}\subset H$ be defined by

$$(3.8)\{\begin{array}{l}{x}_1\in C\mathit{\text{chosen arbitrarily}},\\ y_n=[I-b{A}^{\ast }(I-T_{n(modN)})A]x_n,\\ s_n=a_{n,0}{y}_n+{\sum }_{i=1}^{\mathrm{\infty }}{b}_{n,i}{W}_i{y}_n+{\sum }_{i=1}^{\mathrm{\infty }}{a}_{n,i}{J}_{r_i}^{A_{f_i}}{y}_n,\\ x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)({\beta }_n{\theta }_n+(1-{\beta }_{n}V)s_n)\end{array}$$

for each $n\in \mathbb{N}$, $\{{\alpha }_n,{\beta }_n\}\subset (0,1)$, $\{a_{n,i}:n,i\in \mathbb{N}\}\subset [0,1]$, $\{b_{n,i}:n,i\in \mathbb{N}\}\subset [0,1]$ and $\{r_i\}\subset (0,\mathrm{\infty })$. Assume the following:

• (i) $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\alpha }_n<1$;

• (ii) ${lim}_{n\to \mathrm{\infty }}{\beta }_n=0$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\beta }_n=\mathrm{\infty }$;

• (iii) $a_{n,0}+{\sum }_{i=1}^{\mathrm{\infty }}{b}_{n,i}+{\sum }_{i=1}^{\mathrm{\infty }}{a}_{n,i}=1$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{a}_{n,0}{a}_{n,i}>0$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{a}_{n,0}{b}_{n,i}>0$, $\mathrm{\forall }i\in \mathbb{N}$ and $0<b<\frac{1-\rho }{{\parallel A\parallel }^2}$;

• (iv) ${lim}_{n\to \mathrm{\infty }}{\theta }_n=\theta$ for some $\theta \in H$.

Then ${lim}_{n\to \mathrm{\infty }}{x}_n=\overline{x}$, where $\overline{x}=P_{{\mathrm{\Pi }}_8}(\overline{x}-V\overline{x}+\theta )$. This point $\overline{x}$ is also a unique solution of the following hierarchical variational inequality:

$$〈V\overline{x}-\theta ,q-\overline{x}〉\ge 0,\mathrm{\forall }q\in {\mathrm{\Pi }}_8.$$

Proof For each $i\in \mathbb{N}$, by Lemma 2.7, we know that $EP(f_i)=A_{f_i}^{-1}0$ and $A_{f_i}$ is a maximal monotone operator with the domain of $A_{f_i}\subset C$. For each $i\in \mathbb{N}$, by Lemma 2.3, we know that $J_{{\lambda }_i}^{A_{f_i}}$ is firmly nonexpansive, so $J_{{\lambda }_i}^{A_{f_i}}$ is quasi-nonexpansive. For each $i\in \mathbb{N}$, let $F_{2i}=W_i$, and let $F_{2i-1}=J_{{\lambda }_i}^{A_{f_i}}$ in Theorem 3.1. Then $F_i$ is a quasi-nonexpansive mapping and

$$\bigcap _{i=1}^{\mathrm{\infty }}Fix(F_i)=\bigcap _{i=1}^{\mathrm{\infty }}Fix\left(J_{{\lambda }_i}^{A_{f_i}}\right)\cap \bigcap _{i=1}^{\mathrm{\infty }}Fix(W_i).$$

${\mathrm{\Pi }}_9={\mathrm{\Pi }}_1$. By Theorem 3.1, we have that ${lim}_{n\to \mathrm{\infty }}{x}_n=\overline{x}$, where $\overline{x}=P_{{\mathrm{\Pi }}_8}(\overline{x}-V\overline{x}+\theta )$. This point $\overline{x}$ is also a unique solution of the following hierarchical variational inequality:

$$〈V\overline{x}-\theta ,q-\overline{x}〉\ge 0,\mathrm{\forall }q\in {\mathrm{\Pi }}_8.$$

The proof is completed. □

Remark 3.4 Theorem 3.8 improves Theorem 3.1 in [34] and Theorem 15 in [33] in the sense that our convergence is for the split feasibility problem and for the common solutions of a countable family of equilibrium problems and fixed points for a countable family of quasi-nonexpansive mappings.

• (V) Let $f_i:C\times C\to \mathbb{R}$ be a bifunction satisfying conditions (A1)-(A4). We study split common solutions for a countable family of equilibrium problems, common fixed points for a countable family of ${\delta }_i$-strictly pseudo-nonspreading mappings and common fixed points of a finite family of ${\rho }_i$-strictly pseudo-nonspreading mappings:

  $$\{\begin{array}{l}\text{Find }\overline{x}\in C\text{ such that}\\ \overline{x}\in {\bigcap }_{i=1}^{\mathrm{\infty }}EP(f_i)\cap {\bigcap }_{i=1}^{\mathrm{\infty }}Fix(S_i)\end{array}$$

for all $i\in \mathbb{N}$, and

$$\{\begin{array}{l}\overline{u}=A\overline{x}\in H_2\text{ such that}\\ \overline{u}\in {\bigcap }_{i=1}^{N}Fix(T_i).\end{array}$$

Theorem 3.9 For each $i\in \mathbb{N}$, let $A_{f_i}$ be as in Lemma  2.7, and suppose that ${\mathrm{\Pi }}_9:=\{x\in C:x\in {\bigcap }_{i=1}^{\mathrm{\infty }}EP(f_i)\cap {\bigcap }_{i=1}^{\mathrm{\infty }}Fix(S_i)\mathit{\text{ and }}Ax\in {\bigcap }_{i=1}^{N}Fix(T_i)\}\ne \mathrm{\varnothing }$. Take $\mu \in \mathbb{R}$ as follows:

$$0<\mu <\frac{2\overline{\gamma }}{L^2}.$$

Let $\{x_n\}\subset H$ be defined by

$$(3.9)\{\begin{array}{l}{x}_1\in C\mathit{\text{chosen arbitrarily}},\\ y_n=[I-b{A}^{\ast }(I-T_{n(modN)})A]x_n,\\ s_n=a_{n,0}{y}_n+{\sum }_{i=1}^{\mathrm{\infty }}{b}_{n,i}{S}_{i,\gamma }{y}_n+{\sum }_{i=1}^{\mathrm{\infty }}{a}_{n,i}{J}_{r_i}^{A_{f_i}}{y}_n,\\ x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)({\beta }_n{\theta }_n+(1-{\beta }_{n}V)s_n),\end{array}$$

where $S_{i,\gamma }=\gamma I+(1-\gamma )S_i$, $i\ge 1$, $\gamma \in [\delta ,1)$ for each $n\in \mathbb{N}$, $\{{\alpha }_n,{\beta }_n\}\subset (0,1)$, $\{a_{n,i}:n,i\in \mathbb{N}\}\subset [0,1]$, $\{b_{n,i}:n,i\in \mathbb{N}\}\subset [0,1]$ and $\{r_i\}\subset (0,\mathrm{\infty })$. Assume the following:

• (i) $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\alpha }_n<1$;

• (ii) ${lim}_{n\to \mathrm{\infty }}{\beta }_n=0$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\beta }_n=\mathrm{\infty }$;

• (iii) $a_{n,0}+{\sum }_{i=1}^{\mathrm{\infty }}{b}_{n,i}+{\sum }_{i=1}^{\mathrm{\infty }}{a}_{n,i}=1$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{a}_{n,0}{a}_{n,i}>0$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{a}_{n,0}{b}_{n,i}>0$, $\mathrm{\forall }i\in \mathbb{N}$ and $0<b<\frac{1-\rho }{{\parallel A\parallel }^2}$;

• (iv) ${lim}_{n\to \mathrm{\infty }}{\theta }_n=\theta$ for some $\theta \in H$.

Then ${lim}_{n\to \mathrm{\infty }}{x}_n=\overline{x}$, where $\overline{x}=P_{{\mathrm{\Pi }}_9}(\overline{x}-V\overline{x}+\theta )$. This point $\overline{x}$ is also a unique solution of the following hierarchical variational inequality:

$$〈V\overline{x}-\theta ,q-\overline{x}〉\ge 0,\mathrm{\forall }q\in {\mathrm{\Pi }}_9.$$

Proof For each $i\in \mathbb{N}$, let $F_{2i}=S_{i,\gamma }$ and $F_{2i-1}=J_{{\lambda }_i}^{A_{f_i}}$. Then $F_i$ is a quasi-nonexpansive mapping and

$$\bigcap _{i=1}^{\mathrm{\infty }}Fix(F_i)=\bigcap _{i=1}^{\mathrm{\infty }}Fix\left(J_{{\lambda }_i}^{A_{f_i}}\right)\cap \bigcap _{i=1}^{\mathrm{\infty }}Fix(S_i).$$

${\mathrm{\Pi }}_9={\mathrm{\Pi }}_1$. By Theorem 3.1, we have that ${lim}_{n\to \mathrm{\infty }}{x}_n=\overline{x}$, where $\overline{x}=P_{{\mathrm{\Pi }}_9}(\overline{x}-V\overline{x}+\theta )$. This point $\overline{x}$ is also a unique solution of the following hierarchical variational inequality:

$$〈V\overline{x}-\theta ,q-\overline{x}〉\ge 0,\mathrm{\forall }q\in {\mathrm{\Pi }}_9,$$

and the proof is completed. □

• (VI) For each $i\in \mathbb{N}$, let $M_i:C\to H_1$ be a hemicontinuous monotone mapping. We study split common solutions for a countable family of variational inequality problems, common fixed points of a countable family of quasi-nonexpansive mappings and common fixed points of a finite family of ${\rho }_i$-strictly pseudo-nonspreading mappings:

  $$\{\begin{array}{l}\text{Find }\overline{x}\in C\text{ such that}\\ \overline{x}\in {\bigcap }_{i=1}^{\mathrm{\infty }}VI(C,M_i)\cap {\bigcap }_{i=1}^{\mathrm{\infty }}Fix(W_i),\end{array}$$

where

$$VI(C,M_i)=\{z\in C:〈y-z,M_{i}z〉\ge 0\text{ for all }y\in C\}$$

and

$$\{\begin{array}{l}\overline{u}=A\overline{x}\in H_2\text{ such that}\\ \overline{u}\in {\bigcap }_{i=1}^{N}Fix(T_i).\end{array}$$

For each $i\in \mathbb{N}$, let $f_i(x,y)=〈y-x,M_{i}x〉$ for all $x\in C$ in Theorem 3.9, we have the following theorem.

Theorem 3.10 For each $i\in \mathbb{N}$, let $A_{f_i}$ be as in Lemma  2.7 and suppose that $M_i$ is bounded on any line segment of C; that is, for each $x,y\in C$, there exists ${\kappa }_i(x,y)>0$ such that $M_i(tx+(1-t)y)\le {\kappa }_i(x,y)$ for all $t\in [0,1]$. Suppose that ${\mathrm{\Pi }}_{10}:=\{x\in C:x\in {\bigcap }_{i=1}^{\mathrm{\infty }}VI(C,M_i)\cap {\bigcap }_{i=1}^{\mathrm{\infty }}Fix(W_i)\mathit{\text{ and }}Ax\in {\bigcap }_{i=1}^{N}Fix(T_i)\}\ne \mathrm{\varnothing }$. Take $\mu \in \mathbb{R}$ as follows:

$$0<\mu <\frac{2\overline{\gamma }}{L^2}.$$

Let $\{x_n\}\subset H$ be defined by

$$(3.10)\{\begin{array}{l}{x}_1\in C\mathit{\text{chosen arbitrarily}},\\ y_n=[I-b{A}^{\ast }(I-T_{n(modN)})A]x_n,\\ s_n=a_{n,0}{y}_n+{\sum }_{i=1}^{\mathrm{\infty }}{b}_{n,i}{W}_i{y}_n+{\sum }_{i=1}^{\mathrm{\infty }}{a}_{n,i}{J}_{r_i}^{A_{f_i}},\\ x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)({\beta }_n{\theta }_n+(1-{\beta }_{n}V)s_n),\end{array}$$

where $f_i(x,y)=〈y-x,M_{i}x〉$ for all $x,y\in C$, for each $n\in \mathbb{N}$, $\{{\alpha }_n,{\beta }_n\}\subset (0,1)$, $\{a_{n,i}:n,i\in \mathbb{N}\}\subset [0,1]$, $\{b_{n,i}:n,i\in \mathbb{N}\}\subset [0,1]$, and $\{r_i\}\subset (0,\mathrm{\infty })$. Assume the following:

• (i) $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\alpha }_n<1$;

• (ii) ${lim}_{n\to \mathrm{\infty }}{\beta }_n=0$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\beta }_n=\mathrm{\infty }$;

• (iii) $a_{n,0}+{\sum }_{i=1}^{\mathrm{\infty }}{b}_{n,i}+{\sum }_{i=1}^{\mathrm{\infty }}{a}_{n,i}=1$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{a}_{n,0}{a}_{n,i}>0$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{a}_{n,0}{b}_{n,i}>0$, $\mathrm{\forall }i\in \mathbb{N}$ and $0<b<\frac{1-\rho }{{\parallel A\parallel }^2}$;

• (iv) ${lim}_{n\to \mathrm{\infty }}{\theta }_n=\theta$ for some $\theta \in H$.

Then ${lim}_{n\to \mathrm{\infty }}{x}_n=\overline{x}$, where $\overline{x}=P_{{\mathrm{\Pi }}_{10}}(\overline{x}-V\overline{x}+\theta )$. This point $\overline{x}$ is also a unique solution of the following hierarchical variational inequality:

$$〈V\overline{x}-\theta ,q-\overline{x}〉\ge 0,\mathrm{\forall }q\in {\mathrm{\Pi }}_{10}.$$

Proof Since $M_i:C\to H$ is a hemicontinuous monotone mapping, it is sufficient to show that for each $i\in \mathbb{N}$, $f_i$ satisfies conditions (A1)-(A4). Since $M_i$ is monotone, we have

$$〈y-x,M_{i}y-M_{i}x〉\ge 0$$

and

$$f_i(x,y)+f_i(y,x)=〈y-x,M_{i}x-M_{i}y〉\le 0.$$

Therefore (A2) is satisfied. Since $M_i:C\to H$ is a hemicontinuous mapping, for any $x,y\in H_1$, the mapping $g:[0,1]\to H_1$ defined by $g(t)=M_i(tx+(1-t)y)$ is continuous, where $H_1$ has a weak topology. Since $M_i$ is bounded on any line segment of C and each $x,y,z\in H_1$,

$$\begin{array}{rcl}g(tz+(1-t)x,y)& =& 〈y-tz-(1-t)x,M_i(tz+(1-t)x)〉\\ =& 〈y-tz-(1-t)x-(y-x),M_i(tz+(1-t)x)〉\\ +〈(y-x),M_i(tz+(1-t)x)〉,\end{array}$$

it is easy to see that ${lim}_{t\downarrow 0}{f}_i(tz+(1-t)x,y)={lim}_{t\downarrow 0}〈y-tz-(1-t)x,M_i(tz+(1-t)x)〉=〈y-x,M_i(x)〉=f_i(x,y)$. This shows that condition (A3) is satisfied. It is easy to see that (A1) and (A4) are satisfied. Then Theorem 3.10 follows immediately from Theorem 3.8. □

Remark 3.5 (i) It is easy to see that if $M_i$ is continuous on C, then $M_i$ is hemicontinuous and bounded on any line segment of C.

(ii) It is easy to see that Theorem 3.10 is true if for each $i\in \mathbb{N}$, $M_i:C\to H$ is a ${\kappa }_i$-inverse strongly monotone mapping or is a continuous strongly monotone mapping. Theorem 3.10 improves Theorem 3.1 of Qing and Shang [35] in the sense that our convergence is for the split feasibility problem and for the common solutions of a countable family of variational inequality problems of hemicontinuous monotone mappings and fixed points for a countable family of quasi-nonexpansive mappings.

(VII) For each $i\in \mathbb{N}$, let $M_i:C\to H$ be a hemicontinuous monotone mapping. We study split common solutions for a countable family of variational inequality problems, common fixed points for a countable family of ${\delta }_i$-strictly pseudo-nonspreading mappings and common fixed points of a finite family of ${\rho }_i$-strictly pseudo-nonspreading mappings:

$$\{\begin{array}{l}\text{Find }\overline{x}\in C\text{ such that}\\ \overline{x}\in {\bigcap }_{i=1}^{\mathrm{\infty }}VI(C,M_i)\cap {\bigcap }_{i=1}^{\mathrm{\infty }}Fix(S_i)\end{array}$$

and

$$\{\begin{array}{l}\overline{u}=A\overline{x}\in H_2\text{ such that}\\ \overline{u}\in {\bigcap }_{i=1}^{N}Fix(T_i).\end{array}$$

Applying Theorem 3.9 and following the same argument as in Theorem 3.10, we have the following result.

Theorem 3.11 For each $i\in \mathbb{N}$, let $A_{f_i}$ be as in Lemma  2.7 and suppose that $M_i$ is bounded on any line segment of C. Suppose that ${\mathrm{\Pi }}_{11}:=\{x\in C:x\in {\bigcap }_{i=1}^{\mathrm{\infty }}VI(C,M_i)\cap {\bigcap }_{i=1}^{\mathrm{\infty }}Fix{S}_i\mathit{\text{ and }}Ax\in {\bigcap }_{i=1}^{N}Fix(T_i)\}\ne \mathrm{\varnothing }$. Take $\mu \in \mathbb{R}$ as follows:

$$0<\mu <\frac{2\overline{\gamma }}{L^2}.$$

Let $\{x_n\}\subset H$ be defined by

$$(3.11)\{\begin{array}{l}{x}_1\in C\mathit{\text{chosen arbitrarily}},\\ y_n=[I-b{A}^{\ast }(I-T_{n(modN)})A]x_n,\\ s_n=a_{n,0}{y}_n+{\sum }_{i=1}^{\mathrm{\infty }}{b}_{n,i}{S}_{i,\gamma }{y}_n+{\sum }_{i=1}^{\mathrm{\infty }}{a}_{n,i}{J}_{r_i}^{A_{f_i}}{y}_n,\\ x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)({\beta }_n{\theta }_n+(1-{\beta }_{n}V)s_n),\end{array}$$

where $S_{i,\gamma }=\gamma I+(1-\gamma )S_i$, $\gamma \in [\delta ,1)$, $f_i(x,y)=〈y-x,M_{i}x〉$ for all $x,y\in C$, $i\in \mathbb{N}$, for each $n\in \mathbb{N}$, $\{{\alpha }_n,{\beta }_n\}\subset (0,1)$, $\{a_{n,i}:n,i\in \mathbb{N}\}\subset [0,1]$, $\{b_{n,i}:n,i\in \mathbb{N}\}\subset [0,1]$ and $\{r_i\}\subset (0,\mathrm{\infty })$. Assume the following:

• (i) $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\alpha }_n<1$;

• (ii) ${lim}_{n\to \mathrm{\infty }}{\beta }_n=0$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\beta }_n=\mathrm{\infty }$;

• (iii) $a_{n,0}+{\sum }_{i=1}^{\mathrm{\infty }}{b}_{n,i}+{\sum }_{i=1}^{\mathrm{\infty }}{a}_{n,i}=1$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{a}_{n,0}{a}_{n,i}>0$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{a}_{n,0}{b}_{n,i}>0$, $\mathrm{\forall }i\in \mathbb{N}$ and $0<b<\frac{1-\rho }{{\parallel A\parallel }^2}$;

• (iv) ${lim}_{n\to \mathrm{\infty }}{\theta }_n=\theta$ for some $\theta \in H$.

Then ${lim}_{n\to \mathrm{\infty }}{x}_n=\overline{x}$, where $\overline{x}=P_{{\mathrm{\Pi }}_{11}}(\overline{x}-V\overline{x}+\theta )$. This point $\overline{x}$ is also a unique solution of the following hierarchical variational inequality:

$$〈V\overline{x}-\theta ,q-\overline{x}〉\ge 0,\mathrm{\forall }q\in {\mathrm{\Pi }}_{11}.$$

• (VIII) For each $i\in \mathbb{N}$, let ${\mathrm{\Psi }}_i:C\to C$ be a hemicontinuous pseudo-contractive mapping. We study split common fixed points for a countable family of hemicontinuous pseudo-contractive mappings, a countable family of quasi-nonexpansive mappings and a finite family of ${\rho }_i$-strictly pseudo-nonspreading mappings:

  $$\{\begin{array}{l}\text{Find }\overline{x}\in C\text{ such that}\\ \overline{x}\in {\bigcap }_{i=1}^{\mathrm{\infty }}Fix({\mathrm{\Psi }}_i)\cap {\bigcap }_{i=1}^{\mathrm{\infty }}Fix(W_i)\end{array}$$

and

$$\{\begin{array}{l}\overline{u}=A\overline{x}\in H_2\text{ such that}\\ \overline{u}\in {\bigcap }_{i=1}^{N}Fix(T_i).\end{array}$$

For $i\in \mathbb{N}$, let $f_i(x,y)=〈y-x,(I-{\mathrm{\Psi }}_i)x〉$ for all $x,y\in C$ in Theorem 3.11, we have the following theorem.

Theorem 3.12 For each $i\in \mathbb{N}$, let $A_{f_i}$ be as in Lemma  2.7 and suppose that ${\mathrm{\Psi }}_i$ is bounded on any line segment of C. Suppose that ${\mathrm{\Pi }}_{12}:=\{x\in C:x\in {\bigcap }_{i=1}^{\mathrm{\infty }}Fix({\mathrm{\Psi }}_i){\bigcap }_{i=1}^{\mathrm{\infty }}Fix(W_i)\mathit{\text{ and }}Ax\in {\bigcap }_{i=1}^{N}Fix(T_i)\}\ne \mathrm{\varnothing }$. Take $\mu \in \mathbb{R}$ as follows:

$$0<\mu <\frac{2\overline{\gamma }}{L^2}.$$

Let $\{x_n\}\subset H$ be defined by

$$(3.12)\{\begin{array}{l}{x}_1\in C\mathit{\text{chosen arbitrarily}},\\ y_n=[I-b{A}^{\ast }(I-T_{n(modN)})A]x_n,\\ s_n=a_{n,0}{y}_n+{\sum }_{i=1}^{\mathrm{\infty }}{b}_{n,i}{W}_i{y}_n+{\sum }_{i=1}^{\mathrm{\infty }}{a}_{n,i}{J}_{r_i}^{A_{f_i}},\\ x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)({\beta }_n{\theta }_n+(1-{\beta }_{n}V)s_n),\end{array}$$

where $f_i(x,y)=〈y-x,x-{\mathrm{\Psi }}_{i}x〉$ for all $x,y\in C$, for each $n\in \mathbb{N}$, $\{{\alpha }_n,{\beta }_n\}\subset (0,1)$, $\{a_{n,i}:n,i\in \mathbb{N}\}\subset [0,1]$, $\{b_{n,i}:n,i\in \mathbb{N}\}\subset [0,1]$ and $\{r_i\}\subset (0,\mathrm{\infty })$. Assume the following:

• (i) $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\alpha }_n<1$;

• (ii) ${lim}_{n\to \mathrm{\infty }}{\beta }_n=0$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\beta }_n=\mathrm{\infty }$;

• (iii) $a_{n,0}+{\sum }_{i=1}^{\mathrm{\infty }}{b}_{n,i}+{\sum }_{i=1}^{\mathrm{\infty }}{a}_{n,i}=1$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{a}_{n,0}{a}_{n,i}>0$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{a}_{n,0}{b}_{n,i}>0$, $\mathrm{\forall }i\in \mathbb{N}$ and $0<b<\frac{1-\rho }{{\parallel A\parallel }^2}$;

• (iv) ${lim}_{n\to \mathrm{\infty }}{\theta }_n=\theta$ for some $\theta \in H$.

Then ${lim}_{n\to \mathrm{\infty }}{x}_n=\overline{x}$, where $\overline{x}=P_{{\mathrm{\Pi }}_{12}}(\overline{x}-V\overline{x}+\theta )$. This point $\overline{x}$ is also a unique solution of the following hierarchical variational inequality:

$$〈V\overline{x}-\theta ,q-\overline{x}〉\ge 0,\mathrm{\forall }q\in {\mathrm{\Pi }}_{12}.$$

Proof Let $M_i=I-{\mathrm{\Psi }}_i$. Since, for each $i\in \mathbb{N}$, ${\mathrm{\Psi }}_i$ is a pseudo-contraction, it is easy to see that $M_i$ is monotone. Then Theorem 3.12 follows from Theorem 3.10. □

Remark 3.6 Theorem 3.12 improves Theorem 3.1 of Cheng et al. in [13], Theorem 3.1 of Deng [14], Theorem 3.1 of Zegeye and Shahzad in [16] in the sense that our convergence is for the split feasibility problem and for the common fixed points of a countable family of quasi-nonexpansive mappings and a countable family hemicontinuous pseudo-contractive mappings without any closedness or Lipschitz continuity assumption on these pseudo-contractive mappings.

• (IX) For each $i\in \mathbb{N}$, let ${\mathrm{\Psi }}_i:C\to C$ be a hemicontinuous pseudo-contractive mapping. We study split common fixed points for a countable family of hemicontinuous pseudo-contractive mappings, a countable family of ${\delta }_i$-strictly pseudo-nonspreading mappings and a finite family of ${\rho }_i$-strictly pseudo-nonspreading mappings:

  $$\{\begin{array}{l}\text{Find }\overline{x}\in C\text{ such that}\\ \overline{x}\in {\bigcap }_{i=1}^{\mathrm{\infty }}Fix({\mathrm{\Psi }}_i)\cap {\bigcap }_{i=1}^{\mathrm{\infty }}Fix(S_i)\end{array}$$

and

$$\{\begin{array}{l}\overline{u}=A\overline{x}\in H_2\text{ such that}\\ \overline{u}\in {\bigcap }_{i=1}^{N}Fix(T_i).\end{array}$$

For each $i\in \mathbb{N}$, let $W_i=S_i$ in Theorem 3.12, we have the following theorem.

Theorem 3.13 Suppose that for each $i\in \mathbb{N}$, $M_i$ is bounded on any line segment of C and ${\mathrm{\Pi }}_{13}:=\{x\in C:x\in {\bigcap }_{i=1}^{\mathrm{\infty }}Fix({\mathrm{\Psi }}_i){\bigcap }_{i=1}^{\mathrm{\infty }}Fix(S_i)\mathit{\text{ and }}Ax\in {\bigcap }_{i=1}^{N}Fix(T_i)\}\ne \mathrm{\varnothing }$. Take $\mu \in \mathbb{R}$ as follows:

$$0<\mu <\frac{2\overline{\gamma }}{L^2}.$$

Let $\{x_n\}\subset H$ be defined by

$$(3.13)\{\begin{array}{l}{x}_1\in C\mathit{\text{chosen arbitrarily}},\\ y_n=[I-b{A}^{\ast }(I-T_{n(modN)})A]x_n,\\ s_n=a_{n,0}{y}_n+{\sum }_{i=1}^{\mathrm{\infty }}{b}_{n,i}{S}_{i,\gamma }{y}_n+{\sum }_{i=1}^{\mathrm{\infty }}{a}_{n,i}{J}_{r_i}^{A_{f_i}},\\ x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)({\beta }_n{\theta }_n+(1-{\beta }_{n}V)s_n),\end{array}$$

where $S_{i,\gamma }=\gamma I+(1-\gamma )S_i$, $\gamma \in [\delta ,1)$, $f_i(x,y)=〈y-x,x-{\mathrm{\Psi }}_{i}x〉$ for all $x,y\in C$, for each $n\in \mathbb{N}$, $\{{\alpha }_n,{\beta }_n\}\subset (0,1)$, $\{a_{n,i}:n,i\in \mathbb{N}\}\subset [0,1]$, $\{b_{n,i}:n,i\in \mathbb{N}\}\subset [0,1]$ and $\{r_i\}\subset (0,\mathrm{\infty })$. Assume the following:

• (i) $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\alpha }_n<1$;

• (ii) ${lim}_{n\to \mathrm{\infty }}{\beta }_n=0$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\beta }_n=\mathrm{\infty }$;

• (iii) $a_{n,0}+{\sum }_{i=1}^{\mathrm{\infty }}{b}_{n,i}+{\sum }_{i=1}^{\mathrm{\infty }}{a}_{n,i}=1$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{a}_{n,0}{a}_{n,i}>0$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{a}_{n,0}{b}_{n,i}>0$, $\mathrm{\forall }i\in \mathbb{N}$ and $0<b<\frac{1-\rho }{{\parallel A\parallel }^2}$;

• (iv) ${lim}_{n\to \mathrm{\infty }}{\theta }_n=\theta$ for some $\theta \in H$.

Then ${lim}_{n\to \mathrm{\infty }}{x}_n=\overline{x}$, where $\overline{x}=P_{{\mathrm{\Pi }}_{13}}(\overline{x}-V\overline{x}+\theta )$. This point $\overline{x}$ is also a unique solution of the following hierarchical variational inequality:

$$〈V\overline{x}-\theta ,q-\overline{x}〉\ge 0,\mathrm{\forall }q\in {\mathrm{\Pi }}_{13}.$$

• (X) For each $i\in \mathbb{N}$, let ${\mathrm{\Psi }}_i:C\to C$ be a hemicontinuous pseudo-contractive mapping. We study a split feasibility problem for common fixed points of a countable family of hemicontinuous pseudo-contractive mappings, zeros for a countable family of mappings which are the sum of two monotone mappings and common fixed points of a finite family of ${\rho }_i$-strictly pseudo-nonspreading mappings.

That is,

$$\{\begin{array}{l}\text{Find }\overline{x}\in C\text{ such that}\\ \overline{x}\in {\bigcap }_{i=1}^{\mathrm{\infty }}{(G_i+B_i)}^{-1}0\cap {\bigcap }_{i=1}^{\mathrm{\infty }}Fix({\mathrm{\Psi }}_i)\end{array}$$

and

$$\{\begin{array}{l}\overline{u}=A\overline{x}\in H_2\text{ such that}\\ \overline{u}\in {\bigcap }_{i=1}^{N}Fix(T_i).\end{array}$$

For each $i\in \mathbb{N}$, let $W_i=J_{r_i}^{G_i}(I-r_i{B}_i)$ in Theorem 3.12, we have the following theorem.

Theorem 3.14 Suppose that for each $i\in \mathbb{N}$, ${\mathrm{\Psi }}_i$ is bounded on any line segment of C and ${\mathrm{\Pi }}_{14}:=\{x\in C:x\in {\bigcap }_{i=1}^{\mathrm{\infty }}Fix({\mathrm{\Psi }}_i){\bigcap }_{i=1}^{\mathrm{\infty }}Fix(J_{r_i}^{G_i}(I-r_i{B}_i))\mathit{\text{ and }}Ax\in {\bigcap }_{i=1}^{N}Fix(T_i)\}\ne \mathrm{\varnothing }$. Take $\mu \in \mathbb{R}$ as follows:

$$0<\mu <\frac{2\overline{\gamma }}{L^2}.$$

Let $\{x_n\}\subset H$ be defined by

$$(3.14)\{\begin{array}{l}{x}_1\in C\mathit{\text{chosen arbitrarily}},\\ y_n=[I-b{A}^{\ast }(I-T_{n(modN)})A]x_n,\\ s_n=a_{n,0}{y}_n+{\sum }_{i=1}^{\mathrm{\infty }}{b}_{n,i}{J}_{{\lambda }_i}^{A_{f_i}}{y}_n+{\sum }_{i=1}^{\mathrm{\infty }}{a}_{n,i}{J}_{r_i}^{G_i}(I-r_i{B}_i)y_n,\\ x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)({\beta }_n{\theta }_n+(1-{\beta }_{n}V)s_n),\end{array}$$

where $f_i(x,y)=〈y-x,(I-{\mathrm{\Psi }}_i)x〉$ for all $x,y\in C$, for each $n\in \mathbb{N}$, $\{{\alpha }_n,{\beta }_n\}\subset (0,1)$, $\{a_{n,i}:n,i\in \mathbb{N}\}\subset [0,1]$, $\{b_{n,i}:n,i\in \mathbb{N}\}\subset [0,1]$ and $\{{\lambda }_i,r_i\}\subset (0,\mathrm{\infty })$. Assume the following:

• (i) $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\alpha }_n<1$;

• (ii) ${lim}_{n\to \mathrm{\infty }}{\beta }_n=0$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\beta }_n=\mathrm{\infty }$;

• (iii) $a_{n,0}+{\sum }_{i=1}^{\mathrm{\infty }}{b}_{n,i}+{\sum }_{i=1}^{\mathrm{\infty }}{a}_{n,i}=1$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{a}_{n,0}{a}_{n,i}>0$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{a}_{n,0}{b}_{n,i}>0$, $\mathrm{\forall }i\in \mathbb{N}$ and $0<b<\frac{1-\rho }{{\parallel A\parallel }^2}$;

• (iv) $0<r_i<2{\kappa }_i$ for all $i\in \mathbb{N}$ and ${lim}_{n\to \mathrm{\infty }}{\theta }_n=\theta$ for some $\theta \in H$.

Then ${lim}_{n\to \mathrm{\infty }}{x}_n=\overline{x}$, where $\overline{x}=P_{{\mathrm{\Pi }}_{14}}(\overline{x}-V\overline{x}+\theta )$. This point $\overline{x}$ is also a unique solution of the following hierarchical variational inequality:

$$〈V\overline{x}-\theta ,q-\overline{x}〉\ge 0,\mathrm{\forall }q\in {\mathrm{\Pi }}_{14}.$$

Remark 3.7 Theorem 3.14 improves Theorem 3.1 of Zegeye and Shahzad [10], Theorem 3.1 of Yao et al. [11], Theorem 4.1 of Takahashi et al. [28], Theorem 2.1 of Cho et al. [36] and Theorem 3.1 of Tang [9] in the sense that our convergence is for the split feasibility problem and for the common solutions of a countable family of variational inclusion problems and fixed points of a countable family hemicontinuous pseudo-contractive mappings without any closedness or Lipschitz continuity assumption on these pseudo-contractive mappings.