Weak- and strong-convergence theorems of solutions to split feasibility problem for nonspreading type mapping in Hilbert spaces

Abstract

The purpose of this article is to study the weak- and strong-convergence theorems of solutions to split a feasibility problem for a family of nonspreading-type mapping in Hilbert spaces. The main result presented in this paper improves and extends some recent results of Censor et al., Byrne, Yang, Moudafi, Xu, Censor and Segal, Masad and Reich, and others. As an application, we solve the hierarchical variational inequality problem by using the main theorem.

MSC:47J05, 47H09, 49J25.

1 Introduction

Throughout this paper, we assume that H is a real Hilbert space, D is a nonempty and closed convex subset of H. In the sequel, we denote by ‘$x_n\to x$’ and ‘$x_n\rightharpoonup x$’ the strong and weak convergence of $\{x_n\}$, respectively. Denote by the set of all positive integers and by $F(T)$ the set of fixed points of a mapping $T:D\to D$.

Definition 1.1 Let $T:D\to D$ be a mapping.

• (1) $T:D\to D$ is said to be nonexpansive if

  $$\parallel Tx-Ty\parallel \le \parallel x-y\parallel ,\mathrm{\forall }x,y\in D.$$

• (2) T is said to be quasi-nonexpansive if $F(T)$ is nonempty and

  $$\parallel Tx-p\parallel \le \parallel x-p\parallel ,\mathrm{\forall }x\in D,p\in F(T).$$

  (1.1)

• (3) T is said to be nonspreading if

  $$2{\parallel Tx-Ty\parallel }^2\le {\parallel Tx-y\parallel }^2+{\parallel Ty-x\parallel }^2,\mathrm{\forall }x,y\in D.$$

  (1.2)

It is easy to prove that equation (1.2) is equivalent to

$${\parallel Tx-Ty\parallel }^2\le {\parallel x-y\parallel }^2+2〈x-Tx,y-Ty〉,\mathrm{\forall }x,y\in D.$$

(1.3)

• (4) T is said to be k-strictly pseudo-nonspreading [1], if there exists a constant $k\in [0,1)$ such that

  $${\parallel Tx-Ty\parallel }^2\le {\parallel x-y\parallel }^2+k{\parallel x-Tx-(y-Ty)\parallel }^2+2〈x-Tx,y-Ty〉,\mathrm{\forall }x,y\in D.$$

  (1.4)

Remark 1.2 It follows from Definition 1.1 that

• (1) if T is nonspreading and $F(T)\ne \mathrm{\varnothing }$, then T is quasi-nonexpansive;

• (2) if T is nonspreading, then it is k-strictly pseudo-nonspreading with $k=0$. But the converse is not true from the following example. Thus, we know that the class of k-strictly pseudo-nonspreading mappings is more general than the class of nonspreading mappings.

Example 1.3 [1]

Let ℛ denote the set of real numbers with the usual norm. Let $T:\mathcal{R}\to \mathcal{R}$ be a mapping defined by

$$Tx=\{\begin{array}{ll}x,& x\in (-\mathrm{\infty },0),\\ -2x,& x\in [0,\mathrm{\infty }).\end{array}$$

(1.5)

Then T is a k-strictly pseudo-nonspreading mapping, but it is not nonspreading.

In 2010, Kurokawa and Takahashi [2] obtained a weak mean ergodic theorem of Baillon’s type [3] for nonspreading mappings in Hilbert spaces. They further proved a strong-convergence theorem somewhat related to Halpern’s type [4] for this class of mappings using the idea of mean convergence in Hilbert spaces.

In 2011, Osilike and Isiogugu [1] first introduced the concept of k-strictly pseudo-nonspreading mappings and proved a weak mean convergence theorem of Baillon’s type similar to the ones obtained in [2]. Furthermore, using the idea of mean convergence, a strong-convergence theorem similar to the one obtained in [2] is proved which extends and improves the main theorems of [2] and an affirmative answer given to an open problem posed by Kurokawa and Takahashi [2] for the case where the mapping T is averaged.

On the other hand, the split feasibility problem (SFP) in finitely dimensional spaces was first introduced by Censor and Elfving [5] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction [6]. Recently, it has been found that the (SFP) can also be used in various disciplines such as image restoration, computer tomography and radiation therapy treatment planning [7–9].

The split feasibility problem in an infinitely dimensional Hilbert space can be found in [6, 8, 10–12].

The purpose of this paper is to introduce the following multiple-set split feasibility problem (MSSFP) for an infinite family of k-strictly pseudo-nonspreading mappings and a finite family of ρ-strictly pseudo-nonspreading mappings in infinitely dimensional Hilbert spaces, i.e., to find $x^{\ast }\in C$ such that

$$A{x}^{\ast }\in Q,$$

(1.6)

where $H_1$, $H_2$ are two real Hilbert spaces, $A:H_1\to H_2$ is a bounded linear operator, ${\{S_i\}}_{i=1}^{\mathrm{\infty }}:H_1\to H_1$ is an infinite family of $k_i$-strictly pseudo-nonspreading mappings and ${\{T_i\}}_{i=1}^N:H_2\to H_2$ is a finite family of ${\rho }_i$-strictly pseudo-nonspreading mappings, $C:={\bigcap }_{i=1}^{\mathrm{\infty }}F(S_i)$ and $Q:={\bigcap }_{i=1}^{N}F(T_i)$. Also we wish to study the weak and strong convergence of the following iterative sequence to a solution of problem (1.6):

$$\{\begin{array}{l}{x}_1\in H_1\text{chosen arbitrarily},\\ x_{n+1}={\alpha }_{0,n}{y}_n+{\sum }_{i=1}^{\mathrm{\infty }}{\alpha }_{i,n}{S}_{i,\beta }{y}_n,\\ y_n=x_n+\gamma A^{\ast }(T_{n(modN)}-I)A{x}_n,\mathrm{\forall }n\ge 1,\end{array}$$

where $S_{i,\beta }:=\beta I+(1-\beta )S_i$, $\beta \in (0,1)$ is a constant.

In the sequel we denote Γ the set of solutions of (MSSFP) equation (1.6), i.e.,

$$\mathrm{\Gamma }=\{x\in C,Ax\in Q\}=C\cap A^{-1}(Q).$$

(1.7)

2 Preliminaries

For this purpose, we first recall some definitions, notations and conclusions which will be needed in proving our main results.

Definition 2.1 Let E be a real Banach space, and $T:E\to E$ be a mapping.

• (1) $I-T$ is said to be demiclosed at 0, if, for any sequence $\{x_n\}\subset H$ with $x_n\rightharpoonup x^{\ast }$, $\parallel (I-T)x_n\parallel \to 0$, then $x^{\ast }=T{x}^{\ast }$.

• (2) T is said to be semicompact, if, for any bounded sequence $\{x_n\}\subset E$, ${lim}_{n\to \mathrm{\infty }}\parallel x_n-T{x}_n\parallel =0$, then there exists a subsequence $\{x_{n_i}\}\subset \{x_n\}$ such that $\{x_{n_i}\}$ converges strongly to some point $x^{\ast }\in E$.

Lemma 2.2 [1]

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

• (1) If $F(T)\ne \mathrm{\varnothing }$, then $F(T)$ is closed and convex;

• (2) $I-T$ is demiclosed at zero.

Lemma 2.3 Let H be a real Hilbert space. Then the following statements hold:

• (1) For all $x,y\in H$ and for all $t\in [0,1]$,

  $${\parallel tx+(1-t)y\parallel }^2=t{\parallel x\parallel }^2+(1-t){\parallel y\parallel }^2-t(1-t){\parallel x-y\parallel }^2.$$

  (2.1)

• (2) For all $x,y\in H$,

  $${\parallel x+y\parallel }^2\le {\parallel x\parallel }^2+2〈y,x+y〉.$$

Lemma 2.4 [13]

Let E be a uniformly convex Banach space, $B_r(0):=\{x\in E:\parallel x\parallel \le r\}$ be a closed ball with center 0 and radius $r>0$. Then for any given sequence $\{x_1,x_2,\dots ,x_n,\dots \}\subset B_r(0)$ and any given number sequence $\{{\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n,\dots \}$ with ${\lambda }_i\ge 0$, ${\sum }_{i=1}^{\mathrm{\infty }}{\lambda }_i=1$, there exists a strictly increasing continuous and convex function $g:[0,2r)\to [0,\mathrm{\infty })$ with $g(0)=0$ such that for any $i,j\in \mathcal{N}$, $i<j$,

$${\parallel \sum _{n=1}^{\mathrm{\infty }}{\lambda }_n{x}_n\parallel }^2\le \sum _{n=1}^{\mathrm{\infty }}{\lambda }_n{\parallel x_n\parallel }^2-{\lambda }_i{\lambda }_{j}g(\parallel x_i-x_j\parallel ).$$

(2.2)

Lemma 2.5 [10]

Let $\{a_n\}$, $\{b_n\}$ and $\{{\delta }_n\}$ be sequences of nonnegative real numbers satisfying

$$a_{n+1}\le (1+{\delta }_n)a_n+b_n,\mathrm{\forall }n\ge 1.$$

(2.3)

If ${\sum }_{i=1}^{\mathrm{\infty }}{\delta }_n<\mathrm{\infty }$ and ${\sum }_{i=1}^{\mathrm{\infty }}{b}_n<\mathrm{\infty }$, then the limit ${lim}_{n\to \mathrm{\infty }}{a}_n$ exists.

Lemma 2.6 Let D be a nonempty and closed convex subset of H and $T:D\to D$ be a k-strictly pseudo-nonspreading mapping with $F(T)\ne \mathrm{\varnothing }$. Let $T_{\beta }=\beta I+(1-\beta )T$, $\beta \in [k,1)$. Then the following conclusions hold:

• (1) $F(T)=F(T_{\beta })$;

• (2) $I-T_{\beta }$ is demiclosed at zero;

• (3) ${\parallel T_{\beta }x-T_{\beta }y\parallel }^2\le {\parallel x-y\parallel }^2+\frac{2}{1-\beta }〈x-T_{\beta }x,y-T_{\beta }y〉$;

• (4) $T_{\beta }$ is a quasi-nonexpansive mapping.

Proof Since $(I-T_{\beta })=(1-\beta )(I-T)$, the conclusions (1), (2) are obvious.

Now we prove the conclusion (3). In fact, since T is a k-strictly pseudo-nonspreading mapping, it follows from Lemma 2.3 that

$$\begin{array}{rcl}{\parallel T_{\beta }x-T_{\beta }y\parallel }^2& =& {\parallel \beta (x-y)+(1-\beta )(Tx-Ty)\parallel }^2\\ =& \beta {\parallel x-y\parallel }^2+(1-\beta ){\parallel Tx-Ty\parallel }^2\\ -\beta (1-\beta ){\parallel x-Tx-(y-Ty)\parallel }^2\\ \le & \beta {\parallel x-y\parallel }^2+(1-\beta )\{{\parallel x-y\parallel }^2+k{\parallel x-Tx-(y-Ty)\parallel }^2\\ +2〈x-Tx,y-Ty〉\}-\beta (1-\beta ){\parallel x-Tx-(y-Ty)\parallel }^2\\ =& {\parallel x-y\parallel }^2+2(1-\beta )〈x-Tx,y-Ty〉\\ -(1-\beta )(\beta -k){\parallel x-Tx-(y-Ty)\parallel }^2\\ \le & {\parallel x-y\parallel }^2+2(1-\beta )〈x-Tx,y-Ty〉\\ =& {\parallel x-y\parallel }^2+\frac{2}{(1-\beta )}〈x-T_{\beta }x,y-T_{\beta }y〉,\mathrm{\forall }x,y\in D.\end{array}$$

(2.4)

If $y\in F(T)$, then $y\in F(T_{\beta })$. Hence from equation (2.4),

$$\parallel T_{\beta }x-y\parallel =\parallel T_{\beta }x-T_{\beta }y\parallel \le \parallel x-y\parallel ,\mathrm{\forall }x\in D.$$

(2.5)

This completes the proof of Lemma 2.6. □

Lemma 2.7 [14]

Let H be a Hilbert space and $\{u_n\}$ be a sequence in H such that there exists a nonempty set $W\subset H$ satisfying:

• (1) for every $w\in W$, ${lim}_{n\to \mathrm{\infty }}\parallel u_n-w\parallel$ exists;

• (2) each weak-cluster point of the sequence $\{w_n\}$ is in W.

Then there exists $w^{\ast }\in W$ such that $\{u_n\}$ weakly converges to $w^{\ast }$.

3 Weak- and strong-convergence theorems

For solving the multiple-set split feasibility problem (MSSFP) equation (1.6), we assume that the following conditions are satisfied:

• (1) $H_1$ and $H_2$ are two real Hilbert spaces, $A:H_1\to H_2$ is a bounded linear operator and $A^{\ast }:H_2\to H_1$ is the adjoint of A;

• (2) ${\{S_i\}}_{i=1}^{\mathrm{\infty }}:H_1\to H_1$ is an infinite family of $k_i$-strictly pseudo-nonspreading mappings with $k:={sup}_{i\ge 1}{k}_i\in (0,1)$;

• (3) ${\{T_i\}}_{i=1}^N:H_2\to H_2$ is a finite family of ${\rho }_i$-strictly pseudo-nonspreading mappings with $\rho =max\{{\rho }_i:i=1,2,\dots ,N\}\in (0,1)$;

• (4) $C:={\bigcap }_{i=1}^{\mathrm{\infty }}F(S_i)\ne \mathrm{\varnothing }$ and $Q:={\bigcap }_{i=1}^{N}F(T_i)\ne \mathrm{\varnothing }$.

Now we are in a position to give the following main theorem.

Theorem 3.1 Let $H_1$, $H_2$, A, $A^{\ast }$, ${\{S_i\}}_{i=1}^{\mathrm{\infty }}$, ${\{T_i\}}_{i=1}^N$, C, Q, k, ρ be the same as above. Let $\{x_n\}$ be a sequence generated by

$$\{\begin{array}{l}{x}_1\in H_1\mathit{\text{chosen arbitrarily}},\\ x_{n+1}={\alpha }_{0,n}{y}_n+{\sum }_{i=1}^{\mathrm{\infty }}{\alpha }_{i,n}{S}_{i,\beta }{y}_n,\\ y_n=x_n+\gamma A^{\ast }(T_{n(modN)}-I)A{x}_n,\end{array}\mathrm{\forall }n\ge 1,$$

(3.1)

where $S_{i,\beta }:=\beta I+(1-\beta )S_i$, $i\ge 1$, $\beta \in [k,1)$ is a constant, $\{{\alpha }_{i,n}\}\subset (0,1)$ and $\gamma >0$ satisfy the following conditions:

• (a) ${\sum }_{i=0}^{\mathrm{\infty }}{\alpha }_{i,n}=1$, for each $n\ge 1$;

• (b) for each $i\ge 1$, $lim{inf}_{n\to \mathrm{\infty }}{\alpha }_{0,n}{\alpha }_{i,n}>0$;

• (c) $\gamma \in (0,\frac{1-\rho }{{\parallel A\parallel }^2})$.

Let $\mathrm{\Gamma }=\{x\in C,Ax\in Q\}\ne \mathrm{\varnothing }$ (the set of solutions of (MSSFP) equation (1.6) defined by equation (1.7)). Then we have the following:

• (I) both $\{x_n\}$ and $\{y_n\}$ converge weakly to some point $x^{\ast }\in \mathrm{\Gamma }$;

• (II) in addition, if there exists some positive integer m such that $S_m$ is semicompact, then both $\{x_n\}$ and $\{y_n\}$ converge strongly to $x^{\ast }\in \mathrm{\Gamma }$.

Proof First we prove the conclusion (I).

Step 1. We prove that the sequences $\{x_n\}$, $\{y_n\}$ and $\{S_{i,\beta }{y}_n\}$ are bounded and, for each $p\in \mathrm{\Gamma }$, the following limits exist and

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-p\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel y_n-p\parallel .$$

In fact, for given $p\in \mathrm{\Gamma }$, by the definition of Γ,

$$p\in C=\bigcap _{i=1}^{\mathrm{\infty }}F(S_i)=\bigcap _{i=1}^{\mathrm{\infty }}F(S_{i,\beta })$$

and

$$Ap\in Q:=\bigcap _{i=1}^{N}F(T_i).$$

Therefore, we have

$$Ap=T_{n(modN)}Ap.$$

(3.2)

Since ${\{S_i\}}_{i=1}^{\mathrm{\infty }}$ is a family of k-strictly pseudo-nonspreading mappings, by Lemma 2.2, $C={\bigcap }_{i=1}^{\mathrm{\infty }}F(S_i)$ is closed and convex. It follows from Lemma 2.6 that, for each $n\ge 1$ and $p\in \mathrm{\Gamma }$,

$$\begin{array}{c}\parallel x_{n+1}-p\parallel =\parallel {\alpha }_{0,n}(y_n-p)+\sum _{i=1}^{\mathrm{\infty }}{\alpha }_{i,n}(S_{i,\beta }{y}_n-p)\parallel \hfill \\ \phantom{\parallel x_{n+1}-p\parallel }\le {\alpha }_{0,n}\parallel y_n-p\parallel +\sum _{i=1}^{\mathrm{\infty }}{\alpha }_{i,n}\parallel S_{i,\beta }{y}_n-p\parallel \hfill \\ \phantom{\parallel x_{n+1}-p\parallel }=\parallel y_n-p\parallel ,\hfill \end{array}$$

(3.3)

$$\begin{array}{c}{\parallel y_n-p\parallel }^2={\parallel x_n-p+\gamma A^{\ast }(T_{n(modN)}-I)A{x}_n\parallel }^2\hfill \\ \phantom{{\parallel y_n-p\parallel }^2}={\parallel x_n-p\parallel }^2+2\gamma 〈x_n-p,A^{\ast }(T_{n(modN)}-I)A{x}_n〉\hfill \\ \phantom{{\parallel y_n-p\parallel }^2}+{\gamma }^2{\parallel A^{\ast }(T_{n(modN)}-I)A{x}_n\parallel }^2\hfill \end{array}$$

(3.4)

and

$$\begin{array}{rcl}{\gamma }^2{\parallel A^{\ast }(T_{n(modN)}-I)A{x}_n\parallel }^2& =& {\gamma }^2〈A^{\ast }(T_{n(modN)}-I)A{x}_n,A^{\ast }(T_{n(modN)}-I)A{x}_n〉\\ =& {\gamma }^2〈A{A}^{\ast }(T_{n(modN)}-I)A{x}_n,(T_{n(modN)}-I)A{x}_n〉\\ \le & {\gamma }^2{\parallel A\parallel }^2{\parallel (T_{n(modN)}-I)A{x}_n\parallel }^2.\end{array}$$

(3.5)

Further, since ${\{T_i\}}_{i=1}^N$ is a finite family of ρ-strictly pseudo-nonspreading mappings, we have

$$\begin{array}{c}〈x_n-p,A^{\ast }(T_{n(modN)}-I)A{x}_n〉\hfill \\ =〈A(x_n-p),(T_{n(modN)}-I)A{x}_n〉\hfill \\ =〈A(x_n-p)+(T_{n(modN)}-I)A{x}_n-(T_{n(modN)}-I)A{x}_n,(T_{n(modN)}-I)A{x}_n〉\hfill \\ =〈T_{n(modN)}A{x}_n-Ap,(T_{n(modN)}-I)A{x}_n〉-{\parallel (T_{n(modN)}-I)A{x}_n\parallel }^2\hfill \\ =\frac{1}{2}\{{\parallel T_{n(modN)}A{x}_n-Ap\parallel }^2+{\parallel (T_{n(modN)}-I)A{x}_n\parallel }^2\hfill \\ -{\parallel A{x}_n-Ap\parallel }^2\}-{\parallel (T_{n(modN)}-I)A{x}_n\parallel }^2\hfill \\ =\frac{1}{2}\{{\parallel T_{n(modN)}A{x}_n-T_{n(modN)}Ap\parallel }^2+{\parallel (T_{n(modN)}-I)A{x}_n\parallel }^2\hfill \\ -{\parallel A{x}_n-Ap\parallel }^2\}-{\parallel (T_{n(modN)}-I)A{x}_n\parallel }^2\hfill \\ \le \frac{1}{2}\{{\parallel A{x}_n-Ap\parallel }^2+\rho {\parallel (T_{n(modN)}-I)A{x}_n\parallel }^2\}\hfill \\ +\frac{1}{2}\{{\parallel (T_{n(modN)}-I)A{x}_n\parallel }^2-{\parallel A{x}_n-Ap\parallel }^2\}-{\parallel (T_{n(modN)}-I)A{x}_n\parallel }^2\hfill \\ =\frac{\rho -1}{2}{\parallel (T_{n(modN)}-I)A{x}_n\parallel }^2.\hfill \end{array}$$

(3.6)

Substituting equations (3.5) and (3.6) into equation (3.4) and simplifying, we have

$${\parallel y_n-p\parallel }^2\le {\parallel x_n-p\parallel }^2-\gamma (1-\rho -\gamma {\parallel A\parallel }^2){\parallel (T_{n(modN)}-I)A{x}_n\parallel }^2.$$

(3.7)

By condition (c), $(1-\rho -\gamma {\parallel A\parallel }^2)>0$, therefore we have

$${\parallel y_n-p\parallel }^2\le {\parallel x_n-p\parallel }^2.$$

(3.8)

Substituting equation (3.8) into equation (3.3), we have

$$\parallel x_{n+1}-p\parallel \le \parallel x_n-p\parallel ,\mathrm{\forall }n\ge 1.$$

This implies that the limit ${lim}_{n\to \mathrm{\infty }}\parallel x_n-p\parallel$ exists. It follows from equations (3.8) and (3.3) that the limit ${lim}_{n\to \mathrm{\infty }}\parallel y_n-p\parallel$ exists also, and

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-p\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel y_n-p\parallel ,\mathrm{\forall }p\in \mathrm{\Gamma }.$$

(3.9)

Therefore, $\{x_n\}$ and $\{y_n\}$ are bounded. Since for each $i\ge 1$, $S_{i,\beta }$ is quasi-nonexpansive, we have

$$\parallel S_{i,\beta }{y}_n-p\parallel \le \parallel y_n-p\parallel .$$

Hence $\{S_{i,\beta }{y}_n\}$ is also bounded.

Step 2. Now we prove that for any given positive integer $l\ge 1$, the following conclusions hold:

$$\underset{n\to \mathrm{\infty }}{lim}\parallel y_n-S_{l,\beta }{y}_n\parallel =0;\underset{n\to \mathrm{\infty }}{lim}\parallel T_{n(modN)}A{x}_n-A{x}_n\parallel =0.$$

(3.10)

In fact, for any given $p\in \mathrm{\Gamma }$, it follows from equation (3.1), Lemma 2.4, and equation (3.7) that

$$\begin{array}{rcl}{\parallel x_{n+1}-p\parallel }^2& =& {\parallel {\alpha }_{0,n}(y_n-p)+\sum _{i=1}^{\mathrm{\infty }}{\alpha }_{i,n}(S_{i,\beta }{y}_n-p)\parallel }^2\\ \le & {\alpha }_{0,n}{\parallel y_n-p\parallel }^2+\sum _{i=1}^{\mathrm{\infty }}{\alpha }_{i,n}{\parallel S_{i,\beta }{y}_n-p\parallel }^2-{\alpha }_{0,n}{\alpha }_{l,n}g(\parallel y_n-S_{i,\beta }{y}_n\parallel )\\ \le & {\alpha }_{0,n}{\parallel y_n-p\parallel }^2+\sum _{i=1}^{\mathrm{\infty }}{\alpha }_{i,n}{\parallel y_n-p\parallel }^2-{\alpha }_{0,n}{\alpha }_{l,n}g(\parallel y_n-S_{i,\beta }{y}_n\parallel )\\ =& {\parallel y_n-p\parallel }^2-{\alpha }_{0,n}{\alpha }_{l,n}g(\parallel y_n-S_{i,\beta }{y}_n\parallel )\\ \le & {\parallel x_n-p\parallel }^2-\gamma (1-\rho -\gamma {\parallel A\parallel }^2){\parallel (T_{n(modN)}-I)A{x}_n\parallel }^2\\ -{\alpha }_{0,n}{\alpha }_{l,n}g(\parallel y_n-S_{i,\beta }{y}_n\parallel ),n\ge 1.\end{array}$$

(3.11)

Therefore, we have

$$\begin{array}{c}\gamma (1-\rho -\gamma {\parallel A\parallel }^2){\parallel (T_{n(modN)}-I)A{x}_n\parallel }^2+{\alpha }_{0,n}{\alpha }_{l,n}g(\parallel y_n-S_{l,\beta }{y}_n\parallel )\hfill \\ \le {\parallel x_n-p\parallel }^2-{\parallel x_{n+1}-p\parallel }^2\hfill \\ \to 0(\text{as }n\to \mathrm{\infty }).\hfill \end{array}$$

By conditions (b) and (c) we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel (T_{n(modN)}-I)A{x}_n\parallel =0;\underset{n\to \mathrm{\infty }}{lim}g(\parallel y_n-S_{l,\beta }{y}_n\parallel )=0.$$

(3.12)

Since g is continuous and strictly increasing with $g(0)=0$, from equation (3.12) we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel y_n-S_{l,\beta }{y}_n\parallel =0\text{for each }l\ge 1.$$

(3.13)

Hence conclusion (3.10) is proved.

Step 3. Now, we prove that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_{n+1}-x_n\parallel =0;\underset{n\to \mathrm{\infty }}{lim}\parallel y_{n+1}-y_n\parallel =0.$$

(3.14)

In fact, it follows from equation (3.1) that

$$\begin{array}{rcl}{\parallel x_{n+1}-x_n\parallel }^2& =& {\parallel {\alpha }_{0,n}(y_n-x_n)+\sum _{i=1}^{\mathrm{\infty }}{\alpha }_{i,n}(S_{i,\beta }{y}_n-x_n)\parallel }^2\\ =& {\parallel {\alpha }_{0,n}(\gamma A^{\ast }(T_{n(modN)}-I)A{x}_n)+\sum _{i=1}^{\mathrm{\infty }}{\alpha }_{i,n}(S_{i,\beta }{y}_n-x_n)\parallel }^2\\ \le & {\alpha }_{0,n}{\parallel \gamma A^{\ast }(T_{n(modN)}-I)A{x}_n\parallel }^2+\sum _{i=1}^{\mathrm{\infty }}{\alpha }_{i,n}{\parallel S_{i,\beta }{y}_n-x_n\parallel }^2\\ \le & {\alpha }_{0,n}{\parallel \gamma A^{\ast }(T_{n(modN)}-I)A{x}_n\parallel }^2\\ +\sum _{i=1}^{\mathrm{\infty }}{\alpha }_{i,n}{(\parallel S_{i,\beta }{y}_n-y_n\parallel +\parallel y_n-x_n\parallel )}^2.\end{array}$$

(3.15)

By virtue of equations (3.1) and (3.10), one has

$$\begin{array}{rcl}\parallel y_n-x_n\parallel & =& \gamma \parallel A^{\ast }(T_{n(modN)}-I)A{x}_n\parallel \\ \to & 0(\text{as }n\to \mathrm{\infty }).\end{array}$$

(3.16)

This together with equations (3.10) and (3.15) shows that

$$\parallel x_{n+1}-x_n\parallel \to 0(\text{as }n\to \mathrm{\infty }).$$

Similarly, we have

$$\begin{array}{rcl}\parallel y_{n+1}-y_n\parallel & =& \parallel x_{n+1}+\gamma A^{\ast }(T_{(n+1)(modN)}-I)A{x}_{n+1}\\ -[x_n+\gamma A^{\ast }(T_{n(modN)}-I)A{x}_n]\parallel \\ \le & \parallel x_{n+1}-x_n\parallel +\gamma \parallel A^{\ast }(T_{(n+1)(modN)}-I)A{x}_{n+1}\parallel \\ +\gamma \parallel A^{\ast }(T_{n(modN)}-I)A{x}_n\parallel \\ \to & 0(\text{as }n\to \mathrm{\infty }).\end{array}$$

Step 4. Now we show that every weak-cluster point $x^{\ast }$ of the sequence $\{x_n\}$ is in Γ.

Indeed, since $\{y_n\}$ is a bounded sequence in $H_1$, there exists a subsequence $\{y_{n_i}\}\subset \{y_n\}$ such that $y_{n_i}\rightharpoonup x^{\ast }\in H_1$. It follows from equation (3.10) that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel y_{n_i}-S_{l,\beta }{y}_{n_i}\parallel =0\text{for each }l\ge 1.$$

By Lemma 2.2, $(I-S_i)$ is demiclosed at zero. Since $(I-S_{l,\beta })=(1-\beta )(I-S_i)$, this implies that $(I-S_{l,\beta })$ is also demiclosed at zero. Hence $x^{\ast }\in F(S_{l,\beta })=F(S_l)$. By the arbitrariness of $l\ge 1$, we have

$$x^{\ast }\in \bigcap _{i=1}^{\mathrm{\infty }}F(S_i)=C.$$

On the other hand, it follows from equations (3.1) and (3.10) that

$$x_{n_i}=y_{n_i}-\gamma A^{\ast }(T_{n_i(modN)}-I)A{x}_{n_i}\rightharpoonup x^{\ast }.$$

(3.17)

Since A is a bounded linear operator, this implies that $A{x}_{n_i}\rightharpoonup A{x}^{\ast }$. Also, by equation (3.10)

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

(3.18)

Hence for any given positive integer $j=1,2,\dots ,N$, there exists a subsequence $\{n_{i_k}\}\subset \{n_i\}$ with $n_{i_k}(modN)=j$ such that

$$\underset{n_{i_k}\to \mathrm{\infty }}{lim}\parallel T_{j}A{x}_{n_{i_k}}-A{x}_{n_{i_k}}\parallel =0.$$

Since $A{x}_{n_{i_k}}\rightharpoonup A{x}^{\ast }$, and by Lemma 2.2, $I-T_j$ is demiclosed at 0. This implies that $A{x}^{\ast }\in F(T_j)$. By the arbitrariness of $j=1,2,\dots ,N$,

$$A{x}^{\ast }\in \bigcap _{j=1}^{N}F(T_j)=Q.$$

These show that $x^{\ast }\in \mathrm{\Gamma }$.

Step 5. Summing up the above arguments, we have proved that: (i) for each $p\in \mathrm{\Gamma }$, the limits ${lim}_{n\to \mathrm{\infty }}\parallel x_n-p\parallel$ and ${lim}_{n\to \mathrm{\infty }}\parallel y_n-p\parallel$ exist (see equation (3.2)); (ii) every weak-cluster point $x^{\ast }$ of the sequence $\{x_n\}$ (or $\{y_n\}$) is in Γ. Taking $W=\mathrm{\Gamma }$ and $\{u_n\}=\{x_n\}$ (or $\{y_n\}$) in Lemma 2.7, therefore all conditions in Lemma 2.7 are satisfied. By using Lemma 2.7, $x_n\rightharpoonup x^{\ast }$, $y_n\rightharpoonup x^{\ast }$ and $x^{\ast }\in \mathrm{\Gamma }$. This completes the proof of the conclusion (I).

Next we prove the conclusion (II).

Without loss of generality, we may assume that $S_1$ is semicompact. Since $(I-S_{1,\beta })=(1-\beta )(I-S_1)$, this implies that $S_{1,\beta }$ is also semicompact. In view of equation (3.10), we have

$$\parallel y_n-S_{1,\beta }{y}_n\parallel \to 0(\text{as }n\to \mathrm{\infty }).$$

(3.19)

Therefore, there exists a subsequence of $\{y_{n_i}\}\subset \{y_n\}$ such that $y_{n_i}\to u^{\ast }\in H_1$. Since $y_{n_i}\rightharpoonup x^{\ast }$, we have $x^{\ast }=u^{\ast }$ and so $y_{n_i}\to x^{\ast }\in \mathrm{\Gamma }$. By virtue of equation (3.9), we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel y_n-x^{\ast }\parallel =0,\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-x^{\ast }\parallel =0,$$

i.e., $\{y_n\}$ and $\{x_n\}$ both converge strongly to the point $x^{\ast }\in \mathrm{\Gamma }$. This completes the proof of Theorem 3.1. □

Remark 3.2 Theorem 3.1 improves and extends the corresponding results of Censor et al. [5, 8, 9], Byrne [6], Yang [11], Moudafi [15], Xu [16], Censor and Segal [17], Masad and Reich [18], Deepho and Kumam [19, 20] and others in the following aspects:

• (a) for the mappings, we extend the mappings from nonexpansive mappings, or demi-contractive mappings, to the more general family of k-strictly pseudo-nonspreading mappings;

• (b) for the algorithms, we propose some new hybrid iterative algorithms which are different from the ones given in [5–7, 9, 17, 18, 21, 22]. Under suitable conditions, some weak- and strong-convergence results for the algorithms are proved.

If we put $\gamma =0$ in Theorem 3.1, we immediately get the following.

Corollary 3.3 Let H, ${\{S_i\}}_{i=1}^{\mathrm{\infty }}$, k be the same as above. Let $\{x_n\}$ be a sequence generated by

$$\{\begin{array}{l}{x}_1\in H\mathit{\text{chosen arbitrarily}},\\ x_{n+1}={\alpha }_{0,n}{x}_n+{\sum }_{i=1}^{\mathrm{\infty }}{\alpha }_{i,n}{S}_{i,\beta }{x}_n,\end{array}\mathrm{\forall }n\ge 1,$$

(3.20)

where $S_{i,\beta }:=\beta I+(1-\beta )S_i$, $i\ge 1$, $\beta \in [k,1)$ is a constant, $\{{\alpha }_{i,n}\}\subset (0,1)$ satisfy the following conditions:

• (a) ${\sum }_{i=0}^{\mathrm{\infty }}{\alpha }_{i,n}=1$, for each $n\ge 1$;

• (b) for each $i\ge 1$, $lim{inf}_{n\to \mathrm{\infty }}{\alpha }_{0,n}{\alpha }_{i,n}>0$. Let

  $$\mathcal{F}:=\bigcap _{i=1}^{\mathrm{\infty }}F(S_i)\ne \mathrm{\varnothing }.$$

Then we have the following:

• (I) the sequence $\{x_n\}$ converges weakly to some point $x^{\ast }\in \mathcal{F}$;

• (II) in addition, if there exists some positive integer m such that $S_m$ is semicompact, then the sequence $\{x_n\}$ converges strongly to $x^{\ast }\in \mathcal{F}$.

4 Applications

In this section we utilize the results presented in Section 3 to study the hierarchical variational inequality problem.

Let H be a real Hilbert space, $\{S_i\}:H\to H$, $i=1,2,\dots$ be a countable family of $k_i$-strictly pseudo-nonspreading mappings with $k={sup}_{i\ge 1}{k}_i\in (0,1)$, and

$$\mathcal{F}:=\bigcap _{i=1}^{\mathrm{\infty }}F(S_i)\ne \mathrm{\varnothing }.$$

Let $T:H\to H$ be a nonspreading mapping. The so-called hierarchical variational inequality problem for a countable family of mappings $\{S_i\}$ with respect to mapping T is to find an $x^{\ast }\in \mathcal{F}$ such that

$$〈x^{\ast }-T{x}^{\ast },x^{\ast }-x〉\le 0,\mathrm{\forall }x\in \mathcal{F}.$$

(4.1)

It is easy to see that equation (4.1) is equivalent to the following fixed point problem: to find $x^{\ast }\in \mathcal{F}$ such that

$$x^{\ast }=P_{\mathcal{F}}T{x}^{\ast },$$

(4.2)

where $P_{\mathcal{F}}$ is the metric projection from H onto ℱ. Letting $C=\mathcal{F}$ and $Q=F(P_{\mathcal{F}}T)$ (the fixed point set of $P_{\mathcal{F}}T$) and $A=I$ (the identity mapping on H), then the problem (4.2) is equivalent to the following multi-set split feasibility problem: to find $x^{\ast }\in C$ such that

$$x^{\ast }\in Q.$$

(4.3)

Hence from Theorem 3.1 we have the following theorem.

Theorem 4.1 Let H, $\{S_i\}$, T, C, Q, k be the same as above. Let $\{x_n\}$, $\{y_n\}$ be the sequences defined by

$$\{\begin{array}{l}{x}_1\in H_1\mathit{\text{chosen arbitrarily}},\\ x_{n+1}={\alpha }_{0,n}{y}_n+{\sum }_{i=1}^{\mathrm{\infty }}{\alpha }_{i,n}{S}_{i,\beta }{y}_n,\\ y_n=x_n+\gamma (T-I)x_n,\mathrm{\forall }n\ge 1,\end{array}$$

(4.4)

where $S_{i,\beta }:=\beta I+(1-\beta )S_i$, $i\ge 1$, $\beta \in [k,1)$, $\{{\alpha }_{i,n}\}\subset (0,1)$ and $\gamma >0$ satisfy the following conditions:

• (a) ${\sum }_{i=0}^{\mathrm{\infty }}{\alpha }_{i,n}=1$, for each $n\ge 1$;

• (b) for each $i\ge 1$, $lim{inf}_{n\to \mathrm{\infty }}{\alpha }_{0,n}{\alpha }_{i,n}>0$;

• (c) $\gamma \in (0,1)$.

If $C\cap Q\ne \mathrm{\varnothing }$, then $\{x_n\}$ converges weakly to a solution of the hierarchical variational inequality problem (4.1). In addition, if one of the mappings $S_i$ is semicompact, then both $\{x_n\}$ and $\{y_n\}$ converge strongly to a solution of the hierarchical variational inequality problem (4.1).

Proof In fact, by the assumption that T is a nonspreading mapping, hence by Remark 1.2, T is a ρ-strictly pseudo-nonspreading with $\rho =0$. Taking $N=1$ and $A=I$ in Theorem 3.1, all conditions in Theorem 3.1 are satisfied. The conclusions of Theorem 4.1 can immediately be obtained from Theorem 3.1. □

Remark 4.2 If $T=I$ (the identity mapping), then we can get the results of Corollary 3.3.