Strong convergence theorems for the general split variational inclusion problem in Hilbert spaces

Shih-sen Chang; Lin Wang

doi:10.1186/1687-1812-2014-171

Research
Open Access

Strong convergence theorems for the general split variational inclusion problem in Hilbert spaces

Shih-sen Chang¹Email author and
Lin Wang¹

Fixed Point Theory and Applications20142014:171

https://doi.org/10.1186/1687-1812-2014-171

Received: 1 May 2014
Accepted: 19 July 2014
Published: 18 August 2014

Abstract

The purpose of this paper is to introduce and study a general split variational inclusion problem in the setting of infinite-dimensional Hilbert spaces. Under suitable conditions, we prove that the sequence generated by the proposed new algorithm converges strongly to a solution of the general split variational inclusion problem. As a particular case, we consider the algorithms for a split feasibility problem and a split optimization problem and give some strong convergence theorems for these problems in Hilbert spaces.

Keywords

general split variational inclusion problem
split feasibility problem
split optimization problem
quasi-nonexpansive mapping
zero point
resolvent mapping

1 Introduction

Let C and Q be nonempty closed convex subsets of real Hilbert spaces

H_{1}

and

H_{2}

, respectively. The split feasibility problem

(S F P)

is formulated as

to find x^{*} \in C and A x^{*} \in Q,

(1.1)

where $A : H_{1} \to H_{2}$ is a bounded linear operator. In 1994, Censor and Elfving [1] first introduced the $S F P$ in finite-dimensional Hilbert spaces for modeling inverse problems which arise from phase retrievals and in medical image reconstruction [2]. It has been found that the $S F P$ can also be used in various disciplines such as image restoration, computer tomography and radiation therapy treatment planning [3–5]. The $S F P$ in an infinite-dimensional real Hilbert space can be found in [2, 4, 6–10]. For comprehensive literature, bibliography and a survey on SFP, we refer to [11].

Assuming that the

S F P

is consistent, it is not hard to see that

x^{*} \in C

solves

S F P

if and only if it solves the fixed point equation

x^{*} = P_{C} (I - γ A^{*} (I - P_{Q}) A) x^{*},

where $P_{C}$ and $P_{Q}$ are the metric projection from $H_{1}$ onto C and from $H_{2}$ onto Q, respectively, $γ > 0$ is a positive constant, and $A^{*}$ is the adjoint of A.

A popular algorithm to be used to solves the

S F P

(1.1) is due to Byrne’s CQ-algorithm [2]:

x_{k + 1} = P_{C} (I - γ_{k} A^{*} (I - P_{Q}) A) x_{k}, k \geq 1,

where $γ_{k} \in (0, 2 / λ)$ with λ being the spectral radius of the operator $A^{*} A$ .

On the other hand, let H be a real Hilbert space, and B be a set-valued mapping with domain

D (B) : = {x \in H : B (x) \neq \emptyset}

. Recall that B is called monotone, if

〈 u - v, x, x - y 〉 \geq 0

for any

u \in B x

and

v \in B y

; B is maximal monotone, if its graph

{(x, y) : x \in D (B), y \in B x}

is not properly contained in the graph of any other monotone mapping. An important problem for set-valued monotone mappings is to find

x^{*} \in H

such that

0 \in B (x^{*})

. Here,

x^{*}

is called a zero point of B. A well-known method for approximating a zero point of a maximal monotone mapping defined in a real Hilbert space H is the proximal point algorithm first introduced by Martinet [12] and generated by Rockafellar [13]. This is an iterative procedure, which generates

{x_{n}}

by

x_{1} = x \in H

and

x_{n + 1} = J_{β_{n}}^{B} x_{n}, n \geq 1,

(1.2)

where

{β_{n}} \subset (0, \infty)

, B is a maximal monotone mapping in a real Hilbert space, and

J_{r}^{B}

is the resolvent mapping of B defined by

J_{r}^{B} = {(I + r B)}^{- 1}

for each

r > 0

. Rockafellar [13] proved that if the solution set

B^{- 1} (0)

is nonempty and

{lim inf}_{n \to \infty} β_{n} > 0

, then the sequence

{x_{n}}

in (1.2) converges weakly to an element of

B^{- 1} (0)

. In particular, if B is the sub-differential ∂f of a proper convex and lower semicontinuous function

f : H \to R

, then (1.2) is reduced to

x_{n + 1} = \underset{y \in H}{argmin} {f (y) + \frac{1}{2 β_{n}} {∥ y - x_{n} ∥}^{2}}, \forall n \geq 1 .

(1.3)

In this case, ${x_{n}}$ converges weakly to a minimizer of f. Later, many researchers have studied the convergence problems of the proximal point algorithm in Hilbert spaces (see [14–21] and the references therein).

Motivated by the works in [14–17] and related literature, the purpose of this paper is to introduce and consider the following general split variational inclusion problem.

Let

H_{1}

and

H_{2}

be two real Hilbert spaces,

B_{i} : H_{1} \to H_{1}

and

K_{i} : H_{2} \to H_{2}

,

i = 1, 2, \dots

be two families of set-valued maximal monotone mappings,

A : H_{1} \to H_{2}

be a linear and bounded operator, and

A^{*}

be the adjoint of A. The so-called general split variational inclusion problem is

to find x^{*} \in H_{1} such that 0 \in ⋂_{i = 1}^{\infty} B_{i} (x^{*}) and 0 \in ⋂_{i = 1}^{\infty} K_{i} (A x^{*}) .

(1.4)

The following examples are special cases of (GSVIP) (1.4).

Classical split variational inclusion problem. Let

B : H_{1} \to H_{1}

and

K : H_{2} \to H_{2}

be set-valued maximal monotone mappings. The so-called classical split variational inclusion problem (CSVIP) is

to find x^{*} \in H_{1} such that 0 \in B (x^{*}) and 0 \in K (A x^{*}),

(1.5)

which was introduced by Moudafi [17]. It is obvious that problem (1.5) is a special case of (GSVIP) (1.4). In [17], Moudafi proved that the iteration process

x_{n + 1} = J_{λ}^{B} (x_{n} + γ A^{*} (J_{λ}^{K} - I) A x_{n})

converges weakly to a solution of problem (1.5), where λ and γ are given positive numbers.

Split optimization problem. Let

f : H_{1} \to R

,

g : H_{2} \to R

be two proper convex and lower semicontinuous functions. The so-called split optimization problem (SOP) is

to find x^{*} \in H_{1} such that f (x^{*}) = min_{y \in H_{1}} f (y) and g (A x^{*}) = min_{z \in H_{2}} g (z) .

(1.6)

Denote by

B = \partial (f)

and

K = \partial (g)

, then B and K both are maximal monotone mappings, and problem (1.6) is equivalent to the following classical split variational inclusion problem, i.e.:

to find x^{*} \in H_{1} such that 0 \in \partial (f (x^{*})) and 0 \in \partial (g (A x^{*})) .

(1.7)

Split feasibility problem. As in (1.1), let C and Q be two nonempty closed convex subsets of real Hilbert spaces

H_{1}

and

H_{2}

, respectively and A be the same as above. The split feasibility problem is

to find x^{*} \in C such A x^{*} \in Q .

(1.8)

It is well known that this kind of problems was first introduced by Censor and Elfving [1] for modeling inverse problems arising from phase retrievals and in medical image reconstruction [2]. Also it can be used in various disciplines such as image restoration, computer tomography and radiation therapy treatment planning.

Let

i_{C}

(

i_{Q}

) be the indicator function of C (Q), i.e.,

i_{C} (x) = {\begin{cases} 0, & if x \in C, \\ + \infty, & if x \notin C; \end{cases} i_{Q} (x) = {\begin{cases} 0, & if x \in Q, \\ + \infty, & if x \notin Q . \end{cases}

(1.9)

Then

i_{C}

and

i_{Q}

both are proper convex and lower semicontinuous functions, and its subdifferentials

\partial i_{C}

and

\partial i_{Q}

are maximal monotone operators. Consequently problem (1.8) is equivalent to the following ‘split optimization problem’ and ‘Moudafi’s classical split variational inclusion problem’, i.e.,

\begin{aligned} to find x^{*} \in H_{1} such that i_{C} (x^{*}) = min_{y \in H_{1}} i_{C} (y) and i_{Q} (A x^{*}) = min_{z \in H_{2}} i_{Q} (z) \\ \Leftrightarrow to find x^{*} \in H_{1} such that 0 \in \partial (i_{C} (x^{*})) and 0 \in \partial (i_{Q} (A x^{*})) . \end{aligned}

(1.10)

For solving (GSVIP) (1.4), in our paper we propose the following iterative algorithms:

x_{n + 1} = α_{n} x_{n} + ξ_{n} f (x_{n}) + \sum_{i = 1}^{\infty} γ_{n, i} J_{β_{i}}^{B_{i}} [x_{n} - λ_{n, i} A^{*} (I - J_{β_{i}}^{K_{i}}) A x_{n}], \forall n \geq 0,

(1.11)

where $f : H_{1} \to H_{1}$ is a contraction mapping with a contractive constant $k \in (0, 1)$ , ${α_{n}}$ , ${ξ_{n}}$ and ${γ_{n, i}}$ are sequence in $[0, 1]$ satisfying some conditions. Under suitable conditions, some strong convergence theorems for the sequence proposed by (1.11) to a solution for (GSVIP) (1.4) in Hilbert spaces are proved. As a particular case, we consider the algorithms for a split feasibility problem and a split optimization problem and give some strong convergence theorems for these problems in Hilbert spaces. Our results extend and improve the related results of Censor and Elfving [1], Byrne [2], Censor et al. [3–5], Rockafellar [13], Moudafi [14, 17], Eslamian and Latif [15], Eslamian [21], and Chuang [22].

2 Preliminaries

Throughout the paper, we denote by H a real Hilbert space, C be a nonempty closed and convex subset of H.

F (T)

denote by the set of fixed points of a mapping T. Let

{x_{n}}

be a sequence in H and

x \in H

. Strong convergence of

{x_{n}}

to x is denoted by

x_{n} \to x

, and weak convergence of

{x_{n}}

to x is denoted by

x_{n} ⇀ x

. For every point

x \in H

, there exists a unique nearest point in C, denoted by

P_{C} x

. This point satisfies.

∥ x - P_{C} x ∥ \leq ∥ x - y ∥, \forall y \in C .

The operator

P_{C}

is called the metric projection. The metric projection

P_{C}

is characterized by the fact that

P_{C} x \in C

and

〈 x - P_{C} x, P_{C} x - y 〉 \geq 0, \forall x \in H, y \in C .

Recall that a mapping

T : C \to H

is said to be nonexpansive, if

∥ T x - T y ∥ \leq ∥ x - y ∥

for every

x, y \in C

. T is said to be quasi-nonexpansive, if

F (T) \neq \emptyset

and

∥ T x - p ∥ \leq ∥ x - p ∥

for every

x \in C

and

p \in F (T)

. It is easy to see that

F (T)

is a closed convex subset of C if T is a quasi-nonexpansive mapping. Besides, T is said to be a firmly nonexpansive, if

\begin{aligned} {∥ T x - T y ∥}^{2} \leq 〈 x - y, T x - T y 〉 \forall x, y \in C; \\ \Leftrightarrow {∥ T x - T y ∥}^{2} \leq {∥ x - y ∥}^{2} - {∥ (I - T) x - (I - T) y ∥}^{2} \forall x, y \in C . \end{aligned}

Lemma 2.1 (demi-closed principle)

Let C be a nonempty closed convex subset of a real Hilbert space H. Let $T : C \to H$ be a nonexpansive mapping, and let ${x_{n}}$ be a sequence in C. If $x_{n} ⇀ w$ and ${lim}_{n \to \infty} ∥ x_{n} - T x_{n} ∥ = 0$ , then $T w = w$ .

Lemma 2.2 [23]

Let H be a (real) Hilbert space. Then for all

x, y \in H

,

{∥ x + y ∥}^{2} \leq {∥ x ∥}^{2} + 2 〈 y, x + y 〉 .

(2.1)

Lemma 2.3 [24]

Let H be a Hilbert space and let

{x_{n}}

be a sequence in H. Then, for any given sequence

{λ_{n}} \subset (0, 1)

with

\sum_{n = 1}^{\infty} λ_{n} = 1

and for any positive integers i, j with

i < j

,

{∥ \sum_{n = 1}^{\infty} λ_{n} x_{n} ∥}^{2} \leq \sum_{n = 1}^{\infty} λ_{n} {∥ x_{n} ∥}^{2} - λ_{i} λ_{j} {∥ x_{i} - x_{j} ∥}^{2} .

(2.2)

Lemma 2.4 Let

{a_{n}}

be a sequence of nonnegative real numbers,

{b_{n}}

be a sequence of real numbers in

(0, 1)

with

\sum_{n = 1}^{\infty} b_{n} = \infty

,

{u_{n}}

be a sequence of nonnegative real numbers with

\sum_{n = 1}^{\infty} u_{n} < \infty

,

{t_{n}}

be a real numbers with

{lim sup}_{n \to \infty} t_{n} \leq 0

. If

a_{n + 1} \leq (1 - b_{n}) a_{n} + b_{n} t_{n} + u_{n}, for each n \geq 1,

then ${lim}_{n \to \infty} a_{n} = 0$ .

Lemma 2.5 [25]

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 $i \in N$ . Then there exists a nondecreasing sequence ${m_{k}} \subset N$ such that $m_{k} \to \infty$ , $a_{m_{k}} \leq a_{m_{k} + 1}$ and $a_{k} \leq a_{m_{k} + 1}$ are satisfied by all (sufficiently large) numbers $k \in N$ . In fact, $m_{k} = max {j \leq k : a_{j} < a_{j + 1}}$ .

Lemma 2.6 [22]

Let H be a real Hilbert space,

B : H \to 2^{H}

be a set-valued maximal monotone mapping,

β > 0

, and let

J_{β}^{B}

be the resolvent mapping of B.

(i)

For each $β > 0$ , $J_{β}^{B}$ is a single-valued and firmly nonexpansive mapping;
(ii)

$D (J_{β}^{B}) = H$ and $F (J_{β}^{B}) = B^{- 1} (0) : = {x \in D (B) : 0 \in B x}$ ;
(iii)

$(I - J_{β}^{B})$ is a firmly nonexpansive mapping for each $β > 0$ ;
(iv)

suppose that $B^{- 1} (0) \neq \emptyset$ , then for each $x \in H$ , each $x^{*} \in B^{- 1} (0)$ and each $β > 0$
${∥ x - J_{β}^{B} x ∥}^{2} + ∥ J_{β}^{B} x - x^{*} ∥ \leq {∥ x - x^{*} ∥}^{2};$
(v)

suppose that $B^{- 1} (0) \neq \emptyset$ . Then $〈 x - J_{β}^{B} x, J_{β}^{B} x - w 〉 \geq 0$ for each $x \in H$ and each $w \in B^{- 1} (0)$ , and each $β > 0$ .

Lemma 2.7 Let

H_{1}

,

H_{2}

be two real Hilbert spaces,

A : H_{1} \to H_{2}

be a linear bounded operator and

A^{*}

be the adjoint of A. Let

B : H_{2} \to 2_{2}^{H}

be a set-valued maximal monotone mapping,

β > 0

, and let

J_{β}^{B}

be the resolvent mapping of B, then

(i)

${∥ (I - J_{β}^{B}) A x - (I - J_{β}^{B}) A y ∥}^{2} \leq 〈 (I - J_{β}^{B}) A x - (I - J_{β}^{B}) A y, A x - A y 〉$ ;
(ii)

${∥ A^{*} (I - J_{β}^{B}) A x - A^{*} (I - J_{β}^{B}) A y ∥}^{2} \leq {∥ A ∥}^{2} 〈 (I - J_{β}^{B}) A x - (I - J_{β}^{B}) A y, A x - A y 〉$ ;
(iii)

if $ρ \in (0, \frac{2}{{∥ A ∥}^{2}})$ , then $(I - ρ A^{*} (I - J_{β}^{B}) A)$ is a nonexpansive mapping.

Proof By Lemma 2.6(iii), the mapping $(I - J_{β}^{B})$ is firmly nonexpansive, hence the conclusions (i) and (ii) are obvious.

Now we prove the conclusion (iii).

In fact, for any

x, y \in H_{1}

, it follows from the conclusions (i) and (ii) that

\begin{aligned} {∥ (I - ρ A^{*} (I - J_{β}^{B}) A) x - (I - ρ A^{*} (I - J_{β}^{B}) A) y ∥}^{2} \\ = {∥ x - y ∥}^{2} - 2 ρ 〈 x - y, A^{*} (I - J_{β}^{B}) A x - A^{*} (I - J_{β}^{B}) A y 〉 \\ + ρ^{2} {∥ A^{*} (I - J_{β}^{B}) A x - A^{*} (I - J_{β}^{B}) A y ∥}^{2} \\ \leq {∥ x - y ∥}^{2} - 2 ρ 〈 A x - A y, (I - J_{β}^{B}) A x - (I - J_{β}^{B}) A y 〉 \\ + ρ^{2} {∥ A ∥}^{2} {∥ (I - J_{β}^{B}) A x - (I - J_{β}^{B}) A y ∥}^{2} \\ \leq {∥ x - y ∥}^{2} - ρ (2 - ρ {∥ A ∥}^{2}) {∥ (I - J_{β}^{B}) A x - (I - J_{β}^{B}) A y ∥}^{2} \\ \leq {∥ x - y ∥}^{2} (since ρ (2 - ρ {∥ A ∥}^{2}) \geq 0) . \end{aligned}

This completes the proof of Lemma 2.7. □

3 Main results

The following lemma will be used in proving our main results.

Lemma 3.1 Let

H_{1}

and

H_{2}

be two real Hilbert spaces,

A : H_{1} \to H_{2}

be a linear and bounded operator, and

A^{*}

be the adjoint of A. Let

B_{i} : H_{1} \to 2^{H_{1}}

, and

K_{i} : H_{2} \to 2^{H_{2}}

,

i = 1, 2, \dots

, be two families of set-valued maximal monotone mappings, and let

β > 0

and

γ > 0

. If

Ω \neq \emptyset

(the solution set of (GSVIP) (1.4)), then

x^{*} \in H_{1}

is a solution of (GSVIP) (1.4) if and only if for each

i \geq 1

, for each

γ > 0

and for each

β > 0

x^{*} = J_{β}^{B_{i}} (x^{*} - γ A^{*} (I - J_{β}^{K_{i}}) A x^{*}) .

(3.1)

Proof Indeed, if

x^{*}

is a solution of (GSVIP) (1.4), then for each

i \geq 1

,

γ > 0

and

β > 0

,

x^{*} \in B_{i}^{- 1} (0) and A x^{*} \in K_{i}^{- 1} (0), i.e., x^{*} = J_{β}^{B_{i}} x^{*} and A x^{*} = J_{β}^{K_{i}} A x^{*} .

This implies that $x^{*} = J_{β}^{B_{i}} (x^{*} - γ A x^{*} (I - J_{β}^{K_{i}}) A x^{*})$ .

Conversely, if

x^{*}

solves (3.1), by Lemma 2.6(v), we have

〈 x^{*} - (x^{*} - γ A^{*} (I - J_{β}^{K_{i}}) A x^{*}), y - x^{*} 〉 \geq 0, \forall y \in B_{i}^{- 1} (0) .

Hence we have

〈 (I - J_{β}^{K_{i}}) A x^{*}, A y - A x^{*} 〉 \geq 0, \forall y \in B_{i}^{- 1} (0) .

(3.2)

On the other hand, by Lemma 2.6(v) again

〈 (A x^{*} - J_{β}^{K_{i}} A x^{*}, J_{β}^{K_{i}} A x^{*} - v 〉 \geq 0, \forall v \in K_{i}^{- 1} (0) .

(3.3)

Adding up (3.2) and (3.3), we have

〈 A x^{*} - J_{β}^{K_{i}} A x^{*}, J_{β}^{K_{i}} A x^{*} + A y - A x^{*} - v 〉 \geq 0, \forall y \in B_{i}^{- 1} (0), and v \in K_{i}^{- 1} (0) .

Simplifying it, we have

{∥ A x^{*} - J_{β}^{K_{i}} A x^{*} ∥}^{2} \leq 〈 A x^{*} - J_{β}^{K_{i}} A x^{*}, A y - v 〉 \geq 0, \forall y \in B_{i}^{- 1} (0), and v \in K_{i}^{- 1} (0) .

(3.4)

By the assumption that

Ω \neq \emptyset

. Taking

w \in Ω

, hence for each

i \geq 1

w \in B_{i}^{- 1} (0)

and

A w \in K_{i}^{- 1} (0)

. In (3.4), taking

y = w

and

v = A w

, then we have

{∥ A x^{*} - J_{β}^{K_{i}} A x^{*} ∥}^{2} = 0 .

This implies that $A x^{*} = J_{β}^{K_{i}} A x^{*}$ , and so $A x^{*} \in K_{i}^{- 1} (0)$ for each $i \geq 1$ . Hence from (3.1), $x^{*} = J_{β}^{B_{i}} x^{*}$ , i.e., $x^{*} \in B_{i}^{- 1} (0)$ . Hence $x^{*}$ is a solution of (GSVIP)(1.4).

This completes the proof of Lemma 3.1. □

We are now in a position to prove the following main result.

Theorem 3.2 Let

H_{1}

,

H_{2}

, A,

A^{*}

,

{B_{i}}

,

{K_{i}}

, Ω be the same as in Lemma 3.1. Let

f : H_{1} \to H_{1}

be a contractive mapping with contractive constant

k \in (0, 1)

. Let

{α_{n}}

,

{ξ_{n}}

,

{γ_{n, i}}

be the sequences in

(0, 1)

with

α_{n} + ξ_{n} + \sum_{i = 1}^{\infty} γ_{n, i} = 1

, for each

n \geq 0

. Let

{β_{i}}

be a sequence in

(0, \infty)

, and

{λ_{n, i}}

be a sequence in

(0, \frac{2}{{∥ A ∥}^{2}})

. Let

{x_{n}}

be the sequence defined by (1.11). If

Ω \neq \emptyset

and the following conditions are satisfied:

(i)

${lim}_{n \to \infty} ξ_{n} = 0$ , and $\sum_{n = 0}^{\infty} ξ_{n} = \infty$ ;
(ii)

${lim inf}_{n \to \infty} α_{n} γ_{n, i} > 0$ for each $i \geq 1$ ;
(iii)

$0 < {lim inf}_{n \to \infty} λ_{n, i} \leq {lim sup}_{n \to \infty} λ_{n, i} < \frac{2}{{∥ A ∥}^{2}}$ ,

then $x_{n} \to x^{*} \in Ω$ where $x^{*} = P_{Ω} f (x^{*})$ , where $P_{Ω}$ is the metric projection from $H_{1}$ onto Ω.

Proof (I) First we prove that ${x_{n}}$ is bounded.

In fact, letting

z \in Ω

, by Lemma 3.1, for each

i \geq 1

,

z = J_{β_{i}}^{B_{i}} [z - λ_{n, i} A^{*} (I - J_{β_{i}}^{K_{i}}) A z] .

Hence it follows from Lemma 2.7(iii) that for each

i \geq 1

and each

n \geq 1

we have

\begin{aligned} ∥ x_{n + 1} - z ∥ & = ∥ α_{n} x_{n} + ξ_{n} f (x_{n}) + \sum_{i = 1}^{\infty} γ_{n, i} J_{β_{i}}^{B_{i}} [x_{n} - λ_{n, i} A^{*} (I - J_{β_{i}}^{K_{i}}) A x_{n}] - z ∥ \\ \leq α_{n} ∥ x_{n} - z ∥ + ξ_{n} ∥ f (x_{n}) - z ∥ + \sum_{i = 1}^{\infty} γ_{n, i} ∥ J_{β_{i}}^{B_{i}} [x_{n} - λ_{n, i} A^{*} (I - J_{β_{i}}^{K_{i}}) A x_{n}] - z ∥ \\ \leq α_{n} ∥ x_{n} - z ∥ + ξ_{n} ∥ f (x_{n}) - z ∥ + \sum_{i = 1}^{\infty} γ_{n, i} ∥ J_{β_{i}}^{B_{i}} [x_{n} - λ_{n, i} A^{*} (I - J_{β_{i}}^{K_{i}}) A x_{n}] - z ∥ \\ \leq α_{n} ∥ x_{n} - z ∥ + ξ_{n} ∥ f (x_{n}) - z ∥ + \sum_{i = 1}^{\infty} γ_{n, i} ∥ x_{n} - z ∥ \\ = (1 - ξ_{n}) ∥ x_{n} - z ∥ + ξ_{n} ∥ f (x_{n}) - z ∥ \\ \leq (1 - ξ_{n}) ∥ x_{n} - z ∥ + ξ_{n} ∥ f (x_{n}) - f (z) ∥ + ξ_{n} ∥ f (z) - z ∥ \\ \leq (1 - ξ_{n} (1 - k)) ∥ x_{n} - z ∥ + \frac{ξ_{n} (1 - k)}{1 - k} ∥ f (z) - z ∥ \\ \leq max {∥ x_{n} - z ∥, \frac{1}{1 - k} ∥ f (z) - z ∥} . \end{aligned}

By induction, we can prove that

∥ x_{n} - z ∥ \leq max {∥ x_{0} - z ∥, \frac{1}{1 - k} ∥ f (z) - z ∥}, \forall n \geq 0 .

(3.5)

This implies that ${x_{n}}$ is bounded, so is ${f (x_{n})}$ .

(II) Now we prove that for each

j \geq 1

\begin{aligned} α_{n} γ_{n, j} {∥ x_{n} - J_{β_{i}}^{B_{i}} [x_{n} - λ_{n, i} A^{*} (I - J_{β_{i}}^{K_{i}}) A x_{n}] ∥}^{2} \\ \leq {∥ x_{n} - z ∥}^{2} - {∥ x_{n + 1} - z ∥}^{2} + ξ_{n} {∥ f (x_{n}) - z ∥}^{2}, for each i \geq 1 . \end{aligned}

(3.6)

Indeed, it follows from Lemma 2.3 that for any positive

j \geq 1

\begin{aligned} {∥ x_{n + 1} - z ∥}^{2} = & {∥ α_{n} x_{n} + ξ_{n} f (x_{n}) + \sum_{i = 1}^{\infty} γ_{n, i} J_{β_{i}}^{B_{i}} [x_{n} - λ_{n, i} A^{*} (I - J_{β_{i}}^{K_{i}}) A x_{n}] - z ∥}^{2} \\ \leq & α_{n} {∥ x_{n} - z ∥}^{2} + ξ_{n} {∥ f (x_{n}) - z ∥}^{2} \\ + \sum_{i = 1}^{\infty} γ_{n, i} {∥ J_{β_{i}}^{B_{i}} [x_{n} - λ_{n, i} A^{*} (I - J_{β_{i}}^{K_{i}}) A x_{n}] - z ∥}^{2} \\ - α_{n} γ_{n, j} {∥ x_{n} - J_{β_{i}}^{B_{i}} [x_{n} - λ_{n, i} A^{*} (I - J_{β_{i}}^{K_{i}}) A x_{n}] ∥}^{2} \\ \leq & (1 - ξ_{n}) {∥ x_{n} - z ∥}^{2} + ξ_{n} {∥ f (x_{n}) - z ∥}^{2} \\ - α_{n} γ_{n, j} {∥ x_{n} - J_{β_{i}}^{B_{i}} [x_{n} - λ_{n, i} A^{*} (I - J_{β_{i}}^{K_{i}}) A x_{n}] ∥}^{2} . \end{aligned}

Simplifying it, (3.6) is proved.

By the assumption that

Ω \neq \emptyset

, and it is easy to prove that Ω is closed and convex. This implies that

P_{Ω}

is well defined. Again since

P_{Ω} f : H_{1} \to Ω

is a contraction mapping with contractive constant

k \in (0, 1)

, there exists a unique

x^{*} \in Ω

such that

x^{*} = P_{Ω} f x^{*}

. Since

x^{*} \in Ω

, it solves (GSVIP) (1.4). By Lemma 3.1,

x^{*} = J_{β_{j}}^{B_{j}} (x^{*} - λ_{n, j} A^{*} (I - J_{β_{j}}^{K_{j}}) A x^{*}), \forall j \geq 1, n \geq 0 .

(3.7)

(III)

Now we prove that $x_{n} \to x^{*}$ .

In order to prove that $x_{n} \to x^{*}$ (as $n \to \infty$ ), we consider two cases.

Case 1. Assume that

{∥ x_{n} - x^{*} ∥}

is a monotone sequence. In other words, for

n_{0}

large enough,

{∥ x_{n} - x^{*} ∥}_{n \geq n_{0}}

is either nondecreasing or non-increasing. Since

{∥ x_{n} - x^{*} ∥}

is bounded,

{∥ x_{n} - x^{*} ∥}

is convergence. Again since

{lim}_{n \to \infty} ξ_{n} = 0

, and

{f (x_{n})}

is bounded, from (3.6) we get

lim_{n \to \infty} α_{n} γ_{n, j} {∥ x_{n} - J_{β_{i}}^{B_{i}} [x_{n} - λ_{n, i} A^{*} (I - J_{β_{i}}^{K_{i}}) A x_{n}] ∥}^{2} = 0 .

By condition (ii), we obtain

lim_{n \to \infty} ∥ x_{n} - J_{β_{i}}^{B_{i}} [x_{n} - λ_{n, i} A^{*} (I - J_{β_{i}}^{K_{i}}) A x_{n}] ∥ = 0 .

(3.8)

Now we prove that

\underset{n \to \infty}{lim sup} 〈 f (x^{*}) - x^{*}, x_{n} - x^{*} 〉 \leq 0 .

(3.9)

To show this inequality, we choose a subsequence

{x_{n_{k}}}

of

{x_{n}}

such that

x_{n_{k}} ⇀ w

,

λ_{n_{k}, i} \to λ_{i} \in (0, \frac{2}{{∥ A ∥}^{2}})

for each

i \geq 1

, and

\underset{n \to \infty}{lim sup} 〈 f (x^{*}) - x^{*}, x_{n} - x^{*} 〉 = lim_{n_{k} \to \infty} 〈 f (x^{*}) - x^{*}, x_{n_{k}} - x^{*} 〉 .

(3.10)

It follows from (3.8) that

\begin{aligned} ∥ J_{β_{i}}^{B_{i}} [x_{n} - λ_{i} A^{*} (I - J_{β_{i}}^{K_{i}}) A x_{n}] - x_{n} ∥ \\ \leq ∥ J_{β_{i}}^{B_{i}} [x_{n} - λ_{i} A^{*} (I - J_{β_{i}}^{K_{i}}) A x_{n}] - J_{β_{i}}^{B_{i}} [x_{n} - λ_{n, i} A^{*} (I - J_{β_{i}}^{K_{i}}) A x_{n}] ∥ \\ + ∥ J_{β_{i}}^{B_{i}} [x_{n} - λ_{n, i} A^{*} (I - J_{β_{i}}^{K_{i}}) A x_{n}] - x_{n} ∥ \\ \leq ∥ [x_{n} - λ_{i} A^{*} (I - J_{β_{i}}^{K_{i}}) A x_{n}] - [x_{n} - λ_{n, i} A^{*} (I - J_{β_{i}}^{K_{i}}) A x_{n}] ∥ \\ + ∥ J_{β_{i}}^{B_{i}} [x_{n} - λ_{n, i} A^{*} (I - J_{β_{i}}^{K_{i}}) A x_{n}] - x_{n} ∥ \\ \leq | λ_{i} - λ_{n, i} | ∥ A^{*} (I - J_{β_{i}}^{K_{i}}) A x_{n} ∥ \\ + ∥ J_{β_{i}}^{B_{i}} [x_{n} - λ_{n, i} A^{*} (I - J_{β_{i}}^{K_{i}}) A x_{n}] - x_{n} ∥ \to 0 (as n \to \infty) . \end{aligned}

For each

i \geq 1

,

J_{β_{i}}^{B_{i}} [I - λ_{i} A^{*} (I - J_{β_{i}}^{K_{i}}) A]

is a nonexpansive mapping. Thus from Lemma 2.1,

w = J_{β_{i}}^{B_{i}} [I - λ_{i} A^{*} (I - J_{β_{i}}^{K_{i}}) A] w

. By Lemma 3.1

w \in Ω

, i.e., w is a solution of (GSVIP) (1.4). Consequently we have

\begin{aligned} \underset{n \to \infty}{lim sup} 〈 f (x^{*}) - x^{*}, x_{n} - x^{*} 〉 & = lim_{n_{k} \to \infty} 〈 f (x^{*}) - x^{*}, x_{n_{k}} - x^{*} 〉 \\ = 〈 f (x^{*}) - x^{*}, w - x^{*} 〉 \leq 0 . \end{aligned}

(IV)

Finally, we prove that $x_{n} \to P_{Ω} f (x^{*})$ .

In fact, from Lemma 2.2 we have

\begin{aligned} {∥ x_{n + 1} - x^{*} ∥}^{2} \\ \leq {∥ α_{n} (x_{n} - x^{*}) + \sum_{i = 1}^{\infty} γ_{n, i} J_{β_{i}}^{B_{i}} [x_{n} - λ_{n, i} A^{*} (I - J_{β_{i}}^{K_{i}}) A x_{n}] - x^{*} ∥}^{2} \\ + 2 ξ_{n} 〈 f (x_{n}) - x^{*}, x_{n + 1} - x^{*} 〉 \\ \leq {(1 - ξ_{n})}^{2} {∥ x_{n} - x^{*} ∥}^{2} + 2 ξ_{n} 〈 f (x_{n}) - f (x^{*}), x_{n + 1} - x^{*} 〉 + 2 ξ_{n} 〈 f (x^{*}) - x^{*}, x_{n + 1} - x^{*} 〉 \\ \leq {(1 - ξ_{n})}^{2} {∥ x_{n} - x^{*} ∥}^{2} + 2 ξ_{n} k ∥ x_{n} - x^{*} ∥ ∥ x_{n + 1} - x^{*} ∥ + 2 ξ_{n} 〈 f (x^{*}) - x^{*}, x_{n + 1} - x^{*} 〉 \\ \leq {(1 - ξ_{n})}^{2} {∥ x_{n} - x^{*} ∥}^{2} + ξ_{n} k {{∥ x_{n + 1} - x^{*} ∥}^{2} + {∥ x_{n} - x^{*} ∥}^{2}} \\ + 2 ξ_{n} 〈 f (x^{*}) - x^{*}, x_{n + 1} - x^{*} 〉 . \end{aligned}

Simplifying it, we have

\begin{aligned} {∥ x_{n + 1} - x^{*} ∥}^{2} \leq & \frac{{(1 - ξ_{n})}^{2} + ξ_{n} k}{1 - ξ_{n} k} {∥ x_{n} - x^{*} ∥}^{2} + \frac{2 ξ_{n}}{1 - ξ_{n} k} 〈 f (x^{*}) - x^{*}, x_{n + 1} - x^{*} 〉 \\ \leq & \frac{1 - 2 ξ_{n} + ξ_{n} k}{1 - ξ_{n} k} {∥ x_{n} - x^{*} ∥}^{2} + \frac{ξ_{n}^{2}}{1 - ξ_{n} k} {∥ x_{n} - x^{*} ∥}^{2} \\ + \frac{2 ξ_{n}}{1 - ξ_{n} k} 〈 f (x^{*}) - x^{*}, x_{n + 1} - x^{*} 〉 \\ \leq & (1 - η_{n}) {∥ x_{n} - x^{*} ∥}^{2} + η_{n} δ_{n}, \forall n \geq 0, \end{aligned}

where $δ_{n} = \frac{ξ_{n} M}{2 (1 - k)} + \frac{1}{1 - k} 〈 f (x^{*}) - x^{*}, x_{n + 1} - x^{*} 〉$ , $M = {sup}_{n \geq 0} {∥ x_{n} - x^{*} ∥}^{2}$ , and $η_{n} = \frac{2 (1 - k) ξ_{n}}{1 - ξ_{n} k}$ . It is easy to see that $η_{n} \to 0$ , $\sum_{n = 1}^{\infty} η_{n} = \infty$ , and ${lim sup}_{n \to \infty} δ_{n} \leq 0$ . Hence by Lemma 2.4, the sequence ${x_{n}}$ converges strongly to $x^{*} = P_{Ω} f (x^{*})$ .

Case 2. Assume that

{∥ x_{n} - x^{*} ∥}

is not a monotone sequence. Then, by Lemma 2.3, we can define a sequence of positive integers:

{τ (n)}

,

n \geq n_{0}

(where

n_{0}

large enough) by

τ (n) = max {k \leq n : ∥ x_{k} - x^{*} ∥ \leq ∥ x_{k + 1} - x^{*} ∥} .

(3.11)

Clearly

{τ (n)}

is a nondecreasing sequence such that

τ (n) \to \infty

as

n \to \infty

, and for all

n \geq n_{0}

∥ x_{τ (n)} - x^{*} ∥ \leq ∥ x_{τ (n) + 1} - x^{*} ∥ .

(3.12)

Therefore

{∥ x_{τ (n)} - x^{*} ∥}

is a nondecreasing sequence. According to Case (1),

{lim}_{n \to \infty} ∥ x_{τ (n)} - x^{*} ∥ = 0

and

{lim}_{n \to \infty} ∥ x_{τ (n) + 1} - x^{*} ∥ = 0

. Hence we have

0 \leq ∥ x_{n} - x^{*} ∥ \leq max {∥ x_{n} - x^{*} ∥, ∥ x_{τ (n)} - x^{*} ∥} \leq ∥ x_{τ (n) + 1} - x^{*} ∥ \to 0, as n \to \infty .

This implies that $x_{n} \to x^{*}$ and $x^{*} = P_{Ω} f (x^{*})$ is a solution of (GSVIP) (1.4).

This completes the proof of Theorem 3.2. □

In Theorem 3.2, if $B_{i} = B$ and $K_{i} = K$ , for each $i \geq 1$ , where $B : H_{1} \to 2^{H_{1}}$ and $K : H_{2} \to 2^{H_{2}}$ are two set-valued maximal monotone mappings, then from Theorem 3.2 we have the following.

Theorem 3.3 Let

H_{1}

,

H_{2}

, A,

A^{*}

, B, K, Ω, f be the same as in Theorem 3.2. Let

{α_{n}}

,

{ξ_{n}}

,

{γ_{n}}

be the sequence in

(0, 1)

with

α_{n} + ξ_{n} + γ_{n} = 1

for each

n \geq 0

. Let

β > 0

be any given positive number, and

{λ_{n}}

be a sequence in

(0, \frac{2}{{∥ A ∥}^{2}})

. Let

{x_{n}}

be the sequence defined by

x_{n + 1} = α_{n} x_{n} + ξ_{n} f (x_{n}) + γ_{n} J_{β}^{B} [x_{n} - λ_{n} A^{*} (I - J_{β}^{K}) A x_{n}], \forall n \geq 0 .

(3.13)

If

Ω \neq \emptyset

and the following conditions are satisfied:

(i)

${lim}_{n \to \infty} ξ_{n} = 0$ , and $\sum_{n = 0}^{\infty} ξ_{n} = \infty$ ;
(ii)

${lim inf}_{n \to \infty} α_{n} γ_{n} > 0$ ;
(iii)

$0 < {lim inf}_{n \to \infty} λ_{n} \leq {lim sup}_{n \to \infty} λ_{n} < \frac{2}{{∥ A ∥}^{2}}$ ,

then $x_{n} \to x^{*} \in Ω$ where $x^{*} = P_{Ω} f (x^{*})$ .

4 Applications

In this section we shall utilize the results presented in Theorem 3.2 and Theorem 3.3 to study some problems.

4.1 Application to split optimization problem

Let

H_{1}

and

H_{2}

be two real Hilbert spaces. Let

h : H_{1} \to R

and

g : H_{2} \to R

be two proper, convex and lower semicontinuous functions, and

A : H_{1} \to H_{2}

be a linear and bounded operators. The so-called split optimization problem (SOP) is

to find x^{*} \in H_{1} such that h (x^{*}) = min_{y \in H_{1}} h (y) and g (A x^{*}) = min_{z \in H_{2}} g (z) .

(4.1)

Denote by

\partial h = B

and

\partial g = K

. It is know that

B : H_{1} \to 2^{H_{1}}

(resp.

K : H_{2} \to 2^{H_{2}}

) is a maximal monotone mapping, so we can define the resolvent

J_{β}^{B} = {(I + β B)}^{- 1}

and

J_{β}^{K} = {(I + β K)}^{- 1}

, where

β > 0

. Since

x^{*}

and

A x^{*}

is a minimum of h on

H_{1}

and g on

H_{2}

, respectively, for any given

β > 0

, we have

x^{*} \in B^{- 1} (0) = F (J_{β}^{B}), and A x^{*} \in K^{- 1} (0) = F (J_{β}^{K}) .

(4.2)

This implies that the (SOP) (4.1) is equivalent to the split variational inclusion problem (SVIP) (4.2). From Theorem 3.3 we have the following.

Theorem 4.1 Let

H_{1}

,

H_{2}

, A, B, K, h, g be the same as above. Let f,

{α_{n}}

,

{ξ_{n}}

,

{γ_{n}}

be the same as in Theorem 3.3. Let

β > 0

be any given positive number, and

{λ_{n}}

be a sequence in

(0, \frac{2}{{∥ A ∥}^{2}})

. Let

{x_{n}}

be a sequence generated by

x_{0} \in H_{1}

{\begin{matrix} y_{n} = {argmin}_{z \in H_{2}} {g (z) + \frac{1}{2 β} {∥ z - A x_{n} ∥}^{2}}, \\ z_{n} = x_{n} - λ_{n} A^{*} (A x_{n} - y_{n}), \\ w_{n} = {argmin}_{y \in H_{1}} {h (y) + \frac{1}{2 β} {∥ y - z_{n} ∥}^{2}}, \\ x_{n + 1} = α_{n} x_{n} + ξ_{n} f (x_{n}) + γ_{n} w_{n}, n \geq 0 . \end{matrix}

(4.3)

If

Ω_{1} \neq \emptyset

, the solution set of the split optimization problem (4.1), and the following conditions are satisfied:

(i)

${lim}_{n \to \infty} ξ_{n} = 0$ , and $\sum_{n = 0}^{\infty} ξ_{n} = \infty$ ;
(ii)

${lim inf}_{n \to \infty} α_{n} γ_{n} > 0$ ;
(iii)

$0 < {lim inf}_{n \to \infty} λ_{n} \leq {lim sup}_{n \to \infty} λ_{n} < \frac{2}{{∥ A ∥}^{2}}$ ,

then $x_{n} \to x^{*} \in Ω_{1}$ where $x^{*} = P_{Ω_{1}} f (x^{*})$ .

Proof Since

\partial h = B

,

\partial g : = K

, and

y_{n} = {argmin}_{z \in H_{2}} {g (z) + \frac{1}{2 β} {∥ z - A x_{n} ∥}^{2}}

, we have

0 \in {[K (z) + \frac{1}{β} (z - A x_{n})]}_{z = y_{n}}, i.e., A x_{n} \in (β K + I) (y_{n}) .

This implies that

y_{n} = J_{β}^{K} (A x_{n}) .

(4.4)

Similarly, from (4.3), we have

w_{n} = J_{β}^{B} (z_{n}) .

(4.5)

From (4.3)-(4.5), we have

w_{n} = J_{β}^{B} (x_{n} - λ_{n} A^{*} (I - J_{β}^{K}) A x_{n}) .

(4.6)

Therefore (4.3) can be rewritten as

x_{n + 1} = α_{n} x_{n} + ξ_{n} f (x_{n}) + γ_{n} J_{β}^{B} (x_{n} - λ_{n} A^{*} (I - J_{β}^{K}) A x_{n}), n \geq 0 .

(4.7)

The conclusion of Theorem 4.1 can be obtained from Theorem 3.3 immediately. □

4.2 Application to split feasibility problem

Let

C \subset H_{1}

and

Q \subset H_{2}

be two nonempty closed convex subsets and

A : H_{1} \to H_{2}

be a bounded linear operator. Now we consider the following split feasibility problem, i.e.: to find

x^{*} \in C such that A x^{*} \in Q .

(4.8)

Let

i_{C}

and

i_{Q}

be the indicator functions of C and Q defined by (1.9). Let

N_{C} (u)

be the normal cone at

u \in H_{1}

defined by

N_{C} (u) = {z \in H_{1} : 〈 z, v - u 〉 \leq 0, \forall v \in C} .

Since

i_{C}

and

i_{Q}

both are proper convex and lower semicontinuous functions on

H_{1}

and

H_{2}

, respectively, and the subdifferential

\partial i_{C}

of

i_{C}

(resp.

\partial i_{Q}

of

i_{Q}

) is a maximal monotone operator, we can define the resolvents

J_{β}^{\partial i_{C}}

of

\partial i_{C}

and

J_{β}^{\partial i_{Q}}

of

\partial i_{Q}

by

\begin{matrix} J_{β}^{\partial i_{C}} (x) = {(I + β \partial i_{C})}^{- 1} (x), \forall x \in H_{1}, \\ J_{β}^{\partial i_{Q}} (x) = {(I + β \partial i_{Q})}^{- 1} (x), \forall x \in H_{2}, \end{matrix}

where

β > 0

. By definition, we know that

\begin{aligned} \partial i_{C} (x) & = {z \in H_{1} : i_{C} (x) + 〈 z, y - x 〉 \leq i_{C} (y), \forall y \in H_{1}} \\ = {z \in H_{1} : 〈 z, y - x 〉 \leq 0, \forall y \in C} = N_{C} (x), x \in C . \end{aligned}

Hence, for each

β > 0

, we have

\begin{aligned} u = J_{β}^{\partial i_{C}} (x) \Leftrightarrow x - u \in β N_{C} (u) \\ \Leftrightarrow 〈 x - u, y - u 〉 \leq 0, \forall y \in C \Leftrightarrow u = P_{C} (x) . \end{aligned}

This implies that

J_{β}^{\partial i_{C}} = P_{C}

. Similarly

J_{β}^{\partial i_{Q}} = P_{Q}

. Taking

h (x) = i_{C} (x)

and

g (x) = i_{Q} (x)

in (4.1), then the (SFP) (4.8) is equivalent to the following split optimization problem:

to find x^{*} \in H_{1} such that i_{C} (x^{*}) = min_{y \in H_{1}} i_{C} (y) and i_{Q} (A x^{*}) = min_{z \in H_{2}} i_{Q} (z) .

(4.9)

Hence, the following result can be obtained from Theorem 4.1 immediately.

Theorem 4.2 Let

H_{1}

,

H_{2}

, A,

A^{*}

,

i_{C}

,

i_{Q}

be the same as above. Let f,

{α_{n}}

,

{ξ_{n}}

,

{γ_{n}}

be the same as in Theorem 4.1. Let

{λ_{n}}

be a sequence in

(0, \frac{2}{{∥ A ∥}^{2}})

. Let

{x_{n}}

be the sequence defined by

x_{n + 1} = α_{n} x_{n} + ξ_{n} f (x_{n}) + γ_{n} P_{C} [x_{n} - λ_{n} A^{*} (I - P_{Q}) A x_{n}], \forall n \geq 0 .

(4.10)

If the solution set of the split optimization problem (4.4)

Ω_{2} \neq \emptyset

, and the following conditions are satisfied:

(i)

${lim}_{n \to \infty} ξ_{n} = 0$ , and $\sum_{n = 0}^{\infty} ξ_{n} = \infty$ ;
(ii)

${lim inf}_{n \to \infty} α_{n} γ_{n} > 0$ ;
(iii)

$0 < {lim inf}_{n \to \infty} λ_{n} \leq {lim sup}_{n \to \infty} λ_{n} < \frac{2}{{∥ A ∥}^{2}}$ ,

then $x_{n} \to x^{*} \in Ω_{2}$ where $x^{*} = P_{Ω_{2}} f (x^{*})$ .

Remark 4.3 Theorem 4.2 extends and improves the main results in Censor and Elfving [1] and Byrne [2].

Declarations

Acknowledgements

The authors would like to express their thanks to the referees and the editors for their kind and helpful comments and advice. This work was supported by the National Natural Science Foundation of China (Grant No. 11361070).

Authors’ Affiliations

(1)

College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan, 650221, P.R. China

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