Projection methods of iterative solutions in Hilbert spaces

Feng Gu; Jing Lu

doi:10.1186/1687-1812-2012-162

Research

Open Access

Projection methods of iterative solutions in Hilbert spaces

Feng Gu¹ and
Jing Lu¹Email author

Fixed Point Theory and Applications20122012:162

https://doi.org/10.1186/1687-1812-2012-162

Received: 27 June 2012

Accepted: 6 September 2012

Published: 25 September 2012

Abstract

In this paper, zero points of the sum of two monotone mappings, solutions of a monotone variational inequality, and fixed points of a nonexpansive mapping are investigated based on a hybrid projection iterative algorithm. Strong convergence of the purposed iterative algorithm is obtained in the framework of real Hilbert spaces without any compact assumptions.

MSC:47H05, 47H09, 47J25, 90C33.

Keywords

fixed pointmonotone operatornonexpansive mappingvariational inequalityzero point

1 Introduction and preliminaries

Throughout this paper, we always assume that H is a real Hilbert space with an inner product $〈 \cdot, \cdot 〉$ and a norm $∥ \cdot ∥$ . Let C be a nonempty, closed, and convex subset of H. Let $S : C \to C$ be a nonlinear mapping. $F (S)$ stands for the fixed point set of S; that is, $F (S) : = {x \in C : x = T x}$ .

Recall that S is said to be nonexpansive iff

∥ S x - S y ∥ \leq ∥ x - y ∥, \forall x, y \in C .

If C is a bounded, closed, and convex subset of H, then $F (S)$ is not empty, closed, and convex; see [1].

Let

A : C \to H

be a mapping. Recall that A is said to be inverse-strongly monotone iff there exists a constant

α > 0

such that

〈 A x - A y, x - y 〉 \geq α {∥ A x - A y ∥}^{2}, \forall x, y \in C .

For such a case, A is also said to be α-inverse-strongly monotone.

A is said to be monotone iff

〈 A x - A y, x - y 〉 \geq 0, \forall x, y \in C .

Recall that the classical variational inequality is to find an

x \in C

such that

〈 A x, y - x 〉 \geq 0, \forall y \in C .

(1.1)

In this paper, we use $VI (C, A)$ to denote the solution set of (1.1). It is known that $x \in C$ is a solution of (1.1) if and only if x is a fixed point of the mapping ${Proj}_{C} (I - r A)$ , where $r > 0$ is a constant, I stands for the identity mapping, and ${Proj}_{C}$ stands for the metric projection from H onto C. If A is α-inverse-strongly monotone and $r \in (0, 2 α]$ , then the mapping ${Proj}_{C} (I - r A)$ is nonexpansive; see [2] for more details. It follows that $VI (C, A)$ is closed and convex.

Monotone variational inequality theory has emerged as a fascinating branch of mathematical and engineering sciences with a wide range of applications in industry, finance, economics, ecology, social, regional, pure, and applied sciences. In recent years, much attention has been given to developing efficient numerical methods for treating solution problems of monotone variational inequality. The gradient-projection method is a powerful tool for solving constrained convex optimization problems and has extensively been studied; see [3–5] and the references therein. It has recently been applied to solving split feasibility problems which find applications in image reconstructions and the intensity modulated radiation theory; see [6–9] and the references therein. However, the gradient-projection method requires the operator to be strongly monotone and Lipschitz continuous. These strong conditions rule out many applications. Extra gradient-projection method which was first introduce by Korpelevich [10] in the finite dimensional Euclidean space has been studied recently for relaxing the strong monotonicity of operators; see [11–13] and the references therein.

Recall that a set-valued mapping $M : H ⇉ H$ is said to be monotone iff, for all $x, y \in H$ , $f \in M x$ and $g \in M y$ imply $〈 x - y, f - g 〉 > 0$ . A monotone mapping $M : H ⇉ H$ is maximal iff the graph $Graph (M)$ of R is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping M is maximal if and only if, for any $(x, f) \in H \times H$ , $〈 x - y, f - g 〉 \geq 0$ , for all $(y, g) \in Graph (M)$ implies $f \in R x$ .

For a maximal monotone operator M on H and $r > 0$ , we may define the single-valued resolvent $J_{r} : H \to D (M)$ , where $D (M)$ denotes the domain of M. It is known that $J_{r}$ is firmly nonexpansive, and $M^{- 1} (0) = F (J_{r})$ , where $F (J_{r}) : = {x \in D (M) : x = J_{r} x}$ and $M^{- 1} (0) : {x \in H : 0 \in M x}$ .

Recently, variational inequalities, fixed point problems, and zero point problems have been investigated by many authors based on iterative methods; see, for example, [14–32] and the references therein. In this paper, zero point problems of the sum of a maximal monotone operator and an inverse-strongly monotone mapping, solution problems of a monotone variational inequality, and fixed point problems of a nonexpansive mapping are investigated. A hybrid iterative algorithm is considered for analyzing the convergence of iterative sequences. Strong convergence theorems are established in the framework of real Hilbert spaces without any compact assumptions.

In order to prove our main results, we also need the following definitions and lemmas.

Lemma 1.1 Let C be a nonempty, closed, and convex subset of H. Then the following inequality holds:

{∥ x - {Proj}_{C} x ∥}^{2} + {∥ y - {Proj}_{C} ∥}^{2} \leq {∥ x - y ∥}^{2}, \forall x \in H, y \in C .

Lemma 1.2[1]

Let C be a nonempty, closed, and convex subset of H. Let $S : C \to C$ be a nonexpansive mapping. Then the mapping $I - S$ is demiclosed at zero, that is, if ${x_{n}}$ is a sequence in C such that $x_{n} ⇀ \bar{x}$ and $x_{n} - S x_{n} \to 0$ , then $\bar{x} \in F (S)$ .

Lemma 1.3 Let C be a nonempty, closed, and convex subset of H, $B : C \to H$ be a mapping, and $M : H ⇉ H$ be a maximal monotone operator. Then $F (J_{r} (I - s B)) = {(B + M)}^{- 1} (0)$ .

Proof Notice that

\begin{aligned} p \in F (J_{r} (I - s B)) ⟺ p = J_{r} (I - s B) p ⟺ p - s B p \in p + s M p \\ ⟺ 0 \in {(B + M)}^{- 1} (0) ⟺ p \in {(B + M)}^{- 1} (0) . \end{aligned}

This completes the proof. □

Lemma 1.4[33]

Let C be a nonempty, closed, and convex subset of H,

A : C \to H

be a Lipschitz monotone mapping, and

N_{C} x

be the normal cone to C at

x \in C

; that is,

N_{C} x = {y \in H : 〈 x - u, y 〉, \forall u \in C}

. Define

W x = {\begin{matrix} A x + N_{C} x, & x \in C, \\ \emptyset, & x \notin C . \end{matrix}

Then W is maximal monotone and $0 \in W x$ if and only if $x \in VI (C, A)$ .

2 Main results

Now, we are in a position to give our main results.

Theorem 2.1 Let C be a nonempty, closed, and convex subset of H. Let

S : C \to C

be a nonexpansive mapping with a nonempty fixed point set,

A : C \to H

be an α-Lipschitz continuous and monotone mapping, and

B : C \to H

be a β-inverse-strongly monotone mapping. Let

M : H ⇉ H

be a maximal monotone operator such that

D (M) \subset C

. Assume that

F : = F (S) \cap {(B + M)}^{- 1} (0) \cap VI (C, A)

is not empty. Let

{x_{n}}

be a sequence generated by the following iterative process:

{\begin{matrix} x_{1} \in C, \\ C_{1} = C, \\ z_{n} = {Proj}_{C} (J_{s_{n}} (x_{n} - s_{n} B x_{n}) - r_{n} A J_{s_{n}} (x_{n} - s_{n} B x_{n})), \\ y_{n} = α_{n} x_{n} + (1 - α_{n}) S {Proj}_{C} (J_{s_{n}} (x_{n} - s_{n} B x_{n}) - r_{n} A z_{n}), \\ C_{n + 1} = {v \in C_{n} : ∥ y_{n} - v ∥ \leq ∥ x_{n} - v ∥}, \\ x_{n + 1} = {Proj}_{C_{n + 1}} x_{1}, n \geq 0, \end{matrix}

(2.1)

where

J_{s_{n}} = {(I + s_{n} M)}^{- 1}

,

{r_{n}}

is a sequence in

(0, \frac{1}{α})

,

{s_{n}}

is a sequence in

(0, 2 β)

, and

{α_{n}}

is a sequence in

(0, 1)

. Assume that the following restrictions are satisfied:

(a)

$0 < a \leq r_{n} \leq b < \frac{1}{α}$ ;
(b)

$0 < c \leq s_{n} \leq d < 2 β$ ;
(c)

$0 \leq α_{n} \leq e < 1$ ,

where a, b, c, d, and e are real constants. Then the sequence ${x_{n}}$ converges strongly to ${Proj}_{F} x_{1}$ .

Proof First, we show that

C_{n}

is closed and convex for each

n \geq 1

. From the assumption, we see that

C_{1} = C

is closed and convex. Suppose that

C_{m}

is closed and convex for some

m \geq 1

. We show that

C_{m + 1}

is closed and convex for the same m. Let

v_{1}, v_{2} \in C_{m + 1}

and

v = t v_{1} + (1 - t) v_{2}

, where

t \in (0, 1)

. Notice that

∥ y_{m} - v ∥ \leq ∥ x_{m} - v ∥

is equivalent to

{∥ y_{m} ∥}^{2} - {∥ x_{m} ∥}^{2} - 2 〈 v, y_{m} - x_{m} 〉 \geq 0 .

It is clear that $v \in C_{m + 1}$ . This shows that $C_{n}$ is closed and convex for each $n \geq 1$ .

Next, we show that

F \subset C_{n}

for each

n \geq 1

. Put

u_{n} = {Proj}_{C} (v_{n} - r_{n} A z_{n})

, where

v_{n} = J_{s_{n}} (x_{n} - s_{n} B x_{n})

. From the assumption, we see that

F \subset C = C_{1}

. Suppose that

F \subset C_{m}

for some

m \geq 1

. For any

p \in F \subset C_{m}

, we see from Lemma 1.1 that

\begin{aligned} {∥ u_{m} - p ∥}^{2} \leq & {∥ v_{m} - r_{m} A z_{m} - p ∥}^{2} - {∥ v_{m} - r_{m} A z_{m} - u_{m} ∥}^{2} \\ = & {∥ v_{m} - p ∥}^{2} - {∥ v_{m} - u_{m} ∥}^{2} + 2 r_{m} 〈 A z_{m}, p - u_{m} 〉 \\ = & {∥ v_{m} - p ∥}^{2} - {∥ v_{m} - u_{m} ∥}^{2} + 2 r_{m} (〈 A z_{m} - A p, p - z_{m} 〉 + 〈 A p, p - z_{m} 〉 \\ + 〈 A z_{m}, z_{m} - u_{m} 〉) \\ \leq & {∥ v_{m} - p ∥}^{2} - {∥ v_{m} - z_{m} + z_{m} - u_{m} ∥}^{2} + 2 r_{m} 〈 A z_{m}, z_{m} - u_{m} 〉 \\ = & {∥ v_{m} - p ∥}^{2} - {∥ v_{m} - z_{m} ∥}^{2} - {∥ z_{m} - u_{m} ∥}^{2} \\ + 2 〈 v_{m} - z_{m} - r_{m} A z_{m}, u_{m} - z_{m} 〉 . \end{aligned}

(2.2)

Notice that A is Lipschitz continuous. In view of

z_{m} = {Proj}_{C} (v_{m} - r_{m} A v_{m})

, we find that

\begin{aligned} 〈 v_{m} - z_{m} - r_{m} A z_{m}, u_{m} - z_{m} 〉 \\ = 〈 v_{m} - z_{m} - r_{m} A v_{m}, u_{m} - z_{m} 〉 + 〈 r_{m} A v_{m} - r_{m} A z_{m}, u_{m} - z_{m} 〉 \\ \leq r_{m} α ∥ v_{m} - z_{m} ∥ ∥ u_{m} - z_{m} ∥ . \end{aligned}

(2.3)

Substituting (2.3) into (2.2), we obtain that

\begin{aligned} {∥ u_{m} - p ∥}^{2} & \leq {∥ v_{m} - p ∥}^{2} - {∥ v_{m} - z_{m} ∥}^{2} - {∥ z_{m} - u_{m} ∥}^{2} + 2 r_{m} α ∥ v_{m} - z_{m} ∥ ∥ u_{m} - z_{m} ∥ \\ \leq {∥ v_{m} - p ∥}^{2} - (1 - r_{m}^{2} α^{2}) {∥ v_{m} - z_{m} ∥}^{2} . \end{aligned}

(2.4)

This in turn implies from restriction (a) that

\begin{aligned} {∥ y_{m} - p ∥}^{2} & \leq α_{m} {∥ x_{m} - p ∥}^{2} + (1 - α_{m}) {∥ S u_{m} - p ∥}^{2} \\ \leq α_{m} {∥ x_{m} - p ∥}^{2} + (1 - α_{m}) {∥ u_{m} - p ∥}^{2} \\ \leq α_{m} {∥ x_{m} - p ∥}^{2} + (1 - α_{m}) ({∥ v_{m} - p ∥}^{2} - (1 - r_{m}^{2} α^{2}) {∥ v_{m} - z_{m} ∥}^{2}) \\ \leq {∥ x_{m} - p ∥}^{2} - (1 - α_{m}) (1 - r_{m}^{2} α^{2}) {∥ v_{m} - z_{m} ∥}^{2} \\ \leq {∥ x_{m} - p ∥}^{2} . \end{aligned}

(2.5)

This shows that

p \in C_{m + 1}

. This proves that

F \subset C_{n}

for each

n \geq 1

. Note

x_{n} = {Proj}_{C_{n}} x_{1}

. For each

p \in F \subset C_{n}

, we have

∥ x_{1} - x_{n} ∥ \leq ∥ x_{1} - p ∥

. Since B is inverse-strongly monotone, we see from Lemma 1.3 that

{(B + M)}^{- 1} (0)

is closed and convex. Since A is Lipschitz continuous, we find that

VI (C, A)

is close and convex. This proves that

F

is closed and convex. It follows that

∥ x_{1} - x_{n} ∥ \leq ∥ x_{1} - {Proj}_{F} x_{1} ∥ .

(2.6)

This implies that

{x_{n}}

is bounded. Since

x_{n} = {Proj}_{C_{n}} x_{1}

and

x_{n + 1} = {Proj}_{C_{n + 1}} x_{1} \in C_{n + 1} \subset C_{n}

, we have

\begin{aligned} 0 & \leq 〈 x_{1} - x_{n}, x_{n} - x_{n + 1} 〉 \\ = 〈 x_{1} - x_{n}, x_{n} - x_{1} + x_{1} - x_{n + 1} 〉 \\ \leq - {∥ x_{1} - x_{n} ∥}^{2} + ∥ x_{1} - x_{n} ∥ ∥ x_{1} - x_{n + 1} ∥ . \end{aligned}

It follows that

∥ x_{n} - x_{1} ∥ \leq ∥ x_{n + 1} - x_{1} ∥ .

This proves that

{lim}_{n \to \infty} ∥ x_{n} - x_{1} ∥

exists. Notice that

\begin{aligned} {∥ x_{n} - x_{n + 1} ∥}^{2} \\ = {∥ x_{n} - x_{1} ∥}^{2} + 2 〈 x_{n} - x_{1}, x_{1} - x_{n + 1} 〉 + {∥ x_{1} - x_{n + 1} ∥}^{2} \\ = {∥ x_{n} - x_{1} ∥}^{2} - 2 {∥ x_{n} - x_{1} ∥}^{2} + 2 〈 x_{n} - x_{1}, x_{n} - x_{n + 1} 〉 + {∥ x_{1} - x_{n + 1} ∥}^{2} \\ \leq {∥ x_{1} - x_{n + 1} ∥}^{2} - {∥ x_{n} - x_{1} ∥}^{2} . \end{aligned}

It follows that

lim_{n \to \infty} ∥ x_{n} - x_{n + 1} ∥ = 0 .

(2.7)

In view of

x_{n + 1} = {Proj}_{C_{n + 1}} x_{1} \in C_{n + 1}

, we see that

∥ y_{n} - x_{n + 1} ∥ \leq ∥ x_{n} - x_{n + 1} ∥ .

This implies that

∥ y_{n} - x_{n} ∥ \leq ∥ y_{n} - x_{n + 1} ∥ + ∥ x_{n} - x_{n + 1} ∥ \leq 2 ∥ x_{n} - x_{n + 1} ∥ .

From (2.7), we find that

lim_{n \to \infty} ∥ x_{n} - y_{n} ∥ = 0 .

(2.8)

Since B is β-inverse-strongly monotone, we see from restriction (b) that

\begin{aligned} {∥ (I - s_{n} B) x - (I - s_{n} B) y ∥}^{2} & = {∥ x - y ∥}^{2} - 2 s_{n} 〈 x - y, B x - B y 〉 + s_{n}^{2} {∥ B x - B y ∥}^{2} \\ \leq {∥ x - y ∥}^{2} - s_{n} (2 β - s_{n}) {∥ B x - B y ∥}^{2} \\ \leq {∥ x - y ∥}^{2}, \forall x, y \in C . \end{aligned}

This implies from (2.5) that

\begin{aligned} {∥ y_{n} - p ∥}^{2} & \leq α_{n} {∥ x_{n} - p ∥}^{2} + (1 - α_{n}) {∥ v_{n} - p ∥}^{2} \\ = α_{n} {∥ x_{n} - p ∥}^{2} + (1 - α_{n}) {∥ J_{s_{n}} (x_{n} - s_{n} B x_{n}) - J_{s_{n}} (p - s_{n} B p) ∥}^{2} \\ \leq {∥ x_{n} - p ∥}^{2} - (1 - α_{n}) s_{n} (2 β - s_{n}) {∥ B x_{n} - B p ∥}^{2} . \end{aligned}

It follows that

\begin{aligned} (1 - α_{n}) s_{n} (2 β - s_{n}) {∥ B x_{n} - B p ∥}^{2} & \leq {∥ x_{n} - p ∥}^{2} - {∥ y_{n} - p ∥}^{2} \\ \leq ∥ x_{n} - y_{n} ∥ (∥ x_{n} - p ∥ + ∥ y_{n} - p ∥) . \end{aligned}

In view of restrictions (b) and (c), we find from (2.8) that

lim_{n \to \infty} ∥ B x_{n} - B p ∥ = 0 .

(2.9)

Since

J_{s_{n}}

is firmly nonexpansive, we find that

\begin{aligned} {∥ v_{n} - p ∥}^{2} = & {∥ J_{s_{n}} (x_{n} - s_{n} B x_{n}) - J_{s_{n}} (p - s_{n} B p) ∥}^{2} \\ \leq & 〈 v_{n} - p, (x_{n} - s_{n} B x_{n}) - (p - s_{n} B p) 〉 \\ = & \frac{1}{2} ({∥ v_{n} - p ∥}^{2} + {∥ (x_{n} - s_{n} B x_{n}) - (p - s_{n} B p) ∥}^{2} \\ - {∥ (v_{n} - p) - ((x_{n} - s_{n} B x_{n}) - (p - s_{n} B p)) ∥}^{2}) \\ \leq & \frac{1}{2} ({∥ v_{n} - p ∥}^{2} + {∥ x_{n} - p ∥}^{2} - {∥ v_{n} - x_{n} + s_{n} (B x_{n} - B p) ∥}^{2}) \\ = & \frac{1}{2} ({∥ v_{n} - p ∥}^{2} + {∥ x_{n} - p ∥}^{2} - {∥ v_{n} - x_{n} ∥}^{2} - s_{n}^{2} {∥ B x_{n} - B p ∥}^{2} \\ - 2 s_{n} 〈 v_{n} - x_{n}, B x_{n} - B p 〉) \\ \leq & \frac{1}{2} ({∥ v_{n} - p ∥}^{2} + {∥ x_{n} - p ∥}^{2} - {∥ v_{n} - x_{n} ∥}^{2} + 2 s_{n} ∥ v_{n} - x_{n} ∥ ∥ B x_{n} - B p ∥) . \end{aligned}

This in turn implies that

{∥ v_{n} - p ∥}^{2} \leq {∥ x_{n} - p ∥}^{2} - {∥ v_{n} - x_{n} ∥}^{2} + 2 s_{n} ∥ v_{n} - x_{n} ∥ ∥ B x_{n} - B p ∥ .

(2.10)

Combining (2.5) with (2.10), we arrive at

\begin{aligned} {∥ y_{n} - p ∥}^{2} & \leq α_{n} {∥ x_{n} - p ∥}^{2} + (1 - α_{n}) {∥ v_{n} - p ∥}^{2} \\ \leq {∥ x_{n} - p ∥}^{2} - (1 - α_{n}) {∥ v_{n} - x_{n} ∥}^{2} + 2 s_{n} ∥ v_{n} - x_{n} ∥ ∥ B x_{n} - B p ∥ . \end{aligned}

It follows that

\begin{aligned} (1 - α_{n}) {∥ v_{n} - x_{n} ∥}^{2} & \leq {∥ x_{n} - p ∥}^{2} - {∥ y_{n} - p ∥}^{2} + 2 s_{n} ∥ v_{n} - x_{n} ∥ ∥ B x_{n} - B p ∥ \\ \leq ∥ x_{n} - y_{n} ∥ (∥ x_{n} - p ∥ + ∥ y_{n} - p ∥) + 2 s_{n} ∥ v_{n} - x_{n} ∥ ∥ B x_{n} - B p ∥ . \end{aligned}

In view of (2.8) and (2.9), we see from restriction (c) that

lim_{n \to \infty} ∥ v_{n} - x_{n} ∥ = 0 .

(2.11)

On the other hand, we find from (2.5) that

\begin{aligned} (1 - α_{n}) (1 - r_{n}^{2} α^{2}) {∥ v_{n} - z_{n} ∥}^{2} & \leq {∥ x_{n} - p ∥}^{2} - {∥ y_{n} - p ∥}^{2} \\ \leq ∥ x_{n} - y_{n} ∥ (∥ x_{n} - p ∥ + ∥ y_{n} - p ∥) . \end{aligned}

In view of restrictions (a) and (c), we obtain from (2.8) that

lim_{n \to \infty} ∥ v_{n} - z_{n} ∥ = 0 .

(2.12)

Notice that

\begin{aligned} {∥ u_{n} - z_{n} ∥}^{2} & = {∥ P_{C} (v_{n} - r_{n} A z_{n}) - P_{C} (v_{n} - r_{n} A v_{n}) ∥}^{2} \\ \leq {∥ (v_{n} - r_{n} A z_{n}) - (v_{n} - r_{n} A v_{n}) ∥}^{2} \\ \leq r_{n}^{2} α^{2} {∥ z_{n} - v_{n} ∥}^{2} . \end{aligned}

Thanks to (2.12), we arrive at

lim_{n \to \infty} ∥ u_{n} - z_{n} ∥ = 0 .

(2.13)

Notice that

\begin{aligned} ∥ x_{n} - S x_{n} ∥ & \leq ∥ x_{n} - S u_{n} ∥ + ∥ S u_{n} - S x_{n} ∥ \\ \leq \frac{∥ x_{n} - y_{n} ∥}{1 - α_{n}} + ∥ u_{n} - x_{n} ∥ \\ \leq \frac{∥ x_{n} - y_{n} ∥}{1 - α_{n}} + ∥ u_{n} - z_{n} ∥ + ∥ z_{n} - v_{n} ∥ + ∥ v_{n} - x_{n} ∥ . \end{aligned}

In view of (2.8), (2.11), (2.12), and (2.13), we find from restriction (c) that

lim_{n \to \infty} ∥ x_{n} - S x_{n} ∥ = 0 .

Since ${x_{n}}$ is bounded, we find that there exists a subsequence ${x_{n_{i}}}$ of ${x_{n}}$ such that $x_{n_{i}} ⇀ q \in C$ . From Lemma 1.2, we easily conclude that $q \in F (S)$ .

Now, we are in a position to show that

x \in VI (C, A)

. Define

W x = {\begin{matrix} A x + N_{C} x, & x \in C, \\ \emptyset, & x \notin C . \end{matrix}

For any given

(x, y) \in G (W)

, we have

y - A x \in N_{C} x

. Since

u_{n} \in C

, we see from the definition of

N_{C}

〈 x - u_{n}, y - A x 〉 \geq 0 .

(2.14)

In view of

u_{n} = P_{C} (v_{n} - r_{n} A z_{n})

, we obtain that

〈 x - u_{n}, u_{n} + r_{n} A z_{n} - v_{n} 〉 \geq 0

and hence

〈 x - u_{n}, \frac{u_{n} - v_{n}}{r_{n}} + A z_{n} 〉 \geq 0 .

(2.15)

In view of (2.14) and (2.15), we find that

\begin{aligned} 〈 x - u_{n_{i}}, y 〉 \geq & 〈 x - u_{n_{i}}, A x 〉 \\ \geq & 〈 x - u_{n_{i}}, A x 〉 - 〈 x - u_{n_{i}}, \frac{u_{n_{i}} - v_{n_{i}}}{r_{n_{i}}} + A z_{n_{i}} 〉 \\ = & 〈 x - u_{n_{i}}, A x - A u_{n_{i}} 〉 + 〈 x - u_{n_{i}}, A u_{n_{i}} - A z_{n_{i}} 〉 \\ - 〈 x - u_{n_{i}}, \frac{u_{n_{i}} - v_{n_{i}}}{r_{n_{i}}} 〉 \\ \geq & 〈 x - u_{n_{i}}, A u_{n_{i}} - A z_{n_{i}} 〉 - 〈 x - u_{n_{i}}, \frac{u_{n_{i}} - v_{n_{i}}}{r_{n_{i}}} 〉 . \end{aligned}

(2.16)

Notice that

∥ x_{n} - u_{n} ∥ \leq ∥ x_{n} - v_{n} ∥ + ∥ v_{n} - z_{n} ∥ + ∥ z_{n} - u_{n} ∥ .

It follows from (2.11), (2.12), and (2.13) that

lim_{n \to \infty} ∥ x_{n} - u_{n} ∥ = 0 .

Since A is Lipschitz continuous, we find from (2.16) that

〈 x - q, y 〉 \geq 0 .

Since W is maximal monotone, we conclude that $q \in W^{- 1} (0)$ . This proves that $q \in VI (C, A)$ .

Finally, we prove that

q \in {(B + M)}^{- 1} (0)

. Notice that

x_{n} - s_{n} B x_{n} \in v_{n} + s_{n} M v_{n};

that is,

\frac{x_{n} - v_{n}}{s_{n}} - B x_{n} \in M v_{n} .

(2.17)

Let

μ \in M ν

. Since M is monotone, we find from (2.17)

〈 \frac{x_{n} - v_{n}}{s_{n}} - B x_{n} - μ, v_{n} - ν 〉 \geq 0 .

In view of the restriction (b), we see that

〈 - B q - μ, q - ν 〉 \geq 0 .

This implies that $- B q \in M q$ , that is, $q \in {(B + M)}^{- 1} (0)$ . This completes $q \in F$ . Assume that there exists another subsequence ${x_{n_{j}}}$ of ${x_{n}}$ which converges weakly to $q^{'} \in F$ . We can easily conclude from Opial’s condition that $q = q^{'}$ .

Finally, we show that

q = {Proj}_{F} x_{1}

and

{x_{n}}

converges strongly to q. This completes the proof of Theorem 2.1. In view of the weak lower semicontinuity of the norm, we obtain from (2.6) that

\begin{aligned} ∥ x_{1} - {Proj}_{F} x_{1} ∥ & \leq ∥ x_{1} - q ∥ \leq \underset{n \to \infty}{lim inf} ∥ x_{1} - x_{n} ∥ \\ \leq \underset{n \to \infty}{lim sup} ∥ x_{1} - x_{n} ∥ \leq ∥ x_{1} - {Proj}_{F} x_{1} ∥, \end{aligned}

which yields that ${lim}_{n \to \infty} ∥ x_{1} - x_{n} ∥ = ∥ x_{1} - {Proj}_{F} x_{1} ∥ = ∥ x_{1} - q ∥$ . It follows that ${x_{n}}$ converges strongly to ${Proj}_{F} x_{1}$ . This completes the proof. □

If $B = 0$ , then Theorem 2.1 is reduced to the following.

Corollary 2.2 Let C be a nonempty, closed, and convex subset of H. Let

S : C \to C

be a nonexpansive mapping with a nonempty fixed point set and

A : C \to H

be an α-Lipschitz continuous and monotone mapping. Let

M : H ⇉ H

be a maximal monotone operator such that

D (M) \subset C

. Assume that

F : = F (S) \cap M^{- 1} (0) \cap VI (C, A)

is not empty. Let

{x_{n}}

be a sequence generated by the following iterative process:

{\begin{matrix} x_{1} \in C, \\ C_{1} = C, \\ z_{n} = {Proj}_{C} (J_{s_{n}} x_{n} - r_{n} A J_{s_{n}} x_{n}), \\ y_{n} = α_{n} x_{n} + (1 - α_{n}) S {Proj}_{C} (J_{s_{n}} x_{n} - r_{n} A z_{n}), \\ C_{n + 1} = {v \in C_{n} : ∥ y_{n} - v ∥ \leq ∥ x_{n} - v ∥}, \\ x_{n + 1} = {Proj}_{C_{n + 1}} x_{1}, n \geq 0, \end{matrix}

where

J_{s_{n}} = {(I + s_{n} M)}^{- 1}

,

{r_{n}}

is a sequence in

(0, \frac{1}{α})

,

{s_{n}}

is a sequence in

(0, + \infty)

, and

{α_{n}}

is a sequence in

(0, 1)

. Assume that the following restrictions are satisfied:

(a)

$0 < a \leq r_{n} \leq b < \frac{1}{α}$ ;
(b)

$0 < c \leq s_{n} < \infty$ ;
(c)

$0 \leq α_{n} \leq d < 1$ ,

where a, b, c, and d are real constants. Then the sequence ${x_{n}}$ converges strongly to ${Proj}_{F} x_{1}$ .

If $M = 0$ , then $J_{s_{n}} = I$ . Corollary 2.2 is reduced to the following.

Corollary 2.3 Let C be a nonempty, closed, and convex subset of H. Let

S : C \to C

be a nonexpansive mapping with a nonempty fixed point set and

A : C \to H

be an α-Lipschitz continuous and monotone mapping. Assume that

F : = F (S) \cap VI (C, A)

is not empty. Let

{x_{n}}

be a sequence generated by the following iterative process:

{\begin{matrix} x_{1} \in C, \\ C_{1} = C, \\ z_{n} = {Proj}_{C} (x_{n} - r_{n} A x_{n}), \\ y_{n} = α_{n} x_{n} + (1 - α_{n}) S {Proj}_{C} (x_{n} - r_{n} A z_{n}), \\ C_{n + 1} = {v \in C_{n} : ∥ y_{n} - v ∥ \leq ∥ x_{n} - v ∥}, \\ x_{n + 1} = {Proj}_{C_{n + 1}} x_{1}, n \geq 0, \end{matrix}

where

{r_{n}}

is a sequence in

(0, \frac{1}{α})

, and

{α_{n}}

is a sequence in

(0, 1)

. Assume that the following restrictions are satisfied:

(a)

$0 < a \leq r_{n} \leq b < \frac{1}{α}$ ;
(b)

$0 \leq α_{n} \leq c < 1$ ,

where a, b, and c are real constants. Then the sequence ${x_{n}}$ converges strongly to ${Proj}_{F} x_{1}$ .

If $A = 0$ , then Theorem 2.1 is reduced to the following.

Corollary 2.4 Let C be a nonempty, closed, and convex subset of H. Let

S : C \to C

be a nonexpansive mapping with a nonempty fixed point set and

B : C \to H

be a β-inverse-strongly monotone mapping. Let

M : H ⇉ H

be a maximal monotone operator such that

D (M) \subset C

. Assume that

F : = F (S) \cap {(B + M)}^{- 1} (0)

is not empty. Let

{x_{n}}

be a sequence generated by the following iterative process:

{\begin{matrix} x_{1} \in C, \\ C_{1} = C, \\ y_{n} = α_{n} x_{n} + (1 - α_{n}) S J_{s_{n}} (x_{n} - s_{n} B x_{n}), \\ C_{n + 1} = {v \in C_{n} : ∥ y_{n} - v ∥ \leq ∥ x_{n} - v ∥}, \\ x_{n + 1} = {Proj}_{C_{n + 1}} x_{1}, n \geq 0, \end{matrix}

where

J_{s_{n}} = {(I + s_{n} M)}^{- 1}

,

{s_{n}}

is a sequence in

(0, 2 β)

, and

{α_{n}}

is a sequence in

(0, 1)

. Assume that the following restrictions are satisfied:

(a)

$0 < a \leq s_{n} \leq b < 2 β$ ;
(b)

$0 \leq α_{n} \leq c < 1$ ,

where a, b, and c are real constants. Then the sequence ${x_{n}}$ converges strongly to ${Proj}_{F} x_{1}$ .

Let

f : H \to (- \infty, + \infty]

be a proper convex lower semicontinuous function. Then the subdifferential ∂f of f is defined as follows:

\partial f (x) = {y \in H : f (z) \geq f (x) + 〈 z - x, y 〉, z \in H}, \forall x \in H .

From Rockafellar [34], we know that ∂f is maximal monotone. It is not hard to verify that $0 \in \partial f (x)$ if and only if $f (x) = {min}_{y \in H} f (y)$ .

Let

I_{C}

be the indicator function of C, i.e.,

I_{C} (x) = {\begin{matrix} 0, & x \in C, \\ + \infty, & x \notin C . \end{matrix}

Since

I_{C}

is a proper lower semicontinuous convex function on H, we see that the subdifferential

\partial I_{C}

of

I_{C}

is a maximal monotone operator. It is clear that

J_{s} x = P_{C} x

,

\forall x \in H

. Notice that

{(B + \partial I_{C})}^{- 1} (0) = VI (C, B)

. Indeed,

\begin{aligned} x \in {(B + \partial I_{C})}^{- 1} (0) ⟺ 0 \in B x + \partial I_{C} x \\ ⟺ - B x \in \partial I_{C} x \\ ⟺ 〈 B x, y - x 〉 \geq 0 \\ ⟺ x \in VI (C, B) . \end{aligned}

(2.18)

In the light of the above, the following is not hard to derive from Corollary 2.4.

Corollary 2.5 Let C be a nonempty, closed, and convex subset of H. Let

S : C \to C

be a nonexpansive mapping with a nonempty fixed point set and

B : C \to H

be a β-inverse-strongly monotone mapping. Assume that

F : = F (S) \cap VI (C, B)

is not empty. Let

{x_{n}}

be a sequence generated by the following iterative process:

{\begin{matrix} x_{1} \in C, \\ C_{1} = C, \\ y_{n} = α_{n} x_{n} + (1 - α_{n}) S P_{C} (x_{n} - s_{n} B x_{n}), \\ C_{n + 1} = {v \in C_{n} : ∥ y_{n} - v ∥ \leq ∥ x_{n} - v ∥}, \\ x_{n + 1} = {Proj}_{C_{n + 1}} x_{1}, n \geq 0, \end{matrix}

where

{s_{n}}

is a sequence in

(0, 2 β)

, and

{α_{n}}

is a sequence in

(0, 1)

. Assume that the following restrictions are satisfied:

(a)

$0 < a \leq s_{n} \leq b < 2 β$ ;
(b)

$0 \leq α_{n} \leq c < 1$ ,

where a, b, and c are real constants. Then the sequence ${x_{n}}$ converges strongly to ${Proj}_{F} x_{1}$ .

3 Applications

First, we consider the problem of finding a minimizer of a proper convex lower semicontinuous function.

Theorem 3.1 Let

f : H \to (- \infty, + \infty]

be a proper convex lower semicontinuous function such that

{(\partial f)}^{- 1} (0)

is not empty. Let

{x_{n}}

be a sequence generated by the following iterative process:

{\begin{matrix} x_{1} \in H, \\ C_{1} = H, \\ y_{n} = arg {min}_{s \in H} {f (z) + \frac{{∥ z - x_{n} ∥}^{2}}{2 s_{n}}}, \\ C_{n + 1} = {v \in C_{n} : ∥ y_{n} - v ∥ \leq ∥ x_{n} - v ∥}, \\ x_{n + 1} = {Proj}_{C_{n + 1}} x_{1}, n \geq 0, \end{matrix}

where ${s_{n}}$ is a positive sequence such that $0 < a \leq s_{n}$ , where a is a real constant. Then the sequence ${x_{n}}$ converges strongly to ${Proj}_{{(\partial f)}^{- 1} (0)} x_{1}$ .

Proof Putting $A = B = 0$ , $S = I$ , and $α_{n} \equiv 0$ , we can immediately draw the desired conclusion from Theorem 2.1. □

Second, we consider the problem of approximating a common fixed point of a pair of nonexpansive mappings.

Theorem 3.2 Let C be a nonempty, closed, and convex subset of H. Let

S : C \to C

and

T : C \to C

be a pair of nonexpansive mappings with a nonempty common fixed point set. Let

{x_{n}}

be a sequence generated by the following iterative process:

{\begin{matrix} x_{1} \in C, \\ C_{1} = C, \\ z_{n} = (1 - s_{n}) x_{n} + s_{n} T x_{n}, \\ y_{n} = α_{n} x_{n} + (1 - α_{n}) S z_{n}, \\ C_{n + 1} = {v \in C_{n} : ∥ y_{n} - v ∥ \leq ∥ x_{n} - v ∥}, \\ x_{n + 1} = {Proj}_{C_{n + 1}} x_{1}, n \geq 0, \end{matrix}

where

{s_{n}}

, and

{α_{n}}

are sequences in

(0, 1)

. Assume that the following restrictions are satisfied:

(a)

$0 < a \leq s_{n} \leq b < 1$ ;
(b)

$0 \leq α_{n} \leq c < 1$ ,

where a, b, and c are real constants. Then the sequence ${x_{n}}$ converges strongly to ${Proj}_{F (S) \cap F (T)} x_{1}$ .

Proof Putting $A = 0$ , $M = \partial I_{C}$ , and $B = I - T$ , we see that B is $\frac{1}{2}$ -inverse-strongly monotone. We also have $F (T)$ = $VI (C, B)$ and $P_{C} (x_{n} - s_{n} B x_{n}) = (1 - s_{n}) x_{n} + s_{n} T x_{n}$ . In view of (2.18), we can immediately obtain the desired result. □

Let F be a bifunction of

C \times C

into

R

, where

R

denotes the set of real numbers. Recall the following equilibrium problem in the terminology of Blum and Oettli [35] (see also Fan [36]):

Find x \in C such that F (x, y) \geq 0, \forall y \in C .

(3.1)

To study the equilibrium problem (3.1), we may assume that F satisfies the following conditions:

(A1) $F (x, x) = 0$ for all $x \in C$ ;

(A2) F is monotone, i.e., $F (x, y) + F (y, x) \leq 0$ for all $x, y \in C$ ;

(A3) for each

x, y, z \in C

,

\underset{t ↓ 0}{lim sup} F (t z + (1 - t) x, y) \leq F (x, y);

(A4) for each $x \in C$ , $y \mapsto F (x, y)$ is convex and lower semi-continuous.

Putting $F (x, y) = 〈 A x, y - x 〉$ for every $x, y \in C$ , we see that the equilibrium problem (3.3) is reduced to the variational inequality (1.1).

The following lemma can be found in [35] and [37].

Lemma 3.3 Let C be a nonempty, closed, and convex subset of H and

F : C \times C \to R

be a bifunction satisfying (A1)-(A4). Then, for any

s > 0

and

x \in H

, there exists

z \in C

such that

F (z, y) + \frac{1}{s} 〈 y - z, z - x 〉 \geq 0, \forall y \in C .

Further, define

T_{s} x = {z \in C : F (z, y) + \frac{1}{s} 〈 y - z, z - x 〉 \geq 0, \forall y \in C}

(3.2)

for all

s > 0

and

x \in H

. Then, the following hold:

(a)

$T_{s}$ is single-valued;
(b)

$T_{s}$ is firmly nonexpansive; that is,
${∥ T_{s} x - T_{s} y ∥}^{2} \leq 〈 T_{s} x - T_{s} y, x - y 〉, \forall x, y \in H;$
(c)

$F (T_{s}) = E P (F)$ ;
(d)

$E P (F)$ is closed and convex.

Lemma 3.4[30]

Let C be a nonempty, closed, and convex subset of H, F be a bifunction from

C \times C

to

R

which satisfies (A1)-(A4), and

A_{F}

be a multivalued mapping of H into itself defined by

A_{F} x = {\begin{matrix} {z \in H : F (x, y) \geq 〈 y - x, z 〉, \forall y \in C}, & x \in C, \\ \emptyset, & x \notin C . \end{matrix}

(3.3)

Then

A_{F}

is a maximal monotone operator with the domain

D (A_{F}) \subset C

,

E P (F) = A_{F}^{- 1} (0)

, where

F P (F)

stands for the solution set of (3.1), and

T_{s} x = {(I + s A_{F})}^{- 1} x, \forall x \in H, r > 0,

where $T_{s}$ is defined as in (3.3).

Finally, we consider finding a solution of the equilibrium problem.

Theorem 3.5 Let C be a nonempty, closed, and convex subset of H. Let

F : C \times C \to R

be a bifunction satisfying (A1)-(A4) such that

E P (F) \neq \emptyset

. Let

{x_{n}}

be a sequence generated by the following iterative process:

{\begin{matrix} x_{1} \in C, \\ C_{1} = C, \\ y_{n} = {(I + s_{n} A_{F})}^{- 1} x_{n}, \\ C_{n + 1} = {v \in C_{n} : ∥ y_{n} - v ∥ \leq ∥ x_{n} - v ∥}, \\ x_{n + 1} = {Proj}_{C_{n + 1}} x_{1}, n \geq 0, \end{matrix}

where $A_{F}$ is defined as (3.3), and ${s_{n}}$ is a positive sequence such that $0 < a \leq s_{n} < \infty$ , where a is a real constant Then the sequence ${x_{n}}$ converges strongly to ${Proj}_{F P (F)} x_{1}$ .

Proof Putting $A = B = 0$ , $S = I$ and $α_{n} \equiv 0$ , we immediately reach the desired conclusion from Lemma 3.4. □

Authors’ information

Author’s information

Declarations

Acknowledgements

The first author was supported by the National Natural Science Foundation of China (11071169, 11271105), the Natural Science Foundation of Zhejiang Province (Y6110287, Y12A010095).

Authors’ Affiliations

(1)

Department of Mathematics, Hangzhou Normal University, Hangzhou, China

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