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# On weak convergence of an iterative algorithm for common solutions of inclusion problems and fixed point problems in Hilbert spaces

- Yuan Hecai
^{1}Email author

**2013**:155

https://doi.org/10.1186/1687-1812-2013-155

© Hecai; licensee Springer. 2013

**Received:**10 March 2013**Accepted:**28 May 2013**Published:**17 June 2013

## Abstract

In this paper, a monotone inclusion problem and a fixed point problem of nonexpansive mappings are investigated based on a Mann-type iterative algorithm with mixed errors. Strong convergence theorems of common elements are established in the framework of Hilbert spaces.

**MSC:**47H05, 47H09, 47J25.

## Keywords

- maximal monotone operator
- nonexpansive mapping
- Hilbert space
- fixed point

## 1 Introduction

Variational inclusion has become rich of inspiration in pure and applied mathematics. In recent years, classical variational inclusion problems have been extended and generalized to study a large variety of problems arising in image recovery, economics, and signal processing; for more details, see [1–14]. Based on the projection technique, it has been shown that the variational inclusion problems are equivalent to the fixed point problems. This alternative formulation has played a fundamental and significant part in developing several numerical methods for solving variational inclusion problems and related optimization problems.

The purposes of this paper is to study the zero point problem of the sum of a maximal monotone mapping and an inverse-strongly monotone mapping, and the fixed point problem of a nonexpansive mapping. The organization of this paper is as follows. In Section 2, we provide some necessary preliminaries. In Section 3, a Mann-type iterative algorithm with mixed errors is investigated. A weak convergence theorem is established. Applications of the main results are also discussed in this section.

## 2 Preliminaries

Throughout this paper, we always assume that *H* is a real Hilbert space with the inner product $\u3008\cdot ,\cdot \u3009$ and the norm $\parallel \cdot \parallel $, respectively. Let *C* be a nonempty closed convex subset of *H* and let ${P}_{C}$ be the metric projection from *H* onto *C*.

Let $S:C\to C$ be a mapping. $F(S)$ stands for the fixed point set of *S*; that is, $F(S):=\{x\in C:x=Sx\}$.

*S*is said to be nonexpansive iff

If *C* is a bounded, closed, and convex subset of *H*, then $F(S)$ is not empty, closed, and convex; see [15].

*A*is said to be monotone iff

*A*is said to be strongly monotone iff there exists a constant $\alpha >0$ such that

*A*is also said to be

*α*-strongly monotone.

*A*is said to be inverse-strongly monotone iff there exists a constant $\alpha >0$ such that

For such a case, *A* is also said to be *α*-inverse-strongly monotone. It is not hard to see that inverse-strongly monotone mappings are monotone and Lipschitz continuous.

*ω*is a fixed point of the mapping ${P}_{C}(I-\lambda A)$, where $\lambda >0$ is a constant, and

*I*stands for the identity mapping. If

*A*is

*α*-inverse-strongly monotone and $\lambda \in (0,2\alpha ]$, then the mapping ${P}_{C}(I-\lambda A)$ is nonexpansive. Indeed, we have

This shows that ${P}_{C}(I-\lambda A)$ is nonexpansive. It follows that $VI(C,A)$ is closed and convex.

*T*is said to be maximal if its graph $G(T)=\{(x,y):y\in Tx\}$ is not properly contained in the graph of any other monotone operator. Let

*I*denote the identity operator on

*H*and let $T:H\to {2}^{H}$ be a maximal monotone operator. Then we can define, for each $\lambda >0$, a nonexpansive single-valued mapping ${J}_{\lambda}:H\to H$ by ${J}_{\lambda}={(I+\lambda T)}^{-1}$. It is called the resolvent of

*T*. We know that ${T}^{-1}0=F({J}_{\lambda})$ for all $\lambda >0$ and ${J}_{\lambda}$ is firmly nonexpansive, that is,

for more details, see [16–22] and the references therein.

where $\{{\alpha}_{n}\}$ is a sequence in $(0,1)$, $\{{\lambda}_{n}\}$ is a positive sequence, $T:H\to {2}^{H}$ is maximal monotone and ${J}_{{\lambda}_{n}}={(I+{\lambda}_{n}T)}^{-1}$. They showed that the sequence $\{{x}_{n}\}$ converges weakly to some $z\in {T}^{-1}(0)$ provided that the control sequence satisfies some restrictions. Further, using this result, they also investigated the case that $T=\partial f$, where $f:H\to (-\mathrm{\infty},\mathrm{\infty}]$ is a proper lower semicontinuous convex function. Convergence theorems are established in the framework of real Hilbert spaces.

where $\{{\alpha}_{n}\}$ is a sequence in $(0,1)$, $\{{\lambda}_{n}\}$ is a positive sequence, $S:C\to C$ is a nonexpansive mapping and $A:C\to H$ is an inverse-strongly monotone mapping. They showed that the sequence $\{{x}_{n}\}$ converges weakly to some $z\in VI(C,A)\cap F(S)$ provided that the control sequence satisfies some restrictions.

where $\{{\alpha}_{n}\}$ is a sequence in $(0,1)$, $\{{r}_{n}\}$ is a positive sequence, $S:C\to C$ is a nonexpansive mapping and $F:C\times C\to R$ is a bifunction. They showed that the sequence $\{{x}_{n}\}$ converges weakly to some $z\in VI(C,A)\cap F(S)$ provided that the control sequence satisfies some restrictions.

Recently, fixed point and zero point problems have been studied by many authors based on iterative methods; see, for example, [23–34] and the references therein. In this paper, motivated by the above results, we consider the problem of finding a common solution to the zero point problem and the fixed point problem based on Mann-type iterative methods with errors. Weak convergence theorems are established in the framework of Hilbert spaces.

To obtain our main results in this paper, we need the following lemmas.

**Lemma 2.1** [34]

*Let* *C* *be a nonempty*, *closed*, *and convex subset of* *H*, *let* $A:C\to H$ *be a mapping*, *and let* $B:H\rightrightarrows H$ *be a maximal monotone operator*. *Then* $F({J}_{\lambda}(I-\lambda A))={(A+B)}^{-1}(0)$, *where* ${J}_{\lambda}(I-\lambda A)$ *is the resolvent of* *B* *for* $\lambda >0$.

**Lemma 2.2** [36]

*Let*$\{{a}_{n}\}$, $\{{b}_{n}\}$,

*and*$\{{c}_{n}\}$

*be three nonnegative sequences satisfying the following condition*:

*where* ${n}_{0}$ *is some nonnegative integer*, ${\sum}_{n=1}^{\mathrm{\infty}}{b}_{n}<\mathrm{\infty}$ *and* ${\sum}_{n=1}^{\mathrm{\infty}}{c}_{n}<\mathrm{\infty}$. *Then the limit* ${lim}_{n\to \mathrm{\infty}}{a}_{n}$ *exists*.

**Lemma 2.3** [37]

*Suppose that*

*H*

*is a real Hilbert space and*$0<p\le {t}_{n}\le q<1$

*for all*$n\ge 1$.

*Suppose further that*$\{{x}_{n}\}$

*and*$\{{y}_{n}\}$

*are sequences of*

*H*

*such that*

*and*

*hold for some* $r\ge 0$. *Then* ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-{y}_{n}\parallel =0$.

**Lemma 2.4** [15]

*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* ${x}_{n}\rightharpoonup \overline{x}$ *and* ${x}_{n}-S{x}_{n}\to 0$, *then* $\overline{x}\in F(S)$.

## 3 Main results

**Theorem 3.1**

*Let*

*C*

*be a nonempty closed convex subset of a real Hilbert space*

*H*,

*let*$A:C\to H$

*be an*

*α*-

*inverse*-

*strongly monotone mapping*,

*let*$S:C\to C$

*be a nonexpansive mapping and let*

*B*

*be a maximal monotone operator on*

*H*

*such that the domain of*

*B*

*is included in*

*C*.

*Assume that*$\mathcal{F}=F(S)\cap {(A+B)}^{-1}(0)\ne \mathrm{\varnothing}$.

*Let*$\{{\alpha}_{n}\}$, $\{{\beta}_{n}\}$,

*and*$\{{\gamma}_{n}\}$

*be real number sequences in*$(0,1)$

*such that*${\alpha}_{n}+{\beta}_{n}+{\gamma}_{n}=1$.

*Let*$\{{\lambda}_{n}\}$

*be a positive real number sequence and let*$\{{e}_{n}\}$

*be a bounded error sequence in*

*C*.

*Let*$\{{x}_{n}\}$

*be a sequence in*

*C*

*generated in the following iterative process*:

*for all*$n\in \mathbb{N}$,

*where*${J}_{{\lambda}_{n}}={(I+{\lambda}_{n}B)}^{-1}$.

*Assume that the sequences*$\{{\alpha}_{n}\}$, $\{{\beta}_{n}\}$, $\{{\gamma}_{n}\}$,

*and*$\{{\lambda}_{n}\}$

*satisfy the following restrictions*:

- (a)
$0<a\le {\beta}_{n}\le b<1$,

- (b)
$0<c\le {\lambda}_{n}\le d<2\alpha $,

- (c)
${\sum}_{n=1}^{\mathrm{\infty}}{\gamma}_{n}<\mathrm{\infty}$,

*where* *a*, *b*, *c*, *and* *d* *are some real numbers*. *Then the sequence* $\{{x}_{n}\}$ *generated in* (3.1) *converges weakly to some point in* ℱ.

*Proof*Notice that $I-{\lambda}_{n}A$ is nonexpansive. Indeed, we have

*B*is monotone, we get for any $(u,v)\in B$ that

*n*by ${n}_{i}$ and letting $i\to \mathrm{\infty}$, we obtain from (3.10) that

This means $-A\omega \in B\omega $, that is, $0\in (A+B)(\omega )$. Hence, we get $\omega \in {(A+B)}^{-1}(0)$. This completes the proof that $\omega \in \mathcal{F}$.

This is a contradiction. This shows that $\omega ={\omega}^{\ast}$. This proves that the sequence $\{{x}_{n}\}$ converges weakly to $\omega \in F$. This completes the proof. □

We obtain from Theorem 3.1 the following inclusion problem.

**Corollary 3.2**

*Let*

*C*

*be a nonempty closed convex subset of a real Hilbert space*

*H*,

*let*$A:C\to H$

*be an*

*α*-

*inverse*-

*strongly monotone mapping*,

*and let*

*B*

*be a maximal monotone operator on*

*H*

*such that the domain of*

*B*

*is included in*

*C*.

*Assume that*${(A+B)}^{-1}(0)\ne \mathrm{\varnothing}$.

*Let*$\{{\alpha}_{n}\}$, $\{{\beta}_{n}\}$,

*and*$\{{\gamma}_{n}\}$

*be real number sequences in*$(0,1)$

*such that*${\alpha}_{n}+{\beta}_{n}+{\gamma}_{n}=1$.

*Let*$\{{\lambda}_{n}\}$

*be a positive real number sequence and let*$\{{e}_{n}\}$

*be a bounded error sequence in*

*C*.

*Let*$\{{x}_{n}\}$

*be a sequence in*

*C*

*generated in the following iterative process*:

*for all*$n\in \mathbb{N}$,

*where*${J}_{{\lambda}_{n}}={(I+{\lambda}_{n}B)}^{-1}$.

*Assume that the sequences*$\{{\alpha}_{n}\}$, $\{{\beta}_{n}\}$, $\{{\gamma}_{n}\}$,

*and*$\{{\lambda}_{n}\}$

*satisfy the following restrictions*:

- (a)
$0<a\le {\beta}_{n}\le b<1$,

- (b)
$0<c\le {\lambda}_{n}\le d<2\alpha $,

- (c)
${\sum}_{n=1}^{\mathrm{\infty}}{\gamma}_{n}<\mathrm{\infty}$,

*where* *a*, *b*, *c*, *and* *d* *are some real numbers*. *Then the sequence* $\{{x}_{n}\}$ *converges weakly to some point in* ${(A+B)}^{-1}(0)$.

*∂f*is a maximal monotone operator of

*H*into itself; for more details, see [38]. Let

*C*be a nonempty closed convex subset of

*H*and let ${i}_{C}$ be the indicator function of

*C*, that is,

*C*at

*v*as follows:

*H*and $\partial {i}_{C}$ is a maximal monotone operator. Let ${J}_{\lambda}x={(I+\lambda \partial {i}_{C})}^{-1}x$ for any $\lambda >0$ and $x\in H$. From $\partial {i}_{C}x={N}_{C}x$ and $x\in C$, we get

where ${P}_{C}$ is the metric projection from *H* into *C*. Similarly, we can get that $x\in {(A+\partial {i}_{C})}^{-1}(0)\iff x\in VI(A,C)$. Putting $B=\partial {i}_{C}$ in Theorem 3.1, we find that ${J}_{{\lambda}_{n}}={P}_{C}$. The following results are not hard to derive.

**Theorem 3.3**

*Let*

*C*

*be a nonempty closed convex subset of a real Hilbert space*

*H*,

*let*$A:C\to H$

*be an*

*α*-

*inverse*-

*strongly monotone mapping and let*$S:C\to C$

*be a nonexpansive mapping*.

*Assume that*$\mathcal{F}=F(S)\cap VI(C,A)\ne \mathrm{\varnothing}$.

*Let*$\{{\alpha}_{n}\}$, $\{{\beta}_{n}\}$,

*and*$\{{\gamma}_{n}\}$

*be real number sequences in*$(0,1)$

*such that*${\alpha}_{n}+{\beta}_{n}+{\gamma}_{n}=1$.

*Let*$\{{\lambda}_{n}\}$

*be a positive real number sequence and let*$\{{e}_{n}\}$

*be a bounded error sequence in*

*C*.

*Let*$\{{x}_{n}\}$

*be a sequence in*

*C*

*generated in the following iterative process*:

*for all*$n\in \mathbb{N}$.

*Assume that the sequences*$\{{\alpha}_{n}\}$, $\{{\beta}_{n}\}$, $\{{\gamma}_{n}\}$,

*and*$\{{\lambda}_{n}\}$

*satisfy the following restrictions*:

- (a)
$0<a\le {\beta}_{n}\le b<1$,

- (b)
$0<c\le {\lambda}_{n}\le d<2\alpha $,

- (c)
${\sum}_{n=1}^{\mathrm{\infty}}{\gamma}_{n}<\mathrm{\infty}$,

*where* *a*, *b*, *c*, *and* *d* *are some real numbers*. *Then the sequence* $\{{x}_{n}\}$ *converges weakly to some point in* ℱ.

In view of Theorem 3.3, we have the following result.

**Corollary 3.4**

*Let*

*C*

*be a nonempty closed convex subset of a real Hilbert space*

*H*

*and let*$A:C\to H$

*be an*

*α*-

*inverse*-

*strongly monotone mapping such that*$VI(C,A)\ne \mathrm{\varnothing}$.

*Let*$\{{\alpha}_{n}\}$, $\{{\beta}_{n}\}$,

*and*$\{{\gamma}_{n}\}$

*be real number sequences in*$(0,1)$

*such that*${\alpha}_{n}+{\beta}_{n}+{\gamma}_{n}=1$.

*Let*$\{{\lambda}_{n}\}$

*be a positive real number sequence and let*$\{{e}_{n}\}$

*be a bounded error sequence in*

*C*.

*Let*$\{{x}_{n}\}$

*be a sequence in*

*C*

*generated in the following iterative process*:

*for all*$n\in \mathbb{N}$.

*Assume that the sequences*$\{{\alpha}_{n}\}$, $\{{\beta}_{n}\}$, $\{{\gamma}_{n}\}$,

*and*$\{{\lambda}_{n}\}$

*satisfy the following restrictions*:

- (a)
$0<a\le {\beta}_{n}\le b<1$,

- (b)
$0<c\le {\lambda}_{n}\le d<2\alpha $,

- (c)
${\sum}_{n=1}^{\mathrm{\infty}}{\gamma}_{n}<\mathrm{\infty}$,

*where* *a*, *b*, *c*, *and* *d* *are some real numbers*. *Then the sequence* $\{{x}_{n}\}$ *converges weakly to some point in* $VI(C,A)$.

*F*be a bifunction of $C\times C$ into ℝ, where ℝ denotes the set of real numbers. Recall the following equilibrium problem.

In this paper, we use $\mathit{EP}(F)$ to denote the solution set of the equilibrium problem.

*F*satisfies the following conditions:

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

- (A2)
*F*is monotone,*i.e.*, $F(x,y)+F(y,x)\le 0$ for all $x,y\in C$; - (A3)for each $x,y,z\in C$,$\underset{t\downarrow 0}{lim\hspace{0.17em}sup}F(tz+(1-t)x,y)\le F(x,y);$
- (A4)
for each $x\in C$, $y\mapsto F(x,y)$ is convex and weakly lower semi-continuous.

Putting $F(x,y)=\u3008Ax,y-x\u3009$ for every $x,y\in C$, we see that the equilibrium problem is reduced to the variational inequality (2.1).

The following lemma can be found in [39].

**Lemma 3.5**

*Let*

*C*

*be a nonempty closed convex subset of*

*H*

*and let*$F:C\times C\to \mathbb{R}$

*be a bifunction satisfying*(A1)-(A4).

*Then*,

*for any*$r>0$

*and*$x\in H$,

*there exists*$z\in C$

*such that*

*Further*,

*define*

*for all*$r>0$

*and*$x\in H$.

*Then the following hold*:

- (a)
${T}_{r}$

*is single*-*valued*, - (b)${T}_{r}$
*is firmly nonexpansive*,*i*.*e*.,*for any*$x,y\in H$,${\parallel {T}_{r}x-{T}_{r}y\parallel}^{2}\le \u3008{T}_{r}x-{T}_{r}y,x-y\u3009,$ - (c)
$F({T}_{r})=\mathit{EP}(F)$,

- (d)
$\mathit{EP}(F)$

*is closed and convex*.

**Lemma 3.6** [5]

*Let*

*C*

*be a nonempty closed convex subset of a real Hilbert space*

*H*,

*let*

*F*

*a bifunction from*$C\times C$

*to*ℝ

*which satisfies*(A1)-(A4)

*and let*${A}_{F}$

*be a multivalued mapping of*

*H*

*into itself defined by*

*Then*${A}_{F}$

*is a maximal monotone operator with the domain*$D({A}_{F})\subset C$, $\mathit{EP}(F)={A}_{F}^{-1}(0)$

*and*

*where* ${T}_{r}$ *is defined as in* (3.13).

**Theorem 3.7**

*Let*

*C*

*be a nonempty closed convex subset of a real Hilbert space*

*H*,

*let*$S:C\to C$

*be a nonexpansive mapping and let*

*F*

*be a bifunction from*$C\times C$

*to*ℝ

*which satisfies*(A1)-(A4).

*Assume that*$\mathcal{F}=F(S)\cap \mathit{EP}(F)\ne \mathrm{\varnothing}$.

*Let*$\{{\alpha}_{n}\}$, $\{{\beta}_{n}\}$,

*and*$\{{\gamma}_{n}\}$

*be real number sequences in*$(0,1)$

*such that*${\alpha}_{n}+{\beta}_{n}+{\gamma}_{n}=1$.

*Let*$\{{\lambda}_{n}\}$

*be a positive real number sequence and let*$\{{e}_{n}\}$

*be a bounded error sequence in*

*C*.

*Let*$\{{x}_{n}\}$

*be a sequence in*

*C*

*generated in the following iterative process*:

*for all*$n\in \mathbb{N}$,

*where*${y}_{n}\in C$

*such that*

*Assume that the sequences*$\{{\alpha}_{n}\}$, $\{{\beta}_{n}\}$, $\{{\gamma}_{n}\}$,

*and*$\{{\lambda}_{n}\}$

*satisfy the following restrictions*:

- (a)
$0<a\le {\beta}_{n}\le b<1$,

- (b)
$0<c\le {\lambda}_{n}\le d<\mathrm{\infty}$,

- (c)
${\sum}_{n=1}^{\mathrm{\infty}}{\gamma}_{n}<\mathrm{\infty}$,

*where* *a*, *b*, *c*, *and* *d* *are some real numbers*. *Then the sequence* $\{{x}_{n}\}$ *converges weakly to some point in* ℱ.

## Declarations

### Acknowledgements

The author is grateful to the reviewers for useful suggestions which improved the contents of the article.

## Authors’ Affiliations

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