# On solutions of inclusion problems and fixed point problems

- Yuan Hecai
^{1}Email author

**2013**:11

https://doi.org/10.1186/1687-1812-2013-11

© Hecai; licensee Springer 2013

**Received: **20 November 2012

**Accepted: **27 December 2012

**Published: **14 January 2013

## Abstract

An inclusion problem and a fixed point problem are investigated based on a hybrid projection method. The strong convergence of the hybrid projection method is obtained in the framework of Hilbert spaces. Variational inequalities and fixed point problems of quasi-nonexpansive mappings are also considered as applications of the main results.

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

## Keywords

## 1 Introduction and preliminaries

*A*and

*B*in a Hilbert space

*H*. In this paper, we consider the problem of finding a solution to the following problem: find an

*x*in the fixed point set of the mapping

*S*such that

where *A* and *B* are two monotone operators. The problem has been addressed by many authors in view of the applications in image recovery and signal processing; see, for example, [5–9] and the references therein.

Throughout this paper, we always assume that *H* is a real Hilbert space with the inner product $\u3008\cdot ,\cdot \u3009$ and norm $\parallel \cdot \parallel $, respectively. Let *C* be a nonempty closed convex subset of *H* and ${P}_{C}$ be the metric projection from *H* onto *C*. Let $S:C\to C$ be a mapping. In this paper, we use $F(S)$ to denote 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 [10].

*S*is said to be quasi-nonexpansive iff $F(S)\ne \mathrm{\varnothing}$ and

It is easy to see that nonexpansive mappings are Lipschitz continuous; however, the quasi-nonexpansive mapping is discontinuous on its domain generally. Indeed, the quasi-nonexpansive mapping is only continuous in its fixed point set.

*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

*A*is also said to be

*α*-inverse-strongly monotone. Notice that

clearly shows that *A* is $\frac{1}{\alpha}$-Lipschitz continuous.

In this paper, we use $\mathit{VI}(C,A)$ to denote the solution set of (1.1). It is known that ${x}^{\ast}\in C$ is a solution to (1.1) iff ${x}^{\ast}$ is a fixed point of the mapping ${P}_{C}(I-\lambda A)$, where $\lambda >0$ is a constant, *I* stands for the identity mapping, and ${P}_{C}$ stands for the metric projection from *H* onto *C*.

A multivalued operator $T:H\to {2}^{H}$ with the domain $D(T)=\{x\in H:Tx\ne \mathrm{\varnothing}\}$ and the range $R(T)=\{Tx:x\in D(T)\}$ is said to be monotone if for ${x}_{1}\in D(T)$, ${x}_{2}\in D(T)$, ${y}_{1}\in T{x}_{1}$, and ${y}_{2}\in T{x}_{2}$, we have $\u3008{x}_{1}-{x}_{2},{y}_{1}-{y}_{2}\u3009\ge 0$. A monotone operator *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 $T:H\to {2}^{H}$ 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.

The Mann iterative algorithm is efficient to study fixed point problems of nonlinear operators. Recently, many authors have studied the common solution problem, that is, find a point in a solution set and a fixed point (zero) point set of some nonlinear problems; see, for example, [11–30] 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 a maximal monotone, and ${J}_{{\lambda}_{n}}={(I+{\lambda}_{n}T)}^{-1}$. They showed that the sequence $\{{x}_{n}\}$ generated in (1.2) 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 $\{{x}_{n}\}$ generated in (1.3) converges weakly to some $z\in \mathit{VI}(C,A)\cap F(S)$ provided that the control sequence satisfies some restrictions.

The above convergence theorems are weak. In this paper, motivated by the above results, we consider the problem of finding a common solution to the zero point problems and fixed point problems based on hybrid iterative methods with errors. Strong convergence theorems are established in the framework of Hilbert spaces.

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

Let *C* be a nonempty, closed, and convex subset of *H*. Let $S:C\to C$ be a 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 $\overline{x}\in F(S)$.

**Lemma** [9]

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

## 2 Main results

**Theorem 2.1**

*Let*

*C*

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

*H*, $A:C\to H$

*be an*

*α*-

*inverse*-

*strongly monotone mapping*, $S:C\to C$

*be a quasi*-

*nonexpansive mapping such that*$I-S$

*is demiclosed at zero*,

*and*

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

*be a positive real number sequence*.

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

*be a real number sequence in*$[0,1]$.

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

*be a sequence in*

*C*

*generated in the following iterative process*:

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

*Suppose that the sequences*$\{{\alpha}_{n}\}$

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

*satisfy the following restrictions*:

- (a)
$0\le {\alpha}_{n}\le a<1$;

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

*Then the sequence* $\{{x}_{n}\}$ *converges strongly to* ${P}_{\mathcal{F}}{x}_{1}$.

*Proof*First, we show that ${C}_{n}$ is closed and convex. Notice that ${C}_{1}=C$ is closed and convex. Suppose that ${C}_{i}$ is closed and convex for some $i\ge 1$. We show that ${C}_{i+1}$ is closed and convex for the same

*i*. Indeed, for any $v\in {C}_{i}$, we see that

Thus ${C}_{i+1}$ is closed and convex. This shows that ${C}_{n}$ is closed and convex.

In view of the assumption that *S* is demiclosed at zero, we see that ${x}^{\ast}\in F(S)$.

*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 (2.10) 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 ${x}^{\ast}\in \mathcal{F}$.

This implies ${x}_{{n}_{i}}\to {x}^{\ast}={P}_{\mathcal{F}}{x}_{1}$. Since $\{{x}_{{n}_{i}}\}$ is an arbitrary subsequence of $\{{x}_{n}\}$, we obtain that ${x}_{n}\to {P}_{\mathcal{F}}{x}_{1}$ as $n\to \mathrm{\infty}$. This completes the proof. □

From Theorem 2.1, we have the following results immediately.

**Corollary 2.2**

*Let*

*C*

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

*H*, $A:C\to H$

*be an*

*α*-

*inverse*-

*strongly monotone mapping*,

*and*

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

*be a positive real number sequence*.

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

*be a real number sequence in*$[0,1]$.

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

*be a sequence in*

*C*

*generated in the following iterative process*:

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

*Suppose that the sequences*$\{{\alpha}_{n}\}$

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

*satisfy the following restrictions*:

- (a)
$0\le {\alpha}_{n}\le a<1$;

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

*Then the sequence* $\{{x}_{n}\}$ *converges strongly to* ${P}_{{(A+B)}^{-1}(0)}{x}_{1}$.

*∂f*is a maximal monotone operator of

*H*into itself; see [23] for more details. Let

*C*be a nonempty closed convex subset of

*H*and ${i}_{C}$ 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 $x\in {(A+\partial {i}_{C})}^{-1}(0)\iff x\in \mathit{VI}(A,C)$. Putting $B=\partial {i}_{C}$ in Theorem 2.1, we can see ${J}_{{\lambda}_{n}}={P}_{C}$. The following is not hard to derive.

**Corollary 2.3**

*Let*

*C*

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

*H*, $A:C\to H$

*be an*

*α*-

*inverse*-

*strongly monotone mapping*,

*and*$S:C\to C$

*be a quasi*-

*nonexpansive mapping such that*$I-S$

*is demiclosed at zero*.

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

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

*be a positive real number sequence*.

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

*be a real number sequence in*$[0,1]$.

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

*be a sequence in*

*C*

*generated in the following iterative process*:

*Suppose that the sequences*$\{{\alpha}_{n}\}$

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

*satisfy the following restrictions*:

- (a)
$0\le {\alpha}_{n}\le a<1$;

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

*Then the sequence* $\{{x}_{n}\}$ *converges strongly to* ${P}_{\mathcal{F}}{x}_{1}$.

In view of Corollary 2.3, we have the following corollary on variational inequalities.

**Corollary 2.4**

*Let*

*C*

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

*H*

*and*$A:C\to H$

*be an*

*α*-

*inverse*-

*strongly monotone mapping*.

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

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

*be a positive real number sequence*.

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

*be a real number sequence in*$[0,1]$.

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

*be a sequence in*

*C*

*generated in the following iterative process*:

*Suppose that the sequences*$\{{\alpha}_{n}\}$

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

*satisfy the following restrictions*:

- (a)
$0\le {\alpha}_{n}\le a<1$;

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

*Then the sequence* $\{{x}_{n}\}$ *converges strongly to* ${P}_{\mathit{VI}(C,A)}{x}_{1}$.

## Declarations

### Acknowledgements

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

## Authors’ Affiliations

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