- Research
- Open Access

# 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

- nonexpansive mapping
- inverse-strongly monotone mapping
- maximal monotone operator
- fixed point

## 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

## References

- Peaceman DH, Rachford HH: The numerical solution of parabolic and elliptic differential equations.
*J. Soc. Ind. Appl. Math.*1995, 3: 28–415.MathSciNetView ArticleGoogle Scholar - Douglas J, Rachford HH: On the numerical solution of heat conduction problems in two and three space variables.
*Trans. Am. Math. Soc.*1956, 82: 421–439. 10.1090/S0002-9947-1956-0084194-4MathSciNetView ArticleGoogle Scholar - Kellogg RB: Nonlinear alternating direction algorithm.
*Math. Comput.*1969, 23: 23–28. 10.1090/S0025-5718-1969-0238507-3MathSciNetView ArticleGoogle Scholar - Lions PL, Mercier B: Splitting algorithms for the sum of two nonlinear operators.
*SIAM J. Numer. Anal.*1979, 16: 964–979. 10.1137/0716071MathSciNetView ArticleGoogle Scholar - Qin X, Kang JI, Cho YJ: On quasi-variational inclusions and asymptotically strict pseudo-contractions.
*J. Nonlinear Convex Anal.*2010, 11: 441–453.MathSciNetGoogle Scholar - Zhang M: Iterative algorithms for common elements in fixed point sets and zero point sets with applications.
*Fixed Point Theory Appl.*2012, 2012: 21. 10.1186/1687-1812-2012-21View ArticleGoogle Scholar - Takahashi S, Takahashi W, Toyoda M: Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces.
*J. Optim. Theory Appl.*2010, 147: 27–41. 10.1007/s10957-010-9713-2MathSciNetView ArticleGoogle Scholar - Kamimura S, Takahashi W: Weak and strong convergence of solutions to accretive operator inclusions and applications.
*Set-Valued Anal.*2010, 8: 361–374.MathSciNetView ArticleGoogle Scholar - Aoyama K, Kimura Y, Takahashi W, Toyoda M: On a strongly nonexpansive sequence in Hilbert spaces.
*J. Nonlinear Convex Anal.*2007, 8: 471–489.MathSciNetGoogle Scholar - Browder FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces.
*Proc. Symp. Pure Math.*1976, 18: 78–81.Google Scholar - Kamimura S, Takahashi W: Approximating solutions of maximal monotone operators in Hilbert spaces.
*J. Approx. Theory*2000, 106: 226–240. 10.1006/jath.2000.3493MathSciNetView ArticleGoogle Scholar - Takahashi W, Toyoda M: Weak convergence theorems for nonexpansive mappings and monotone mappings.
*J. Optim. Theory Appl.*2003, 118: 417–428. 10.1023/A:1025407607560MathSciNetView ArticleGoogle Scholar - Ye J, Huang J: Strong convergence theorems for fixed point problems and generalized equilibrium problems of three relatively quasi-nonexpansive mappings in Banach spaces.
*J. Math. Comput. Sci.*2011, 1: 1–18.MathSciNetView ArticleGoogle Scholar - Cho SY, Kang SM: Approximation of fixed points of pseudocontraction semigroups based on a viscosity iterative process.
*Appl. Math. Lett.*2011, 24: 224–228. 10.1016/j.aml.2010.09.008MathSciNetView ArticleGoogle Scholar - Zegeye H, Shahzad N: Strong convergence theorem for a common point of solution of variational inequality and fixed point problem.
*Adv. Fixed Point Theory*2012, 2: 374–397.Google Scholar - Qin X, Cho SY, Kang SM: Strong convergence of shrinking projection methods for quasi-
*ϕ*-nonexpansive mappings and equilibrium problems.*J. Comput. Appl. Math.*2010, 234: 750–760. 10.1016/j.cam.2010.01.015MathSciNetView ArticleGoogle Scholar - Lu H, Wang Y: Iterative approximation for the common solutions of a infinite variational inequality system for inverse-strongly accretive mappings.
*J. Math. Comput. Sci.*2012, 2(6):1660–1670.MathSciNetGoogle Scholar - Husain S, Gupta S: A resolvent operator technique for solving generalized system of nonlinear relaxed cocoercive mixed variational inequalities.
*Adv. Fixed Point Theory*2012, 2: 18–28.Google Scholar - Noor MA, Huang Z: Some resolvent iterative methods for variational inclusions and nonexpansive mappings.
*Appl. Math. Comput.*2007, 194: 267–275. 10.1016/j.amc.2007.04.037MathSciNetView ArticleGoogle Scholar - Qin X, Cho YJ, Kang SM: Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces.
*J. Comput. Appl. Math.*2009, 225: 20–30. 10.1016/j.cam.2008.06.011MathSciNetView ArticleGoogle Scholar - Kim JK, Tuyen TM: Regularization proximal point algorithm for finding a common fixed point of a finite family of nonexpansive mappings in Banach spaces.
*Fixed Point Theory Appl.*2011, 2011: 52. 10.1186/1687-1812-2011-52MathSciNetView ArticleGoogle Scholar - Wei Z, Shi G: Convergence of a proximal point algorithm for maximal monotone operators in Hilbert spaces.
*J. Inequal. Appl.*2012, 2012: 137. 10.1186/1029-242X-2012-137MathSciNetView ArticleGoogle Scholar - Qin X, Chang SS, Cho YJ: Iterative methods for generalized equilibrium problems and fixed point problems with applications.
*Nonlinear Anal.*2010, 11: 2963–2972. 10.1016/j.nonrwa.2009.10.017MathSciNetView ArticleGoogle Scholar - Qin X, Shang M, Su Y: Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems.
*Math. Comput. Model.*2008, 48: 1033–1046. 10.1016/j.mcm.2007.12.008MathSciNetView ArticleGoogle Scholar - He XF, Xu YC, He Z: Iterative approximation for a zero of accretive operator and fixed points problems in Banach space.
*Appl. Math. Comput.*2011, 217: 4620–4626. 10.1016/j.amc.2010.11.014MathSciNetView ArticleGoogle Scholar - Wu C, Liu A: Strong convergence of a hybrid projection iterative algorithm for common solutions of operator equations and of inclusion problems.
*Fixed Point Theory Appl.*2012., 2012: Article ID 90Google Scholar - Qin X, Su Y: Approximation of a zero point of accretive operator in Banach spaces.
*J. Math. Anal. Appl.*2007, 329: 415–424. 10.1016/j.jmaa.2006.06.067MathSciNetView ArticleGoogle Scholar - Abdel-Salam HS, Al-Khaled K: Variational iteration method for solving optimization problems.
*J. Math. Comput. Sci.*2012, 2: 1457–1497.MathSciNetGoogle Scholar - Qin X, Cho SY, Kang SM: On hybrid projection methods for asymptotically quasi-
*ϕ*-nonexpansive mappings.*Appl. Math. Comput.*2010, 215: 3874–3883. 10.1016/j.amc.2009.11.031MathSciNetView ArticleGoogle Scholar - Zegeye H, Shahzad N, Alghamdi M: Strong convergence theorems for a common point of solution of variational inequality, solutions of equilibrium and fixed point problems.
*Fixed Point Theory Appl.*2012., 2012: Article ID 119Google Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.