# A General Iterative Method for Solving the Variational Inequality Problem and Fixed Point Problem of an Infinite Family of Nonexpansive Mappings in Hilbert Spaces

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## Abstract

We introduce an iterative scheme for finding a common element of the set of common fixed points of a family of infinitely nonexpansive mappings, and the set of solutions of the variational inequality for an inverse-strongly monotone mapping in a Hilbert space. Under suitable conditions, some strong convergence theorems for approximating a common element of the above two sets are obtained. As applications, at the end of the paper we utilize our results to study the problem of finding a common element of the set of fixed points of a family of infinitely nonexpansive mappings and the set of fixed points of a finite family of Open image in new window -strictly pseudocontractive mappings. The results presented in the paper improve some recent results of Qin and Cho (2008).

### Keywords

Hilbert Space Variational Inequality Monotone Mapping Nonexpansive Mapping Iterative Scheme## 1. Introduction

We denote by Open image in new window the set of fixed points of Open image in new window . Recall that a mapping Open image in new window is said to be

(i)monotone if Open image in new window , for all Open image in new window ;

(ii) Open image in new window -Lipschitz if there exists a constant Open image in new window such that Open image in new window , for all Open image in new window ;

Remark 1.1.

It is obvious that any Open image in new window -inverse-strongly monotone mapping Open image in new window is monotone and Open image in new window -Lipschitz continuous.

The set of solutions of variational inequality (1.3) is denoted by Open image in new window . The variational inequality has been extensively studied in the literature; see, for example, [3, 4] and the references therein.

where Open image in new window is a potential function for Open image in new window (i.e., Open image in new window for Open image in new window ).

where the sequence Open image in new window is in the interval Open image in new window .

where Open image in new window is a contraction, Open image in new window is a nonexpansive mapping, and Open image in new window is a strongly positive linear bounded self-adjoint operator, proved that, under certain appropriate assumptions on the parameters, Open image in new window defined by (1.12) converges strongly to a fixed point of Open image in new window , which solves some variational inequality and is also the optimality condition for the minimization problem (1.9).

where Open image in new window is an Open image in new window -inverse strongly monotone mapping, Open image in new window and Open image in new window satisfy some parameters controlling conditions. They showed that if Open image in new window is nonempty, then the sequence Open image in new window generated by (1.14) converges strongly to some Open image in new window .

The existence of common fixed points for a finite family of nonexpansive mappings has been considered by many authors (see [17, 18, 19, 20] and the references therein). The well-known convex feasibility problem reduces to finding a point in the intersection of the fixed point sets of a family of nonexpansive mappings (see [21, 22]). The problem of finding an optimal point that minimizes a given cost function over the common set of fixed points of a family of nonexpansive mappings is of wide interdisciplinary interest and practical importance (see [18, 23]). A simple algorithmic solution to the problem of minimizing a quadratic function over the common set of fixed points of a family of nonexpansive mappings is of extreme value in many applications including set theoretic signal estimation (see [23, 24]).

where Open image in new window is a nonnegative real sequence with Open image in new window , for all Open image in new window , Open image in new window , form a family of infinitely nonexpansive mappings of Open image in new window into itself. Nonexpansivity of each Open image in new window ensures the nonexpansivity of Open image in new window . Such a Open image in new window is nonexpansive from Open image in new window to Open image in new window and it is called a Open image in new window -mapping generated by Open image in new window and Open image in new window .

where Open image in new window is a mapping defined by (1.15), Open image in new window is a contraction, Open image in new window is strongly positive linear bounded self-adjoint operator, Open image in new window is a Open image in new window -inverse strongly monotone, and we prove that under certain appropriate assumptions on the sequences Open image in new window , Open image in new window , Open image in new window , and Open image in new window , the sequences Open image in new window defined by (1.16) converge strongly to a common element of the set of common fixed points of a family of Open image in new window and the set of solutions of the variational inequality for an inverse-strongly monotone mapping, which solves some variational inequality and is also the optimality condition for the minimization problem (1.9).

## 2. Preliminaries

It is well known that each Hilbert space satisfies the Opial's condition.

Then Open image in new window is the maximal monotone and Open image in new window if and only if Open image in new window ; see [26].

Now we collect some useful lemmas for proving the convergence result of this paper.

Lemma 2.1.

Lemma 2.2 (see [27]).

Let Open image in new window and Open image in new window be bounded sequences in a Banach space Open image in new window and let Open image in new window be a sequence in Open image in new window with Open image in new window Suppose Open image in new window for all integers Open image in new window and Open image in new window Then, Open image in new window

Lemma 2.3 (see [28]).

where Open image in new window is a sequence in Open image in new window and Open image in new window is a sequence in Open image in new window such that

(2) Open image in new window or Open image in new window

Lemma 2.4 (see [9]).

Assume that Open image in new window is a strongly positive linear bounded self-adjoint operator on a Hilbert space Open image in new window with coefficient Open image in new window and Open image in new window . Then Open image in new window .

Throughout this paper, we will assume that Open image in new window , for all Open image in new window . Concerning Open image in new window defined by (1.15), we have the following lemmas which are important to prove our main result.

Lemma 2.5 (see [29]).

Let Open image in new window be a nonempty closed convex subset of a Hilbert space Open image in new window , let Open image in new window be a family of infinitely nonexpansive mapping with Open image in new window , and let Open image in new window be a real sequence such that Open image in new window , for all Open image in new window . Then

(1) Open image in new window is nonexpansive and Open image in new window for each Open image in new window ;

(2)for each Open image in new window and for each positive integer Open image in new window , the limit Open image in new window exists;

is a nonexpansive mapping satisfying Open image in new window and it is called the Open image in new window -mapping generated by Open image in new window and Open image in new window

Lemma 2.6 (see [30]).

## 3. Main Results

Now we are in a position to state and prove the main result in this paper.

Theorem 3.1.

Let Open image in new window be a closed convex subset of a real Hilbert space Open image in new window , let Open image in new window be a contraction of Open image in new window into itself, let Open image in new window be an Open image in new window -inverse strongly monotone mapping of Open image in new window into Open image in new window , and let Open image in new window be a family of infinitely nonexpansive mappings with Open image in new window . Let Open image in new window be a strongly positive linear bounded self-adjoint operator with the coefficient Open image in new window such that Open image in new window . Assume that Open image in new window . Let Open image in new window , Open image in new window , Open image in new window , and Open image in new window be sequences in Open image in new window satisfying the following conditions:

(C5) Open image in new window .

Proof.

which gives that the sequence Open image in new window is bounded, and so are Open image in new window and Open image in new window .

where Open image in new window is a constant such that Open image in new window . Similarly, there exists Open image in new window such that Open image in new window .

Putting Open image in new window , we get, Open image in new window .

Banach's Contraction Mapping Principle guarantees that Open image in new window has a unique fixed point, say Open image in new window . That is, Open image in new window .

Next we prove that Open image in new window .

First, we prove that Open image in new window .

This is a contradiction, which shows that Open image in new window .

Since Open image in new window is maximal monotone, we have Open image in new window , and hence Open image in new window .

The conclusion Open image in new window is proved.

Using (C1), (3.54), and (3.55), we get Open image in new window . Now applying Lemma 2.3 to (3.58), we conclude that Open image in new window . This completes the proof.

Remark 3.2.

Theorem 3.1 mainly improve the results of Qin and Cho [14] from a single nonexpansive mapping to an infinite family of nonexpansive mappings.

## 4. Applications

In this section, we obtain two results by using a special case of the proposed method.

Theorem 4.1.

where Open image in new window , Open image in new window , Open image in new window , and Open image in new window are sequences in Open image in new window satisfying the following conditions:

(C5) Open image in new window .

Proof.

We have Open image in new window and Open image in new window . Applying Theorem 3.1, we obtain the desired result.

Next, we will apply the main results to the problem for finding a common element of the set of fixed points of a family of infinitely nonexpansive mappings and the set of fixed points of a finite family of Open image in new window -strictly pseudocontractive mappings.

Definition 4.2.

The following lemmas can be obtained from [31, Proposition 2.6] by Acedo and Xu, easily.

Lemma 4.3.

Let Open image in new window be a Hilbert space, let Open image in new window be a closed convex subset of Open image in new window . For any integer Open image in new window , assume that, for each Open image in new window is a Open image in new window -strictly pseudocontractive mapping for some Open image in new window . Assume that Open image in new window is a positive sequence such that Open image in new window . Then Open image in new window is a Open image in new window -strictly pseudocontractive mapping with Open image in new window .

Lemma 4.4.

Let Open image in new window and Open image in new window be as in Lemma 4.3. Suppose that Open image in new window has a common fixed point in Open image in new window . Then Open image in new window .

This shows that Open image in new window is Open image in new window -inverse-strongly monotone.

Theorem 4.5.

where Open image in new window , Open image in new window , Open image in new window , and Open image in new window are the sequences in Open image in new window satisfying the following conditions:

Proof.

The conclusion of Theorem 4.5 can be obtained from Theorem 3.1.

Remark 4.6.

Theorem 4.5 is a generalization and improvement of the theorems by Qin and Cho [14], Iiduka and Takahashi [16, Thorem 3.1], and Takahashi and Toyoda [15].

## Notes

### Acknowledgments

The authors would like to thank the referee for the comments which improve the manuscript. R. Wangkeeree was supported for CHE-PhD-THA-SUP/2551 from the Commission on Higher Education and the Thailand Research Fund under Grant TRG5280011.

### References

- 1.Browder FE, Petryshyn WV:
**Construction of fixed points of nonlinear mappings in Hilbert space.***Journal of Mathematical Analysis and Applications*1967,**20:**197–228. 10.1016/0022-247X(67)90085-6MathSciNetCrossRefMATHGoogle Scholar - 2.Liu F, Nashed MZ:
**Regularization of nonlinear ill-posed variational inequalities and convergence rates.***Set-Valued Analysis*1998,**6**(4):313–344. 10.1023/A:1008643727926MathSciNetCrossRefMATHGoogle Scholar - 3.Yao J-C, Chadli O:
**Pseudomonotone complementarity problems and variational inequalities.**In*Handbook of Generalized Convexity and Generalized Monotonicity, Nonconvex Optimization and Its Applications*.*Volume 76*. Springer, New York, NY, USA; 2005:501–558.CrossRefGoogle Scholar - 4.Zeng LC, Schaible S, Yao JC:
**Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities.***Journal of Optimization Theory and Applications*2005,**124**(3):725–738. 10.1007/s10957-004-1182-zMathSciNetCrossRefMATHGoogle Scholar - 5.Deutsch F, Yamada I:
**Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings.***Numerical Functional Analysis and Optimization*1998,**19**(1–2):33–56.MathSciNetCrossRefMATHGoogle Scholar - 6.Xu H-K:
**Iterative algorithms for nonlinear operators.***Journal of the London Mathematical Society. Second Series*2002,**66**(1):240–256. 10.1112/S0024610702003332MathSciNetCrossRefMATHGoogle Scholar - 7.Xu HK:
**An iterative approach to quadratic optimization.***Journal of Optimization Theory and Applications*2003,**116**(3):659–678. 10.1023/A:1023073621589MathSciNetCrossRefMATHGoogle Scholar - 8.Yamada I:
**The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings.**In*Inherently Parallel Algorithm for Feasibility and Optimization and Their Applications, Studies in Computational Mathematics*.*Volume 8*. Edited by: Butnariu D, Censor Y, Reich S. North-Holland, Amsterdam, The Netherlands; 2001:473–504.Google Scholar - 9.Marino G, Xu H-K:
**A general iterative method for nonexpansive mappings in Hilbert spaces.***Journal of Mathematical Analysis and Applications*2006,**318**(1):43–52. 10.1016/j.jmaa.2005.05.028MathSciNetCrossRefMATHGoogle Scholar - 10.Moudafi A:
**Viscosity approximation methods for fixed-points problems.***Journal of Mathematical Analysis and Applications*2000,**241**(1):46–55. 10.1006/jmaa.1999.6615MathSciNetCrossRefMATHGoogle Scholar - 11.Mann WR:
**Mean value methods in iteration.***Proceedings of the American Mathematical Society*1953,**4:**506–510. 10.1090/S0002-9939-1953-0054846-3MathSciNetCrossRefMATHGoogle Scholar - 12.Ishikawa S:
**Fixed points by a new iteration method.***Proceedings of the American Mathematical Society*1974,**44:**147–150. 10.1090/S0002-9939-1974-0336469-5MathSciNetCrossRefMATHGoogle Scholar - 13.Genel A, Lindenstrauss J:
**An example concerning fixed points.***Israel Journal of Mathematics*1975,**22**(1):81–86. 10.1007/BF02757276MathSciNetCrossRefMATHGoogle Scholar - 14.Qin X, Cho Y:
**Convergence of a general iterative method for nonexpansive mappings in Hilbert spaces.***Journal of Computational and Applied Mathematics*. In pressGoogle Scholar - 15.Takahashi W, Toyoda M:
**Weak convergence theorems for nonexpansive mappings and monotone mappings.***Journal of Optimization Theory and Applications*2003,**118**(2):417–428. 10.1023/A:1025407607560MathSciNetCrossRefMATHGoogle Scholar - 16.Iiduka H, Takahashi W:
**Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings.***Nonlinear Analysis: Theory, Methods & Applications*2005,**61**(3):341–350. 10.1016/j.na.2003.07.023MathSciNetCrossRefMATHGoogle Scholar - 17.Aoyama K, Kimura Y, Takahashi W, Toyoda M:
**Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space.***Nonlinear Analysis: Theory, Methods & Applications*2007,**67**(8):2350–2360. 10.1016/j.na.2006.08.032MathSciNetCrossRefMATHGoogle Scholar - 18.Bauschke HH:
**The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space.***Journal of Mathematical Analysis and Applications*1996,**202**(1):150–159. 10.1006/jmaa.1996.0308MathSciNetCrossRefMATHGoogle Scholar - 19.Shang M, Su Y, Qin X:
**Strong convergence theorems for a finite family of nonexpansive mappings.***Fixed Point Theory and Applications*2007, Article ID 76971**2007:**-9.Google Scholar - 20.Shimoji K, Takahashi W:
**Strong convergence to common fixed points of infinite nonexpansive mappings and applications.***Taiwanese Journal of Mathematics*2001,**5**(2):387–404.MathSciNetMATHGoogle Scholar - 21.Bauschke HH, Borwein JM:
**On projection algorithms for solving convex feasibility problems.***SIAM Review*1996,**38**(3):367–426. 10.1137/S0036144593251710MathSciNetCrossRefMATHGoogle Scholar - 22.Combettes PL:
**The foundations of set theoretic estimation.***Proceedings of the IEEE*1993,**81**(2):182–208.CrossRefGoogle Scholar - 23.Youla DC:
**Mathematical theory of image restoration by the method of convex projections.**In*Image Recovery: Theory and Application*. Edited by: Star H. Academic Press, Orlando, Fla, USA; 1987:29–77.Google Scholar - 24.Iusem AN, De Pierro AR:
**On the convergence of Han's method for convex programming with quadratic objective.***Mathematical Programming Series B*1991,**52**(1–3):265–284.MathSciNetCrossRefMATHGoogle Scholar - 25.Su Y, Shang M, Qin X:
**A general iterative scheme for nonexpansive mappings and inverse-strongly monotone mappings.***Journal of Applied Mathematics and Computing*2008,**28**(1–2):283–294. 10.1007/s12190-008-0103-yMathSciNetCrossRefMATHGoogle Scholar - 26.Rockafellar RT:
**On the maximality of sums of nonlinear monotone operators.***Transactions of the American Mathematical Society*1970,**149**(1):75–88. 10.1090/S0002-9947-1970-0282272-5MathSciNetCrossRefMATHGoogle Scholar - 27.Suzuki T:
**Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals.***Journal of Mathematical Analysis and Applications*2005,**305**(1):227–239. 10.1016/j.jmaa.2004.11.017MathSciNetCrossRefMATHGoogle Scholar - 28.Xu H-K:
**Viscosity approximation methods for nonexpansive mappings.***Journal of Mathematical Analysis and Applications*2004,**298**(1):279–291. 10.1016/j.jmaa.2004.04.059MathSciNetCrossRefMATHGoogle Scholar - 29.Takahashi S, Takahashi W:
**Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces.***Journal of Mathematical Analysis and Applications*2007,**331**(1):506–515. 10.1016/j.jmaa.2006.08.036MathSciNetCrossRefMATHGoogle Scholar - 30.Chang SS:
*Variational Inequalities and Related Problems*. Chongqing Publishing House, Chongqing, China; 2007.Google Scholar - 31.Acedo GL, Xu H-K:
**Iterative methods for strict pseudo-contractions in Hilbert spaces.***Nonlinear Analysis: Theory, Methods & Applications*2007,**67**(7):2258–2271. 10.1016/j.na.2006.08.036MathSciNetCrossRefMATHGoogle Scholar

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