Research

# New iterative methods for a common solution of fixed points for pseudo-contractive mappings and variational inequalities

Rabian Wangkeeree1 and Kamonrat Nammanee2*

Author Affiliations

1 Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok, 65000, Thailand

2 Department of Mathematics, School of Science, University of Phayao, Phayao, 56000, Thailand

For all author emails, please log on.

Fixed Point Theory and Applications 2013, 2013:233  doi:10.1186/1687-1812-2013-233

 Received: 29 May 2013 Accepted: 18 August 2013 Published: 5 September 2013

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.

### Abstract

In this paper, we introduce three iterative methods for finding a common element of the set of fixed points for a continuous pseudo-contractive mapping and the solution set of a variational inequality problem governed by continuous monotone mappings. Strong convergence theorems for the proposed iterative methods are proved. Our results improve and extend some recent results in the literature.

MSC: 47H05, 47H09, 47J25, 65J15.

##### Keywords:
pseudo-contractive mapping; monotone mapping; strong convergence theorem

### 1 Introduction

The theory of variational inequalities represents, in fact, a very natural generalization of the theory of boundary value problems and allows us to consider new problems arising from many fields of applied mathematics, such as mechanics, physics, engineering, the theory of convex programming, and the theory of control. While the variational theory of boundary value problems has its starting point in the method of orthogonal projection, the theory of variational inequalities has its starting point in the projection on a convex set.

Let C be a nonempty closed and convex subset of a real Hilbert space H. The classical variational inequality problem is to find such that for all , where A is a nonlinear mapping. The set of solutions of the variational inequality is denoted by . The variational inequality problem has been extensively studied in the literature; see [1-9] and the references therein. In the context of the variational inequality problem, this implies that , , where is a metric projection of H into C.

Let A be a mapping from C to H, then A is called monotone if and only if for each ,

(1.1)

An operator A is said to be strongly positive on H if there exists a constant such that

A mapping A of C into itself is called L-Lipschitz continuous if there exits a positive number L such that

A mapping A of C into H is called α-inverse-strongly monotone if there exists a positive real number α such that

for all ; see [9-14]. If A is an α-inverse strongly monotone mapping of C into H, then it is obvious that A is -Lipschitz continuous, that is, for all . Clearly, the class of monotone mappings includes the class of α-inverse strongly monotone mappings.

A mapping A of C into H is called -strongly monotone if there exists a positive real number such that

for all ; see [15]. Clearly, the class of -strongly monotone mappings includes the class of strongly positive mappings.

Recall that a mapping T of C into H is called pseudo-contractive if for each , we have

(1.2)

T is said to be a k-strict pseudo-contractive mapping if there exists a constant such that

A mapping T of C into itself is called nonexpansive if for all . We denote by the set of fixed points of T. Clearly, the class of pseudo-contractive mappings includes the class of nonexpansive and strict pseudo-contractive mappings.

For a sequence of real numbers in and arbitrary , let the sequence in C be iteratively defined by and

(1.3)

where T is a nonexpansive mapping of C into itself. Halpern [16] was first to study the convergence of algorithm (1.3) in the framework of Hilbert spaces. Lions [17] and Wittmann [18] improved the result of Halpern by proving strong convergence of to a fixed point of T if the real sequence satisfies certain conditions. Reich [19], Shioji and Takahashi [20], and Zegeye and Shahzad [21] extended the result of Wittmann [18] to the case of a Banach space.

In 2000, Moudafi [22] introduced a viscosity approximation method and proved that if H is a real Hilbert space, for given , the sequence generated by the algorithm

(1.4)

where is a contraction mapping and satisfies certain conditions, converges strongly to the unique solution in C of the variational inequality

(1.5)

Moudafi [22] generalized Halpern’s theorems in the direction of viscosity approximations. In [23], Zegeye et al. extended Moudafi’s result to the case of Lipschitz pseudo-contractive mappings in Banach spaces more general that Hilbert spaces.

In 2006, Marino and Xu [24] introduced the following general iterative method:

(1.6)

They proved that if the sequence of parameters satisfies appropriate conditions, then the sequence generated by (1.6) converges strongly to the unique solution of the variational inequality

(1.7)

which is the optimality condition for the minimization problem

where h is a potential function for γf (i.e., for ).

Recently, Zegeye and Shahzad [25] introduced an iterative method and proved that if C is a nonempty subset of a real Hilbert space H, is a pseudo-contractive mapping and is a continuous monotone mapping such that . For defined by the following: for and , define

(1.8)

(1.9)

Then the sequence generated by and

(1.10)

where is a contraction mapping and and satisfy certain conditions, converges strongly to , where .

In this paper, motivated and inspired by the method of Marino and Xu [24] and the work of Zegeye and Shahzad [25], we introduce a viscosity approximation method for finding a common fixed point of a set of fixed points of continuous pseudo-contractive mappings more general than nonexpansive mappings and a solution set of the variational inequality problem for continuous monotone mappings more general than α-inverse strongly monotone mappings in a real Hilbert space. Our result extend and unify most of the results that have been proved for important classes of nonlinear operators.

Let C be a nonempty closed and convex subset of a real Hilbert space H. Let be a continuous pseudo-contractive mapping and a continuous monotone mapping, respectively. For and , let be defined by (1.8) and (1.9).

We consider the three iterative methods given as follows:

(1.11)

(1.12)

(1.13)

where A is a -strongly monotone and L-Lipschitzian continuous operator and is a contraction mapping. We prove in Section 3 that if and of parameters satisfy appropriate conditions, then the sequences , and converge strongly to .

### 2 Preliminaries

Let C be a closed and convex subset of a real Hilbert space H. For every , there exists a unique nearest point in C, denoted by , such that

(2.1)

is called the metric projection of H onto C. We know that is a nonexpansive mapping of H onto C. In connection with metric projection, we have the following lemma.

Lemma 2.1Let H be a real Hilbert space. The following identity holds:

Lemma 2.2LetCbe a nonempty closed convex subset of a Hilbert spaceH. Letand. Thenif and only if

(2.2)

Lemma 2.3[26]

Letbe a sequence of nonnegative real numbers such that

where

(i) , and

(ii) or.

Thenas.

Lemma 2.4[27]

LetCbe a nonempty closed and convex subset of a real Hilbert spaceH. Letbe a continuous monotone mapping. Then, forand, there existssuch that

(2.3)

Moreover, by a similar argument as in the proof of Lemmas 2.8 and 2.9 in[28], Zegeye[27]obtained the following lemmas.

Lemma 2.5[27]

LetCbe a nonempty closed and convex subset of a real Hilbert spaceH. Letbe a continuous monotone mapping. Forand, define a mappingas follows:

for all. Then the following hold:

(1) is single-valued;

(2) is a firmly nonexpansive type mapping, i.e., for all,

(3) ;

(4) is closed and convex.

In the sequel, we shall make use of the following lemmas.

Lemma 2.6[27]

LetCbe a nonempty closed and convex subset of a real Hilbert spaceH. Letbe a continuous pseudo-contractive mapping. Then, forand, there existssuch that

(2.4)

Lemma 2.7[27]

LetCbe a nonempty closed and convex subset of a real Hilbert spaceH. Letbe a continuous pseudo-contractive mapping. Forand, define a mappingas follows:

for all. Then the following hold:

(1) is single-valued;

(2) is a firmly nonexpansive type mapping, i.e., for all,

(3) ;

(4) is closed and convex.

Lemma 2.8[15]

Letand letfbe anα-contraction of a real Hilbert spaceHinto itself, and letAbe a-strongly monotone andL-Lipschitzian continuous operator ofHinto itself withand. Takeμ, γto be real numbers as follows:

If, and, then

### 3 Main results

Now, we prove our main theorems.

Theorem 3.1LetHbe a real Hilbert space, be a continuous pseudo-contractive mapping andbe a continuous monotone mapping such that. Letand letfbe anα-contraction ofHinto itself, and letAbe a-strongly monotone andL-Lipschitzian continuous operator ofCintoHwithand. Takeμ, γto be real numbers as follows:

For, letbe a sequence generated by (1.11), whereandare such that, , , and. Then the sequenceconverges strongly to, where.

Proof Since as , we may assume, without loss of generality, for all n. For , it implies that is a contraction of H into itself. Since H is a real Hilbert space, there exists a unique element such that .

Let , and let , where . Then we have from Lemma (2.5) and (2.7) that

(3.1)

Moreover, from (1.11) and (3.1), we get that

(3.2)

It follows from induction that

(3.3)

Thus is bounded, and hence so are , and . Next, to show that , we have

(3.4)

where .

Moreover, since , , we get that

(3.5)

(3.6)

Putting in (3.5) and in (3.6), we get that

(3.7)

(3.8)

Adding (3.7) and (3.8), we have

(3.9)

which implies that

(3.10)

Now, using the fact that is monotone, we get that

(3.11)

and hence

(3.12)

Without loss of generality, let us assume that there exists a real number b such that for all . Then we have

(3.13)

and hence from (3.13) we obtain that

(3.14)

where .

Furthermore, from (3.4) and (3.14) we have that

(3.15)

Hence by Lemma 2.3, we have

(3.16)

Consequently, from (3.14) and (3.16), we have that

(3.17)

Moreover, since , , we get that

(3.18)

and

(3.19)

Putting in (3.18) and in (3.19), we get that

(3.20)

and

(3.21)

Adding (3.20) and (3.21), we have

(3.22)

which implies that

(3.23)

Now, using the fact that is pseudo-contractive, we get that

(3.24)

and hence

(3.25)

Thus, using the method in (3.13) and (3.14), we have that

(3.26)

where .

Therefore, from (3.16) and the property of , we have that

(3.27)

Furthermore, since , we have that

(3.28)

From , we have .

Now, for , using Lemma 2.5, we get that

(3.29)

and hence

(3.30)

Therefore, we have

(3.31)

Hence

(3.32)

So, we have

(3.33)

Since , it follows that

Next, we show that

where .

To show this equality, we choose a subsequence of such that

Since is bounded, there exists a subsequence of and such that . Without loss of generality, we may assume that . Since and H is closed and convex, we get that . Moreover, since as , we have that .

Now, we show that . Note that from the definition of , we have

(3.34)

Put for all and . Consequently, we get that . From (3.34) it follows that

From the fact that as , we obtain that as .

Since is monotone, we have that . Thus, it follows that

and hence

(3.35)

Letting and using the fact that is continuous, we obtain that

(3.36)

This implies that .

Furthermore, from the definition of we have that

(3.37)

Put for all and . Consequently, we get that . From (3.33) and pseudo-contractivity of , it follows that

(3.38)

Then, since as , we obtain that as . Thus, as , it follows that

(3.39)

and hence

(3.40)

Letting and using the fact that is continuous, we obtain that

(3.41)

Now, let . Then we obtain that and hence .

Therefore, and since , by Lemma 2.2, implies that

(3.42)

Now, we show that as . From , we have that

(3.43)

This implies that

(3.44)

where , and . We put . It is easy to see that , and by (3.42). Hence, by Lemma 2.3, the sequence converges strongly to z. This completes the proof. □

Theorem 3.2LetHbe a real Hilbert space, be a continuous pseudo-contractive mapping andbe a continuous monotone mapping such that. Letand letfbe anα-contraction ofHinto itself, and letAbe a-strongly monotone andL-Lipschitzian continuous operator ofHinto itselfHwithand. Takeμ, γto be real numbers as follows:

For, letbe a sequence generated by (1.12), whereandare such that, , , and. The sequenceconverges strongly to, where.

Proof Let be the sequence given by and

From Theorem 3.1, . We claim that . Indeed, we estimate

(3.45)

It follows from , and Lemma 2.3 that .

Consequently, as required. □

Theorem 3.3LetHbe a real Hilbert space, be a continuous pseudo-contractive mapping andbe a continuous monotone mapping such that. Letand letfbe anα-contraction ofHinto itself, and letAbe a-strongly monotone andL-Lipschitzian continuous operator ofHinto itselfHwithand. Takeμ, γto be real numbers as follows:

For, letbe a sequence generated by (1.13), whereandare such that, , , and. The sequenceconverges strongly to, where.

Proof Define the sequences and by

Taking , we have

(3.46)

It follows from induction that

(3.47)

Thus both and are bounded. We observe that

Thus Theorem 3.2 implies that converges to some point z. In this case, we also have

Hence the sequence converges to some point z. This completes the proof. □

Setting , , where I is the identity mapping in Theorem 3.1, we have the following result.

Corollary 3.4LetHbe a real Hilbert space, be a continuous pseudo-contractive mapping andbe a continuous monotone mapping such that. Letfbe a contraction ofHinto itself and letbe a sequence generated byand

(3.48)

whereandare such that, , , and. The sequenceconverges strongly to, where.

In Theorem 3.1, , , is a constant mapping, then we get . In fact, we have the following corollary.

Corollary 3.5LetHbe a real Hilbert space, be a continuous pseudo-contractive mapping andbe a continuous monotone mapping such that. Letbe a sequence generated byand

(3.49)

whereandare such that, , , and. The sequenceconverges strongly to, where.

In Theorem 3.1, and , where I is the identity mapping, then we have the following corollary.

Corollary 3.6LetHbe a real Hilbert space andbe a continuous monotone mapping such that. Letfbe a contraction ofHinto itself, and letbe a sequence generated byand

(3.50)

whereandare such that, , , and. The sequenceconverges strongly to, where.

Remark 3.7 Our results extend and unify most of the results that have been proved for these important classes of nonlinear operators. In particular, Theorem 3.1 extends Theorem 3.1 of Iiduka and Takahashi [12] and Zegeye et al.[25], Corollary 3.2 of Su et al.[29] in the sense that our convergence is for the more general class of continuous pseudo-contractive and continuous monotone mappings. Corollary 3.4 also extends Theorem 4.2 of Iiduka and Takahashi [12] in the sense that our convergence is for the more general class of continuous pseudo-contractive and continuous monotone mappings.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

### Acknowledgements

The authors would like to thank Naresuan University and the university of Phayao. Moreover, the authors would like to thank the referees for reading this paper carefully, providing valuable suggestions and comments, and pointing out a major error in the original version of this paper.

### References

1. Borwein, FE: Nonlinear monotone operators and convex sets in Banach spaces. Bull. Am. Math. Soc.. 71, 780–785 (1965). Publisher Full Text

2. Bruck, RE: On the weak convergence of an ergodic iteration for the solution of variational inequalities for monotone operators in Hilbert space. J. Math. Anal. Appl.. 61, 159–164 (1977). Publisher Full Text

3. Noor, MA, Noor, KI, Al-Said, E: Iterative methods for solving general quasi-variational inequalities. Optim. Lett.. 4, 513–530 (2010). Publisher Full Text

4. Takahashi, W: Nonlinear complementarity problem and systems of convex inequalities. J. Optim. Theory Appl.. 24, 493–508 (1978)

5. Yao, Y, Liou, YC, Yao, JC: An extragradient method for fixed point problems and variational inequality problems. J. Inequal. Appl. (2007). PubMed Abstract | Publisher Full Text

6. Yao, Y, Liou, YC, Yao, JC: An iterative algorithm for approximating convex minimization problem. Appl. Math. Comput.. 188, 648–656 (2007). Publisher Full Text

7. Yao, Y, Noor, MA, Liou, YC: Strong convergence of a modified extragradient method to the minimum-norm solution of variational inequalities. Abstr. Appl. Anal. (2012). Publisher Full Text

8. Yao, Y, Cho, YJ, Chen, R: An iterative algorithm for solving fixed point problems, variational inequality problems and mixed equilibrium problems. Nonlinear Anal.. 71, 3363–3373 (2009). Publisher Full Text

9. Zegeye, H, Shahzad, N: Strong convergence for monotone mappings and relatively weak nonexpansive mappings. Nonlinear Anal.. 70, 2707–2716 (2009). Publisher Full Text

10. Borwein, JM: Fifty years of maximal monotonicity. Optim. Lett.. 4, 473–490 (2010). Publisher Full Text

11. Browder, FE, Petryshyn, WV: Construction of fixed points of nonlinear mappings in Hilbert spaces. J. Math. Anal. Appl.. 20, 197–228 (1967). Publisher Full Text

12. Iiduka, H, Takahashi, W, Toyoda, M: Approximation of solutions of variational inequalities for monotone mappings. Panam. Math. J.. 14, 49–61 (2004)

13. Liu, F, Nashed, MZ: Regularization of nonlinear ill-posed variational inequalities and convergence rates. Set-Valued Anal.. 6, 313–344 (1998). Publisher Full Text

14. Nakajo, K, Takahashi, W: Strong and weak convergence theorems by an improved splitting method. Commun. Appl. Nonlinear Anal.. 9, 99–107 (2002)

15. Lin, LJ, Takahashi, W: A general iterative method for hierarchical variational inequality problems in Hilbert spaces and applications. Positivity (2012). Publisher Full Text

16. Halpern, B: Fixed points of nonexpansive maps. Bull. Am. Math. Soc.. 73, 957–961 (1967). Publisher Full Text

17. Lions, PL: Approximation de points fixes de contractions. C. R. Acad. Sci. Paris Sér. A-B. 284, 1357–1359 (1977). PubMed Abstract

18. Wittmann, R: Approximation of fixed points of nonexpansive mappings. Arch. Math.. 58, 486–491 (1991)

19. Reich, S: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. Math. Anal. Appl.. 75, 287–292 (1980). Publisher Full Text

20. Shioji, N, Takahashi, W: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces. Proc. Am. Math. Soc.. 70, 45–57 (2009)

21. Zegeye, H, Shahzad, N: Viscosity approximation methods for fixed-points problems. Appl. Math. Comput.. 191, 155–163 (2007). Publisher Full Text

22. Moudafi, A: Viscosity approximation methods for fixed-points problems. J. Math. Anal. Appl.. 241, 46–55 (2000). Publisher Full Text

23. Zegeye, H, Shahzad, N, Mekkonen, T: Viscosity approximation methods for pseudocontractive mappings in Banach spaces. Appl. Math. Comput.. 185, 538–546 (2007). Publisher Full Text

24. Marino, G, Xu, HK: A general iterative method for nonexpansive mapping in Hilbert spaces. J. Math. Anal. Appl.. 318, 43–52 (2006). Publisher Full Text

25. Zegeye, H, Shahzad, N: Strong convergence of an iterative method for pseudo-contractive and monotone mappings. J. Glob. Optim.. 54, 173–184 (2012). Publisher Full Text

26. Xu, HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc.. 66, 240–256 (2002). Publisher Full Text

27. Zegeye, H: An iterative approximation methods for a common fixed point of two pseudo-contractive mappings. ISRN Math. Anal.. 2011, Article ID 621901 (2011)

Article ID 621901

Publisher Full Text

28. Takahashi, W, Zembayashi, K: Strong and weak convergence theorems for equilibriums problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Anal.. 70, 45–57 (2009). Publisher Full Text

29. Su, Y, Shang, M, Qin, X: An iterative method of solution for equilibrium and optimization problems. Nonlinear Anal.. 69, 2709–2719 (2008). Publisher Full Text