# Iterative Methods for Family of Strictly Pseudocontractive Mappings and System of Generalized Mixed Equilibrium Problems and Variational Inequality Problems

### Article metrics

• 1128 Accesses

• 2 Citations

## Abstract

We introduce a new iterative scheme by hybrid method for finding a common element of the set of common fixed points of infinite family of -strictly pseudocontractive mappings and the set of common solutions to a system of generalized mixed equilibrium problems and the set of solutions to a variational inequality problem in a real Hilbert space. We then prove strong convergence of the scheme to a common element of the three above described sets. We give an application of our results. Our results extend important recent results from the current literature.

## 1. Introduction

Let be a nonempty closed and convex subset of a real Hilbert space . A mapping is called monotone if

(1.1)

A mapping is called inverse-strongly monotone (see, e.g., [1, 2]) if there exists a positive real number such that , for all . For such a case, is called -inverse-strongly monotone. A -inverse-strongly monotone is sometime called -cocoercive. A mapping is said to be relaxed-cocoercive if there exists such that

(1.2)

is said to be relaxed-cocoercive if there exist such that

(1.3)

A mapping is said to be -Lipschitzian if there exists such that

(1.4)

Let be a nonlinear mapping. The variational inequality problem is to find an such that

(1.5)

(See, e.g., [3, 4].) We will denote the set of solutions of the variational inequality problem (1.5) by .

A monotone mapping is said to be maximal if the graph is not properly contained in the graph of any other monotone map, where for a multivalued mapping . It is also known that is maximal if and only if for for every implies . Let be a monotone mapping defined from into and a normal cone to at , that is, . Define a mapping by

(1.6)

Then, is maximal monotone and (see, e.g., [5]).

A mapping is said to be -strictly pseudocontractive if there exists a constant such that

(1.7)

for all . If , then the mapping is nonexpansive. A point is called a fixed point of if . The fixed points set of is the set . Iterative approximation of fixed points of -strictly pseudocontractive mappings have been studied extensively by many authors (see, e.g., [1, 69] and the references contained therein).

Let be a real-valued function and a nonlinear mapping. Suppose into is an equilibrium bifunction. That is, , forall . The generalized mixed equilibrium problem is to find (see, e.g., [1012]) such that

(1.8)

for all . We shall denote the set of solutions of this generalized mixed equilibrium problem by . Thus,

(1.9)

If , then problem (1.8) reduces to equilibrium problem studied by many authors (see, e.g., [8, 1317]), which is to find such that

(1.10)

for all . The set of solutions of (1.10) is denoted by .

If , then problem (1.8) reduces to generalized equilibrium problem studied by many authors (see, e.g., [1820]), which is to find such that

(1.11)

for all . The set of solutions of (1.11) is denoted by EP.

If , then problem (1.8) reduces to mixed equilibrium problem considered by many authors (see, e.g., [2123]), which is to find such that

(1.12)

for all . The set of solutions of (1.12) is denoted by MEP.

The generalized mixed equilibrium problems include fixed-point problems, optimization problems, variational inequality problems, Nash equilibrium problems, and equilibrium problems as special cases (see, e.g., [24]). Numerous problems in Physics, optimization, and economics reduce to find a solution of problem (1.8). Several methods have been proposed to solve the fixed-point problems, variational inequality problems and equilibrium problems in the literature (see, e.g., [5, 11, 12, 20, 2530]).

Recently, Ceng and Yao [25] introduced a new iterative scheme of approximating a common element of the set of solutions to mixed equilibrium problem and set of common fixed points of finite family of nonexpansive mappings in a real Hilbert space . In their results, they imposed the following condition on a nonempty closed and convex subset of :

(E) is -strongly convex and its derivative is sequentially continuous from weak topology to the strong topology.

We remark here that this condition (E) has been used by many authors for approximation of solution to mixed equilibrium problem in a real Hilbert space (see, e.g., [31, 32]). However, it is observed that the condition (E) does not include the case and . Furthermore, Peng and Yao [21], R. Wangkeeree and R. Wangkeeree [30], and many other authors replaced condition (E) with the following conditions:

(B1) for each and , there exists a bounded subset and such that for any ,

(1.13)

or

(B2) is a bounded set.

Consequently, conditions (B1) and (B2) have been used by many authors in approximating solution to generalized mixed equilibrium (mixed equilibrium) problems in a real Hilbert space (see, e.g., [21, 30]).

Recently, Takahashi et al. [33] proved the following convergence theorem using hybrid method.

Theorem 1.1 (Takahashi et al. [33]).

Let be a nonempty closed and convex subset of a real Hilbert space . Let be a nonexpansive mapping of into itself such that . For , define sequences and of as follows:

(1.14)

Assume that satisfies . Then, converges strongly to .

Motivated by the results of Takahashi et al. [33], Kumam [28] studied the problem of approximating a common element of set of solutions to an equilibrium problem, set of solutions to variational inequality problem and the set of fixed points of a nonexpansive mapping in a real Hilbert space. In particular, he proved the following theorem.

Theorem 1.2 (Kumam, [28]).

Let be a nonempty closed convex subset of a real Hilbert space . Let be a bifunction from satisfying (A1)–(A4) and let be a -inverse-strongly monotone mapping of into . Let be a nonexpansive mapping of into such that . For , define sequences and of as follows:

(1.15)

Assume that and satisfy

(1.16)

Then, converges strongly to .

Motivated by the ongoing research and the above-mentioned results, we introduce a new iterative scheme for finding a common element of the set of fixed points of an infinite family of-strictly pseudocontractive mappings, the set of common solutions to a system of generalized mixed equilibrium problems and the set of solutions to a variational inequality problem in a real Hilbert space. Furthermore, we show that our new iterative scheme converges strongly to a common element of the three afore mentioned sets. In our results, we use conditions (B1) and (B2) mentioned above. Our result extends many important recent results. Finally, we give some applications of our results.

## 2. Preliminaries

Let be a real Hilbert space with inner product and norm and let be a nonempty closed and convex subset of . The strong convergence of to is denoted by as .

For any point , there exists a unique point such that

(2.1)

is called the metric projection of onto . We know that is a nonexpansive mapping of onto . It is also known that satisfies

(2.2)

for all . Furthermore, is characterized by the properties and

(2.3)

for all and

(2.4)

In the context of the variational inequality problem, (2.3) implies that

(2.5)

If is -inverse-strongly monotone mapping of into , then it is obvious that is -Lipschitz continuous. We also have that for all and ,

(2.6)

So, if , then is a nonexpansive mapping of into .

For solving the generalized mixed equilibrium problem for a bifunction , let us assume that satisfies the following conditions:

(A1) for all ,

(A2) is monotone, that is, for all ,

(A3) for each ,

(A4)for each is convex and lower semicontinuous.

We need the following technical result.

Lemma 2.1 (R. Wangkeeree and R. Wangkeeree [30]).

Assume that satisfies (A1)–(A4) and let be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For and , define a mapping as follows:

(2.7)

for all . Then, the following hold:

(1)for each ,

(2) is single-valued,

(3) is firmly nonexpansive, that is, for any ,

(2.8)

(4),

(5) is closed and convex.

## 3. Main Results

Theorem 3.1.

Let be a nonempty closed and convex subset of a real Hilbert space . For each , let be a bifunction from satisfying (A1)–(A4), a proper lower semicontinuous and convex function with assumption () or (), an -inverse-strongly monotone mapping of into , a -inverse-strongly monotone mapping of into and for each , let be a -strictly pseudocontractive mapping for some such that . Let be a μ-Lipschitzian, relaxed -cocoercive mapping of into . Suppose . Let , , , and be generated by , , ,

(3.1)

Assume that and satisfy

(i),

(ii),

(iii),

(iv).

Then, converges strongly to .

Proof.

For all and , we obtain

(3.2)

This shows that is nonexpansive for each . Let . Then

(3.3)

Since both and are nonexpansive for each and , from (2.6), we have

(3.4)

Therefore,

(3.5)

Let , then is closed convex for each . Now assume that is closed convex for some . Then, from definition of , we know that is closed convex for the same . Hence, is closed convex for and for each . This implies that is closed convex for . Furthermore, we show that . For , . For , let . Then,

(3.6)

which shows that , for all , for all . Thus, , for all , for all . Hence, it follows that , for all . Therefore, is well defined. Since , for all and , for all , we have

(3.7)

Also, as , by (2.1) it follows that

(3.8)

From (3.7) and (3.8), we have that exists. Hence, is bounded and so are , , , , , , and , . For , we have that . By (2.4), we obtain

(3.9)

Letting and taking the limit in (3.9), we have , , which shows that is Cauchy. In particular, . Since, is Cauchy and is closed, there exists such that , . Since , therefore

(3.10)

and it follows that

(3.11)

Thus,

(3.12)

Furthermore,

(3.13)

Since , , we have

(3.14)

Hence, . From (3.1), we have

(3.15)

On the other hand,

(3.16)

and hence

(3.17)

Putting (3.17) into (3.15), we have

(3.18)

It follows that

(3.19)

Therefore, . Furthermore,

(3.20)

Since and , we have

(3.21)

Hence, . From (3.1), we have

(3.22)

On the other hand,

(3.23)

and hence

(3.24)

Putting (3.24) into (3.22), we have

(3.25)

It follows that

(3.26)

Therefore, . Then, we obtain that

(3.27)

Since , then

(3.28)

But implies that

(3.29)

Putting (3.29) into (3.28), we have

(3.30)

Thus, we get

(3.31)

But

(3.32)

Putting (3.32) into (3.31) and rearranging, we have

(3.33)

Hence, , . Now,

(3.34)

Furthermore,

(3.35)

Thus,

(3.36)

By conditions (iii) and (iv), we have that . Now, (2.2), we obtain

(3.37)

Thus,

(3.38)

Using this last inequality, we obtain from (3.1) that

(3.39)

This implies that

(3.40)

Since , we have . Also since and , we have that . By and , , we have that .Since , , we have for any that

(3.41)

Furthermore, from the last inequality and using (A2), we obtain

(3.42)

Let for all and . This implies that . Then, we have

(3.43)

Since , , we obtain , . Furthermore, by the monotonicity of , we obtain . Then, by (A4) we obtain (noting that , since ),

(3.44)

Using (A1), (A4) and (3.44), we also obtain

(3.45)

and hence

(3.46)

Letting , we have, for each ,

(3.47)

This implies that . By following the same arguments, we can show that .

Next, we show . Put

(3.48)

Since is relaxed -cocoercive and by condition (iv), we have

(3.49)

which shows that is monotone. Thus, is maximal monotone. Let . Since and , we have

(3.50)

On the other hand, from , we have

(3.51)

and hence

(3.52)

It follows that

(3.53)

which implies that . We have and hence . Therefore, .

Noting that , we have by (2.3) that

(3.54)

for all . Since and by the continuity of inner product, we obtain from the above inequality that

(3.55)

for all . By (2.3) again, we conclude that . This completes the proof.

Corollary 3.2.

Let be a nonempty closed and convex subset of a real Hilbert space . For each , let be a bifunction from satisfying (A1)–(A4), a proper lower semicontinuous and convex function with assumption (B1) or (B2), an -inverse-strongly monotone mapping of into , a -inverse-strongly monotone mapping of into and for each , let be a nonexpansive mapping such that . Let be a -Lipschitzian, relaxed -cocoercive mapping of into . Suppose . Let , , , and be generated by , , , ,

(3.56)

Assume that , and satisfy

(i),

(ii),

(iii),

(iv).

Then, converges strongly to .

Let be a nonempty closed and convex cone in and an operator of into . We define the polar of in to be the set

(3.57)

Then, the element is called a solution of the complementarity problem if

(3.58)

The set of solutions of the complementarity problem is denoted by . We shall assume that satisfies the following conditions:

(E1) is -inverse strongly monotone,

(E2).

Also, we replace conditions (B1) and (B2) with

(D1) for each and there exist a bounded subset and such that for any ,

(3.59)

(D2) is a bounded set.

Theorem 3.3.

Let be a nonempty closed and convex cone of a real Hilbert space . For each , let be a bifunction from satisfying (A1)–(A4), a proper lower semicontinuous and convex function with assumption () or (), an -inverse-strongly monotone mapping of into , a -inverse-strongly monotone mapping of into and for each , let be a -strictly pseudocontractive mapping for some such that . Let be a -Lipschitzian, relaxed -cocoercive mapping of into . Suppose . Let , and be generated by ,

(3.60)

Assume that , and satisfy

(i),

(ii),

(iii),

(iv).

Then, converges strongly to .

Proof.

Using Lemma  7.1.1 of [34], we have that . Hence, by Theorem 3.1, we obtain the desired conclusion.

Remark 3.4.

Our Corollary 3.2 extends Theorems 1.1 and 1.2.

Remark 3.5.

Our iterative scheme (3.1) is simpler than the iterative schemes (5.1) and (5.11) of Acedo and Xu [6]. Furthermore, in our results, we use iterative scheme (3.1) to approximate a common fixed point of an infinite family of-strictly pseudocontractive mappings while the iterative schemes (5.1) and (5.11) of Acedo and Xu [6] are used to approximate a common fixed point of a finite family of-strictly pseudocontractive mappings.

Remark 3.6.

Our results also hold for infinite family of uniformly continuous quasistrict pseudocontractions. Hence, we can adapt our results for an infinite family of uniformly continuous quasi-nonexpansive mappings in a real Hilbert space.

## References

1. 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)90085-6

2. 2.

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.2003.07.023

3. 3.

Bruck RE Jr.: On the weak convergence of an ergodic iteration for the solution of variational inequalities for monotone operators in Hilbert space. Journal of Mathematical Analysis and Applications 1977,61(1):159–164. 10.1016/0022-247X(77)90152-4

4. 4.

Ceng L-C, Khan AR, Ansari QH, Yao J-C: Viscosity approximation methods for strongly positive and monotone operators. Fixed Point Theory 2009,10(1):35–72.

5. 5.

Rockafellar RT: Monotone operators and the proximal point algorithm. SIAM Journal on Control and Optimization 1976,14(5):877–898. 10.1137/0314056

6. 6.

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.2006.08.036

7. 7.

Cho YJ, Qin X, Kang JI: Convergence theorems based on hybrid methods for generalized equilibrium problems and fixed point problems. Nonlinear Analysis: Theory, Methods & Applications 2009,71(9):4203–4214. 10.1016/j.na.2009.02.106

8. 8.

Liu Y: A general iterative method for equilibrium problems and strict pseudo-contractions in Hilbert spaces. Nonlinear Analysis: Theory, Methods & Applications 2009,71(10):4852–4861. 10.1016/j.na.2009.03.060

9. 9.

Zhou H: Convergence theorems of fixed points for -strict pseudo-contractions in Hilbert spaces. Nonlinear Analysis: Theory, Methods & Applications 2008,69(2):456–462. 10.1016/j.na.2007.05.032

10. 10.

Liu M, Chang S, Zuo P: On a hybrid method for generalized mixed equilibrium problem and fixed point problem of a family of quasi--asymptotically nonexpansive mappings in Banach spaces. Fixed Point Theory and Applications 2010, 2010:-18.

11. 11.

Petrot N, Wattanawitoon K, Kumam P: A hybrid projection method for generalized mixed equilibrium problems and fixed point problems in Banach spaces. Nonlinear Analysis: Hybrid Systems 2010,4(4):631–643. 10.1016/j.nahs.2010.03.008

12. 12.

Zhang S: Generalized mixed equilibrium problem in Banach spaces. Applied Mathematics and Mechanics. English Edition 2009,30(9):1105–1112. 10.1007/s10483-009-0904-6

13. 13.

Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces. Journal of Nonlinear and Convex Analysis 2005,6(1):117–136.

14. 14.

Plubtieng S, Punpaeng R: A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings. Applied Mathematics and Computation 2008,197(2):548–558. 10.1016/j.amc.2007.07.075

15. 15.

Qin X, Shang M, Su Y: Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems. Mathematical and Computer Modelling 2008,48(7–8):1033–1046. 10.1016/j.mcm.2007.12.008

16. 16.

Su Y, Shang M, Qin X: An iterative method of solution for equilibrium and optimization problems. Nonlinear Analysis: Theory, Methods & Applications 2008,69(8):2709–2719. 10.1016/j.na.2007.08.045

17. 17.

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):506–518. 10.1016/j.jmaa.2006.08.036

18. 18.

Qin X, Cho YJ, Kang SM: Viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Analysis: Theory, Methods & Applications 2010,72(1):99–112. 10.1016/j.na.2009.06.042

19. 19.

Shehu Y: Fixed point solutions of generalized equilibrium problems for nonexpansive mappings. Journal of Computational and Applied Mathematics 2010,234(3):892–898. 10.1016/j.cam.2010.01.055

20. 20.

Takahashi S, Takahashi W: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Analysis: Theory, Methods & Applications 2008,69(3):1025–1033. 10.1016/j.na.2008.02.042

21. 21.

Peng J-W, Yao J-C: Strong convergence theorems of iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems. Mathematical and Computer Modelling 2009,49(9–10):1816–1828. 10.1016/j.mcm.2008.11.014

22. 22.

Plubtieng S, Sombut K: Weak convergence theorems for a system of mixed equilibrium problems and nonspreading mappings in a Hilbert space. Journal of Inequalities and Applications 2010, 2010:-12.

23. 23.

Yao Y, Liou Y-C, Yao J-C: A new hybrid iterative algorithm for fixed-point problems, variational inequality problems, and mixed equilibrium problems. Fixed Point Theory and Applications 2008, 2008:-15.

24. 24.

Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. The Mathematics Student 1994,63(1–4):123–145.

25. 25.

Ceng L-C, Yao J-C: A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. Journal of Computational and Applied Mathematics 2008,214(1):186–201. 10.1016/j.cam.2007.02.022

26. 26.

Nadezhkina N, Takahashi W: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. Journal of Optimization Theory and Applications 2006,128(1):191–201. 10.1007/s10957-005-7564-z

27. 27.

Noor MA: General variational inequalities and nonexpansive mappings. Journal of Mathematical Analysis and Applications 2007,331(2):810–822. 10.1016/j.jmaa.2006.09.039

28. 28.

Kumam P: A new hybrid iterative method for solution of equilibrium problems and fixed point problems for an inverse strongly monotone operator and a nonexpansive mapping. Journal of Applied Mathematics and Computing 2009,29(1–2):263–280. 10.1007/s12190-008-0129-1

29. 29.

Qin X, Cho YJ, Kang SM: Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces. Journal of Computational and Applied Mathematics 2009,225(1):20–30. 10.1016/j.cam.2008.06.011

30. 30.

Wangkeeree R, Wangkeeree R: A general iterative method for variational inequality problems, mixed equilibrium problems, and fixed point problems of strictly pseudocontractive mappings in Hilbert spaces. Fixed Point Theory and Applications 2009, 2009:-32.

31. 31.

Liu M, Chang SS, Zuo P: An algorithm for finding a common solution for a system of mixed equilibrium problem, quasi-variational inclusion problem, and fixed point problem of nonexpansive semigroup. Journal of Inequalities and Applications 2010, 2010:-23.

32. 32.

Yao Y, Noor MA, Zainab S, Liou Y-C: Mixed equilibrium problems and optimization problems. Journal of Mathematical Analysis and Applications 2009,354(1):319–329. 10.1016/j.jmaa.2008.12.055

33. 33.

Takahashi W, Takeuchi Y, Kubota R: Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces. Journal of Mathematical Analysis and Applications 2008,341(1):276–286. 10.1016/j.jmaa.2007.09.062

34. 34.

Takahashi W: Nonlinear Functional Analysis, Fixed Point Theory and Its Application. Yokohama Publishers, Yokohama, Japan; 2000:iv+276.

## Acknowledgments

The author is extremely grateful to Professor S. Al-Homidan and the anonymous referees for their valuable comments and useful suggestions which improve the presentation of this paper. This research work is dedicated to Professor C. E. Chidume with admiration and respect.

## Author information

Correspondence to Yekini Shehu.

## Rights and permissions

Reprints and Permissions

• #### DOI

https://doi.org/10.1155/2011/852789

### Keywords

• Convex Subset
• Equilibrium Problem
• Nonexpansive Mapping
• Iterative Scheme
• Maximal Monotone