Journal of Mathematics

Volume 2013, Article ID 254821, 7 pages

http://dx.doi.org/10.1155/2013/254821

## Strong Convergence Theorems for a Pair of Strictly Pseudononspreading Mappings

School of Management, Tianjin University, Tianjin 300072, China

Received 18 December 2012; Accepted 17 June 2013

Academic Editor: Krassimir T. Atanassov

Copyright © 2013 Bin-Chao Deng and Tong Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Let be a real Hilbert space. Let be -, -strictly pseudononspreading mappings; let and be two real sequences in (0,1). For given , the sequence is generated iteratively by , , where with and is strongly monotone and Lipschitzian. Under some mild conditions on parameters and , we prove that the sequence converges strongly to the set of fixed points of a pair of strictly pseudononspreading mappings and .

#### 1. Introduction

Let be a real Hilbert space whose inner product and norm are denoted by and , and let be a nonempty, closed, and convex subset of , respectively. Recall the following definitions.

*Definition 1. *Let be a nonlinear mapping. (1)is said to be monotone if
(2) is said to be strongly monotone if there exists a constant such that
For such a case, is said to be -strongly-monotone.(3) is said to be inverse strongly if there exists a constant such that
For such a case, is said to be -inverse-strongly monotone.

The classical variational inequality which is denoted by is to find such that
The variational inequality has been extensively studied in the literature; see, for example, [1, 2] and the reference therein. Recall that is a nonexpansive mapping of into itself; that is,
The set of fixed points of is the set .

In 2011, Osilike and Isiogugu [3] introduced a new class of mappings, the so-called -strictly pseudononspreading; that is, a mapping is -strictly pseudononspreading if there exists such that
for all . They showed that the class of nonspreading mappings is properly contained in the class of strictly pseudononspreading mappings.

The iteration procedure of Mann’s type for approximating fixed points of a nonexpansive mapping is the following: and
where is a sequence in ; see [4]. For two nonexpansive mappings and , Takahashi and Tamura [5] considered the following iteration procedure: and
where and are sequences in .

In 2010, Tian [6] introduced the following general viscosity iterative scheme for finding an element of set of solutions to the fixed point of nonexpansive mapping in Hilbert space. Define sequence by
where is -Lipschitzian and -strongly monotone operator and is a nonexpansive mapping on ; then he proved that if the sequence satisfies appropriate conditions, the sequence generated by (9) converges strongly to the unique solution of the variational inequality
where .

In this paper, motivated by Takahashi and Tamura [5] and Tian [6], we introduce the following iteration scheme for finding a common point of the set of fixed points of a pair of strictly pseudononspreading mappings and :
where with and is -strongly monotone and -Lipschitzian on with , . Under suitable conditions, we prove a strong convergence theorem, which is different from the results of general viscosity iterative scheme in [6].

#### 2. Preliminaries

We need some facts and tools in Hilbert space which are listed as in the following lemmas.

*Definition 2. *A mapping is said to be demiclosed, if for any sequence which weakly converges to , and if the sequence strongly converges to , then .

*Definition 3. * is called demicontractive on , if there exists a constant such that

*Remark 4. *Every -strictly pseudononspreading mapping with a nonempty fixed point set is demicontractive (see [7, 8]).

*Remark 5 (see [9]). *Let be a -demicontractive mapping on with and for . (A1) -demicontractive is equivalent to
(A2) if .

*Remark 6. *According to with being a -strictly pseudononspreading mapping, we obtain

Proposition 7 (see [3]). * Let be a nonempty closed convex subset of a real Hilbert space , and let be a -strictly pseudononspreading mapping. If , then it is closed and convex. *

Proposition 8 (see [3]). * Let be a nonempty closed convex subset of a real Hilbert space , and let be a -strictly pseudononspreading mapping. Then is demiclosed at . *

Lemma 9. *Let be a real Hilbert space. The following expressions hold: *(i)*, , .*(ii)*, . *

Lemma 10 (see [10]). * Let be a closed convex subset of a Hilbert space , and let be a -strictly pseudononspreading mapping with a nonempty fixed point set. Let be fixed, and define by
**
Then . *

Lemma 11 (see [11]). * Assume is a sequence of nonnegative real numbers such that
**
where is a sequence in and is a sequence in such that *(i)*,
*(ii)* or . ** Then . *

Lemma 12 (see [12]). *Let be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence of which satisfies for all . Also consider that the sequence of integers is defined by
**
Then is a nondecreasing sequence verifying , ; it holds that and one has
*

Lemma 13. *Let be a closed convex subset of a real Hilbert space , given and . Then if and only if there holds the inequality
*

#### 3. Main Results

Let be a nonempty closed convex subset of a real Hilbert space , and let be , -strictly pseudononspreading mappings with nonempty fixed point set . Let be a -Lipschitz mapping on with coefficient . Assume that the set is nonempty. Since is closed and convex, the nearest point projection from onto is well defined. Recall that is -strongly monotone and -Lipschitzian on with , .

Lemma 14 (see [13]). * Let be a real Hilbert space and let be a -Lipschitzian and -strongly monotone operator with , . Let and . Then for , is in contradiction with constant . *

Lemma 15. *Let be a -strictly pseudononspreading mapping on , and with . Then
*

*Proof. *For we have

*Remark 16. *If and from Lemma 15, we can easily claim that , .

Theorem 17. *Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be , -strictly pseudononspreading mappings and demiclosed. Let be -Lipschitz mapping on with coefficient and let be -strongly monotone and -Lipschitzian on with , , , and . Let the sequence of and satisfy the following conditions: **,** and ,* *. **Let , , and let the following sequence be a sequence in generated from an arbitrary by
**
where with , . Then converges strongly to the unique element in verifying
**
which equivalently solves the following variational inequality problem:
*

*Proof. *Let . According to Remark 16, we have
Let . From (22), we obtain
And, from Lemma 14, we also obtain that
Together with (26) and (27), we obtain
Putting , we clearly obtain . By induction, we can deduce that is bounded and the sequences and , , are also bounded.

Next, we show the following estimation:
where and is chosen so that
From Lemma 15, we obtain
By (25) and (31), we get
Consequently,
and the desired inequality (29) follows.

Finally, we show by considering two possible cases.

*Case 1. * is nonincreasing sequence with some . In this case, is then convergent because it is also nonnegative (hence it is bounded form blow). From (29), we have
where . Consequently, both and converge to zero. From (22) and (25), and the sequence is bounded, we obtain
which implies
From , , and , we obtain
Using the demiclosedness principle (Proposition 8) and (35), we know that ; hence
It then follows from (29) that
According to Lemma 11, we obtain .

*Case 2. *Suppose there exists a subsequence of such that for all . In this situation, we consider the sequence of indices as defined in Lemma 12. It follows that , which by (29) amounts to
In a similar way to Case 1, we obtain
From Lemma 12 and (29)
for all . Taking in this inequality, we obtain . Moreover, it follows from (22) that
which together with (40) implies . Consequently, from Lemma 12, we obtain .

In addition, from (38), we obtain
then variational inequality (44) can be written as
So, by Lemma 13, it is equivalent to the fixed point equation

*Remark 18. *For a nonspreading mapping , we have in Theorem 17 to obtain the following corollary.

Corollary 19. *Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be nonspreading mappings and demiclosed. Let be -Lipschitz mapping on with coefficient and let be -strongly monotone and -Lipschitzian on with , . Let the sequence of and satisfy the following conditions: **,** and ,**. **Let the following sequence be a sequence in generated from an arbitrary by
**
where , , . Then converges strongly to the unique element in verifying
**
which equivalently solves the following variational inequality problem:
*

*Remark 20. *If and in Corollary 19, we obtain the following corollary.

Corollary 21. *Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be nonspreading mappings and demiclosed. Let be -Lipschitz mapping on with coefficient . Let the sequence of and satisfy the following conditions: **,** and ,**. **Let the following sequence be a sequence in generated from an arbitrary by
**
where , , . Then converges strongly to the unique element in verifying
**
which equivalently solves the following variational inequality problem:
*

#### 4. Application

In this section, we constructed a numerical example to illustrate that our main results are well defined. The following example is introduced by Deng et al. [14]. They prove that is -strictly pseudononspreading.

*Example 22. *Let with the norm defined by
, and let be an orthogonal subspace of (i.e., , we have ). Then it is obvious that is a nonempty closed convex subset of . Now, for any , define a mapping as follows:

From the definition of this example, let and be a Lipschitz mapping on . Assume that and be -strictly pseudononspreading mappings. Let , , , and . According to (22), we can obtain the following algorithm: From Theorem 17, we can easily know that algorithm (55) converges to the unique point in . Let and , and let be the fixed point of the algorithm (55). Using the software of MATLAB, we obtain .

#### Acknowledgments

This work is supported in part by National Natural Science Foundation of China (71272148), the Ph.D. Programs Foundation of Ministry of Education of China (20120032110039), and China Postdoctoral Science Foundation (Grant no. 20100470783).

#### References

- S. Plubtieng and R. Punpaeng, “A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings,”
*Applied Mathematics and Computation*, vol. 197, no. 2, pp. 548–558, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. C. Zeng, S. Schaible, and J. C. Yao, “Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities,”
*Journal of Optimization Theory and Applications*, vol. 124, no. 3, pp. 725–738, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. O. Osilike and F. O. Isiogugu, “Weak and strong convergence theorems for nonspreading-type mappings in Hilbert spaces,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 74, no. 5, pp. 1814–1822, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - W. R. Mann, “Mean value methods in iteration,”
*Proceedings of the American Mathematical Society*, vol. 4, pp. 506–510, 1953. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. Takahashi and T. Tamura, “Convergence theorems for a pair of nonexpansive mappings,”
*Journal of Convex Analysis*, vol. 5, no. 1, pp. 45–56, 1998. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Tian, “A general iterative algorithm for nonexpansive mappings in Hilbert spaces,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 73, no. 3, pp. 689–694, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. L. Hicks and J. D. Kubicek, “On the Mann iteration process in a Hilbert space,”
*Journal of Mathematical Analysis and Applications*, vol. 59, no. 3, pp. 498–504, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. A. Naimpally and K. L. Singh, “Extensions of some fixed point theorems of Rhoades,”
*Journal of Mathematical Analysis and Applications*, vol. 96, no. 2, pp. 437–446, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. E. Maing, “The viscosity approximation process for quasi-nonexpansive mappings in Hilbert spaces,”
*Computers & Mathematics with Applications*, vol. 59, no. 1, pp. 74–79, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. Petrot and R. Wangkeeree, “A general iterative scheme for strict pseudononspreading mapping related to optimization problem in Hilbert spaces,”
*Journal of Nonlinear Analysis and Optimization*, vol. 2, no. 2, pp. 329–336, 2011. View at Google Scholar · View at MathSciNet - H. K. Xu, “Iterative algorithms for nonlinear operators,”
*Journal of the London Mathematical Society*, vol. 66, no. 1, pp. 240–256, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. E. Maing, “Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization,”
*Set-Valued Analysis*, vol. 16, no. 7-8, pp. 899–912, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Piri, “A general iterative method for finding common solutions of system of equilibrium problems, system of variational inequalities and fixed point problems,”
*Mathematical and Computer Modelling*, vol. 55, no. 3-4, pp. 1622–1638, 2012. View at Publisher · View at Google Scholar - B. C. Deng, T. Chen, and Z. F. Li, “Viscosity iteration algorithm for a $\varrho $-strictly pseudononspreading mapping in a Hilbert space,”
*Journal of Inequalities and Applications*, vol. 2013, article 80, 2013. View at Publisher · View at Google Scholar