Abstract and Applied Analysis

Volume 2014 (2014), Article ID 393512, 5 pages

http://dx.doi.org/10.1155/2014/393512

## Strong Convergence Theorems of the Algorithm for -Monotone Operators in Hilbert Spaces

School of Mathematics and Statistics, Xidian University, Xi’an 710071, China

Received 19 February 2014; Accepted 6 April 2014; Published 30 April 2014

Academic Editor: Yisheng Song

Copyright © 2014 Huimin He and Sanyang Liu. 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

The aim of this paper is to show the strong convergence theorems of the algorithm for -monotone operators in Hilbert spaces by hybrid method in the mathematical programming. The main results extend and improve the corresponding results. Moreover, the assumption conditions of our results are weaker than those of the corresponding results.

#### 1. Introduction

In this paper, we show a algorithm for solving the inclusion problem in Hilbert space . The inclusion problem is finding the zero solutions of ; that is, This problem is closely related to many problems, such as variational inequalities, fixed points problem, and complementarity problem of mathematical programming, and it plays an important role in convex analysis and some partial differential equations. The inclusion problem (1) on monotone operator and maximal monotone operators is extensively investigated by so many researchers. See Tseng [1], Kamimura et al. [2–19], and so on.

In 2003, Fang and Huang [20] firstly introduced -monotone operators and discussed some properties of this class of operators.

Motivated by Fang and Huang [20], very recently, we firstly consider the inclusion problem of -monotone operator for finding the solutions of it in a Hilbert space [21].

In [21], we mainly presented strong and weak convergence theorems for Halpern type and Mann type algorithms, respectively, and the relations between maximal monotone operators and -monotone operators are analyzed in detail. Simultaneously, we apply these results to the minimization problem for and provide some numerical examples to support the theoretical findings. These results start a new branch of research for the inclusion problem , and we do further extending study for this subject.

Motivated by the main results of Nakajo and Takahashi [22], we propose a so-called iteration algorithm as follows: where and .

The aim of this paper is to establish the strong convergence theorems to approximate the zero point of -monotone operator, namely, finding the such that .

#### 2. Preliminaries

*Definition 1. *A multivalued operator is said to be(i)monotone if
(ii)maximal monotone if is monotone and for all , where denotes the identity mapping on .

We note that the is maximal monotone if and only if is monotone and the graph is not properly contained in the graph of any other monotone operator .

*Definition 2 (see [20]). *Let be a single mapping and a multivalued mapping. is said to be(i)-monotone if is monotone and holds for every ;(ii)strongly -monotone if is strongly monotone and holds for every .

*Definition 3. *Let be a single-valued operator. is said to be(i)strictly monotone if is monotone and
(ii)strongly monotone if there exists some constant such that
(iii)Lipschitz continuous if there exists some constant such that

*Remark 4. *We note that if is strongly monotone, then is strictly monotone, but vice is not. If is strongly monotone with constant and Lipschitz continuous with constant , then we have . As , then satisfies

Let be a single-valued operator and let be a strongly monotone and Lipschitz continuous operator with constant . Let be an -monotone operator and the resolvent operator is defined by
for each . We can define the following operators which are called* Yosida approximations*:
We give some elementary properties of and .

Lemma 5 (Proposition 4.1 in [21]). *Let be a strongly monotone and Lipschitz continuous operator with constant and let be an -monotone operator. Then the following properties hold:*(i)* , for all ;*(ii)

*(iii)*

*, for all*, or*, for all**;**is monotone and*(iv)

*, for all*.*Lemma 6 (Proposition 4.2 in [21]). Consider if and only if satisfies the relation
where is a constant and is the resolvent operator defined by (9).*

*Lemma 7 (Proposition 2.1 in [20]). Let be a strictly monotone single-valued operator and an -monotone operator. Then is maximal monotone.*

*Lemma 8 (see [23]). Let be a real Banach space. Then for all *

*3. Strong Convergence Theorems for Algorithm*

*3. Strong Convergence Theorems for Algorithm*

*We consider the following algorithm, and the sequence is generated by
where and . Motivated by Nakajo and Takahashi [22] and Fang et al. [12, 13, 20], we get the following results.*

*Theorem 9. Let be a strongly monotone and Lipschitz continuous operator with constant . Let be an -monotone operator; let and be a sequence defined by (14), where and satisfy , , , and . If , then converges strongly to , where is the metric projection of onto .*

*Proof. *The proofs can be divided into three steps.*Step 1 (** is well defined and **).* Based on the definitions of and , we can get that is a closed and convex set of for every .

And since the inequality
is equivalent to
hence, is closed and convex, so is for every .

From Lemma 6, there exists such that for all , and based on Lemma 5, we know is a nonexpansive mapping from into itself.

For all , it follows that
Then for each . Therefore, for every .

Next, we show that is well defined and by mathematical induction.

For , we have and . Hence, , because .

For , suppose that is given and for all . Then, there exists a unique such that , because is closed and convex.

Based on the property of the projection operators, we can get that is equivalent to
for all . By the assumption , we have .

Hence,
*Step 2 (** is bounded and **).* By Lemma 5, we get that is nonexpansive. And from Lemma 6, we have that is equivalent to . So, is a closed and convex subset of .

The rest of the proofs of this step can follow Lemmas 3.2 and 3.3 of [22]. *Step 3 (**).* From , we get that is bounded. So, is bounded.

Now, we suppose a subsequence of converges weakly to . Since , we have
From Step 2 and the assumption , we obtain
Hence,
And similarly, it follows from and Step 2 that
Therefore,
This implies that
Thus,
Next, we show that . Since is monotone and due to for all , we obtain
for all . Due to the assumption and inequality (24), we have
So, and
By Lemma 5, we know is a maximal monotone operator; from the maximality property of ; we have
If , by the lower semicontinuity of the norm, we get
Therefore, we obtain
Adding , we can get that .

Thus, .

*Remark 10. *In the main results, Theorem 9 of this paper, the assumption condition is , and we can obtain the conclusion . However, the conclusion is obtained by assuming the condition that appeared in [2, 20].

It is worth noting that condition is weaker than condition

by observing

*Conflict of Interests*

*Conflict of Interests*

*The authors declare that there is no conflict of interests regarding the publication of this paper.*

*Acknowledgments*

*Acknowledgments*

*This work was supported by the Fundamental Research Funds for the Central Universities, no. K5051370004, and the National Science Foundation for Young Scientists of China, nos. 11101320, 61202178, and 61373174.*

*References*

*References*

- P. Tseng, “Alternating projection-proximal methods for convex programming and variational inequalities,”
*SIAM Journal on Optimization*, vol. 7, no. 4, pp. 951–965, 1997. View at Publisher · View at Google Scholar · View at MathSciNet - S. Kamimura and W. Takahashi, “Approximating solutions of maximal monotone operators in Hilbert spaces,”
*Journal of Approximation Theory*, vol. 106, no. 2, pp. 226–240, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. T. Rockafellar, “Monotone operators and the proximal point algorithm,”
*SIAM Journal on Control and Optimization*, vol. 14, no. 5, pp. 877–898, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Brézis and P.-L. Lions, “Produits infinis de résolvantes,”
*Israel Journal of Mathematics*, vol. 29, no. 4, pp. 329–345, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Reich, “Strong convergence theorems for resolvents of accretive operators in Banach spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 75, no. 1, pp. 287–292, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Alvarez, “Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert space,”
*SIAM Journal on Optimization*, vol. 14, no. 3, pp. 773–782, 2003. View at Publisher · View at Google Scholar · View at MathSciNet - R. S. Burachik, J. O. Lopes, and B. F. Svaiter, “An outer approximation method for the variational inequality problem,”
*SIAM Journal on Control and Optimization*, vol. 43, no. 6, pp. 2071–2088, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T.-H. Kim and H.-K. Xu, “Strong convergence of modified Mann iterations,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 61, no. 1-2, pp. 51–60, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S.-Y. Matsushita and W. Takahashi, “A strong convergence theorem for relatively nonexpansive mappings in a Banach space,”
*Journal of Approximation Theory*, vol. 134, no. 2, pp. 257–266, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. Nadezhkina and W. Takahashi, “Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings,”
*SIAM Journal on Optimization*, vol. 16, no. 4, pp. 1230–1241, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Plubtieng and K. Ungchittrakool, “Strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space,”
*Journal of Approximation Theory*, vol. 149, no. 2, pp. 103–115, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Yao and R. Chen, “Iterative algorithm for approximating solutions of maximal monotone operators in Hilbert spaces,”
*Fixed Point Theory and Applications*, vol. 2007, Article ID 32870, 8 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L.-C. Zeng, S.-M. Guu, and J.-C. Yao, “An iterative method for generalized nonlinear set-valued mixed quasi-variational inequalities with $H$-monotone mappings,”
*Computers & Mathematics with Applications*, vol. 54, no. 4, pp. 476–483, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - T. Ibaraki and W. Takahashi, “A new projection and convergence theorems for the projections in Banach spaces,”
*Journal of Approximation Theory*, vol. 149, no. 1, pp. 1–14, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P.-E. Maingé, “A hybrid extragradient-viscosity method for monotone operators and fixed point problems,”
*SIAM Journal on Control and Optimization*, vol. 47, no. 3, pp. 1499–1515, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. E. Bruck Jr., “A strongly convergent iterative solution of $0\in U(x)$ for a maximal monotone operator $U$ in Hilbert space,”
*Journal of Mathematical Analysis and Applications*, vol. 48, pp. 114–126, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. He, S. Liu, and H. Zhou, “An explicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of an infinite family of nonexpansive mappings,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 72, no. 6, pp. 3124–3135, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Eckstein and D. P. Bertsekas, “On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators,”
*Mathematical Programming*, vol. 55, no. 3, pp. 293–318, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Kamimura and W. Takahashi, “Strong convergence of a proximal-type algorithm in a Banach space,”
*SIAM Journal on Optimization*, vol. 13, no. 3, pp. 938–945, 2002. View at Publisher · View at Google Scholar · View at MathSciNet - Y.-P. Fang and N.-J. Huang, “$H$-monotone operator and resolvent operator technique for variational inclusions,”
*Applied Mathematics and Computation*, vol. 145, no. 2-3, pp. 795–803, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Liu, H. He, and R. Chen, “Approximating solution of $0\in T(x)$ for an $H$-monotone operator in Hilbert spaces,”
*Acta Mathematica Scientia B: English Edition*, vol. 33, no. 5, pp. 1347–1360, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - K. Nakajo and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups,”
*Journal of Mathematical Analysis and Applications*, vol. 279, no. 2, pp. 372–379, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. V. Petryshyn, “A characterization of strict convexity of Banach spaces and other uses of duality mappings,”
*Journal of Functional Analysis*, vol. 6, pp. 282–291, 1970. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet

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