Abstract and Applied AnalysisVolume 2012 (2012), Article ID 517460, 13 pagesdoi:10.1155/2012/517460

Research Article

An Implicit Algorithm for Maximal Monotone Operators and Pseudocontractive Mappings

Hong-Jun Li,¹ Yeong-Cheng Liou,² Cun-Lin Li,³ Muhammad Aslam Noor,⁴ and Yonghong Yao¹

¹Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China²Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan³School of Management, North University for Nationalities, Yinchuan 750021, China⁴Mathematics Department, COMSATS Institute of Information Technology, Islamabad, Pakistan

Received 21 February 2012; Accepted 28 February 2012

Academic Editor: Khalida Inayat Noor

Copyright © 2012 Hong-Jun Li et al. 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 purpose of this paper is to construct an implicit algorithm for finding the common solution of maximal monotone operators and strictly pseudocontractive mappings in Hilbert spaces. Some applications are also included.

1. Introduction

Let be a real Hilbert space with inner product and norm , respectively. Let be a nonempty closed convex subset of .

Recall that is said to be a strictly pseudo contractive mapping if there exists a constant such that For such case, we also say that is a -strictly pseudo-contractive mapping. When , is said to be nonexpansive. It is clear that (1.1) is equivalent to We denote by the set of fixed points of .

A mapping is said to be -inverse strongly monotone if for some and for all , . It is known that if is an -inverse strongly monotone, then for all , .

Let be a mapping of into . The effective domain of is denoted by , that is, . A multi valued mapping is said to be a monotone operator on iff for all , , , and . A monotone operator on is said to be maximal if its graph is not strictly contained in the graph of any other monotone operator on . Let be a maximal monotone operator on , and let .

For a maximal monotone operator on and , we may define a single-valued operator , which is called the resolvent of for . It is known that the resolvent is firmly nonexpansive, that is, for all , and for all .

Algorithms for finding the fixed points of nonlinear mappings or for finding the zero points of maximal monotone operators have been studied by many authors. The reader can refer to [1–24]. Especially, Takahashi et al. [6] recently gave the following convergence result.

Theorem 1.1. Let be a closed and convex subset of a real Hilbert space . Let be an -inverse strongly monotone mapping of into and let be a maximal monotone operator on , such that the domain of is included in . Let be the resolvent of for , and let be a nonexpansive mapping of into itself, such that . Let and let , be a sequence generated by for all , where , and satisfy then generated by (1.6) converges strongly to a point of .

Motivated and inspired by the works in this field, the purpose of this paper is to construct an implicit algorithm for finding the common solution of maximal monotone operators and strictly-pseudocontractive mappings in Hilbert spaces. Some applications are also included.

2. Preliminaries

The following resolvent identity is well known: for and , there holds the identity We use the following notation:(i) stands for the weak convergence of to ;(ii) stands for the strong convergence of to .

We need the following lemmas for the next section.

Lemma 2.1 (see[14]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a -strict pseudo contraction. Define by for each , then, as , is nonexpansive such that .

Lemma 2.2 (see[15]). Let be a nonempty closed convex subset of a real Hilbert space . Let the mapping be -inverse strongly monotone and a constant, then one has In particular, if , then is nonexpansive.

Lemma 2.3 (see[14]). Let be a nonempty, closed and convex of a real Hilbert space . Let be a -strictly pseudo-contractive mapping, then is demi closed at 0, that is, if and , then .

Lemma 2.4 (see[16]). Let and be bounded sequences in a Banach space , and let be a sequence in with . Suppose that for all and , then .

Lemma 2.5 (see[17]). Assume that is a sequence of nonnegative real numbers such that where is a sequence in and is a sequence such that (1),(2) or , then .

3. Main Results

In this section, we will prove our main results.

Theorem 3.1. Let be a nonempty closed and convex subset of a real Hilbert space . Let be an -inverse strongly monotone mapping of into , and let be a maximal monotone operator on , such that the domain of is included in . Let be the resolvent of for which satisfies where . Let be a constant and a -strict pseudocontraction with such that . For , let be a net defined by then the net converges strongly, as , to a point , where is the metric projection.

Remark 3.2. Now, we show that the net defined by (3.1) is well defined. For any , we define a mapping . Note that (by Lemma 2.1), , and (by Lemma 2.2) are nonexpansive. For any , , we have which implies the mapping is a contraction on . We use to denote the unique fixed point of in . Therefore, is well defined. We can rewrite (3.1) as

In order to prove Theorem 3.1, we need the following propositions.

Proposition 3.3. Under the assumptions of Theorem 3.1, the net defined by (3.1) and hence (3.3) is bounded.

Proof. Let . It follows that for all . We can write as , for all . Since is nonexpansive for all , we have By using the convexity of and the -inverse strong monotonicity of , we derive By the assumption, we have , for all . Then, from (3.5) and (3.6), we obtain It follows from (3.3) and (3.7) that It follows that Therefore, is bounded.

Remark 3.4. Since is -inverse strongly monotone, it is -Lipschitz continuous. At the same time, is nonexpansive. So, from the boundedness, we deduce immediately that , , and are also bounded.

Proposition 3.5. Assume that all conditions in Theorem 3.1 hold. Let be the net defined by (3.1), then one has and .

Proof. By (3.7) and (3.8), we obtain So, Since , we obtain Next, we show . By using the firm nonexpansivity of , we have By the nonexpansivity of , we have It follows that Thus, This together with (3.8) implies that Hence, Since (by (3.12)), we deduce Therefore, Hence,

Finally, we prove Theorem 3.1.

Proof. From (3.5) and (3.8), we have It follows that where is some constant such that Next we show that is relatively norm compact as . Assume that is such that as . Put . From (3.23), we have Since is bounded, without loss of generality, we may assume that . From (3.21), we have We can apply Lemma 2.3 to (3.26) to deduce . Further, we show that is also in . Let . Set , for all , then we have Since is monotone, we have, for , It follows that Since , and , we have . We also observe that , and . Then, from (3.29), we derive Since is maximal monotone, we have . This shows that . So, we have . Hence, because of . Therefore, we can substitute for in (3.25) to get Consequently, the weak convergence of to actually implies that . This has proved the relative norm compactness of the net as .Now we return to (3.25) and take the limit as to get Equivalently, This clearly implies that Therefore, is the minimum norm element in . This completes the proof.

Corollary 3.6. Let be a nonempty closed and convex subset of a real Hilbert space . Let be an -inverse strongly monotone mapping of into H, and let be a maximal monotone operator on , such that the domain of is included in . Let be the resolvent of for which satisfies where . Let be a constant and a nonexpansive mapping such that . For , let be a net defined by then the net converges strongly, as , to a point .

Corollary 3.7. Let be a nonempty closed and convex subset of a real Hilbert space . Let be an -inverse strongly monotone mapping of into H, and let be a maximal monotone operator on , such that the domain of is included in . Let be the resolvent of for such that . Let be a constant satisfying where . For , let be a net generated by then the net converges strongly, as , to a point .

4. Applications

Next, we consider the problem for finding the minimum norm solution of a mathematical model related to equilibrium problems. Let be a nonempty, closed, and convex subset of a Hilbert space, and let be a bifunction satisfying the following conditions:(E1), for all ,(E2) is monotone, that is, , for all , ,(E3) for all , , , ,(E4) for all , is convex and lower semicontinuous.

Then, the mathematical model related to equilibrium problems (with respect to ) is to find such that for all . The set of such solutions is denoted by . The following lemma appears implicitly in Blum and Oettli [19].

Lemma 4.1. Let be a nonempty, closed, and convex subset of , and let be a bifunction of into satisfying (E1)–(E4). Let and , then there exists such that

The following lemma was given in Combettes and Hirstoaga [20].

Lemma 4.2. Assume that satisfies (E1)–(E4). For and , define a mapping as follows: for all . Then, the following hold: (1) is single valued,(2) is a firmly nonexpansive mapping, that is, for all , (3), (4) is closed and convex. We call such the resolvent of for . Using Lemmas 4.1 and 4.2, we have the following lemma. See [18] for a more general result.

Lemma 4.3. Let be a Hilbert space, and let be a nonempty, closed, and convex subset of . Let satisfy (E1)–(E4). Let be a multivalued mapping of into itself defined by then, , and is a maximal monotone operator with . Further, for any and , the resolvent of coincides with the resolvent of , that is, From Lemma 4.3, Theorem 3.1, and Lemma 4.2, one has the following results.

Corollary 4.4. Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be a bifunction from satisfying (E1)–(E4), and let be the resolvent of for . Let be a constant and a -strict pseudocontraction with such that . For , let be a net defined by then the net converges strongly, as , to a point .

Corollary 4.5. Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be a bifunction from satisfying (E1)–(E4), and let be the resolvent of for . Let be a constant and be a nonexpansive mapping such that . For , let be a net defined by then the net converges strongly, as , to a point .

Corollary 4.6. Let be a nonempty, closed and convex subset of a real Hilbert space . Let be a bifunction from satisfying (E1)–(E4), and let be the resolvent of for . Suppose . For , let be a net generated by then the net converges strongly, as , to a point .

Acknowledgments

H.-J. Li was supported in part by Colleges and Universities Science and Technology Development Foundation (20110322) of Tianjin. Y.-C. Liou was supported in part by NSC 100-2221-E-230-012. C.-L. Li was supported in part by NSFC 71161001-G0105. Y. Yao was supported in part by Colleges and Universities Science and Technology Development Foundation (20091003) of Tianjin, NSFC 11071279 and NSFC 71161001-G0105.

References

M. A. Noor and K. I. Noor, “Sensitivity analysis for quasi-variational inclusions,” Journal of Mathematical Analysis and Applications, vol. 236, no. 2, pp. 290–299, 1999.

M. A. Noor, “Generalized set-valued variational inclusions and resolvent equations,” Journal of Mathematical Analysis and Applications, vol. 228, no. 1, pp. 206–220, 1998.

Y. Yao and J. C. Yao, “On modified iterative method for nonexpansive mappings and monotone mappings,” Applied Mathematics and Computation, vol. 186, no. 2, pp. 1551–1558, 2007.

R. Chen, Y. Su, and H. K. Xu, “Regularization and iteration methods for a class of monotone variational inequalities,” Taiwanese Journal of Mathematics, vol. 13, no. 2B, pp. 739–752, 2009.

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.

S. Takahashi, W. Takahashi, and M. Toyoda, “Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces,” Journal of Optimization Theory and Applications, vol. 147, no. 1, pp. 27–41, 2010.

Y. Yao and N. Shahzad, “New methods with perturbations for non-expansive mappings in Hilbert spaces,” Fixed Point Theory and Applications, vol. 2011, article 79, 2011.

R. T. Rockafellar, “On the maximal monotonicity of subdifferential mappings,” Pacific Journal of Mathematics, vol. 33, pp. 209–216, 1970.

M. Aslam Noor and T. M. Rassias, “Projection methods for monotone variational inequalities,” Journal of Mathematical Analysis and Applications, vol. 237, no. 2, pp. 405–412, 1999.

M. A. Noor and Z. Huang, “Some resolvent iterative methods for variational inclusions and nonexpansive mappings,” Applied Mathematics and Computation, vol. 194, no. 1, pp. 267–275, 2007.

Y. Yao, Y.-C. Liou, and S. M. Kang, “Two-stepprojection methods for a system of variational inequality problems in Banach spaces,” Journal of Global Optimization. In press

Y. Yao, R. Chen, and Y.-C. Liou, “A unified implicit algorithm for solving the triple-hierarchical constrained optimization problem,” Mathematical & Computer Modelling, vol. 55, pp. 1506–1515, 2012.

S. Adly, “Perturbed algorithms and sensitivity analysis for a general class of variational inclusions,” Journal of Mathematical Analysis and Applications, vol. 201, no. 2, pp. 609–630, 1996.

G. Marino and H.-K. Xu, “Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 336–349, 2007.

W. Takahashi and M. Toyoda, “Weak convergence theorems for nonexpansive mappings and monotone mappings,” Journal of Optimization Theory and Applications, vol. 118, no. 2, pp. 417–428, 2003.

T. Suzuki, “Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces,” Fixed Point Theory and Applications, vol. 2005, no. 1, pp. 103–123, 2005.

H.-K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society. Second Series, vol. 66, no. 1, pp. 240–256, 2002.

K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda, “On a strongly nonexpansive sequence in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol. 8, no. 3, pp. 471–489, 2007.

E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,” The Mathematics Student, vol. 63, no. 1–4, pp. 123–145, 1994.

P. L. Combettes and S. A. Hirstoaga, “Equilibrium programming in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol. 6, no. 1, pp. 117–136, 2005.

Y. Yao, Y.-C. Liou, and J.-C. Yao, “An extragradient method for fixed point problems and variational inequality problems,” Journal of Inequalities and Applications, vol. 2007, Article ID 38752, 12 pages, 2007.

Y. Yao, Y.-C. Liou, and Y.-J. Wu, “An extragradient method for mixed equilibrium problems and fixed point problems,” Fixed Point Theory and Applications, vol. 2009, Article ID 632819, 15 pages, 2009.

Y. Yao, Y. C. Liou, and G. Marino, “Strong convergence of two iterative algorithms for nonexpansive mappings in Hilbert spaces,” Fixed Point Theory and Applications, vol. 2009, Article ID 279058, 7 pages, 2009.

F. Cianciaruso, G. Marino, L. Muglia, and Y. Yao, “On a two-step algorithm for hierarchical fixed point problems and variational inequalities,” Journal of Inequalities and Applications, vol. 2009, Article ID 208692, 13 pages, 2009.