© Xing Wei et al. 2009
Received: 2 January 2009
Accepted: 19 March 2009
Published: 5 April 2009
We introduce and study a new class of nonlinear general -monotone operator equations with multivalued operator. By using Alber's inequalities, Nalder's results, and the new proximal mapping technique, we construct some new perturbed iterative algorithms with mixed errors for solving the nonlinear general -monotone operator equations and study the approximation-solvability of the nonlinear operator equations in Banach spaces. The results presented in this paper improve and generalize the corresponding results on strongly monotone quasivariational inclusions and nonlinear implicit quasivariational inclusions.
Let be a real Banach space with the topological dual space of , let be the pairing between and , let denote the family of all subsets of and let denote the family of all nonempty closed bounded subsets of . We denote by the for all and . Let , and be nonlinear operators, and let be a general -monotone operator such that . We will consider the following nonlinear general -monotone operator equation with multivalued operator.
where is a constant and is the proximal mapping associated with the general -monotone operator due to Cui et al. .
where is defined by for all . The problem (1.5) was studied by Xia and Huang  when M is a general H-monotone mapping. Further, the problem (1.5) was studied by Peng et al.  if , the identity operator, and M is a multivalued maximal monotone mapping.
We note that for appropriate and suitable choices of , , , , , , and , it is easy to see that the problem (1.1) includes a number of quasivariational inclusions, generalized quasivariational inclusions, quasivariational inequalities, implicit quasivariational inequalities, complementarity problems, and equilibrium problems studied by many authors as special cases; see, for example, [4–7] and the references therein.
The study of such types of problems is motivated by an increasing interest to study the behavior and approximation of the solution sets for many important nonlinear problems arising in mechanics, physics, optimization and control, nonlinear programming, economics, finance, regional structural, transportation, elasticity, engineering, and various applied sciences in a general and unified framework. It is well known that many authors have studied a number of nonlinear variational inclusions and many systems of variational inequalities, variational inclusions, complementarity problems, and equilibrium problems by using the resolvent operator technique, which is a very important method to find solutions of variational inequality and variational inclusion problems; see, for example, [1–15] and the references therein.
On the other hand, Verma [4, 5] introduced the concept of -monotone mappings, which generalizes the well-known general class of maximal monotone mappings and originates way back from an earlier work of the Verma . Furthermore, motivated and inspired by the works of Xia and Huang , Cui et al.  introduced first a new class of general -monotone operators in Banach spaces, studied some properties of general -monotone operator, and defined a new proximal mapping associated with the general -monotone operator.
Inspired and motivated by the research works going on this field, the purpose of this paper is to introduce the new class of nonlinear general -monotone operator equation with multivalued operator. By using Alber's inequalities, Nalder's results, and the new proximal mapping technique, some new perturbed iterative algorithms with mixed errors for solving the nonlinear general -monotone operator equations will be constructed, and applications of general -monotone operators to the approximation-solvability of the nonlinear operator equations in Banach spaces will be studied. The results presented in this paper improve and extend some corresponding results in recent literature.
In this paper, we will use the following definitions and lemmas.
The notion of the cocoercivity is applied in several directions, especially to solving variational inequality problems using the auxiliary problem principle and projection methods , while the notion of the relaxed cocoercivity is more general than the strong monotonicity as well as cocoercivity. Several classes of relaxed cocoercive variational inequalities have been studied in [4, 5].
We also remark that the space is uniformly convex if and only if for all , and it is uniformly smooth if and only if Moreover, is uniformly convex if and only if is uniformly smooth. In this case, is reflexive by the Milman theorem. A Hilbert space is uniformly convex and uniformly smooth. The proof of the following inequalities can be found, for example, in page 24 of Alber .
If is a Hilbert space, then the general -monotone operator reduces to the -monotone operator in Verma . Therefore, the class of general -monotone operators provides a unifying frameworks for classes of maximal monotone operators, -monotone operators, -monotone operators, and general -monotone operators. For details about these operators, we refer the reader to [1, 2, 7] and the references therein.
Let be a reflexive Banach space with the dual space , a maximal monotone mapping, and a bounded, coercive, hemicontinuous, and relaxed monotone mapping. Then for any given , it follows from Theorem 3.1 in page 401 of Guo  that . This shows that is a general -monotone operator.
Example 2.10 (see ).
Let be a reflexive Banach space with its dual, and let be -strongly monotone. Let be locally Lipschitz such that is -relaxed monotone. Then is -monotone, which is equivalent to stating that is pseudomonotone (and in fact, maximal monotone).
Lemma 2.11 (see ).
3. Perturbed Algorithms and Convergence
Now we will consider some new perturbed algorithms for solving the nonlinear general -monotone operator equation problem (1.1) or (1.2) by using the proximal mapping technique associated with the general -monotone operators and the convergence of the sequences given by the algorithms.
Lemma 3.2 (see ).
Choose sequences , , and such that for , are two sequences in and and are error sequences in to take into account a possible inexact computation of the operator point, which satisfies the following conditions:
Based on Theorem 3.3 in , we have the following comment.
then (3.6) holds.
If or or in Algorithms 3.3 and 3.4, then the conclusions of Theorems 3.5 and 3.7 also hold, respectively. The results of Theorems 3.5 and 3.7 improve and generalize the corresponding results of [2, 4–9, 15, 17]. For other related works, we refer to [1–16] and the references therein.
The authors are grateful to Professor Ram U. Verma and the referees for their valuable comments and suggestions. This work was supported by the National Natural Science Foundation of China (60804065), the Scientific Research Fund of Sichuan Provincial Education Department (2006A106, 07ZB151), and the Sichuan Youth Science and Technology Foundation (08ZQ026-008).
- Cui Y-S, Lan H-Y, Chen Y-C: On implicit fuzzy proximal dynamical systems involving general -monotone operators in Banach spaces. Proceedings of the 5th International Conference on Fuzzy Systems and Knowledge Discovery (FSKD '08), October 2008, Jinan, China 1: 615–620.Google Scholar
- Xia F-Q, Huang N-J: Variational inclusions with a general -monotone operator in Banach spaces. Computers & Mathematics with Applications 2007,54(1):24–30. 10.1016/j.camwa.2006.10.028MathSciNetView ArticleMATHGoogle Scholar
- Ceng L-C, Guu S-M, Yao J-C: Iterative approximation of solutions for a class of completely generalized set-valued quasi-variational inclusions. Computers & Mathematics with Applications 2008,56(4):978–987. 10.1016/j.camwa.2008.01.026MathSciNetView ArticleMATHGoogle Scholar
- Verma RU: Approximation-solvability of a class of -monotone variational inclusion problems. Journal of the Korean Society for Industrial and Applied Mathematics 2004,8(1):55–66.Google Scholar
- Verma RU: -monotonicity and applications to nonlinear variational inclusion problems. Journal of Applied Mathematics and Stochastic Analysis 2004,2004(2):193–195. 10.1155/S1048953304403013View ArticleMathSciNetMATHGoogle Scholar
- Cho YJ, Lan H-Y: A new class of generalized nonlinear multi-valued quasi-variational-like inclusions with -monotone mappings. Mathematical Inequalities & Applications 2007,10(2):389–401.MathSciNetView ArticleMATHGoogle Scholar
- Verma RU: Nonlinear variational and constrained hemivariational inequalities involving relaxed operators. Zeitschrift für Angewandte Mathematik und Mechanik 1997,77(5):387–391. 10.1002/zamm.19970770517View ArticleMathSciNetMATHGoogle Scholar
- Peng J-W, Wang Y, Shyu DS, Yao J-C: Common solutions of an iterative scheme for variational inclusions, equilibrium problems, and fixed point problems. Journal of Inequalities and Applications 2008, 2008:-15.Google Scholar
- Ahmad R, Ansari QH: Generalized variational inclusions and -resolvent equations with -accretive operators. Taiwanese Journal of Mathematics 2007,11(3):703–716.MathSciNetMATHGoogle Scholar
- Guo DJ: Nonlinear Functional Analysis. 2nd edition. Shandong Science and Technology Press, Shandong, China; 2001.Google Scholar
- Lan H-Y, Kim JH, Cho YJ: On a new system of nonlinear -monotone multivalued variational inclusions. Journal of Mathematical Analysis and Applications 2007,327(1):481–493. 10.1016/j.jmaa.2005.11.067MathSciNetView ArticleMATHGoogle Scholar
- Lan H-Y, Cho YJ, Verma RU: On solution sensitivity of generalized relaxed cocoercive implicit quasivariational inclusions with -monotone mappings. Journal of Computational Analysis and Applications 2006,8(1):75–87.MathSciNetMATHGoogle Scholar
- Lan H-Y: Projection iterative approximations for a new class of general random implicit quasi-variational inequalities. Journal of Inequalities and Applications 2006, 2006:-17.Google Scholar
- Lan H-Y, Cho YJ, Xie W: General nonlinear random equations with random multi-valued operator in Banach spaces. Journal of Inequalities and Applications 2009, 2009:-17.Google Scholar
- Wang Z, Wu C: A system of nonlinear variational inclusions with -monotone mappings. Journal of Inequalities and Applications 2008, 2008:-6.Google Scholar
- Alber YI: Metric and generalized projection operators in Banach spaces: properties and applications. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure and Applied Mathematics. Volume 178. Edited by: Kartsatos A. Marcel Dekker, New York, NY, USA; 1996:15–50.Google Scholar
- Liu LS: Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces. Journal of Mathematical Analysis and Applications 1995,194(1):114–125. 10.1006/jmaa.1995.1289MathSciNetView ArticleMATHGoogle Scholar
- Nadler SB Jr.: Multi-valued contraction mappings. Pacific Journal of Mathematics 1969, 30: 475–488.MathSciNetView ArticleMATHGoogle Scholar
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