- Research Article
- Open Access
General Nonlinear Random Equations with Random Multivalued Operator in Banach Spaces
© Heng-You Lan et al. 2009
- Received: 16 December 2008
- Accepted: 27 February 2009
- Published: 8 March 2009
We introduce and study a new class of general nonlinear random multivalued operator equations involving generalized -accretive mappings in Banach spaces. By using the Chang's lemma and the resolvent operator technique for generalized -accretive mapping due to Huang and Fang (2001), we also prove the existence theorems of the solution and convergence theorems of the generalized random iterative procedures with errors for this nonlinear random multivalued operator equations in -uniformly smooth Banach spaces. The results presented in this paper improve and generalize some known corresponding results in iterature.
- Lipschitz Continuity
- Smooth Banach Space
- Resolvent Operator
- Measurable Selection
- Random Operator
The variational principle has been one of the major branches of mathematical sciences for more than two centuries. It is a tool of great power that can be applied to a wide variety of problems in pure and applied sciences. It can be used to interpret the basic principles of mathematical and physical sciences in the form of simplicity and elegance. During this period, the variational principles have played an important and significant part as a unifying influence in pure and applied sciences and as a guide in the mathematical interpretation of many physical phenomena. The variational principles have played a fundamental role in the development of the general theory of relativity, gauge field theory in modern particle physics and soliton theory. In recent years, these principles have been enriched by the discovery of the variational inequality theory, which is mainly due to Hartman and Stampacchia . Variational inequality theory constituted a significant extension of the variational principles and describes a broad spectrum of very interesting developments involving a link among various fields of mathematics, physics, economics, regional, and engineering sciences. The ideas and techniques are being applied in a variety of diverse areas of sciences and prove to be productive and innovative. In fact, many researchers have shown that this theory provides the most natural, direct, simple, unified, and efficient framework for a general treatment of a wide class of unrelated linear and nonlinear problems.
Variational inclusion is an important generalization of variational inequality, which has been studied extensively by many authors (see, e.g., [2–14] and the references therein). In 2001, Huang and Fang  introduced the concept of a generalized -accretive mapping, which is a generalization of an -accretive mapping, and gave the definition of the resolvent operator for the generalized -accretive mapping in Banach spaces. Recently, Huang et al. [6, 7], Huang , Jin and Liu  and Lan et al.  introduced and studied some new classes of nonlinear variational inclusions involving generalized -accretive mappings in Banach spaces. By using the resolvent operator technique in , they constructed some new iterative algorithms for solving the nonlinear variational inclusions involving generalized -accretive mappings. Further, they also proved the existence of solutions for nonlinear variational inclusions involving generalized -accretive mappings and convergence of sequences generated by the algorithms.
On the other hand, It is well known that the study of the random equations involving the random operators in view of their need in dealing with probabilistic models in applied sciences is very important. Motivated and inspired by the recent research works in these fascinating areas, the random variational inequality problems, random quasi-variational inequality problems, random variational inclusion problems and random quasi-complementarity problems have been introduced and studied by Ahmad and Bazán , Chang , Chang and Huang , Cho et al. , Ganguly and Wadhwa , Huang , Huang and Cho , Huang et al. , and Noor and Elsanousi .
Inspired and motivated by recent works in these fields (see [3, 11, 12, 16, 25–28]), in this paper, we introduce and study a new class of general nonlinear random multivalued operator equations involving generalized -accretive mappings in Banach spaces. By using the Chang's lemma and the resolvent operator technique for generalized -accretive mapping due to Huang and Fang , we also prove the existence theorems of the solution and convergence theorems of the generalized random iterative procedures with errors for this nonlinear random multivalued operator equations in -uniformly smooth Banach spaces. The results presented in this paper improve and generalize some known corresponding results in literature.
for all , where is a constant. In particular, is the usual normalized duality mapping. It is well known that, in general, for all and is single-valued if is strictly convex (see, e.g., ). If is a Hilbert space, then becomes the identity mapping of . In what follows we will denote the single-valued generalized duality mapping by .
Now, we consider the following problem.
then (1.6) becomes the following problem.
for all and . The problem (1.8) has been studied by Cho et al.  when for all , .
For appropriate and suitable choices of , , , , , , , and for the space , a number of known classes of random variational inequality, random quasi-variational inequality, random complementarity, and random quasi-complementarity problems were studied previously by many authors (see, e.g., [17–20, 22–24] and the references therein).
In this paper, we will use the following definitions and lemmas.
An operator is called a random operator if for any , is measurable. A random operator is said to be continuous (resp., linear, bounded) if, for any , the operator is continuous (resp., linear, bounded).
It is well known that a measurable operator is necessarily a random operator.
If is a Hilbert space, then (a)–(d) of Definition 1.9 reduce to the definition of -monotonicity, strict -monotonicity, strong -monotonicity, and maximal -monotonicity, respectively; if is uniformly smooth and , then (a)–(d) of Definition 1.9reduces to the definitions of accretive, strictly accretive, strongly accretive, and -accretive operators in uniformly smooth Banach spaces, respectively.
In the study of characteristic inequalities in -uniformly smooth Banach spaces, Xu  proved the following result.
In this section, we suggest and analyze a new class of iterative methods and construct some new random iterative algorithms with errors for solving the problems (1.2)–(1.4), respectively.
Lemma 2.1 ().
Lemma 2.2 ().
Based on Lemma 2.3, we can develop a new iterative algorithm for solving the general nonlinear random equation (1.2) as follows.
From Algorithm 2.4, we can get the following algorithms.
In this section, we will prove the convergence of the iterative sequences generated by the algorithms in Banach spaces.
Since and for all , it follows from (2.6) and (3.15) that and so is a Cauchy sequence. Setting as for all . From (3.8)–(3.10), we know that , , are also Cauchy sequences. Hence there exist such that , , as .
From Theorem 3.1, we can get the following theorems.
For an appropriate choice of the mappings and the space , Theorems 3.1–3.4 include many known results of generalized variational inclusions as special cases (see [2, 4–9, 12, 17–23, 25, 26, 29] and the references therein).
This work was supported by the Scientific Research Fund of Sichuan Provincial Education Department (2006A106) and the Sichuan Youth Science and Technology Foundation (08ZQ026-008). This work was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050).
- Hartman P, Stampacchia G: On some non-linear elliptic differential-functional equations. Acta Mathematica 1966,115(1):271–310. 10.1007/BF02392210MathSciNetView ArticleMATHGoogle Scholar
- Agarwal RP, Cho YJ, Huang N-J: Generalized nonlinear variational inclusions involving maximal -monotone mappings. In Nonlinear Analysis and Applications: to V. Lakshmikantham on His 80th Birthday. Vol. 1, 2. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2003:59–73.View ArticleGoogle Scholar
- Agarwal RP, Khan MF, O'Regan D, Salahuddin : On generalized multivalued nonlinear variational-like inclusions with fuzzy mappings. Advances in Nonlinear Variational Inequalities 2005,8(1):41–55.MathSciNetMATHGoogle Scholar
- Ding XP: Generalized quasi-variational-like inclusions with nonconvex functionals. Applied Mathematics and Computation 2001,122(3):267–282. 10.1016/S0096-3003(00)00027-8MathSciNetView ArticleMATHGoogle Scholar
- Huang N-J: Generalized nonlinear variational inclusions with noncompact valued mappings. Applied Mathematics Letters 1996,9(3):25–29. 10.1016/0893-9659(96)00026-2MathSciNetView ArticleMATHGoogle Scholar
- Huang N-J, Fang Y-P: A new class of general variational inclusions involving maximal -monotone mappings. Publicationes Mathematicae Debrecen 2003,62(1–2):83–98.MathSciNetMATHGoogle Scholar
- Huang NJ, Fang YP, Deng CX: Nonlinear variational inclusions involving generalized -accretive mappings. Proceedings of the 9th Bellman Continuum International Workshop on Uncertain Systems and Soft Computing, July 2002, Beijing, China 323–327.Google Scholar
- Huang N-J: Nonlinear implicit quasi-variational inclusions involving generalized -accretive mappings. Archives of Inequalities and Applications 2004,2(4):413–425.MathSciNetMATHGoogle Scholar
- Jin M-M, Liu Q-K: Nonlinear quasi-variational inclusions involving generalized -accretive mappings. Nonlinear Functional Analysis and Applications 2004,9(3):485–494.MathSciNetMATHGoogle Scholar
- Lan H-Y, Kim JK, Huang NJ: On the generalized nonlinear quasi-variational inclusions involving non-monotone set-valued mappings. Nonlinear Functional Analysis and Applications 2004,9(3):451–465.MathSciNetMATHGoogle Scholar
- Lan H-Y, Liu Q-K, Li J: Iterative approximation for a system of nonlinear variational inclusions involving generalized -accretive mappings. Nonlinear Analysis Forum 2004,9(1):33–42.MathSciNetMATHGoogle Scholar
- Liu L-W, Li Y-Q: On generalized set-valued variational inclusions. Journal of Mathematical Analysis and Applications 2001,261(1):231–240. 10.1006/jmaa.2001.7493MathSciNetView ArticleMATHGoogle Scholar
- Verma RU, Khan MF, Salahuddin : Generalized setvalued nonlinear mixed quasivariational-like inclusions with fuzzy mappings. Advances in Nonlinear Variational Inequalities 2005,8(2):11–37.MathSciNetMATHGoogle Scholar
- Lan H-Y, He Z-Q, Li J: Generalized nonlinear fuzzy quasi-variational-like inclusions involving maximal -monotone mappings. Nonlinear Analysis Forum 2003,8(1):43–54.MathSciNetMATHGoogle Scholar
- Huang NJ, Fang YP: Generalized -accretive mappings in Banach spaces. Journal of Sichuan University 2001,38(4):591–592.MATHGoogle Scholar
- Ahmad R, Bazán FF: An iterative algorithm for random generalized nonlinear mixed variational inclusions for random fuzzy mappings. Applied Mathematics and Computation 2005,167(2):1400–1411. 10.1016/j.amc.2004.08.025MathSciNetView ArticleMATHGoogle Scholar
- Chang SS: Variational Inequality and Complementarity Problem Theory with Applications. Shanghai Scientific and Technological Literature, Shanghai, China; 1991.Google Scholar
- Chang SS, Huang NJ: Generalized random multivalued quasi-complementarity problems. Indian Journal of Mathematics 1993,35(3):305–320.MathSciNetMATHGoogle Scholar
- Cho YJ, Huang NJ, Kang SM: Random generalized set-valued strongly nonlinear implicit quasi-variational inequalities. Journal of Inequalities and Applications 2000,5(5):515–531. 10.1155/S1025583400000308MathSciNetMATHGoogle Scholar
- Ganguly A, Wadhwa K: On random variational inequalities. Journal of Mathematical Analysis and Applications 1997,206(1):315–321. 10.1006/jmaa.1997.5194MathSciNetView ArticleMATHGoogle Scholar
- Huang N-J: Random generalized nonlinear variational inclusions for random fuzzy mappings. Fuzzy Sets and Systems 1999,105(3):437–444. 10.1016/S0165-0114(97)00222-4MathSciNetView ArticleMATHGoogle Scholar
- Huang N-J, Cho YJ: Random completely generalized set-valued implicit quasi-variational inequalities. Positivity 1999,3(3):201–213. 10.1023/A:1009784323320MathSciNetView ArticleMATHGoogle Scholar
- Huang N-J, Long X, Cho YJ: Random completely generalized nonlinear variational inclusions with non-compact valued random mappings. Bulletin of the Korean Mathematical Society 1997,34(4):603–615.MathSciNetMATHGoogle Scholar
- Noor MA, Elsanousi SA: Iterative algorithms for random variational inequalities. Panamerican Mathematical Journal 1993,3(1):39–50.MathSciNetMATHGoogle Scholar
- Cho YJ, Shim SH, Huang NJ, Kang SM: Generalized strongly nonlinear implicit quasi-variational inequalities for fuzzy mappings. In Set Valued Mappings with Applications in Nonlinear Analysis, Series in Mathematical Analysis and Applications. Volume 4. Taylor & Francis, London, UK; 2002:63–77.Google Scholar
- Cho YJ, Lan H-Y: Generalized nonlinear random -accretive equations with random relaxed cocoercive mappings in Banach spaces. Computers & Mathematics with Applications 2008,55(9):2173–2182. 10.1016/j.camwa.2007.09.002MathSciNetView ArticleMATHGoogle Scholar
- Lan H-Y, Huang N-J, Cho YJ: A new method for nonlinear variational inequalities with multi-valued mappings. Archives of Inequalities and Applications 2004,2(1):73–84.MathSciNetMATHGoogle Scholar
- Verma RU: A class of projection-contraction methods applied to monotone variational inequalities. Applied Mathematics Letters 2000,13(8):55–62. 10.1016/S0893-9659(00)00096-3MathSciNetView ArticleMATHGoogle Scholar
- Hassouni A, Moudafi A: A perturbed algorithm for variational inclusions. Journal of Mathematical Analysis and Applications 1994,185(3):706–712. 10.1006/jmaa.1994.1277MathSciNetView ArticleMATHGoogle Scholar
- Xu HK: Inequalities in Banach spaces with applications. Nonlinear Analysis: Theory, Methods & Applications 1991,16(12):1127–1138. 10.1016/0362-546X(91)90200-KMathSciNetView ArticleMATHGoogle Scholar
- Chang SS: Fixed Point Theory with Applications. Chongqing Publishing, Chongqing, China; 1984.Google Scholar
- Himmelberg CJ: Measurable relations. Fundamenta Mathematicae 1975, 87: 53–72.MathSciNetMATHGoogle Scholar
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