- Research Article
- Open Access
A Generalized Nonlinear Random Equations with Random Fuzzy Mappings in Uniformly Smooth Banach Spaces
© N. Onjai-Uea and P. Kumam. 2010
- Received: 26 July 2010
- Accepted: 31 October 2010
- Published: 4 November 2010
We introduce and study the general nonlinear random -accretive equations with random fuzzy mappings. By using the resolvent technique for the -accretive operators, we prove the existence theorems and convergence theorems of the generalized random iterative algorithm for this nonlinear random equations with random fuzzy mappings in -uniformly smooth Banach spaces. Our result in this paper improves and generalizes some known corresponding results in the literature.
- Smooth Banach Space
- Measurable Selection
- Random Operator
- Accretive Operator
- Multivalued Operator
Fuzzy Set Theory was formalised by Professor Lofti Zadeh at the University of California in 1965 with a view to reconcile mathematical modeling and human knowledge in the engineering sciences. The concept of fuzzy sets is incredible wide range of areas, from mathematics and logics to traditional and advanced engineering methodologies. Applications are found in many contexts, from medicine to finance, from human factors to consumer products, and from vehicle control to computational linguistics.
Random variational inequality theories is an important part of random function analysis. These topics have attracted many scholars and exports due to the extensive applications of the random problems (see, e.g., [1–17]). In 1997, Huang  first introduced the concept of random fuzzy mapping and studied the random nonlinear quasicomplementarity problem for random fuzzy mappings. Further, Huang studied the random generalized nonlinear variational inclusions for random fuzzy mappings in Hilbert spaces. Ahmad and Bazán  studied a class of random generalized nonlinear mixed variational inclusions for random fuzzy mappings and constructed an iterative algorithm for solving such random problems.
Very recently, Lan et al. , introduced and studied a class of general nonlinear random multivalued operator equations involving generalized -accretive mappings in Banach spaces and an iterative algorithm with errors for this nonlinear random multivalued operator equations.
Inspired and motivated by recent works in these fields (see [2, 13, 14, 18–29]), in this paper, we introduce and study a class of general nonlinear random equations with random fuzzy mappings in Banach spaces. By using Chang's lemma and the resolvent operator technique for -accretive mapping. We prove the existence and convergence theorems of the generalized random iterative algorithm for this nonlinear random equations with random fuzzy mappings in -uniformly smooth Banach spaces. Our results improve and extend the corresponding results of recent works.
Throughout this paper, let be a complete -finite measure space and be a separable real Banach space. We denote by , and the class of Borel -fields in , the inner product and the norm on , respectively. In the sequel, we denote , and by , is nonempty, bounded and closed} and the Hausdorff metric on , respectively.
Next, we will use the following definitions and lemmas.
An operator is said to be measurable if, for any , .
A 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).
Similarly, we can define a random operator . We will write and for all and .
It is well known that a measurable operator is necessarily a random operator.
A multivalued operator is said to be measurable if, for any , .
A operator is called a measurable selection of a multivalued measurable operator if is measurable and for any , .
Lemma 2.5 (see ).
Let be a -continuous random multivalued operator. Then, for any measurable operator , the multivalued operator is measurable.
Lemma 2.6 (see ).
Let be the family of all fuzzy sets over . A mapping is called a fuzzy mapping over .
is called a -cut set of fuzzy set .
(i)A fuzzy mapping is called measurable if, for any given , is a measurable multivalued mapping.
(ii)A fuzzy mapping is called a random fuzzy mapping if, for any , is a measurable fuzzy mapping.
It easy to see that , and are the random multivalued mappings. We call , and are random multivalued mappings induced by fuzzy mappings and , respectively.
Suppose that and with , and be two single-valued mappings. Let be three random fuzzy mappings satisfying the condition (C). Given mappings . Now, we consider the following problem:
The problem (2.7) is called random variational inclusion problem for random fuzzy mappings in Banach spaces. The set of measurable mappings is called a random solution of (2.7).
for all and . The problem (2.8) was considered and studied by Agarwal et al. , when .
If for all , and, for all , is a -accretive mapping, then (2.7) reduces to the following generalized nonlinear random multivalued operator equation involving -accretive mapping in Banach spaces:
for all and .
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 .
A Banach space is called uniformly smooth if and is called -uniformly smooth if there exists a constant such that , where is a real number.
It is well known that Hilbert spaces, (or ) spaces, and the Sobolev spaces , , are all -uniformly smooth.
In the study of characteristic inequalities in a -uniformly smooth Banach space, Xu  proved the following result.
A random operator is said to be:
for all and , where is a real-valued random variable;
for all and .
Let be a random operator. A operator is said to be:
for all and , where is a real-valued random variable;
for all and .
Similarly, we can define the Lipschitz continuity in the second argument and third argument of .
Let be a random operator be a random operator and be a random multivalued operator. Then is said to be:
for all , and where , for all ;
for all , , and and the equality holds if and only if for all ;
for all , , and .
Let be a single-valued mapping, be a single-valued mapping, be a multivalued mapping.
(i) is said to be m-accretive if is accretive and for all and , where is identity operator on ;
(ii) is said to be generalized m-accretive if is -accretive and for all and ;
(iii) is said to be -accretive if is accretive and for all and ;
(iv) is said to be -accretive if is -accretive and for all and .
If is a Hilbert space, then (a)–(c) of Definition 2.11 reduce to the definition of -monotonicity, strict -monotonicity and strong -monotonicity, respectively, if is uniformly smooth and , then (a)–(c) of Definition 2.11 reduce to the definitions of accretive, strictly accretive and strongly accretive in uniformly smooth Banach spaces, respectively.
for all and .
for all .
for all and , where is a measurable function and is a strictly monotone operator.
Lemma 2.17 (see ).
In this section, we suggest and analyze a new class of iterative methods and construct some new random iterative algorithms for solving (2.7).
From Algorithm 3.2, we can get the following algorithms.
In this section, we prove the existence and convergence theorems of the generalized random iterative algorithm for this nonlinear random equations with random fuzzy mappings in -uniformly smooth Banach spaces.
as , where , , and are iterative sequences generated by Algorithm 3.2.
where is the same as in Lemma 2.8.
Thus , as . From (4.2), we know that , for all . Using the same arguments as those used in the proof of (Lan et al. [11, Theorem 3.1, page 14]) it follows that and are Cauchy sequences. Thus by the completeness of , there exist such that as .
By Lemma 3.1, we know that is a solution of (2.7). This completes the proof.
If the fuzzy mapping , and are multivalued operators, , and multivalued is generalized -accretive mapping, is -strongly monotone, is -strongly accretive with respect to , then the result is Theorem 3.1 of Lan et al. .
From Theorem 4.1, we can get the following theorems.
for all , where is the same as in Lemma 2.8, for any , the iterative sequences , and defined by Algorithm 3.3 converge strongly to the solution of (2.8).
where is the same as in Lemma 2.8, for any , the iterative sequences , and defined by Algorithm 3.4 converge strongly to the solution of (2.9).
We note that for suitable choices of the mappings and space . Theorems 4.1–4.4 reduces to many known results of generalized variational inclusions as special cases (see [1, 3, 4, 20–25, 32] and the references therein).
This research is supported by the "Centre of Excellence in Mathematics", the Commission on High Education, Thailand. Moreover, N. Onjai-Uea is supported by the "Centre of Excellence in Mathematics", the Commission on High Education, Thailand for Ph.D. Program at King Mongkuts University of Technology Thonburi (KMUTT).
- 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. Volume 1, 2. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2003:59–73.Google 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
- 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 N-J, Fang YP, Deng CX: Nonlinear variational inclusions involving generalized -accretive mappings. Proceedings of the Bellman Continuum: International Workshop on Uncertain Systems and Soft Computing, July 2002, Beijing, China 323–327.Google Scholar
- Himmelberg CJ: Measurable relations. Fundamenta Mathematicae 1975, 87: 53–72.MathSciNetMATHGoogle 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
- 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: Approximation solvability of nonlinear random -resolvent operator equations with random relaxed cocoercive operators. Computers & Mathematics with Applications 2009, 57(4):624–632. 10.1016/j.camwa.2008.09.036MathSciNetView ArticleMATHGoogle Scholar
- Lan H-Y: A class of nonlinear -monotone operator inclusion problems with relaxed cocoercive mappings. Advances in Nonlinear Variational Inequalities 2006, 9(2):1–11.MathSciNetMATHGoogle Scholar
- Lan H-Y, Cho YJ, Xie W: General nonlinear random equations with random multivalued operator in Banach spaces. Journal of Inequalities and Applications 2009, 2009:-17.Google 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 HY, Kim JK, Huang N-J: 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
- 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
- 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
- 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: Fixed Point Theory with Applications. Chongqing, Chongqing, China; 1984.Google Scholar
- Chang SS: Variational Inequality and Complementarity Problem Theory with Applications. Shanghai Scientific and Technological Literature, Shanghai, China; 1991.Google Scholar
- Chang SS, Huang N-J: Generalized random multivalued quasi-complementarity problems. Indian Journal of Mathematics 1993, 35(3):305–320.MathSciNetMATHGoogle Scholar
- Cho YJ, Shim SH, Huang N-J, Kang SM: Generalized strongly nonlinear implicit quasi-variational inequalities for fuzzy mappings. In Set Valued Mappings with Applications in Nonlinear Analysis, Mathematical Analysis and Applications. Volume 4. Taylor & Francis, London, UK; 2002:63–77.Google Scholar
- Cho YJ, Huang N-J, 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
- 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
- Feng HR, Ding XP: A new system of generalized nonlinear quasi-variational-like inclusions with -monotone operators in Banach spaces. Journal of Computational and Applied Mathematics 2009, 225(2):365–373. 10.1016/j.cam.2008.07.048MathSciNetView 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
- 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
- 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
- Ahmad R, Farajzadeh AP: On random variational inclusions with random fuzzy mappings and random relaxed cocoercive mappings. Fuzzy Sets and Systems 2009, 160(21):3166–3174. 10.1016/j.fss.2009.01.002MathSciNetView 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
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