# A Generalized Nonlinear Random Equations with Random Fuzzy Mappings in Uniformly Smooth Banach Spaces

- Nawitcha Onjai-Uea
^{1, 2}and - Poom Kumam
^{1, 2}Email author

**2010**:728452

https://doi.org/10.1155/2010/728452

© N. Onjai-Uea and P. Kumam. 2010

**Received: **26 July 2010

**Accepted: **31 October 2010

**Published: **4 November 2010

## Abstract

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.

## Keywords

## 1. Introduction

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 [3] 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 [18] 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. [11], 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.

## 2. Preliminaries

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.

Definition 2.1.

An operator
is said to be *measurable* if, for any
,
.

Definition 2.2.

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.

Definition 2.3.

A multivalued operator
is said to be *measurable* if, for any
,
.

Definition 2.4.

A operator
is called a *measurable selection* of a multivalued measurable operator
if
is measurable and for any
,
.

Lemma 2.5 (see [19]).

Let be a -continuous random multivalued operator. Then, for any measurable operator , the multivalued operator is measurable.

Lemma 2.6 (see [19]).

Definition 2.7.

*random multivalued operator*if, for any , is measurable. A random multivalued operator is said to be -

*continuous*if, for any , is continuous in , where is

*the Hausdorff metric*on defined as follows: for any given ,

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).

- (1)

for all and . The problem (2.8) was considered and studied by Agarwal et al. [1], 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 , 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., [29]). 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 [30] proved the following result.

Lemma 2.8.

Definition 2.9.

A random operator is said to be:

for all and , where is a real-valued random variable;

Definition 2.10.

Let be a random operator. A operator is said to be:

for all and , where is a real-valued random variable;

*-Lipschitz continuous*in the first argument if there exists a real-valued random variable such that

Similarly, we can define the Lipschitz continuity in the second argument and third argument of .

Definition 2.11.

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 ;

Definition 2.12.

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
.

Remark 2.13.

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.

Definition 2.14.

*-Lipschitz continuous*if there exists a real-valued random variable such that

Definition 2.15.

*-*

*-Lipschitz continuous*if there exists a measurable function such that, for any ,

Definition 2.16.

*proximal operator*is defined as follows:

for all and , where is a measurable function and is a strictly monotone operator.

Lemma 2.17 (see [31]).

## 3. Random Iterative Algorithms

In this section, we suggest and analyze a new class of iterative methods and construct some new random iterative algorithms for solving (2.7).

Lemma 3.1.

Proof.

Algorithm 3.2.

From Algorithm 3.2, we can get the following algorithms.

Algorithm 3.3.

Algorithm 3.4.

## 4. Existence and Convergence Theorems

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.

Theorem 4.1.

as , where , , and are iterative sequences generated by Algorithm 3.2.

Proof.

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.

Remark 4.2.

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. [11].

From Theorem 4.1, we can get the following theorems.

Theorem 4.3.

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).

Theorem 4.4.

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).

Remark 4.5.

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).

## Declarations

### Acknowledgments

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).

## Authors’ Affiliations

## References

- 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

## Copyright

This article is published under license to BioMed Central Ltd. 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.