Open Access

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

Journal of Inequalities and Applications20102010:728452

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

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.

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., [117]). 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, 1829]), 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]).

Let be two measurable multivalued operators, be a constant and be a measurable selection of . Then there exists a measurable selection of such that, for any ,
(2.1)

Definition 2.7.

A multivalued operator is called a 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 ,
(2.2)

Let be the family of all fuzzy sets over . A mapping is called a fuzzy mapping over .

If is a fuzzy mapping over , then (denoted by ) is fuzzy set on , and is the membership degree of the point in . Let , . Then the set
(2.3)

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.

Let be three random fuzzy mappings satisfying the following condition (C): there exists three mappings , such that
(2.4)
By using the random fuzzy mappings , and , we can define the three multivalued mappings , and as follows, respectively.
(2.5)
It means that
(2.6)

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:

Find measurable mappings such that for all , , , , and
(2.7)

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).

Some special cases of (2.7):
  1. (1)
    If is a single-valued operator, , where is the identity mapping and for all and , then (2.7) is equivalent to finding such that and
    (2.8)
     

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:

Find such that and
(2.9)

for all and .

The generalized duality mapping is defined by
(2.10)

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 .

The modules of smoothness of is the function defined by
(2.11)

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.

Let be a given real number and be a real uniformly smooth Banach space. Then is -uniformly smooth if and only if there exists a constant such that, for all and , the following inequality holds:
(2.12)

Definition 2.9.

A random operator is said to be:

(a) -strongly accretive if there exists such that
(2.13)

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

(b) -Lipschitz continuous if there exists a real-valued random variable such that
(2.14)

for all and .

Definition 2.10.

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

(a) -strongly accretive with respect to in the first argument if there exists such that
(2.15)

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

(b) -Lipschitz continuous in the first argument if there exists a real-valued random variable such that
(2.16)

for all and .

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:

(a) -accretive if
(2.17)

for all , and where , for all ;

(b)strictly -accretive if
(2.18)

for all , , and and the equality holds if and only if for all ;

(c) -strongly -accretive if there exists a real-valued random variable such that, for any ,
(2.19)

for all , , and .

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.

The operator is said to be: -Lipschitz continuous if there exists a real-valued random variable such that
(2.20)

for all and .

Definition 2.15.

A multivalued measurable operator is said to be - -Lipschitz continuous if there exists a measurable function such that, for any ,
(2.21)

for all .

Definition 2.16.

Let be a -accretive random operator and be -strongly monotone random operator. Then the proximal operator is defined as follows:
(2.22)

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

Lemma 2.17 (see [31]).

Let be a -Lipschitz continuous operator, be a r-strongly -accretive operator and be an -accretive operator. Then, the proximal operator is -Lipschitz continuous, that is,
(2.23)

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.

The set of measurable mapping a random solution of problem (2.7) if and only if for all , and
(3.1)

Proof.

The proof directly follows from the definition of as follows:
(3.2)

Algorithm 3.2.

Suppose that be three random fuzzy mappings satisfying the condition (C). Let be -continuous random multivalued mappings induced by , , and , respectively. Let , and be random single-valued operators. Let be a random multivalued operator such that for each fixed and , be a -accretive mapping and . For any given measurable mapping , the multivalued mappings are measurable by Lemma 2.5. Then, there exists measurable selections and by Himmelberg [6]. Set
(3.3)
where and are the same as in Lemma 3.1. Then it is easy to know that is measurable. Since , and , by Lemma 2.6, there exist measurable selections and such that for all ,
(3.4)
By induction, we can define the sequences , , and inductively satisfying
(3.5)

From Algorithm 3.2, we can get the following algorithms.

Algorithm 3.3.

Suppose that , , , , and are the same as in Algorithm 3.2. Let be a random single-valued operator, and for all and . Then, for given measurable , we have
(3.6)

Algorithm 3.4.

Let be a random multivalued operator such that for each fixed , is a -accretive mapping and . If , , , , and are the same as in Algorithm 3.2, then for given measurable , we have
(3.7)

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.

Suppose that is a -uniformly smooth and separable real Banach space, is -strongly accretive and -Lipschitz continuous, be -Lipschitz continuous, is r-strongly -accretive and -Lipschitz continuous and is a random multivalued operator such that for each fixed and , is a -accretive mapping and . Let be a -Lipschitz continuous random operator and be -Lipschitz continuous in the first argument, -Lipschitz continuous in the second argument and -Lipschitz continuous in the third argument, respectively. Let be three random fuzzy mappings satisfying the condition (C), be three random multivalued mappings induced by the mappings , respectively, and are -Lipschitz continuous with constants , and , respectively. If for each real-valued random variables and such that, for any , ,
(4.1)
and the following conditions hold:
(4.2)
where is the same as in Lemma 2.8 for any . If there exist real-valued random variables , then there exist and such that is solution of (2.7) and
(4.3)

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

Proof.

From Algorithm 3.2, Lemma 2.17 and (4.1), we compute
(4.4)
By using is a strongly accretive and Lipschitz continuous, we have
(4.5)
that is
(4.6)

where is the same as in Lemma 2.8.

By Lipschitz continuity of in the first, second and third argument, , , is -Lipschitz continuous, -Lipschitz continuous, -Lipschitz continuous, respectively, and , , and are -Lipschitz continuous, we have
(4.7)
Using (4.6)–(4.7) in (4.4), we obtain for all ,
(4.8)
where
(4.9)
Letting
(4.10)

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 .

Next, we show that , we have
(4.11)
This implies that . Similarly, we can get and for all . Therefore, from Algorithm 3.2 and the continuity of , and , we obtain
(4.12)

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.

Let , , , , and are the same as in Theorem 4.1. Assume that is a random multivalued operator such that, for each fixed and , is a -accretive mapping. Let be -Lipschitz continuous, be a -Lipschitz continuous random operator, be -Lipschitz continuous and be -Lipschitz continuous in the first argument and -Lipschitz continuous in the second argument, respectively. If there exist real-valued random variables and such that (4.13) holds:
(4.13)

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.

Suppose that , , , , , and are the same as in Algorithm 3.2 Let be a random multivalued operator such that, for each fixed , is a -accretive mapping and . If there exists a real-valued random variable such that, for any , and the following conditions hold:
(4.14)

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, 2025, 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

(1)
Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT)
(2)
Centre of Excellence in Mathematics, CHE

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© N. Onjai-Uea and P. Kumam. 2010

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