- Research Article
- Open Access

# General Nonlinear Random Equations with Random Multivalued Operator in Banach Spaces

- Heng-You Lan
^{1}, - Yeol Je Cho
^{2}Email author and - Wei Xie
^{1}

**2009**:865093

https://doi.org/10.1155/2009/865093

© Heng-You Lan et al. 2009

**Received:**16 December 2008**Accepted:**27 February 2009**Published:**8 March 2009

## Abstract

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.

## Keywords

- Lipschitz Continuity
- Smooth Banach Space
- Resolvent Operator
- Measurable Selection
- Random Operator

## 1. Introduction and Preliminaries

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 [1]. 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 [15] 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 [8], Jin and Liu [9] and Lan et al. [11] introduced and studied some new classes of nonlinear variational inclusions involving generalized -accretive mappings in Banach spaces. By using the resolvent operator technique in [6], 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 [16], Chang [17], Chang and Huang [18], Cho et al. [19], Ganguly and Wadhwa [20], Huang [21], Huang and Cho [22], Huang et al. [23], and Noor and Elsanousi [24].

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 [15], 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., [28]). If is a Hilbert space, then becomes the identity mapping of . In what follows we will denote the single-valued generalized duality mapping by .

Suppose that is a random multivalued operator such that for each fixed and , is a generalized -accretive mapping and . Let , and be single-valued operators, and let be three multivalued operators.

Now, we consider the following problem.

for all
and
. The problem (1.2) is called the *general nonlinear random equation* with multivalued operator involving generalized
-accretive mapping in Banach spaces.

- (1)
If is a single-valued operator, , the identity mapping and for all and , then problem (1.2) is equivalent to finding such that and

- (2)
If for all , and, for all , is a generalized -accretive mapping, then the problem (1.2) reduces to the following generalized nonlinear random multivalued operator equation involving generalized -accretive mapping in Banach spaces.

- (3)
If is a Hilbert space and for all , where denotes the subdifferential of a lower semicontinuous and -subdifferetiable function , then the problem (1.4) becomes the following problem.

*generalized nonlinear random variational inclusions*for random multivalued operators in Hilbert spaces. The determinate form of the problem (1.5) was studied by Agarwal et al. [3] when for all , where is a single-valued operator.

- (4)
If for all , , then the problem (1.5) reduces to the following nonlinear random variational inequalities.

- (5)

then (1.6) becomes the following problem.

for all and . The problem (1.8) has been studied by Cho et al. [19] when for all , .

Remark 1.1.

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.

Definition 1.2.

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

Definition 1.3.

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).

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 1.4.

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

Definition 1.5.

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

Definition 1.6.

*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 ,

Definition 1.7.

A random operator is said to be

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

for all and .

Definition 1.8.

Let be a random operator. An 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

for all and .

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

Definition 1.9.

Let be a random operator and be a random multivalued operator. Then is said to be

for all , , and , where ;

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

for all , , and ;

(d)*generalized*
*accretive* if
is
-accretive and
for all
and (equivalently, for some)
.

Remark 1.10.

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.

Definition 1.11.

The operator is said to be

for all and ;

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

for all and ;

for all and .

Definition 1.12.

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

for all .

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 -uniformly smooth Banach spaces, Xu [30] proved the following result.

Lemma 1.13.

Definition 1.14.

*resolvent operator*for is defined as follows:

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

From Huang et al. [6, 15], we can obtain the following lemma.

Lemma 1.15.

for all and .

## 2. Random Iterative Algorithms

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 ([31]).

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

Lemma 2.2 ([31]).

Lemma 2.3.

where and is a real-valued random variable.

Proof.

The proof directly follows from the definition of and so it is omitted.

Based on Lemma 2.3, we can develop a new iterative algorithm for solving the general nonlinear random equation (1.2) as follows.

Algorithm 2.4.

for all .

From Algorithm 2.4, we can get the following algorithms.

Algorithm 2.5.

where is the same as in Algorithm 2.4.

Algorithm 2.6.

where is the same as in Algorithm 2.4.

Remark 2.7.

Algorithms 2.4–2.6 include several known algorithms of [2, 4–9, 12, 17–23, 25, 26, 29] as special cases.

## 3. Existence and Convergence Theorems

In this section, we will prove the convergence of the iterative sequences generated by the algorithms in Banach spaces.

Theorem 3.1.

as , where , , and are iterative sequences generated by Algorithm 2.4.

Proof.

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 .

By Lemma 2.3, now we know that is a solution of the problem (1.2). This completes the proof.

Remark 3.2.

then (3.2) holds. We note that Hilbert spaces and (or ) spaces, , are 2-uniformly smooth.

From Theorem 3.1, we can get the following theorems.

Theorem 3.3.

for all , where is the same as in Lemma 1.13, then, for any , the iterative sequences , and defined by Algorithm 2.5 converge strongly to the solution of the problem (1.3).

Theorem 3.4.

where is the same as in Lemma 1.13, then, for any , the iterative sequences , and defined by Algorithm 2.6 converge strongly to the solution of the problem (1.4).

Remark 3.5.

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).

## Declarations

### Acknowledgment

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).

## Authors’ Affiliations

## References

- 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

## 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.