General Nonlinear Random Equations with Random Multivalued Operator in Banach Spaces

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.

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.

Throughout this paper, we suppose that is a complete -finite measure space and is a separable real Banach space endowed with dual space , the norm and the dual pair between and . We denote by the class of Borel -fields in . Let and denote the family of all the nonempty subsets of , the family of all the nonempty bounded closed sets of , respectively. The generalized duality mapping is defined by

(11)

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.

Find such that , and

(12)

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

Some special cases of the problem (1.2) are as follows.

• (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

(13)

for all and . The determinate form of the problem (1.3) was considered and studied by Agarwal et al. [2] when .

• (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.

Find such that and

(14)

for all and .

• (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.

Find such that and

(15)

for all , and , which is called the 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.

Find such that , and

(16)

for all and , whose determinate form is a generalization of the problem considered in [4, 5, 29].

• (5) If, in the problem (1.6), is the indictor function of a nonempty closed convex set in defined in the form

  

  (17)

then (1.6) becomes the following problem.

Find such that , and

(18)

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.

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 ,

(19)

Definition 1.7.

A random operator is said to be

(a)strongly accretive if there exists such that

(110)

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

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

(111)

for all and .

Definition 1.8.

Let be a random operator. An operator is said to be

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

(112)

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

(113)

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

(a)accretive if

(114)

for all , , and , where ;

(b)strictlyaccretive if

(115)

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

(c)stronglyaccretive if there exists a real-valued random variable such that

(116)

for all , , and ;

(d)generalizedaccretive 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

(a)monotone if

(117)

for all and ;

(b)strictly monotone if

(118)

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

(c)strongly monotone if there exists a measurable function such that

(119)

for all and ;

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

(120)

for all and .

Definition 1.12.

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

(121)

for all .

The modules of smoothness of is the function defined by

(122)

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.

Let be a given real number and let 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:

(123)

Definition 1.14.

Let be a generalized -accretive mapping. Then the resolvent operator for is defined as follows:

(124)

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.

Let be -strongly monotone and -Lipschitz continuous. Let be a generalized -accretive mapping. Then the resolvent operator for is Lipschitz continuous with constant , that is,

(125)

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]).

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

(21)

Lemma 2.3.

Measurable operators are a solution of the problem (1.2) if and only if

(22)

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.

Let be 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, and let be a measurable step size function. Then, by Lemma 2.1 and Himmelberg [32], it is known that, for given , the multivalued operators and are measurable and there exist measurable selections and . Set

(23)

where and are the same as in Lemma 2.3 and is a measurable function. Then it is easy to know that is measurable. Since and , by Lemma 2.2, there exist measurable selections and such that, for all ,

(24)

By induction, one can define sequences , , and inductively satisfying

(25)

where is an error to take into account a possible inexact computation of the resolvent operator point, which satisfies the following conditions:

(26)

for all .

From Algorithm 2.4, we can get the following algorithms.

Algorithm 2.5.

Suppose that , , , , and are the same as in Algorithm 2.4. Let be a random single-valued operator, and for all and . Then, for given measurable , one has

(27)

where is the same as in Algorithm 2.4.

Algorithm 2.6.

Let be a random multivalued operator such that for each fixed , is a generalized -accretive mapping, and . If , , , , and are the same as in Algorithm 2.4, then, for given measurable , we have

(28)

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.

Suppose that is a -uniformly smooth and separable real Banach space, is -strongly accretive and -Lipschitz continuous, and is a random multivalued operator such that for each fixed and , is a generalized -accretive mapping and . Let be -strongly monotone and -Lipschitz continuous, and let be a -Lipschitz continuous random operator, and let be -strongly accretive with respect to and -Lipschitz continuous in the first argument, and -Lipschitz continuous in the second argument, -Lipschitz continuous in the third argument, respectively. Let multivalued operators be --Lipschitz continuous, --Lipschitz continuous, --Lipschitz continuous, respectively. If there exist real-valued random variables and such that, for any , ,

(31)

(32)

where is the same as in Lemma 1.13, then, for any , there exist , , and such that is a solution of the problem (1.2) and

(33)

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

Proof.

It follows from (2.5), Lemma 1.15 and (3.1) that

(34)

Since is strongly accretive and Lipschitz continuous,

(35)

that is,

(36)

where is the same as in Lemma 1.13. Also from the strongly accretivity of with respect to and the Lipschitz continuity of in the first argument, we have

(37)

By Lipschitz continuity of in the second and third argument, and -Lipschitz continuity of , we obtain

(38)

(39)

(310)

Using (3.6)–(3.10) in (3.4), we have, for all

(311)

where

(312)

Let

(313)

Then , as . From the condition (3.2), we know that for all and so there exists a positive measurable function such that for all and . Therefore, for all , by (3.11), we now know that, for all ,

(314)

which implies that, for any ,

(315)

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 .

Now, we show that . In fact, we have

(316)

This implies that . Similarly, we have and . Therefore, from (2.5), (2.6) and the continuity of , and , we have

(317)

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

Remark 3.2.

If is a 2-uniformly smooth Banach space and there exists a measurable function such that

(318)

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.

Let , , , and be the same as in Theorem 3.1. Assume that is a random multivalued operator such that, for each fixed and , is a generalized -accretive mapping. Let be -Lipschitz continuous, let be a -Lipschitz continuous random operator, let be -Lipschitz continuous, and let be -strongly accretive with respect to and -Lipschitz continuous in the first argument and -Lipschitz continuous in the second argument, respectively. If there exist real-valued random variables and such that (3.1) holds and

(319)

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.

Suppose that , , , , , and are the same as in Algorithm 2.4. Let be a random multivalued operator such that, for each fixed , is a generalized -accretive mapping and . If there exists a real-valued random variable such that, for any ,

(320)

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).