#### Abstract

The purpose of this paper is to introduce a new -cocoercive operator, which generalizes many existing monotone operators. The resolvent operator associated with -cocoercive operator is defined, and its Lipschitz continuity is presented. By using techniques of resolvent operator, a new iterative algorithm for solving generalized variational inclusions is constructed. Under some suitable conditions, we prove the convergence of iterative sequences generated by the algorithm. For illustration, some examples are given.

#### 1. Introduction

Various concepts of generalized monotone mappings have been introduced in the literature. Cocoercive mappings which are generalized form of monotone mappings are defined by Tseng [1], Magnanti and Perakis [2], and Zhu and Marcotte [3]. The resolvent operator techniques are important to study the existence of solutions and to develop iterative schemes for different kinds of variational inequalities and their generalizations, which are providing mathematical models to some problems arising in optimization and control, economics, and engineering sciences. In order to study various variational inequalities and variational inclusions, Fang and Huang, Lan, Cho, and Verma investigated many generalized operators such as -monotone [4], -accretive [5], -accretive [6], -monotone[7, 8], -accretive mappings [9]. Recently, Zou and Huang [10] introduced and studied -accretive operators and Xu and Wang [11] introduced and studied -monotone operators.

Motivated and inspired by the excellent work mentioned above, in this paper, we introduce and discuss new type of operators called -cocoercive operators. We define resolvent operator associated with -cocoercive operators and prove the Lipschitz continuity of the resolvent operator. We apply -cocoercive operators to solve a generalized variational inclusion problem. Some examples are constructed for illustration.

#### 2. Preliminaries

Throughout the paper, we suppose that is a real Hilbert space endowed with a norm and an inner product , is the metric induced by the norm (resp., ) is the family of all nonempty (resp., closed and bounded) subsets of , and is the Hausdorff metric on defined by where .

*Definition 2.1. *A mapping is said to be (i)Lipschitz continuous if there exists a constant such that
(ii)monotone if
(iii)strongly monotone if there exists a constant such that
(iv)-expansive if there exists a constant such that
if , then it is expansive.

*Definition 2.2. *A mapping is said to be cocoercive if there exists a constant such that

*Note 1. *Clearly is -Lipschitz continuous and also monotone but not necessarily strongly monotone and Lipschitz continuous (consider a constant mapping). Conversely, strongly monotone and Lipschitz continuous mappings are cocoercive, and it follows that cocoercivity is an intermediate concept that lies between simple and strong monotonicity.

*Definition 2.3. *A multivalued mapping is said to be cocoercive if there exists a constant such that

*Definition 2.4. *A mapping is said to be relaxed cocoercive if there exists a constant such that

*Definition 2.5. *Let and be the mappings. (i) is said to be cocoercive with respect to if there exists a constant such that
(ii) is said to be relaxed cocoercive with respect to if there exists a constant such that
(iii) is said to be -Lipschitz continuous with respect to if there exists a constant such that
(iv) is said to be -Lipschitz continuous with respect to if there exists a constant such that

*Example 2.6. *Let with usual inner product. Let be defined by
Suppose that is defined by
Then is -cocoercive with respect to and -relaxed cocoercive with respect to since
which implies that
That is, is -cocoercive with respect to .
which implies that
that is, is -relaxed cocoercive with respect to .

#### 3. -Cocoercive Operator

In this section, we define a new -cocoercive operator and discuss some of its properties.

*Definition 3.1. *Let be three single-valued mappings. Let be a set-valued mapping. is said to be -cocoercive with respect to mappings and (or simply -cocoercive in the sequel) if is cocoercive and , for every .

*Example 3.2. *Let , , , and be the same as in Example 2.6, and let be define by . Then it is easy to check that is cocoercive and , that is, is -cocoercive with respect to and .

*Remark 3.3. *Since cocoercive operators include monotone operators, hence our definition is more general than definition of -monotone operator [10]. It is easy to check that -cocoercive operators provide a unified framework for the existing -monotone, -monotone operators in Hilbert space and -accretive, -accretive operators in Banach spaces.

Since -cocoercive operators are more general than maximal monotone operators, we give the following characterization of -cocoercive operators.

Proposition 3.4. *Let be -cocoercive with respect to , -relaxed cocoercive with respect to , is -expansive, is -Lipschitz continuous, and , . Let be -cocoercive operator. If the following inequality
**
holds for all , then , where
*

*Proof. *Suppose that there exists some such that
Since is -cocoercive, we know that holds for every , and so there exists such that
It follows from (3.3) and (3.4) that
which gives since . By (3.4), we have . Hence and so .

Theorem 3.5. *Let be a Hilbert space and a maximal monotone operator. Suppose that is a bounded cocoercive and semicontinuous with respect to and . Let be also -cocoercive with respect to and -relaxed cocoercive with respect to . The mapping is -expansive, and is -Lipschitz continuous. If and , then is -cocoercive with respect to and .*

*Proof. *For the proof we refer to [10].

Theorem 3.6. *Let be a -cocoercive with respect to and -relaxed cocoercive with respect to , is -expansive, and is -Lipschitz continuous, and . Let be an -cocoercive operator with respect to and . Then the operator is single-valued.*

*Proof. *For any given , let . It follows that
As is cocoercive (thus monotone), we have
Since is -cocoercive with respect to and -relaxed cocoercive with respect to , is -expansive and is -Lipschitz continuous, thus (3.7) becomes
since . Thus, we have and so is single-valued.

*Definition 3.7. *Let be -cocoercive with respect to and -relaxed cocoercive with respect to , is -expansive, is -Lipschitz continuous, and , . Let be an -cocoercive operator with respect to and . The resolvent operator is defined by

Now, we prove the Lipschitz continuity of resolvent operator defined by (3.9) and estimate its Lipschitz constant.

Theorem 3.8. *Let be -cocoercive with respect to , -relaxed cocoercive with respect to , is -expansive, is -Lipschitz continuous, and , . Let be an -cocoercive operator with respect to and . Then the resolvent operator is -Lipschitz continuous, that is,
*

*Proof. *Let and be any given points in . It follows from (3.9) that
This implies that
For the sake of clarity, we take
Since is cocoercive (hence monotone), we have
which implies that
Further, we have
and so
thus,
This completes the proof.

#### 4. Application of -Cocoercive Operators for Solving Variational Inclusions

We apply -cocoercive operators for solving a generalized variational inclusion problem.

We consider the problem of finding and such that where , and are the mappings. Problem (4.1) is introduced and studied by Huang [12] in the setting of Banach spaces.

Lemma 4.1. *The , where , is a solution of the problem (4.1), if and only if is a solution of the following:
**
where is a constant.*

*Proof. *By using the definition of resolvent operator , the conclusion follows directly.

Based on (4.2), we construct the following algorithm.

*Algorithm 4.2. *For any , compute the sequences and by iterative schemes such that
for all , and is a constant.

Theorem 4.3. *Let be a real Hilbert space and the single-valued mappings. Let be a multi-valued mapping and the multi-valued -cocoercive operator. Assume that *(i)* is -Lipschitz continuous in the Hausdorff metric ;*(ii)* is -cocoercive with respect to and -relaxed cocoercive with respect to ;*(iii)* is -expansive; *(iv)* is -Lipschitz continuous; *(v)* is -Lipschitz continuous and -strongly monotone; *(vi)* is -Lipschitz continuous with respect to and -Lipschitz continuous with respect to ; *(vii)*. *

Then the generalized variational inclusion problem (4.1) has a solution with , and the iterative sequences and generated by Algorithm 4.2 converge strongly to and , respectively.

*Proof. *Since is -Lipschitz continuous, it follows from Algorithm 4.2 that
for .

Using the -strong monotonicity of , we have
which implies that
Now we estimate by using the Lipschitz continuity of ,
Since is -Lipschitz continuous with respect to and -Lipschitz continuous with respect to , is -Lipschitz continuous and using (4.4), (4.7) becomes
or
Using (4.9), (4.6) becomes
where
Let
We know that and . From assumption (vii), it is easy to see that . Therefore, it follows from (4.10) that is a Cauchy sequence in . Since is a Hilbert space, there exists such that as . From (4.4), we know that is also a Cauchy sequence in , thus there exists such that and . By the continuity of , and and Algorithm 4.2, we have
Now, we prove that . In fact, since , we have
which implies that . Since , it follows that . By Lemma 4.1, we know that is a solution of problem (4.1). This completes the proof.

#### Acknowledgments

This work is partially done during the visit of the first author to National Sun-Yat Sen University, Kaohsiung, Taiwan. The first and second authors are supported by Department of Science and Technology, Government of India under Grant no. SR/S4/MS: 577/09. The fourth author is supported by Grant no. NSC 99-2221-E-037-007-MY3.