#### Abstract

We investigate a class of new -mixed cocoercive operators in Hilbert spaces. We extend the concept of resolvent operators associated with -cocoercive operators to the -mixed cocoercive operators and prove that the resolvent operator of -mixed cocoercive operator is single valued and Lipschitz continuous. Some examples are given to justify the definition of -mixed cocoercive operators. Further, by using resolvent operator technique, we discuss the approximate solution and suggest an iterative algorithm for the generalized mixed variational inclusions involving -mixed cocoercive operators in Hilbert spaces. We also discuss the convergence criteria for the iterative algorithm under some suitable conditions. Our results can be viewed as a generalization of some known results in the literature.

#### 1. Introduction

In recent past, monotone mappings have a large number of applications, especially in differential equations, integral equations, mathematical economics, optimal control, and so forth. There are various kinds of generalized monotonicity such as pseudomonotone, quasimonotone, and paramonotone; see for example, [1–4]. The cocoercive mappings were studied by Tseng [5], Magnanti and Perakis [6], and Zhu and Marcotte [7] which are also the generalized forms of monotone mappings.

The resolvent operator techniques are important to study the existence of solutions and to design 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, Kazmi and Khan, and Lan et al. investigated many generalized operators such as -monotone [8], -accretive [9], -proximal point [10], -accretive [11], -monotone [12], and -accretive mappings [13]. Recently, Zou and Huang [14] introduced and studied -accretive operators; Kazmi et al. [15–17] introduced and studied generalized -accretive operators and --proximal point mapping; Xu and Wang [18] introduced and studied -monotone operators; Ahmad et al. [19] introduced and studied -cocoercive operators and Husain and Gupta [20, 21] introduced and studied -mixed operator and generalized --cocoercive operators.

Motivated by the recent work going in this direction, we consider a class of -mixed cocoercive operators, a natural generalization of monotone (accretive) operators in Hilbert (Banach) spaces. For details, we refer to see [8, 9, 12–14, 18–22]. We extend the concept of resolvent operators associated with -cocoercive operators to the -mixed cocoercive operators and prove that the resolvent operator of -mixed cocoercive operator is single valued and Lipschitz continuous. Further, we consider the generalized mixed variational inclusion problem involving -mixed cocoercive operator in Hilbert spaces. Using new resolvent operator technique, we prove the existence of solutions and suggest an iterative algorithm for the generalized mixed variational inclusions. Furthermore, we discuss the convergence criteria of the iterative algorithm under some suitable conditions. Our results can be viewed as an extension and generalization of some known results [14–22]. For illustration of Definitions 4 and 7 and Theorem 20, Examples 5, 8, and 21 are given, respectively.

#### 2. Preliminaries

Let be a real Hilbert space endowed with a norm and an inner product , the metric induced by the norm , (resp., ) the family of all nonempty (resp., closed and bounded) subsets of , and the Hausdorff metric on defined by where and .

In the sequel, we recall important basic concepts and definitions, which will be used in this work.

*Definition 1. *A mapping is said to be(i)*monotone* if
(ii)-*strongly monotone* if there exists a constant such that
(iii)-*cocoercive* if there exists a constant such that
(iv)-*relaxed cocoercive* if there exists a constant such that
(v)-*Lipschitz continuous* if there exists a constant such that
(vi)-*expansive* if there exists a constant such that
if , then it is *expansive*.

*Definition 2. *A set-valued mapping is said to be -*relaxed monotone* if there exists a constant such that

*Definition 3 (see [14, 19]). *Let and be the single-valued mappings. Then(i) is said to be -*strongly monotone* with respect to if there exists a constant such that
(ii) is said to be -*relaxed monotone* with respect to if there exists a constant such that
(iii) is said to be *-cocoercive* with respect to if there exists a constant such that
(iv) is said to be -*relaxed cocoercive* with respect to if there exists a constant such that
(v) is said to be -*Lipschitz continuous* with respect to if there exists a constant such that
(vi) is said to be *-Lipschitz continuous* with respect to if there exists a constant such that

*Definition 4. *Let and be the single-valued mappings. Then(i) is said to be -*mixed cocoercive* with respect to if there exist constants such that
(ii) is said to be -*relaxed mixed cocoercive* with respect to if there exist constants such that
(iii) is said to be -*mixed Lipschitz continuous* with respect to , , , and if there exists a constant such that

*Example 5. *Let with usual inner product. Let be defined by
Suppose that is defined by
Then is -*mixed cocoercive* with respect to , -*relaxed mixed cocoercive* with respect to , and -*mixed Lipschitz continuous* with respect to , , , and .

Indeed, let, for any ,
that is, is -*mixed cocoercive* with respect to . Consider
that is, is -*relaxed mixed cocoercive* with respect to .
that is, is -*mixed Lipschitz continuous* with respect to , , , and .

*Definition 6. *Let , and be the single valued mappings and the set-valued mappings. The mapping is said to be -*mixed strongly monotone* with respect to , , and if and only if there exists a constant such that

#### 3. -Mixed Cocoercive Operators

This section deals with a new concept and properties of -mixed cocoercive mappings, which provides a unifying framework for the existing cocoercive operators, monotone operators in Hilbert spaces, and accretive operators in Banach space.

*Definition 7. *Let and be five single-valued mappings. Let be -mixed cocoercive with respect to and -relaxed mixed cocoercive with respect to . Then the set-valued mapping is said to be -*mixed cocoercive* with respect to and (or simply *-mixed cocoercive* in the sequel) if(i) is -*relaxed monotone*;(ii), for all .

*Example 8. *Let , , , , , and be the same as in Example 5, and let be defined by .

We claim that is 2*-relaxed monotone mapping*. Indeed, for any , we have
Furthermore, is also an -mixed cocoercive operator since for any .

*Remark 9. *(i)If and is cocoercive, then -mixed cocoercive operator reduces to -cocoercive operator which was studied in [19].(ii)If and is --relaxed monotone, then -mixed cocoercive operator reduces to -monotone operator which was studied in [18].(iii)If and is accretive, then -mixed cocoercive operator reduces to -accretive operator which was studied in [14].(iv)If , then -mixed cocoercive operator reduces to -monotone operator which was studied in [8, 9].

Since -mixed cocoercive operator is a generalization of the maximal monotone operator, it is sensible that there are similar properties between them. The following result confirms this expectation.

Proposition 10. *Let set-valued mapping be a -mixed cocoercive operator with respect to and . If is -expansive, is -Lipschitz continuous, and , with , then the following inequality
**
holds for all , implies , where
*

*Proof. *Suppose on contrary that there exists such that
Since is -mixed cocoercive, we know that holds for all , and so there exists such that

Now,
Setting in (27) and then from the resultant, (28) and -relaxed monotonicity of , we obtain
Since is -mixed cocoercive with respect to and -relaxed mixed cocoercive with respect to and is -expansive and is -Lipschitz continuous; thus, (30) becomes
which gives since . By (27), we have , a contradiction. This completes the proof.

Theorem 11. *Let the set-valued mapping be a -mixed cocoercive operator with respect to and . If is -expansive, is -Lipschitz continuous, and , with , then is single valued.*

*Proof. *For any given , let . It follows that
Since is -relaxed monotone, we have

Since is -mixed cocoercive with respect to and -relaxed mixed cocoercive with respect to and is -expansive and is -Lipschitz continuous; thus, (33) becomes
since . Hence, it follows that . This implies that , and so is single valued.

*Definition 12. *Let set-valued mapping be a -mixed cocoercive operator with respect to and . If is -expansive, is -Lipschitz continuous, and , with , then the *resolvent operator * is defined by

Now we prove that the resolvent operator defined by (35) is Lipschitz continuous.

Theorem 13. *Let set-valued mapping be a -mixed cocoercive operator with respect to and . If is -expansive, is -Lipschitz continuous, and , with , then the resolvent operator is -Lipschitz continuous; that is,
*

*Proof. *Let be any given points. It follows from (35) that
Let and . Since is -relaxed monotone, we have
which implies that
Further, we have
and hence
that is
This completes the proof.

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

In this section, we shall show that, under suitable assumptions, the -mixed cocoercive operator can also play important roles for solving the generalized mixed variational inclusions in Hilbert spaces.

Let be set-valued mappings, and let , and be single-valued mappings. Suppose that is a set-valued mapping such that for each fixed is -mixed cocoercive operator with respect to and , and . We consider the following generalized mixed variational inclusion: for given , find , , , and such that The problem of type (43) was studied by Xu and Wang [18] in the setting of Banach spaces when is -monotone. Problem (43) includes many variational inequalities (inclusions) and complementarity problems as special cases as follows.

If , then problem (43) reduces to a generalized mixed quasi-variational inclusion with -mixed cocoercive operators in Hilbert spaces: find , , , and such that If is --proximal point mapping, then the problem of type (44) was studied by Kazmi and Khan [10].

If , , and , , , , , then problem (43) reduces to generalized variational inclusion problem: find such that which was studied by many authors in the setting of Hilbert spaces when is maximal monotone and is strongly monotone operator.

Lemma 14. *Let , , , , , , , , , , , , , and be the same as in problem (43). Then is a solution of problem (43) if and only if satisfies the following relation:
**
where is a constant and is the resolvent operator defined by (35).*

*Proof. *Observe that, for ,

*Remark 15. *To develop a fixed point algorithm for (43), we rewrite (46) as follows:
This fixed point formulation allows us to suggest the following iterative algorithm.

*Algorithm 16. ** **Step **0*. For given and , choose , , , and . Set .*Step **1*. Let
*Step **2*. Let , , , and satisfy
for all .*Step **3*. If and satisfy (50) to sufficient accuracy, stop; otherwise, set and return to Step 1.

We need the following definitions which will be used to state and prove the main result.

*Definition 17. *A set-valued mapping is said to be -*Lipschitz continuous* if there exists a constant such that

*Definition 18. *Let be a single-valued mapping. Then(i) is said to be -*Lipschitz continuous* in the first argument if there exists a constant such that
(ii) is said to be -*Lipschitz continuous* in the second argument if there exists a constant such that

*Definition 19. *Let be a single-valued mapping. Then(i) is said to be -*Lipschitz continuous* in the first argument if there exists a constant such that
(ii) is said to be -*Lipschitz continuous* in the second argument if there exists a constant such that

Theorem 20. *Let be set-valued mappings, let , , , , , , and be single-valued mappings, and let be a set-valued mapping such that for each , is -mixed cocoercive operator with respect to and , and . Assume that*(i)*, , and are -Lipschitz continuous with constants , and , respectively;*(ii)* is -expansive, and is -Lipschitz continuous;*(iii)* is -mixed Lipschitz continuous with constant ;*(iv)* is -Lipschitz continuous and -strongly monotone;*(v)* is -Lipschitz continuous in the first argument and -Lipschitz continuous in the second argument;*(vi)* is -Lipschitz continuous in the first argument and -Lipschitz continuous in the second argument;*(vii)* is -mixed strongly monotone with respect to , , and with constant ;*(viii)*Suppose that there exist constants , such that, for each and ,
* *
where and .**Then generalized mixed variational inclusion problem (43) has a solution , where , , and , and the iterative sequences , generated by Algorithm 16 converges strongly to , , , and , respectively.*

*Proof. *Since , , and are -Lipschitz continuous with constants ,