Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 378364, 13 pages

http://dx.doi.org/10.1155/2013/378364

Research Article

-Mixed Cocoercive Operators with an Application for Solving Variational Inclusions in Hilbert Spaces

Shamshad Husain and Sanjeev Gupta

Department of Applied Mathematics, Faculty of Engineering and Technology, Aligarh Muslim University, Aligarh 202002, India

Received 29 May 2013; Accepted 17 September 2013

Academic Editor: William Ziemer

Copyright © 2013 Shamshad Husain and Sanjeev Gupta. 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.

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 , , and , respectively, it follows from Algorithm 16 that

for .

Now, we estimate , by Lipschitz continuity and strong monotonicity of , we have

Now, we estimate by using Algorithm 16 and the Lipschitz continuity of :

Now, by Algorithm 16 and conditions (i), (vi), and (vii), we get

Since is -mixed Lipschitz continuous and -mixed strongly monotone with respect to , , and and is -Lipschitz continuous and from (61), we have

Using (59)–(63), we have

which implies

where

Let

Since , so . By assumption (ix), it is easy to see that . Therefore, (67) implies that is a Cauchy sequence in . Since is a Hilbert space, there exists such that as . From (58), we know that , , and are also Cauchy sequences in ; thus, there exists such that , , and as , respectively. By the continuity of , and and Algorithm 16, we have

Now, we prove that . In fact, since , we have

which implies that . Since , it follows that . Similarly, it is easy to see that . By Lemma 14, is the solution of problem (43).

Example 21. Let with usual inner product.

Let be defined by

Then, it is easy to check that

(i), and are -Lipschitz continuous for . Let be defined by

 Suppose that is defined by

 Then it is easy to cheek that(ii) is -mixed cocoercive with respect to and -relaxed mixed cocoercive with respect to , is -expansive for , and is -Lipschitz continuous for ;(iii) is -mixed Lipschitz continuous with respect to , and for . Let be defined by

 Then it is easy to check that

(iv) is -strongly monotone for and -Lipschitz continuous for . Let be defined by

 Then it is easy to check that(v) is -Lipschitz continuous in the first argument for and -Lipschitz continuous in the second argument for ;(vi) is -Lipschitz continuous in the first argument for and -Lipschitz continuous in the second argument for ;(vii) is -mixed strongly monotone with respect to , , and for , 1000.  In addition, Let be defined by

where, .

Then it is easy to check that

(viii)

Therefore, for the constants

obtained in (i) to (viii), all the conditions of Theorem 20 are satisfied for the generalized mixed variational inclusion problem (43) for .

Remark 22. If the set-valued mapping is -monotone, is Lipschitz continuous, then becomes a new monotone operator; see, for example, [18]. We leave the proofs to readers who are interested in this area.

Acknowledgment

The authors are extremely grateful to the referee for the useful suggestions that improved the content of the paper.

References

D. Aussel and N. Hadjisavvas, “On quasimonotone variational inequalities,” Journal of Optimization Theory and Applications, vol. 121, no. 2, pp. 445–450, 2004.

N. Hadjisavvas, “Continuity and maximality properties of pseudomonotone operators,” Journal of Convex Analysis, vol. 10, no. 2, pp. 465–475, 2003.

N. Hadjisavvas and S. Schaible, “On a generalization of paramonotone maps and its application to solving the Stampacchia variational inequality,” Optimization, vol. 55, no. 5-6, pp. 593–604, 2006.

N. Hadjisavvas, S. Schaible, and N.-C. Wong, “Pseudomonotone operators: a survey of the theory and its applications,” Journal of Optimization Theory and Applications, vol. 152, no. 1, pp. 1–20, 2012.

P. Tseng, “Further applications of a splitting algorithm to decomposition in variational inequalities and convex programming,” Mathematical Programming, vol. 48, no. 2, pp. 249–263, 1990.

T. L. Magnanti and G. Perakis, “Convergence condition for variational inequality algorithms,” Working Paper OR-282-93, Massachusetts Institute of Technology, 1993.

D. L. Zhu and P. Marcotte, “Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities,” SIAM Journal on Optimization, vol. 6, no. 3, pp. 714–726, 1996.

Y.-P. Fang and N.-J. Huang, “-monotone operator and resolvent operator technique for variational inclusions,” Applied Mathematics and Computation, vol. 145, no. 2-3, pp. 795–803, 2003.

Y.-P. Fang and N.-J. Huang, “-accretive operators and resolvent operator technique for solving variational inclusions in Banach spaces,” Applied Mathematics Letters, vol. 17, no. 6, pp. 647–653, 2004.

K. R. Kazmi and F. A. Khan, “Iterative approximation of a solution of multi-valued variational-like inclusion in Banach spaces: a --proximal-point mapping approach,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 665–674, 2007.

K. R. Kazmi and F. A. Khan, “Sensitivity analysis for parametric generalized implicit quasi-variational-like inclusions involving --accretive mappings,” Journal of Mathematical Analysis and Applications, vol. 337, no. 2, pp. 1198–1210, 2008.

Y.-P. Fang, N.-J. Huang, and H. B. Thompson, “A new system of variational inclusions with -monotone operators in Hilbert spaces,” Computers & Mathematics with Applications, vol. 49, no. 2-3, pp. 365–374, 2005.

H.-Y. Lan, Y. J. Cho, and R. U. Verma, “Nonlinear relaxed cocoercive variational inclusions involving -accretive mappings in Banach spaces,” Computers & Mathematics with Applications, vol. 51, no. 9-10, pp. 1529–1538, 2006.

Y.-Z. Zou and N.-J. Huang, “-accretive operator with an application for solving variational inclusions in Banach spaces,” Applied Mathematics and Computation, vol. 204, no. 2, pp. 809–816, 2008.

K. R. Kazmi, N. Ahmad, and M. Shahzad, “Convergence and stability of an iterative algorithm for a system of generalized implicit variational-like inclusions in Banach spaces,” Applied Mathematics and Computation, vol. 218, no. 18, pp. 9208–9219, 2012.

K. R. Kazmi, M. I. Bhat, and N. Ahmad, “An iterative algorithm based on -proximal mappings for a system of generalized implicit variational inclusions in Banach spaces,” Journal of Computational and Applied Mathematics, vol. 233, no. 2, pp. 361–371, 2009.

K. R. Kazmi, F. A. Khan, and M. Shahzad, “A system of generalized variational inclusions involving generalized -accretive mapping in real -uniformly smooth Banach spaces,” Applied Mathematics and Computation, vol. 217, no. 23, pp. 9679–9688, 2011.

Z. Xu and Z. Wang, “A generalized mixed variational inclusion involving -monotone operators in Banach spaces,” Journal of Mathematics Research, vol. 2, no. 3, pp. 47–56, 2010.

R. Ahmad, M. Dilshad, M.-M. Wong, and J.-C. Yao, “-cocoercive operator and an application for solving generalized variational inclusions,” Abstract and Applied Analysis, vol. 2011, Article ID 261534, 12 pages, 2011.

S. Husain, S. Gupta, and V. N. Mishra, “Graph convergence for the -mixed mapping with an application for solving the system of generalized variational inclusions,” Fixed Point Theory and Applications, vol. 2013, article 304, 2013.

S. Husain, S. Gupta, and V. N. Mishra, “Generalized -cocoercive operators and generalized set-valued Variational-Like Inclusions,” Journal of Mathematics, vol. 2013, Article ID 738491, 10 pages, 2013.

S. Husain and S. Gupta, “Algorithm for solving a new system of generalized variational inclusions in Hilbert spaces,” Journal of Calculus of Variations, vol. 2013, Article ID 461371, 8 pages, 2013.