An iterative algorithm for a system of generalized implicit variational inclusions.

In this paper, we introduce a system of generalized implicit variational inclusions which consists of three variational inclusions. We design an iterative algorithm with error terms based on relaxed resolvent operator due to Ahmad et al. (Stat Optim Inf Comput 4:183-193, 2016) for approximating the solution of our system. The convergence of the iterative sequences generated by the iterative algorithm is also discussed. An example is given which satisfy all the conditions of our main result.

Background

A widely studied problem known as variational inclusion problem have many applications in the fields of optimization and control, economics and transportation equilibrium, engineering sciences, etc.. Several researches used different approaches to develop iterative algorithms for solving various classes of variational inequality and variational inclusion problems. For details see Ansari et al. (2000), Cho et al. (2004), Chang et al. (2005), Ding (2003), Fang and Huang (2004), Kim and Kim (2004), Kassay and Kolumbán (1999), Kassay et al. (2002), Kazmi et al. (2009), Lan et al. (2007), Noor (2001), Siddiqi et al. (1998), Sun et al. (2008), Yan et al. (2005) and the references therein.

A problem of much more interest called system of variational inequalities (inclusions) were introduced and studied in the literature. Peng (2003), Cohen and Chaplais (1988), Bianchi (1993), and Ansari and Yao (1999) considered a system of scalar variational inequalities and Pang showed that the traffic equilibrium problem, the spatial equilibrium problem, the Nash equilibrium, and the general equilibrium problem can be modeled as a system of variational inequalities. Verma (1999, 2001, 2004a, b) introduced and studied some systems of variational inequalities and developed some iterative algorithms for approximating the solutions of system of variational inequalities in Hilbert spaces.

As generalization of system of variational inequalities, Agarwal et al. (2004) introduced a system of generalized nonlinear mixed quasi-variational inclusions and studied the sensitivity analysis of solutions. After that, Fang and Huang (2004), Verma (2005), and Fang et al. (2005) introduced and studied different system of variational inclusions involving H-monotone operators, A-monotone operators, and -monotone operators, respectively.

In this paper, we introduced and study a system of three variational inclusions and we call it system of generalized implicit variational inclusions in real Hilbert spaces. We design an iterative algorithm with error terms based on relaxed resolvent operator for solving system of generalized implicit variational inclusions. Convergence criteria is also discussed. The approach of this paper is different then the methods discussed above. An example is given in support of our main result.

Preliminaries

Let X be a real Hilbert space endowed with a norm and an inner product d is the metric induced by the norm (respectively,CB(X)) is the family of all nonempty (respectively, closed and bounded) subsets of X,  and is the Hausdörff metric on CB(X) defined bywhere and .

Let us recall the known definitions needed in the sequel.

Definition 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) relaxed Lipschitz continuous if, there exists a constant such that

Definition 2

A mapping is said to be Lipschitz continuous with respect to first argument if, there exists a constant such thatSimilarly, we can define the Lipschitz continuity of F in rest of the arguments.

Definition 3

A set-valued mapping is said to be D-Lipschitz continuous if, there exists a constant such that

Definition 4

Ahmad et al. (2016) Let be a mapping and be an identity mapping. Then, a set-valued mapping is a said to be -monotone if, M is monotone, H is relaxed Lipschitz continuous andwhere is a constant.

Definition 5

Ahmad et al. (2016) Let be relaxed Lipschitz continuous mapping and be an identity mapping. Suppose that is a set-valued, -monotone mapping. The relaxed resolvent operator associated with I,H and M is defined bywhere is a constant.

For the sake of convenience of readers, we give the proof following two theorems which can be found in Ahmad et al. (2016).

Theorem 1

Letbe anr-relaxed Lipschitz continuous mapping,be an identity mapping andbe a set-valued, -monotone mapping. Then the operatoris single-valued, whereis a constant.

Proof

For any and a constant let Then,Since M is monotone, we haveSince H is r-relaxed Lipschitz continuous, we haveit follows that which implies that Thus is single-valued.

Theorem 2

Letbe anr-relaxed Lipschitz continuous mapping, be an identity mapping andbe a set-valued, -monotone mapping. Then the relaxed resolvent operatoris-Lipschitz continuous. i.e.,

Let x and y be any given point in X. If follow from (1) thati.e.,Since M is -monotone i.e., M is monotone, we haveIt follows thatBy Cauchy-Schwartz inequality, (5) and r-relaxed Lipschitz continuity of H,  we haveThus, we havei.e., the relaxed resolvent operator is -Lipschitz continuous.

System of generalized implicit variational inclusions and iterative algorithm

In this section, we introduce a system of generalized implicit variational inclusions and design an iterative algorithm with error terms for solving the system of generalized implicit variational inclusions in Hilbert spaces.

For each let be a real Hilbert space, be the single-valued mappings and be the set-valued mappings. Let be the identity mappings and be the set-valued, -monotone mappings. We consider the following system of generalized implicit variational inclusions (in short, SGIVI):

Find such that for each such thatLet us see some special cases of SGIVI (7) below.

(i) If then problem (7) reduces to the system of generalized mixed quasi-variational inclusions with -monotone operators, which is to find such that Problem (8) was introduced and studied by Peng and Zhu (2007).

(ii) If (the identity map on ), (the identity map on ) then problem (7) reduces to the system of variational inclusions with -monotone operators, which is to find such that Problem (9) was introduced and studied by Fang et al. (2005).

Now, we mention the following fixed point formulation of SGIVI (7).

Lemma 1

For eachletbe a real Hilbert space, be single-valued mappings andbe the set-valued mappings. Letbe the identity mappings andbe the set-valued, -monotone mappings. Thenwithis a solution of SGIVI (7), if and only if the following equations are satisfied:whereare the relaxed resolvent operators andare constants.

The proof is a direct consequence of the definition of the relaxed resolvent operator.

We design the following iterative algorithm with error terms to approximate the solution of SGIVI (7).

Iterative Algorithm 1

For each given take and letSince by Nadler’s (1992) theorem, there exist such thatAgain, letBy Nadler’s (1992) theorem, there exist such thatContinuing the above process inductively, we can obtain the sequences by the following iterative schemes:where for are constants, are errors to take into account a possible inexact computation of the resolvent operator point and are the Hausdorff metrics on

An existence and convergence result

In this section, we will prove an existence result for SGIVI (7) and we show the convergence of iterative sequences generated by Algorithm 1, which is our main motive.

Theorem 3

For eachletbe a Hilbert space, be the identity mappings andbe the single-valued mappings such thatis-strongly monotone, -Lipschitz continuous andis-Lipschitz continuous, -relaxed Lipschitz continuous. Suppose thatare the set-valued mappings such thatis--Lipschitz continuous, is--Lipschitz continuous andis--Lipschitz continuous, respectively. Letbe the single-valued mappings such that’sare Lipschitz continuous in all three arguments with constantsrespectively and’s are Lipschitz continuous in all three arguments with constantsrespectively. Suppose thatare the set-valued, -monotone mappings. Assume that there exist constantsandsuch that the following conditions hold:andThen, the SGIVI (7) admits a solutionand the iterative sequencesgenerated by iterative Algorithm1strongly converge torespectively, for each

For each let

Using Algorithm 1, condition (14) and Theorem 2, we haveAs is -strongly monotone and -Lipschitz continuous, we obtainAs is -Lipschitz continuous, is Lipschitz continuous in all three arguments with constants and respectively, is Lipschitz continuous in all three arguments with constants and respectively, is --Lipschitz continuous, is --Lipschitz continuous and is --Lipschitz continuous, respectively, we obtainUsing (17) and (18), (16) becomesUsing the same arguments as for (19), we haveUsing the same arguments as for (19), we haveCombining (19) to (21), we havewhich implies thatwhere and It follows from (22) thatwhereLetting wherethen and as for each From condition (15), we know that and hence there exist and such that for all Therefore, it follows from (23) thatwhich implies thatwhere for all Hence, for any we haveSince and for all and it follows from (24) that and as and so and are Cauchy sequences in and respectively. Thus, there exist and such that and as

Now, we prove that for each In fact, it follows from the Lipschitz continuity of and (11)–(13) thatFrom (25)–(27), we know that and are also Cauchy sequences. Therefore, there exist and such that as

Further, for each Since is closed, we have Similarly, respectively. By continuity of the mappings and iterative Algorithm 1, we know that satisfy the following relation:By Lemma 1, is a solution of SGIVI (7). This completes the proof.

Remark 1

It is to be noted that the techniques used to prove the convergence result Theorem 3 is different than others. For more details, we refer to Shang and Bouffanais (2014a, b).

The following example ensures that all the conditions of Theorem 3 are fulfilled.

Example 1

For each let and be the mappings defined bySuppose that the mappings are defined byand the mappings are defined byThen, it is easy to check that are -Lipschitz continuous and -strongly monotone, ’s are i-Lipschitz continuous and i-relaxed Lipschitz continuous, and ’s are monotone mappings.

In addition, it is easy to verify that for which shows that ’s are -monotone mappings. Hence, the relaxed resolvent operators associated with , and are of the form:It is easy to see that the relaxed resolvent operators defined above are single-valued.

Now,Hence, the resolvent operators are -Lipschitz continuous.

Let the mappings be defined byand the mappings be defined byIt can be verified that ’s are -Lipschitz continuous in first argument, -Lipschitz continuous in second argument and -Lipschitz continuous in third argument, ’s are -Lipschitz continuous in first argument, -Lipschitz continuous in second argument and -Lipschitz continuous in third argument. Suppose that be the identity mappings. Then, clearly ’s, ’s and ’s are 1--Lipschitz continuous mappings. Hence, all the conditions of Theorem 3 are satisfied.

Remark 2

We choose one can easily verify that for and the condition (15) of Theorem 3 is satisfied.

Remark 3

We remark that our results can be further considered in Banach spaces and also the techniques of this paper may be helpful for solving a system of n-variational inclusions.

Conclusion

System of variational inclusions can be viewed as natural and innovative generalizations of the system of variational inequalities. Two of the most difficult and important problems related to inclusions are the establishment of generalized inclusions and the development of an iterative algorithm. In this article, a new system of three variational inclusions is introduced and studied which is more general than many existing system of variational inclusions in the literature. An iterative algorithm is established with error terms to approximate the solution of our system, and convergence criteria is also discussed.

We remark that our results are new and useful for further research and one can extend these results in higher dimensional spaces. Much more work is needed in all these areas to develop a sound basis for applications of the system of general variational inclusions in engineering and physical sciences.