Graph Convergence and Iterative Approximation of Solution of a Set-Valued Variational Inclusions

Khan, Faizan Ahmad; Gupta, Sanjeev

doi:https://doi.org/10.1155/2022/4540369

Journal of Mathematics

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Abstract Introduction and Preliminaries Conclusion Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 4540369 | https://doi.org/10.1155/2022/4540369

Graph Convergence and Iterative Approximation of Solution of a Set-Valued Variational Inclusions

Faizan Ahmad Khan¹and Sanjeev Gupta²

Academic Editor: Valerii Obukhovskii

Received19 Sept 2022

Accepted11 Oct 2022

Published18 Nov 2022

Abstract

In this article, first, we introduce a class of proximal-point mapping associated with generalized --accretive mapping. Further, we discuss the graph convergence of generalized --accretive mapping. As an application, we consider a set-valued variational inclusion problem (SVIP) in real Banach spaces. Furthermore, we propose an iterative scheme involving the above class of proximal-point mapping to find a solution of SVIP and discuss its convergence under some convenient assumptions. An example is constructed and demonstrated few graphics in support of our main results.

1. Introduction and Preliminaries

Variational inclusions (a generalization of variational inequalities) have become very interesting concern for researchers due to its important applications in differential equations, operation research, contact problem in elasticity, general equilibrium problem, economics and optimization, etc. It is observed that the accretive property of proximal-point mapping has a remarkable role in the area of variational inequalities theory. The concept of -accretive mappings was established by Huang and Fang [1] in 2001. For many important generalizations of -accretive mappings, we refer to see [2–25].

Recently, many researchers have studied and introduced -accretive mappings and generalized --accretive mappings [13, 24, 25]. Very recently, Nazemi [26] and Guan and Hu [27] investigated and introduced -monotone mappings and --monotone mappings in Banach spaces to study some classes of variational inclusions.

The presentation of graphical convergence related to -accretive mappings and the equivalence between the proximal-point mapping and graphical convergence of a sequence of -accretive mapping studied and analyzed by Li and Huang [15]. Recently, graphical convergence related to -maximal relaxed monotone and -maximal -relaxed -accretive and the equivalence between the proximal-point mapping and graphical convergence of the sequences of their investigated mappings were studied and analyzed by Verma [28] and Balooee et al. [29]. Since then, many researchers have studied graph convergence in the context of their proximal-point mappings and extensive graph convergence results obtained by Attouch [30]. For the related works, see [2, 3, 10, 15, 31].

Inspired and motivated by the work discussed above, we consider and study a class of proximal-point mapping associated with generalized --accretive mapping. This class of accretive mapping is the generalization of generalized --accretive mappings [13]. Further, we discuss graph convergence of the generalized --accretive mappings. An iterative algorithm involving the above class of proximal-point mapping is constructed for the SVIP in real Banach space. Furthermore, we discuss the existence of a solution of SVIP and discuss the convergence analysis of the proposed iterative algorithm. Some illustrations are constructed and shown some graphics in support of our main results.

Throughout this paper, let be a real Banach space endowed with a norm , and let be the duality pairing between and . Let be the family of all nonempty, closed, and bounded subsets of and be the power set of . is the Hausdorff metric on . Set .

Definition 1. “The normalized duality mapping is defined byIf is equivalent to real Hilbert space , then become identity mapping on . Let be a selection of normalized duality mapping,” see [32].

Lemma 1. “Let be a real Banach space and be the normalized duality mapping. Then, for any ,

Definition 2. A set-valued mapping is said to be -Lipschitz continuous with , if

Lemma 2. “Let two non-negative real sequences and , satisfying with and . Then, ,” see [33].

Definition 3. Let be a single-valued mapping. Then, is called(i)accretive if(ii)-strongly accretive if there exists non-negative such that(iii)-Lipschitz continuous if there exists non-negative such that(iv)-expansive if there exists non-negative such that(v) becomes expansive if .

Throughout this paper, we consider , where , otherwise specified.

Definition 4. Let be single-valued mappings, then(i) is -strongly accretive with respect to if there exists non-negative such that(ii) is -relaxed accretive with respect to mapping if there exists non-negative such that(iii) is -Lipschitz continuous with respect to mapping if there exists non-negative such that(iv) is -symmetric accretive with respect to mappings iff for ; is -strongly accretive with respect to mapping and for ; is -relaxed accretive with respect to mapping , where is even, satisfying and iff ;(v) is -symmetric accretive with respect to mappings iff for ; is -strongly accretive with respect to mapping and for ; is -relaxed accretive with respect to mapping , where is odd, satisfying and iff .

Definition 5. Let be a single-valued mapping and be a set-valued mapping, then(i) is -strongly accretive with respect to mapping if there exists non-negative such that(ii) is -relaxed accretive with respect to mapping if there exists non-negative such that(iii)-symmetric accretive with respect to mappings iff for ; is -strongly accretive with respect to mapping and for ; is -relaxed accretive with respect to mapping , where is even, satisfying and iff ;(iv)-symmetric accretive with respect to mappings iff for ; is -strongly accretive with respect to mapping and for ; is -relaxed accretive with respect to mapping , where is odd, satisfying and iff .

Definition 6. Let be a mapping and , be set-valued mappings, then(i) is -strongly accretive w.r.t in the -argument if there exists non-negative such that(ii) is -Lipschitz continuous in the -argument if there exists non-negative such that

2. Generalized- Accretive Mappings

Let , be the single-valued mappings, and let be a set-valued mapping. We consider the following assumptions (–) to define generalized --accretive mappings: : is -symmetric accretive with respect to mappings if p is even; : is -symmetric accretive with respect to mappings if p is odd; : is -symmetric accretive mapping with respect to mappings if p is even; : is -symmetric accretive mapping with respect to mappings if p is odd.

Definition 7. Let , then is called a generalized --accretive mapping with respect to mappings and :(i)iff is -symmetric accretive with respect to (assumption ) and , when p is even and(ii)iff is -symmetric accretive with respect to (assumption ) and , when p is odd and

Proposition 1. Let and be satisfied and let be a generalized --accretive mapping with respect to mappings and , and . If the following inequality is satisfiedfor every , , where .

Proof. Assume that there exists such that

Case 1. Let is an even number. Since is a generalized --accretive with respect to mappings and , then is -symmetric accretive with respect to mappings and , where , then there exists such thatFrom (17) and (18), we haveSet in (17) and using in (18), we obtainThen, we havewhereThus, we have due to , and . From (17), we have . Hence, . In case of is odd, we can prove the result by the above similar process. This completes the proof.

Theorem 1. Let and be satisfied and let be a generalized --accretive mapping with respect to mappings and , and , then is single-valued.

Proof. Let any , and . It follows that

Case 2. Let p is an even number. Since is -symmetric accretive with respect to mappings , thenUsing (21), thenThus, we have , which gives due to , and . Hence, is single-valued. In case of is odd, we can prove the result by the above similar process. This completes the proof.

Definition 8. Let and be satisfied and let be a generalized --accretive mapping with respect to mappings and, and . The proximal-point mapping is defined by

Next, we have shown the Lipschitz continuity of the proximal-point mapping.

Theorem 2. Let and be satisfied and let be a generalized --accretive mapping with respect to mappings and , and. Then, the proximal-point mapping is -Lipschitz continuous.

Proof. Let and using (26), thenIt follows thatLet and .

Case 3. Let is an even number. Since is -symmetric accretive with respect to mappings , thenIt implies thatWe havethat is,In case of is odd, we can prove the result by the above similar process. This completes the proof.

3. Graph Convergence for --Accretive Mappings

Definition 9. Let be a set-valued mapping, then the graph of is given asNow, we discuss graph convergence of generalized --accretive mapping.

Definition 10. For ., let be the generalized --accretive mappings with respect to mappings and . Graph convergence of a sequence to expressed as , if for each , there exists a sequence such that

Theorem 3. Let and be satisfied. For ., be the generalized --accretive mappings with respect to mappings and and . We assume that is -Lipschitz continuous with respect to mapping , and is -expansive in the -argument. Then, if and only ifwhere and .

Proof. By Theorem 2, we know that and are -Lipschitz continuous.
If part: let . Given for any , let , . Then, , which implies thatBy definition of , then is a sequence such thatas . Since , we haveTherefore,Since proximal-point mapping is -Lipschitz continuous, we haveSince is -Lipschitz continuous with respect to mapping , we haveUsing (40) and (41), we haveSince is -expansive, we haveSince . From (40), (43), we get andFrom (42), we have ; that is,Only if part: let us consider . For any given , we have andTherefore, .
Let . Then, we haveLet . Now, we evaluateAs for given any , we have . Let , equation (48) gives . Therefore, .
Now, we are giving an example to show that is a --accretive mapping with respect to mappings and , and , . Through the MATLAB programming, we demonstrate some graphics for the convergence of the proximal-point mappings.

Example 1. For every , assume that be even and . Let and be the mappings given assuch that the inequality is satisfied for each .
Let be the set-valued mappings given as

Assume that for any given ,

Therefore, is -strongly accretive with respect to mapping . Similarly, we can show that for each , is -strongly accretive with respect to mapping .

Assume that for any given ,

Therefore, is -relaxed accretive with respect to mapping . Similarly, we can show that for each , is -relaxed accretive with respect to mapping .

Assume that for any given ,

Therefore, is -strongly accretive with respect to mapping . Similarly, we can show that for each , is -strongly accretive with respect to mapping .

Assume that for any given ,

Therefore, is -relaxed accretive with respect to mapping . Similarly, we can show that for each , is -relaxed accretive with respect to mapping .

For , it is easily obtained that

Figure 1presents the convergence of proximal-point mapping to as graphical convergent to , where and .

Figure 2 presents the graph of proximal-point mapping , where .

Now, the aim is to show that is graphically convergent to .

Since, for every , there exists , such that . Let us consider

Since . Thus, we have .

Now, we compute

Thus, , and hence, graphically convergent to . Next, we will show that as graphically convergent to . Since

We compute , which shows that as . graphically convergent to .

4. Set-Valued Variational Inclusions

Let be a real Banach space. Let be the single-valued mappings and be a set-valued mapping. Let set-valued mapping be a generalized --accretive mapping.

Now, problem is to find such that

Problem (60) is called a set-valued variational inclusion problem (for short, SVIP).

Special cases are as follows:(i)If and , then SVIP (60) reduced to find such that(ii)If , , and , then SVIP (61) reduced to find such that SVIP (62) has been studied by Huang [34] when is a -accretive mapping.(iii)If , , and is single-valued mapping, then SVIP (40) reduced to find such that

VIP (63) has been studied by Zou and Huang [24] when is an -accretive mapping. For the generalized -accretive mapping, VIP (63) was studied by Bi et al. [35].

Theorem 4. For any given , is a solution of SVIP (60) if and only if satisfieswhere is non-negative constant.

Proof. Let be the solution of SVIP (60), we have

Algorithm 1. For any given , select and obtain , , , …, , by an iterative schemewhere and is non-negative constant.

Now, we establish the result in the context of the existence and uniqueness of the solution of SVIP (60).

Theorem 5. Let SVIP (60) satisfies the assumptions - and be the generalized --accretive mappings with respect to mappings and and . For each , is -Lipschitz continuous with respect to mapping , is --Lipschitz, and is -strongly accretive and -Lipschitz continuous in the -argument. In addition, the following condition is satisfied:

Then,(1)SVIP (60) has a unique solution in .(2)Iterative sequences developed by Algorithm 1 converge strongly to a solution of SVIP (60).

Proof. Let be given asLet , we haveUsing (64) and Lipschitz continuity of the proximal-point mapping, thenFrom Lemma 1, we haveSince is -Lipschitz continuous with respect to mapping , thenNow, we computeSince is -Lipschitz continuous and is --Lipschitz continuous, we haveUsing equations (71)–(74) in equation (70), we haveFrom condition (67), we haveThus, we have . Therefore, (75) implies thatwhich is a contraction mapping and has a unique fixed point in . Hence, is a unique solution of SVIP (60).
Now, we will prove that strongly converges to . Using Algorithm 1, we haveUsing Theorem 3, we haveLetIn the light of equations (71)–(73), one can obtainwhere .
Using (79)–(81) in (78), we getFrom (67), we have . From (80), we have . By using Lemma 2, . By -Lipschitz continuity of and Algorithm 1, we haveThus, are the Cauchy sequences, and then, there exists such that , with . We will prove that . Since , we haveSince is closed, thus . Similarly, we can prove . By the Lipschitz continuity, we know that satisfy the following relation:By Theorem 4, SVIP (60) has a solution .

5. Conclusion

In this manuscript, we used proximal-point mapping linked with generalized --accretive mapping to discuss the graph convergence of this class of accretive mappings. Furthermore, we have discussed convergence analysis of the proposed iterative algorithm to solve SVIP (60) in real Banach space. As an extension of --accretive mapping introduced by Kazmi et al. [13], the generalized --accretive mapping has some important applications in physics, economics, and physical sciences. In the future, we can apply the technique used in this article to study the Yosida inclusion problems in the framework of Banach spaces.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

All authors declare that there are no conflicts of interest.

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Copyright © 2022 Faizan Ahmad Khan 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.

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