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 [225].

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 wit