Approximation solution for system of generalized ordered variational inclusions with ⊕ operator in ordered Banach space.

The resolvent operator approach is applied to address a system of generalized ordered variational inclusions with ⊕ operator in real ordered Banach space. With the help of the resolvent operator technique, Li et al. (J. Inequal. Appl. 2013:514, 2013; Fixed Point Theory Appl. 2014:122, 2014; Fixed Point Theory Appl. 2014:146, 2014; Appl. Math. Lett. 25:1384-1388, 2012; Fixed Point Theory Appl. 2013:241, 2013; Eur. J. Oper. Res. 16(1):1-8, 2011; Fixed Point Theory Appl. 2014:79, 2014; Nonlinear Anal. Forum 13(2):205-214, 2008; Nonlinear Anal. Forum 14: 89-97, 2009) derived an iterative algorithm for approximating a solution of the considered system. Here, we prove an existence result for the solution of the system of generalized ordered variational inclusions and deal with a convergence scheme for the algorithms under some appropriate conditions. Some special cases are also discussed.

Introduction

The theory of variational inequalities (inclusions) is quite application oriented and thus developed much in recent years in many different disciplines. This theory provides us with a framework to understand and solve many problems arising in the field of economics, optimization, transportation, elasticity and applied sciences. A lot of work considered with the ordered variational inequalities and ordered equations was done by Li et al.; see [4, 6, 8, 9].

The fundamental goal in the theory of variational inequality is to develop a streamline algorithm for solving a variational inequality and its various forms. These methods include the projection method and its novel forms, approximation techniques, Newton’s methods and the methods derived from the auxiliary principle techniques.

It is widely known that the projection technique cannot be applied to solve variational inclusion problems and thus one has to use resolvent operator techniques to solve them. The beauty of the iterative methods involving the resolvent operator is that the resolvent step involves the maximal monotone operator only, while other parts facilitate the problem’s decomposition. Most of the problems related to variational inclusions are solved by maximal monotone operators and their generalizations such as H-accretivity [10], H-monotonicity [11] and many more; see e.g., [12-17] and the references therein.

Essentially, using the resolvent technique, one can show that the variational inclusions are commensurate to the fixed point problems. This equivalent formation has played a great job in designing some exotic techniques for solving variational inclusions and related optimization problems.

We initiate a study of a system of generalized ordered variational inclusions in real ordered Banach space. We design an iterative algorithm based on the resolvent operator for solving system of generalized ordered variational inclusions. We prove an existence as well as a convergence result for our problem. For more details of related work, we refer to [2, 18] and the references therein.

Prelude

In the paper, assume that X be a real ordered Banach space endowed with norm , an inner product , a zero element θ and partial order ≤ defined by the normal cone C with a normal constant . The greatest lower bound and least upper bound for the set with partial order relation ≤ are denoted by and , respectively. Assume that and both exist.

The following well-known definitions and results are essential to achieve the goal of this paper.

Definition 2.1

Let C (≠ϕ) be a closed, convex subset of X. C is said to be a cone if

for and , ;

if p and , then .

Definition 2.2

[19]

C is called a normal cone iff there exists a constant such that implies , where is called the normal constant of C.

Definition 2.3

For arbitrary elements , iff , then the relation ≤ is a partial ordered relation in X. The real Banach space X endowed with the ordered relation ≤ defined by C is called an ordered real Banach space.

Definition 2.4

[20]

For arbitrary elements , if (or ) holds, then p and q are called comparable to each other and this is denoted by .

Definition 2.5

[18]

A map is called a β-ordered comparison map, if it is a comparison mapping and

Lemma 2.1

[19]

If p and q are comparable to each other, then and exist, , and .

Lemma 2.2

[19]

Let C be a normal cone with normal constant in X, then for each , we have the relations:

;

;

;

if , then .

Lemma 2.3

[1, 4–6]

Let ≤ be a partial order relation defined by the cone C with a normal constant in X in Definition  2.3. Then hereinafter relations survive:

, ;

;

allow λ to be real, then ;

if p, q and w can be comparative to each other, then ;

presume exists, and if and , then ;

if p, q, r, w can be compared with each other, then ;

if and , then ;

if , then ;

if , then ;

;

if and , and , then and , for all and .

Definition 2.6

[4]

Allow to be a single-valued map.

A is called a γ-order non-extended mapping if there exists a constant such that

A is called a strongly comparison map if it is a comparison mapping and iff , for all .

Definition 2.7

[7]

Allow and to be single-valued and set-valued mappings, respectively.

M is called a weak-comparison map, if for , , and if , then and such that , for all ;

M is called an α-weak-non-ordinary difference map associated with A, if it is weak comparison and for each , and and such that

M is called a λ-order different weak-comparison map associated with A if ∃ a , for all and there exist , such that

M, a weak-comparison map, is called an ordered -weak-ANODM map, if it is an α-weak-non-ordinary difference map and a λ-order different weak-comparison map associated with A, and , for .

Definition 2.8

[7]

Let and be a γ-order non-extended map and an α-non-ordinary difference mapping with respect to A, respectively. The resolvent operator associated with both A and M is defined by where are constants.

Definition 2.9

[8]

A map is called -restricted-accretive map, if it is comparison and ∃ constants such that where I is the identity map on X.

Lemma 2.4

[7]

If and are an α-weak-non-ordinary difference map associated with A and a γ-order non-extended map, respectively, with , then is an α-weak-non-ordinary difference map associated with A and the resolvent operator of is a single-valued for , i.e., of holds.

Lemma 2.5

[7]

Let and be a -weak-ANODD set-valued map and a strongly comparison map associated with , respectively. Then the resolvent operator is a comparison map.

Lemma 2.6

[7]

Let be an ordered -weak-ANODD map and be a γ-ordered non-extended map associated with , for , respectively. Then the following relation holds:

Formulation of the problem and existence results

Allow X to be a real ordered Banach space and C a normal cone having the normal constant . Let be set-valued mappings. Suppose () and are single-valued mappings. Now we look at the problem:

For some and , find such that

This problem is called a system of generalized implicit ordered variational inclusions (in short SGIOVI). Here, we discuss some special cases of SGIOVI (3).

If , (the identity mapping on X), and M and N are single-valued mappings and , then problem (3) reduces to the problem as for , find such that Problem (4) was initiated and studied by [1].

If , , , , M is a single-valued mapping, then problem (3) is to find such that Problem (5) was initiated and studied by [21].

If , , , and , then problem (3) became the problem to find such that Problem (6) was initiated and studied by [7].

If , , , and , then problem (3) is converted to the problem of finding such that Problem (7) was initiated and studied by [5].

If , , and , then the problem (3) converted to the problem of finding such that Problem (8) was initiated and studied by [3].

Now, we mention the fixed point formulation of SGIOVI (3).

Lemma 3.1

The set of elements become a solution of SGIOVI (3) iff fulfill the relations:

Proof

The proof follows from the definition of the resolvent operator (1). □

Main results

In this section, we present an existence result for the system of generalized implicit ordered variational inclusions (in short SGIOVI), under some apt conditions.

Theorem 4.1

Let C be a normal cone having a normal constant in a real ordered Banach space X. Let be single-valued mappings such that A is a -compression mapping, is a -compression and is a -compression and , are comparison mappings, respectively. Let be single-valued mappings such that is an -restricted-accretive mapping w.r.t. and is an -restricted-accretive mapping w.r.t. , respectively. Suppose are the set-valued mappings such that M is a -weak-ANODD set-valued mapping and is a -weak-ANODD set-valued mapping, respectively.

In addition, if , , and for all , the following conditions are satisfied: and Then the SGIOVI (3) grants a solution .

By Lemma 2.6, we know that the resolvent operator and are -Lipschitz continuous and -Lipschitz continuous, respectively.

Here and .

Now, define a map by where and are the mappings defined as For any and (). By using (12), Definition 2.5, Definition 2.9, Lemma 2.6 and Lemma 2.3, we have By Definition 2.2 and Lemma 2.2, we have That is,

For any , (), and by using (13), Definition 2.5, Definition 2.6, Lemma 2.3 and Lemma 2.6, we have

By Definition 2.2 and Lemma 2.2, we have That is,

From (15) and (17), we have where and Now, we define on by One can easily show that is a Banach space. Hence from (11), (18) and (19), we have By (10), we know that . It follows from (20) that G is a contraction. Hence ∃ unique such that This leads to and It is determined by Lemma 3.1 that is a solution of SGIOVI (3). □

Convergence analysis and iterative algorithm

This part of the article is associated with the construction of an iterative scheme as well as the strong convergence of the sequences achieved by the said scheme to the exact solution of SGIOVI (3).

Allow C to be a normal cone with the normal constant in a real ordered Banach space X. Let and be set-valued maps. Assume that and are single-valued maps.

For the initial guess , assume that , . We define an iterative sequence and let , such that For  , where with .

Lemma 5.1

[17]

Allow and to be sequences of nonnegative real numbers such that they satisfy Then approaches zero as n moves to ∞.

, and ;

,  .

Theorem 5.2

Allow X, C, M, N, , , , , , and to be as in Theorem  4.1 such that all the assertions of Theorem  4.1 are valid. Then the sequence formulated by Algorithm (21) and (22) converges strongly to the unique solution of SGIOVI (3).

By Theorem 4.1, the SGIOVI (3) admits a unique solution . It follows from Lemma 3.1 that and

By (21), (23) and Lemma 2.3, we get

By using the same argument as in Theorem 4.1, for (14), we have Similarly, it follows from (22) and (24) that Importing the same logic as in Theorem 4.1 for (16), we have From (26) and (28) we have By (10), we know that . Then (29) becomes where and , . Let and , then (30) can be rewritten as Choosing , we know that . It follows from Lemma 5.1 that . Therefore, converge strongly to the unique solution of the SGIOVI (3). □

Conclusion

System of generalized ordered variational inclusions can be viewed as natural and innovative generalizations of the system of generalized ordered 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 system of generalized ordered variational inclusions is introduced and studied which is more general than many existing systems of ordered variational inclusions in the literature. An iterative algorithm is established with the ⊕ operator to approximate the solution of our system, and a convergence criterion 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 ordered variational inclusions in engineering and physical sciences.