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 [, , , ].
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 [], Hmonotonicity [] and many more; see e.g., [-] 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 [, ] 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 λ C . The greatest lower bound and least upper bound for the set {p, q} with partial order relation ≤ are denoted by glb{p, q} and lub{p, q}, respectively. Assume that glb{p, q} and lub{p, q} both exist.
The following well-known definitions and results are essential to achieve the goal of this paper.
Definition . Let C ( = φ) be a closed, convex subset of X. C is said to be a cone if (i) for p ∈ C and λ > , λp ∈ C; (ii) if p and -p ∈ C, then p = θ .

Definition . ([]
) C is called a normal cone iff there exists a constant λ C >  such that  ≤ p ≤ q implies p ≤ λ C q , where λ C is called the normal constant of C.
Definition . For arbitrary elements p, q ∈ X, p ≤ q iff pq ∈ C, 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 . ([]) For arbitrary elements p, q ∈ X, if p ≤ q (or q ≤ p) holds, then p and q are called comparable to each other and this is denoted by p ∝ q.
Lemma . ([]) Let C be a normal cone with normal constant λ C in X, then for each p, q ∈ X, we have the relations:

Definition . ([]) Allow
A : X → X to be a single-valued map.
() A is called a γ -order non-extended mapping if there exists a constant γ >  such that , for all p, q ∈ X; () A is called a strongly comparison map if it is a comparison mapping and A(p) ∝ A(q) iff p ∝ q, for all p, q ∈ X.

Definition . ([]) Allow
A : X → X and M : X →  X to be single-valued and set-valued mappings, respectively. () M is called a weak-comparison map, if for t p ∈ M(p), p ∝ t p , and if p ∝ q, then ∃ t p ∈ M(p) and t q ∈ M(q) such that t p ∝ t q , for all p, q ∈ X; () M is called an α-weak-non-ordinary difference map associated with A, if it is weak comparison and for each p, q ∈ X, ∃ α >  and t p ∈ M(A(p)) and t q ∈ M(A(q)) such that () M, a weak-comparison map, is called an ordered (α A , λ)-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 (A + λM)(X) = X, for α, λ > .

Definition . ([]) Let
A : X → X and M : X →  X be a γ -order non-extended map and an α-non-ordinary difference mapping with respect to A, respectively. The resolvent operator R M A,λ : X → X associated with both A and M is defined by where γ , α, λ >  are constants.
where I is the identity map on X.

Lemma . ([])
If M : X →  X and A : X → X are an α-weak-non-ordinary difference map associated with A and a γ -order non-extended map, respectively, with αλ = , then

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 This problem is called a system of generalized implicit ordered variational inclusions (in short SGIOVI). Here, we discuss some special cases of SGIOVI ().
() If ρ = , g  = I (the identity mapping on X), f  = I and M and N are single-valued mappings and M(g  (p), q) = M(q, p), then problem () reduces to the problem as for ).
Problem () was initiated and studied by [].
Problem () was initiated and studied by [].
N(p, g  (q)) = N(p), then problem () is converted to the problem of finding p ∈ X such that Lemma . The set of elements (p, q) ∈ X × X become a solution of SGIOVI () iff (p, q) ∈ X × X fulfill the relations: Proof The proof follows from the definition of the resolvent operator ().

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 . Let C be a normal cone having a normal constant λ C in a real ordered Banach space X. Let A, f  , f  , g  , g  : X → X be single-valued mappings such that A is a λ Acompression mapping, f  is a λ f  -compression and f  is a λ f  -compression and g  , g  are comparison mappings, respectively. Let F  , F  : X × X → X be single-valued mappings such that F  is an (α  , α  )-restricted-accretive mapping w.r.t. f  and F  is an (α  , α  )-restrictedaccretive mapping w.r.t. f  , respectively. Suppose M, N  : X × X →  X are the set-valued mappings such that M is a (α A , λ)-weak-ANODD set-valued mapping and N  is a (α A , λ)weak-ANODD set-valued mapping, respectively.
Proof By Lemma ., we know that the resolvent operator R M A,λ (·) and R N  A,λ (·) are μ  -Lipschitz continuous and μ  -Lipschitz continuous, respectively. Here where T : X × X → X and S : X × X → X are the mappings defined as For any p i , q i ∈ X and p i ∝ q j (i, j = , ). By using (), Definition ., Definition ., Lemma . and Lemma ., we have By Definition . and Lemma ., we have That is, For any p i , q j ∈ X, p i ∝ q j (i, j = , ), and by using (), Definition ., Definition ., Lemma . and Lemma ., we have By Definition . and Lemma ., we have That is, From () and (), we have One can easily show that (X × X, · * ) is a Banach space. Hence from (), () and (), we have By (), we know that max{  ,  } < . It follows from () that G is a contraction. Hence ∃ unique (p, q) ∈ X × X such that G(p, q) = (p, q).

This leads to
It is determined by Lemma . that (p, q) is a solution of SGIOVI ().

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 (). Allow C to be a normal cone with the normal constant λ C in a real ordered Banach space X. Let M : X × X →  X and N : X × X →  X be set-valued maps. Assume that f  , f  , g  , g  : X → X and F  , F  : X × X → X are single-valued maps.
Choosing ς n , we know that lim sup n ς n < . It follows from Lemma . that  ≤ ς n ≤ . Therefore, {(p n , q n )} converge strongly to the unique solution {(p, q)} of the SGIOVI ().

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.