System of Multi-Valued Mixed Variational Inclusions with XOR-Operation in Real Ordered Uniformly Smooth Banach Spaces

: In this paper, we consider and study a system of multi-valued mixed variational inclusions with XOR-operation ⊕ in real ordered uniformly smooth Banach spaces. This system consists of bimappings, multi-valued mappings and Cayley operators. An iterative algorithm is suggested to ﬁnd the solution to a system of multi-valued mixed variational inclusions with XOR-operation ⊕ and consequently an existence and convergence result is proved. In support of our main result, an example is constructed. 49H10;


Introduction
In 1964, Stampacchia [1] investigated the theory of variational inequality which provides us a lenient way for solving perplexities occurring in industry, finance, economics, operation research, optimization, decision sciences and several other branches of pure and applied sciences, and so forth, see, for example, [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Hassouni and Moudafi [18] studied a mixed type variational inequality which involves a nonlinear term called variational inclusion. They used the resolvent operator technique in order to find the solution to their problem as the projection method does not work due to the nonlinear term.
A natural generalization of variational inequalities called the system of variational inequalities (inclusions) were considered and studied by several authors. Cohen and Chaplais [19], Ansari and Yao [20] and many more researchers considered various system of variational inequalities (inclusions), see also [21][22][23][24][25][26][27][28][29]. It has been shown by Pang [30] that not only the Nash equilibrium problem but also various equilibrium type problems, like the traffic equilibrium problem, spatial equilibrium problem and the general equilibrium programming problems from operation research, game theory, mathematical physics, and so forth, can be formulated as a variational inequality problem defined over a product of sets, which is equivalent to a system of variational inequalities.
In this paper, we consider and study a system of multi-valued mixed variational inclusions with XOR-operation ⊕ in real ordered uniformly smooth Banach spaces. We prove the existence of solutions to a system of multi-valued mixed variational inclusions with XOR-operation ⊕ and we discuss the convergence of the iterative sequences generated by the proposed algorithm. An example is provided.

Preliminaries
Let E be a real ordered uniformly smooth Banach space with norm · and E * be its topological dual. We denote by d the metric induced by the norm · on E, by CB(E) (respectively, 2 E ) the family of all nonempty closed and bounded subsets (respectively, the set of all nonempty subsets) of E and by D(·, ·) the Hausdörff metric on CB(E). Let C ⊆ E be a cone. For arbitrary elements x, y ∈ E, x ≤ y holds if and only if y − x ∈ C, then the relation " ≤ " in E is called partial order relation induced by the cone C.
Let ·, · be the duality pairing between E and E * , and J : E → 2 E * be the normalized duality mapping defined by We recall some well known concepts and results for the presentation of this paper.
The modulus of smoothness of a Banach space E is a function Definition 1 ([29]). A mapping g : E → E is said to be (i) accretive, if for any x, y ∈ E, there exists j(x − y) ∈ J(x − y) such that (ii) strongly accretive, if for any x, y ∈ E, there exists j(x − y) ∈ J(x − y) and a constant δ g > 0 such that (iii) Lipschitz continuous, if for any x, y ∈ E, there exists a constant λ g > 0 such that Proposition 1 ([42]). Let E be a uniformly smooth Banach space and J : E → 2 E * be a normalized duality mapping. Then, for any x, y ∈ E,

Definition 2.
A multi-valued mapping G : E → CB(E) is said to be D-Lipschitz continuous, if for any x, y ∈ E, there exists a constant λ D G > 0 such that

Definition 3.
A cone C is said to be normal if there exists a constant λ N > 0 such that for 0 ≤ x ≤ y, x ≤ λ N y , where λ N is normal constant of C.

Definition 4.
For arbitrary element x, y ∈ E, x ≤ y (or y ≤ x) holds, then x and y said to be comparable to each other (denoted by x ∝ y).
Most of the following definitions can be found in [43].

Definition 5.
For arbitrary elements x,y of E, lub{x, y} and glb{x, y} mean the least upper bound and the greatest lower bound of the set {x, y}. Suppose lub{x, y} and glb{x, y} exist. Then some binary operations are defined as follows: The operations ∨, ∧, ⊕ and are called OR, AND, XOR and XNOR operations, respectively.

Proposition 2.
Let ⊕ be an XOR-operation and be an XNOR -operation. Then the following relations hold:

Proposition 3 ([43]
). Let C ⊆ E be a normal cone with normal constant λ N . Then for each x,y of E, the following relations hold: (i) A is said to be a comparison mapping, if for all x, y ∈ E, x ∝ y then A(x) ∝ A(y), x ∝ A(x) and y ∝ A(y), (ii) A is said to be strongly comparison mapping, if A is a comparison mapping and A(x) ∝ A(y) if and only if x ∝ y, for all x, y ∈ E, (iii) A is said to be β -ordered compression mapping, if A is a comparison mapping, and  and any v y ∈ M(y), v x ∝ v y for all x, y ∈ E, (ii) M is said to be α M -non-ordinary difference mapping, if for all x, y ∈ E, M is a comparison mapping and v x ∈ M(x) and v y ∈ M(y) such that (iii) M is said to λ-XOR-ordered strongly monotone mapping, if x ∝ y then there exists a constant λ > 0 such that

Definition 8. Let
A : E → E be a strong comparison and β -ordered compression mapping. Then, a comparison multi-valued mapping M : Definition 9. Let A : E → E be a strongly comparison and β -ordered compression mapping and let M : It is proved in [39] that the resolvent operator defined by (1) is a single-valued comparison as well Definition 10. The Cayley operator C M A,λ associated with M is defined as where R M A,λ is defined by (1) and λ > 0.
One can easily prove that the Cayley operator defined by (2) is single-valued, a comparison as well as (2θ + 1)-Lipschitz-type continuous, where θ is same as in Definition 9, for more details see [40].

A System of Multi-Valued Mixed Variational Inclusions with XOR-Operation ⊕ and an Iterative Algorithm
Let E be a real ordered uniformly smooth Banach space. Let G, F : E → CB(E) be multi-valued mappings and A, P, q : E → E; S, T : E × E → E be single-valued mappings. Let M, N : E → 2 E be multi-valued mappings and C M A,λ ; C N A,ρ : E → E be Cayley operators. We deal with the following problem. Find where λ > 0 and ρ > 0 are constants. Problem (3) is called system of multi-valued mixed variational inclusions with XOR-operation ⊕.
If P(x) = 0 = q(y), then we encounter with the following problem, that is, find Problem (4) appears to be the new one.
, and ⊕ is replaced by +, then problem (4) reduces to the problem of finding x, y ∈ E, u ∈ G(x), v ∈ F(y) such that Problem (5) is considered in [26] in the setting of Hilbert spaces. It is easy to check that problem (3) includes many previously studied problems related to variational inclusions.
The following Lemma is a fixed point formulation of problem (3).
is a solution to a system of multi-valued mixed variational inclusions with XOR-operation ⊕ (3), if and only if the following equations are satisfied: where, λ > 0 and ρ > 0 are constants.

Existence of Solutions and Convergence of Iterative Sequences
We prove the following existence and convergence result for problem (3).

Theorem 1.
Let E be a real ordered uniformly smooth Banach space with modulus of smoothness τ E (t) ≤ Ct 2 for some C > 0 and C ⊆ E be a normal cone with normal constant λ N . Let A : E → E; S, T : E × E → E be single-valued mappings such that A is strongly comparison and β -ordered compression mapping; S is Lipschitz continuous in both the arguments with constant λ S 1 and λ S 2 , respectively; T is Lipschitz continuous in both the arguments with constant λ T 1 and λ T 2 , respectively. Let F, G : E → CB(E) be multi-valued mappings such that F is λ D F -D-Lipschitz continuous and G is λ G D -D-Lipschitz continuous. Suppose that P, q : E → E be single-valued mappings such that P is δ P -strongly accretive and λ P -Lipschitz continuous; q is δ q -strongly accretive and λ q -Lipschitz continuous. Let M : E → 2 E be (α M , λ)-XOR-NODSM mapping and N : E → 2 E be (α N , ρ)-XOR-NODSM mapping. Suppose that the resolvent operators R M A,λ , R N A,ρ : E → E are θ-Lipschitz-type continuous and θ -Lipschitz-type continuous, respectively, and the Cayley operators C M A,λ , C M A,ρ : E → E are (2θ + 1) and (2θ + 1)-Lipschitz-type continuous, respectively. Let x n+1 ∝ x n , y n+1 ∝ y n and for some λ, ρ > 0 the following conditions are satisfied: where Then, the system of multi-valued mixed variational inclusions with XOR-operation ⊕ (3) have a solution(x, y, u, v), where x, y ∈ E, u ∈ G(x), v ∈ F(y) such that x n → x, y n → y, u n → u and v n → v strongly, where {x n }, {y n }, {u n } and {v n } are the sequences generated by Algorithm 1.
Proof. As x n+1 ∝ x n , using (iii) of Proposition 2 and (8) of Algorithm 1, we have Since the resolvent operator R M A,λ is Lipschitz-type-continuous with constant θ and A is β -compression mapping, we evaluate Combining (18) and (19), we have Using (iii) of Proposition 3 and (20), we have Using the Lipschitz continuity of S in both the arguments with constants λ s 1 and λ s 2 , respectively, and using (iii) of Proposition 3, we obtain Using D-Lipschitz continuity of F, we have v n − v n−1 ≤ 1 + n −1 D(F(y n ), F(y n−1 )) ≤ 1 + n −1 λ D F y n − y n−1 .
Since P is strongly accretive with constant δ p and Lipschitz continuous with constant λ p , using the techniques of Alber and Yao [45] and Proposition 1, for j(x n − x n−1 ) ∈ J(x n − x n−1 ), we have where B(p) = 1 − 2δ P + 64Cλ P 2 .
Since the Cayley operator C N A,ρ is Lipschitz-type-continuous with constant (2θ + 1), we obtain where θ = 1 As y n+1 ∝ y n and combining (31) to (34) with (30), we have Combining (26) and (35), we have where (1) Clearly, A is strongly comparison mapping and That is, A is 2 5 -ordered compression mapping. (2) It is easy to check that S is Lipschitz continuous in both the arguments with constants 3 2 and 1, respectively and T is Lipschitz continuous in both the arguments with constants 1 and 1 3 , respectively.
Similarly, we can show that q is strongly accretive with constant 1 3 and Lipschitz continuous with constant 2 3 . (4) One can easily show that the resolvent operators R M A,λ is 5 12 -Lipschitz-type-continuous, R N A,ρ is 10 19 -Lipschitz type continuous, the Cayley operators C M A,λ is 11 6 -Lipschitz-type continuous and C N A,ρ is 39 19 -Lipschitz-type-continuous. Also, M is a comparison mapping and 2-non-ordinary difference mapping, N is a comparison mapping and 3-non-ordinary difference mapping.
Thus, all the assumptions and conditions of Theorem 1 are satisfied.
Author Contributions: All the authors have contributed equally to this paper. All the authors read and approved the final manuscript.
Funding: This research received no external funding.