Generalized Complementarity Problem with Three Classes of Generalized Variational Inequalities Involving �

cited. In this study, we introduce and study a generalized complementarity problem involving XOR operation and three classes of generalized variational inequalities involving XOR operation. Under certain appropriate conditions, we establish equivalence between them. An iterative algorithm is deﬁned for solving one of the three generalized variational inequalities involving XOR operation. Finally, an existence and convergence result is proved, supported by an


Introduction
It is well known that the many unrelated free boundary value problems related to mathematical and engineering sciences can be solved by using the techniques of variational inequalities. In a variational inequality formulation, the location of the free boundary becomes an intrinsic part of the solution, and no special devices are needed to locate it. Complementarity theory is an equally important area of operations research and application oriented. e linear as well as nonlinear programs can be distinguished by a family of complementarity problems. e complementarity theory have been elongated for the purpose of studying several classes of problems occurring in fluid flow through porous media, economics, financial mathematics, machine learning, optimization, and transportation equilibrium, for example, [1][2][3][4][5].
e correlations between the variational inequality problem and complementarity problem were recognized by Lions [6] and Mancino and Stampacchia [7]. However, Karamardian [8,9] showed that both the problems are equivalent if the convex set involved is a convex cone. For more details on variational inequalities and complementarity problems, refer to [6,[10][11][12]. e exclusive "XOR," sometimes also exclusive disjunction (short: XOR) or antivalence, is a Boolean operation which only outputs true if only exactly one of its both inputs is true (so, if both inputs differ). ere are many applications of XOR terminology, that is, it is used in cryptography, gray codes, parity, and CRC checks. Commonly, the ⊕ symbol is used to denote the XOR operation. Some problems related to variational inclusions involving XOR operation were studied by [13][14][15][16].
Influenced by the applications of all the above discussed concepts in this study, we introduce and study a generalized complementarity problem involving XOR operation with three classes of generalized variational inequalities involving XOR operation. Some equivalence relations are established between them. An existence and convergence result is proved for one of the three types of generalized variational inequalities involving XOR operation. For illustration, an example is provided. closed and bounded) subsets of E. e Hausdorff metric D(., .) on CB(E) is defined as Let C be a pointed closed convex positive cone in E, and 〈t, x〉 denotes the value of the linear continuous function t ∈ E * at x. e following definitions and concepts are required to achieve the goal of this study, and most of them can be found in [17,18].
e relation " ≤ " is called the partial order relation induced by the cone C, that is, x ≤ y if and only if y − x ∈ C.
Definition 2. For arbitrary elements x, y ∈ E, if x ≤ y (or y ≤ x) holds, then x and y are said to be comparable to each other (denoted by x ∝ y).
Definition 3. For arbitrary elements x, y ∈ E, lub x, y and glb x, y mean the least upper bound and the greatest upper bound of the set x, y . Suppose lub x, y and glb x, y exist, then some binary operations are defined as e operations ∨, ∧, ⊕, and ⊙ are called OR, AND, XOR, and XNOR operations, respectively. Proposition 1. Let ⊕ be an XOR operation and ⊙ be an XNOR operation. en, the following relations hold: for all x, y, u, v ∈ E and λ ∈ R Proposition 2. Let C be a cone in E; then, for each x, y ∈ E, the following relations hold: e set of all subgradients of f at x is denoted by zf(x). e mapping zf : is called subdifferential of f.

Definition 6.
e resolvent operator J zf ρ associated with zf is given by where ρ > 0 is a constant, and I is the identity operator. It is well known that the resolvent operator J zf ρ is single-valued as well as nonexpansive.
Definition 10. A multivalued mapping F: E ⟶ 2 E is said to be relaxed Lipschitz continuous, if there exists a constant k > 0 such that Let F: C ⟶ 2 E * ∖ ∅ { } be a multivalued mapping with nonempty values and f: C ⟶ R ∪ +∞ { } be a proper functional. We consider the following generalized complementarity problem involving XOR operation. Find We denote by S C⊕ the solution set of generalized complementarity problem involving XOR operation (9).
We mention some special cases of problem (9) as follows.
(i) If we replace ⊕ by + and f by f: C ⟶ R, then problem (9) reduces to the problem of finding x ∈ C and t ∈ F(x) such that Problem (10) is called generalized f complementarity problem, introduced and studied by Huang et al. [19]. (ii) If f ≡ 0, then problems (9) as well as (10) reduce to the problem of finding x ∈ C and t ∈ F(x) such that Problem (11) can be found in [20,21].
We remark that for suitable choices of operators involved in the formulation of (9), a number of known complementarity problems can be obtained easily, for example, [17,[22][23][24].
Simultaneously, we also study the following three types of generalized variational inequalities involving XOR operation.
{ } be a multivalued mapping with nonempty values and f: C ⟶ R ∪ +∞ { } be a proper functional. en, the following statements are true: Since 〈t, tx〉 ∝ f(x), by (iv) of Proposition 1, we have Also as 〈t, ty〉⊕f(y) ≥ 0, which implies that Journal of Mathematics 〈t, ty〉 ≥ f(y).

(17)
By using (16) and (17), we have that is, Since C is a pointed closed convex positive cone, clearly y � 2x ∈ C and y � (1/2)x ∈ C. Putting y � 2x in generalized variational inequality involving XOR operation (12) and using positive homogenity of f, we get Now, putting y � (1/2)x in generalized variational inequality involving XOR operation ((12)) and using positive homogenity of f, we get which implies that thus, that is, Combining (21) and (25), we have From generalized variational inequality involving XOR operations (12) and (16), we have (27) which implies that e following statements are true.
Since F is monotone, for every y ∈ C, t ∈ F(y), and using the above inequality, we have us, x ∈ S 3⊕ . (iii) Suppose that the conclusion is not true. en, there exists x ∈ C such that x ∈ S 3⊕ and x ∉ S 2⊕ . en, for some y ∈ C and t ∈ F(x), we have Since F is upper hemicontinuous and f is convex, setting x λ � λy + (1 − λ)x and taking λ ⟶ 0, we have which implies that thus, which contradicts that x ∈ S 3⊕ . us, x ∈ S 2⊕ , and (iii) is true.
□ Remark 1. If we replace ⊕ by + and dropping the concepts related to ⊕ operation, then with slight modification in eorems 1 and 2, one can obtain some results of Huang et al. [19]. Additionally, for suitable choices of operators in eorems 1 and 2, one can obtain some results of Farajzadeh and Harandi [30].

Existence and Convergence Result
In this section, we first establish the equivalence between the generalized variational inequality problem involving XOR operation (12) and a nonlinear equation. Based on this equivalence, we construct an iterative algorithm for solving generalized variational inequality problem involving XOR operation (12).

Lemma 1.
e generalized variational inequality problem involving XOR operation (12) admits a solution (x, tt), x ∈ C and t ∈ F(x), if and only if the following relation is satisfied: where ρ > 0 is a constant, J zf ρ � [I + ρ zf] − 1 is the resolvent operator associated with f, and I is the identity operator.
Proof. From the definition of resolvent operator J zf ρ associated with f and relation (35), we have By the definition of subdifferential operator zf(x) and (37), we have Using (vi) of Proposition 1, we have us, the generalized variational inequality problem involving XOR operation (12) is satisfied.
Conversely, suppose that generalized variational inequality problem involving XOR operation (12) is satisfied.
at is, that is, the relation (35) is satisfied. Based on Lemma 1, we develop the following iterative algorithm for solving the generalized variational inequality problem involving XOR operation (12). □ Iterative Algorithm 1. Let C ⊂ E be a pointed closed convex positive cone. Suppose that t n ∝ t n− 1 , for n � 1, 2, . . .. Let for x 0 ∈ C, there exists t 0 ∈ F(x 0 ), such that Since t 0 ∈ F(x 0 ) ∈ CB(E), by Nadler [31], there exists t 1 ∈ F(x 1 ), using (iv) of Proposition 2, and as t 0 ∝ t 1 , we have Continuing this way, compute the sequences x n and t n by the following scheme: for n � 1, 2, . . ., where x n ∈ C, t n ∈ F(x n ) can be chosen arbitrarily, α ∈ [0, 1], D(., .) is the Hausdorff metric on CB(E), and ρ > 0 is a constant. Now, we prove our main result.
en, the sequences x n and t n strongly converge to x * and t * , respectively, the solutions of generalized variational inequality problem involving XOR operation (12).
Proof. Since x n+1 ∝ x n , for n � 1, 2, . . ., using (iii) of Proposition 1, we evaluate From (47), it follows that As x n ∝ x n− 1 , t n ∝ t n− 1 , obviously, x n + ρt n ∝ x n− 1 + ρt n− 1 , for n � 1, 2, . . .. Since the resolvent operator J zf ρ is strongly comparison, we have Using above facts, (iv) of Proposition 2 and nonexpansiveness of J zf ρ , (48) becomes Since the multivalued mapping F is the relaxed Lipschitz continuous with constant k > 0, D-Lipschitz continuous with constant λ D F > 0, and using (45) of Iterative Algorithm 1, we have thus, where θ � ������������� 1 − 2ρk + ρ 2 λ 2 D F . Combining (50) and (52), we have thus, we have where c � (1 − α + αθ). Hence, for m > n > 0, we have It is clear from condition (46) that 0 < c < 1, and consequently, we have ‖x n − x m ‖ ⟶ 0, as n ⟶ ∞. us, x n is a Cauchy sequence in E, and as E is complete, x n ⟶ x * ∈ E, as n ⟶ ∞. From (45) of Iterative Algorithm 1, we have thus, t n is also a Cauchy sequence in E such that t n ⟶ t * ∈ E, as n ⟶ ∞. Now, we will show that (x * , t * ) is a solution of generalized variational inequality problem involving XOR operation (12). As x n ⟶ x * , t n ⟶ t * , and resolvent operator J zf ρ is continuous, we can write 6 Journal of Mathematics us, the relation (35) is satisfied. It remains to show that t * ∈ F(x * ). Since t n ∈ F(x n ), we have Hence d(t * , F(x * )) ⟶ 0, t * ∈ F(x * ) as F(x * ) ∈ CB(E). By Lemma 1, x * ∈ C, t * ∈ F(x * ) is a solution of generalized variational inequality problem involving XOR operation (12). is completes the proof. □ Remark 2. Combining eorems 1 and 3, we assert that the solution x ∈ C, t ∈ F(x) of generalized variational inequality involving XOR operation (12) is also a solution of generalized complementarity problem involving XOR operation (9).

Numerical Example
In this section, we construct a numerical example in support of eorem 3. Finally, the convergence graphs and the computation tables are provided for the sequences generated by Iterative Algorithm 1.

Example 1.
Let E � E * � R with the usual inner product and norm. Let C � x ∈ tRn: q0h ≤ xx ≤ 71 be a pointed closed convex positive cone in R. Let f: C ⟶ R ∪ +∞ { } be a functional, zf : R ⟶ 2 R be the subdifferential of f, F: C ⟶ 2 R ∖ ∅ { } be a multivalued mapping, and J zf ρ be the resolvent operator associated with f such that en, , ∀x ∈ C. (60) One can easily verify that the resolvent operator J zf ρ is a strongly comparison mapping and continuous.
For x, y ∈ C, w 1 ∈ F(x), and w 2 ∈ F(y), we have that is, us, F is the relaxed Lipschitz continuous with constant k � (1/10). Also, that is, us, F is the D-Lipschitz continuous with constant λ D F � (1/5).
Let us take ρ � 1, then for k � (1/10) and λ D F � (1/5), the condition (46) is satisfied. Furthermore, for ρ � 1 and α � (1/3), we obtain the sequences x n and t n generated by the Iterative Algorithm 1 as where t n ∈ F(x n ), and thus, t n � − (x n /7). It is clear that the sequence x n converges to x * � 0, and consequently, the sequence t n also converges to t * � 0.
For initial values x 0 � 5, 10, and 15, we have the following convergence graphs, which ensure that the sequences x n and t n converge to 0. Two computation tables are    Journal of Mathematics provided for the iterations (Tables 1 and 2) of the sequences x n and t n (Figures 1, and 2).

Conclusion
In this study, we introduce and study a generalized complementarity problem involving XOR operation with three classes of generalized variational inequalities involving XOR operation. Some equivalence relations are established between them. Finally, a generalized variational inequality problem involving XOR operation (12) is solved in real ordered Banach spaces. A numerical example is constructed with convergence graphs and computation tables for illustration of our main result. We remark that our results may be further extended using other tools of functional analysis.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.