Sensitivity Analysis of Mixed Cayley Inclusion Problem with XOR-Operation

: In this paper, we consider the parametric mixed Cayley inclusion problem with Exclusive or (XOR)-operation and show its equivalence with the parametric resolvent equation problem with XOR-operation. Since the sensitivity analysis, Cayley operator, inclusion problems, and XOR-operation are all applicable for solving many problems occurring in basic and applied sciences, such as ﬁnancial modeling, climate models in geography, analyzing “Black Box processes”, computer programming, economics, and engineering, etc., we study the sensitivity analysis of the parametric mixed Cayley inclusion problem with XOR-operation. For this purpose, we use the equivalence of the parametric mixed Cayley inclusion problem with XOR-operation and the parametric resolvent equation problem with XOR-operation, which is an alternative approach to study the sensitivity analysis. In support of some of the concepts used in this paper, an example is provided.


Introduction
It is well known that the variational inequalities (inclusions) and their generalizations are applicable for dealing with a large number of problems related to mechanics, physics, optimization and control, nonlinear programming, economics, transportation equilibrium, engineering sciences, etc. (for example, see [1][2][3][4][5][6][7][8][9][10] and references therein). On the other hand, ordered variational inequalities (inclusions) are also very important from the application point of view. A suitable amount of work related to generalized variational inequalities (inclusions) in Hilbert spaces as well as in Banach spaces can be found in [11][12][13][14][15][16][17][18][19][20][21]. Let S be a symmetric operator on H and express an operator C s as C s = (S − iI)(S + iI) −1 , then C s is the Cayley transformation of S. Cayley transformation is a mapping between skew-symmetric matrices and special orthogonal metrics. Recently, the Cayley inclusion problem with XOR-operation was considered and studied by Ali et al. [22]. They proved some properties of the Cayley operator involved in their problem. "Exclusive or" or exclusive disjunction (XOR) is a logical operation that outputs true only when inputs differ (one is true, the other is false). It gained the name "exclusive or" because the meaning of "or" is ambiguous when both operands are true, the exclusive or operator excludes that case. This is sometimes thought of as "one or the other but not both". This could be written as "A or B, but not A and B". It is to be noted that XOR and symmetric difference for sets use the same symbol ⊕ and the same structure, which are shown by Venn diagrams below in Figure 1.  That is, XOR and symmetric difference for sets are the same. Moreover, XOR is the symmetric difference for truth values.
Let A B denote the symmetric difference of the sets A and B, given an object x, then x ∈ A B, if and only if (x ∈ A)XOR(x ∈ B).
For example, {1, 2, 3}XOR{3, 4} is {1, 2, 4}, and the symmetric difference of {1, 2, 3} and {3, 4} is also {1, 2, 4}. It is well known that symmetric difference as well as XOR-operation are both symmetric. The graphical representation of XOR of x and y for truth values is given by the following Figure 2. XOR-operations depict interesting facts, observations, and results to form several real-time applications; one can find its applications in digital communication, neural network, addition operation in CPU, programming contests, binary search, etc. The study of qualitative behavior of a solution of a variational inequality when the given operator and the feasible convex set vary with a parameter is known as sensitivity analysis. As sensitivity analysis is one of the important tools of functional analysis, we will mention some of its applications. It helps decision makers with more than a solution to a problem, it is used to indicate sensitivity of a simulation to uncertainties in the input values of the model; helps in assessing the risk of a strategy; helps to predict the future changes in the governing system; provides useful information for designing or planning various equilibrium systems, etc. Some applications of sensitivity analysis related to real problems are optimal control of anthroponotic cutaneous leishmania, driving conditions for active vehicle control systems, influenza model with treatment and vaccination, dengue epidemiological model, Stochastic human exposure, dose simulation models, etc., for examples, see [23][24][25][26]. Sensitivity analysis for variational inequalities (inclusions) have been studied by Tobin [27], Kyparises [28,29], Dafermos [30], Qui and Magnanti [31], Li et al. [32], Yen and Lee [33], and Agarwal et al. [34], etc. For the sensitivity analysis related to real-life problems, we refer to [35,36] and references therein.
Due to importance of the Cayley operator, inclusion problems, and XOR-operation, we combine all these concepts to introduce two new equivalent problems, that is, parametric mixed Cayley inclusion problem with XOR-operation and parametric resolvent equation problem with XOR-operation. We have established equivalence between parametric mixed Cayley inclusion problem with XOR-operation and parametric resolvent equation problem with XOR-operation. The motivation of this work is to use this equivalence to develop sensitivity analysis for parametric mixed Cayley inclusion problem with XOR-operation without using differentiability of the given data.

Preliminaries
Throughout the paper, we assume H to be a real ordered Hilbert space with norm . and inner product ·, · . Let C ⊂ H be a cone and "≤" be the ordering induced by the cone C. For arbitrary element x, y ∈ H, x ≤ y holds if and only if, y − x ∈ C. The relation "≤" in H is called a partial ordered relation. The arbitrary elements x, y ∈ H are said to be comparable to each other if x ≤ y or y ≤ x holds and is denoted by x ∝ y.

Definition 1 ([37]
). For arbitrary elements x, y ∈ H, lub{x, y} and glb{x, y} mean the least upper bound and the greatest lower bound for the set {x, y}. Suppose lub{x, y} and glb{x, y} exist, then we have The operations ∨, ∧, ⊕, and are called OR, AND, XOR, and XNOR operations, respectively. Proposition 1 ([37]). Let ⊕ be an XOR-operation, be an XNOR-operation, and C be a cone. Then, for each x, y ∈ H, the following relations hold: (iii) M is said to be a comparison mapping if, for any v x ∈ M(x), x ∝ v x , and if x ∝ y, then for any v y ∈ M(y), v x ∝ v y , f or all x, y ∈ H; (iv) M is said to be weak comparison mapping if, for any x, y ∈ H, x ∝ y, then there exists v x ∈ M(x) and v y ∈ M(y) such that x ∝ v x , y ∝ v y , and v x ∝ v y ; (v) a comparison mapping M is said to be γ-ordered rectangular if there exists a constant γ > 0, and, for any is called the resolvent operator, where ρ > 0 is a constant and I is an identity operator.

Definition 4.
The Cayley operator C M I,ρ associated with M is defined as It is easy to check that when x ∝ y, the resolvent operator defined by (1) and the Cayley operator defined by (2) are single valued as well as comparison mappings, see [22].

Lemma 1 ([22]
). Let M : H → 2 H be a γ-ordered rectangular multivalued mapping with respect to R M I,ρ . Then, the resolvent operator R M I,ρ is Lipschitz-type continuous, that is,  (2) is (2θ + 1)-Lipschitz-type-continuous. That is, The resolvent operator of a multivalued mapping plays a big role in several nonlinear programmings, see [38][39][40][41] for recent related results. If x ∝ y, then R M I,ρ (x) ∝ R M I,ρ (y) and C M I,ρ (x) ∝ C M I,ρ (y). By using (viii) of Proposition 1, we can write

Definition 5. A mapping g : H → H is called
(i) strongly monotone, if there exists a constant α > 0 such that (iii) relaxed Lipschitz continuous, if there exists a constant k > 0 such that Let Ω be a nonempty open subset of H in which the parameter λ takes values. We define the following operators. (3)

Formulation of the Problem
Let H be a real ordered Hilbert space. Let M : H → 2 H be the maximal monotone multivalued mapping and C M I,ρ be the Cayley operator defined by (2). Let T, g : H → H be the single valued mappings. We mention the following mixed Cayley inclusion problem (5) as well as its corresponding resolvent equation problem (6) with XOR-operation. Find where J M I,ρ = [I − R M I,ρ ], I is the identity operator, and ρ > 0 is a constant. A similar version of (5) is considered by Ali et al. [22]. It is easy to show that (5) and (6) are equivalent. The following equation is a fixed point formulation of (5), which can be shown easily by using the definition of resolvent operator given by (1).
Now, we consider the parametric version of (5)  Find (x, λ) ∈ H × Ω such that Problem (8) is called the parametric mixed Cayley inclusion problem with XOR-operation. We assume that problem (8) has a unique solutionx for someλ ∈ Ω.
The following Lemma is a fixed point formulation of (8). (8) has a solution (x, λ) ∈ H × Ω, if and only if it satisfies the following relation:

Lemma 3. The parametric mixed Cayley inclusion Problem
Proof. The proof is a direct consequence of the definition of parametric resolvent operator R M(·,λ) I,ρ and hence, is omitted.
In connection with the parametric mixed Cayley inclusion problem with XOR-operation (8), we mention its corresponding parametric resolvent equation problem with XOR-operation. Find where J M(.,λ) , I is the identity operator, and ρ > 0 is a constant. The following Lemma ensures the equivalence between problems (8) and (10).
That is, Applying Lemma 3, we conclude that the parametric mixed Cayley inclusion problem with XOR-operation (8) has a solution.
For illustration of some of the concepts used in this paper, we construct the following conjoin example.
, f or all x ∈ H and f or f ixed λ ∈ Ω, We claim that the following hold: (1) The mapping T is relaxed Lipschitz continuous, Lipschitz continuous with respect to first argument, and continuous with respect to second argument.
that is, T(x, λ) is 1 4 -relaxed Lipschitz continuous with respect to the first argument.
that is, T(x, λ) is 3 4 -Lipschitz continuous with respect to the first argument.
that is, T(x, λ) is continuous with respect to the second argument. (2) The mapping g is Lipschitz continuous.
In order to show that the mapping M is ordered rectangular mapping, let v Thus, M is 4 5 -ordered rectangular mapping. (4) Now, we will show that the solution z(λ) of the parametric resolvent equation with XOR-operation (10) is continuous (or Lipschitz continuous). We take ρ = 2 and evaluate implicit resolvent operator as well as the implicit Cayley operator as R M(·,λ) I,ρ One can easily show that the implicit resolvent operator R M(·,λ) I,ρ (29) and the implicit Cayley operator C M(·,λ) I,ρ (30) are Lipschitz continuous. Using Equation (14), we evaluate which is a solution of the parametric resolvent equation problem with XOR-operation (10). We verify our claim below: Finally, we evaluate which shows that the solution z(λ) of the parametric resolvent equation problem with XOR-operation (10) is continuous (or Lipschitz continuous) at λ =λ.

Conclusions
Sensitivity analysis is instrumental in solving many problems occurring in day-to-day life, such as global climate models, quantifying uncertainty in corporate finance, econometric models in social sciences, optimal experimental design, engineering design, multicriteria decision making, physics, chemistry, human control models, etc. Keeping in view the above mentioned applications of sensitivity analysis, in this paper, we have introduced and studied a parametric mixed Cayley inclusion problem with XOR-operation and a parametric resolvent equation problem with XOR-operation. We have shown that both the problems are equivalent and this equivalent formulation is used to develop the general framework of sensitivity analysis for parametric mixed Cayley inclusion problem with XOR-operation without using differentiability of the given data. We expect that this study will motivate and inspire other researchers of related fields to explore its applications in modern sciences.
Author Contributions: All the authors have contributed equally to this paper. All authors have read and agreed to the published version of the manuscript.