Yosida Complementarity Problem with Yosida Variational Inequality Problem and Yosida Proximal Operator Equation Involving XOR-Operation

Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India Department of Mathematics, School of Advanced Sciences, Kalasalingam Academy of Research and Education, Anand Nagar, Krishnankoil 626126, India Center for Fundamental Science, Research Center for Nonlinear Analysis and Optimization, Kaohsiung Medical University, Kaohsiung 80708, Taiwan Department of Medical Research, Kaohsiung Medical University Hospital, Kaohsiung 80708, Taiwan


Introduction
Stampacchia [1] and Ficchera [2] originated the study of variational inequalities, separately. Variational inequalities are mathematical models for many problems occurring in physics, engineering sciences, transportation planning, financial problems, and in many industrial strategies, etc. (see, for example, [3][4][5][6][7][8][9][10][11]). In 1968, Cottle and Dantzig [12] proposed linear complementarity problem which appear continually in computational mechanics. It is interesting to note that finding the solution of linear complementarity problem is associated with minimizing some quadratic function. However, in 1964, Cottle [13] in his Ph. D thesis introduced nonlinear complementarity problem which is closely related to Hartman and Stampacchia variational inequality problem. e proximal operator technique is useful to establish equivalence between variational inequalities and proximal operator equations. e proximal operator equation approach is used to solve variational inequalities and related optimization problems. XOR is a logical operation and represents the inequality function, that is, the output is true if the inputs are not alike; otherwise, the output is false. An easy way to remember XOR is "must have one or the other but not both." It is important to note that XOR does not leak information about the original plain text. e inner XOR is the encryption and the outer XOR is the decryption, that is, the exact XOR function can be used for both encryption and decryption. Consider a string of binary digits 10101 and XOR the string 10111 with it to get 00010. at is, the original string is encoded and the second string becomes key; if we XOR our key with our encoded string, we get our original string back. XOR allows to easily encrypt and decrypt a string; the other logical operations do not. e possible strategy of solving stochastic notion of multivalued differential equation in finite dimensional space is based on Yosida approximation approach. e existence of multivalued stochastic differential equation in finite dimensional space with a time-independent, deterministic maximal monotone operator through Yosida approximation approach was first discussed by Petterson [14]. Yosida approximation operators are used to solve wave equations, heat equations, etc. For more details and recent past developments about complementarity problems, variational inequalities, proximal operator equations, Yosida approximation operator, and related topics, we refer to [15][16][17][18][19][20][21][22][23][24][25][26][27][28] and references therein.
Motivated by all the above discussed concepts, in this paper, we consider and study a Yosida complementarity problem, a Yosida variational inequality problem, and a Yosida proximal operator equation involving XOR-operation. Some equivalence results are proved. To obtain the solution of Yosida proximal operator equation involving XOR-operation, we define an algorithm based on fixed point formulation. Convergence criteria are also discussed. In support of our main result, an example is provided using MATLAB program R2018a. A comparison of different iterations is assembled in the form of a computational table, and the convergence of the iterative sequences is shown by some graphs for different initial values.

Preliminaries and Basic Results
We suppose that H is a real ordered positive Hilbert Space with its norm ‖ · ‖ and inner product 〈·, ·〉, C ⊆ H is a closed convex pointed cone, d is the metric induced by the norm ‖ · ‖, CB(H) is the family of nonempty, closed, and bounded subsets of H, and D(·, ·) is the Hausdorff metric on CB(H). e following definitions, concepts, and results are required for the presentation of this paper.

Definition 1.
A convex cone is a subset of a vector space over an ordered field that is closed under linear combinations with positive coefficients.

Definition 2.
Two elements x and y of a set X are said to be comparable with respect to a binary operation ≤ , if at least one of x ≤ y or y ≤ x is true. Comparable elements x and y are denoted by x ∝ y.
Definition 3. A partial order is any binary relation which is reflexive, antisymmetric, and transitive.

Definition 4.
Suppose lub x, y and glb x, y for the set x, y exist; then, XOR and XNOR operations denoted by ⊕ and ⊙ are defined as follows: x ∧ y � glb x, y , lub means the least upper bound, and glb means the greatest lower bound Proposition 1 (see [29]). Let ⊕ be an XOR-operation and ⊙ be an XNOR operation. en, the following axioms are true: (ii) N is said to be Lipschitz continuous in the second argument if there exists a constant λ N 2 > 0 such that Similarly, we can define the Lipschitz continuity of N in the third argument.
Definition 7 (see [30]). Let ψ: e set of all subgradients of ψ at x is denoted by zψ(x). e mapping zψ : H ⟶ 2 H defined by is called subdifferential of ψ. Definition 8. Let P: C ⟶ C be a mapping and ψ: where ρ > 0 is a constant.

Definition 9.
e Yosida approximation operator of ψ is defined by where ρ > 0 is a constant.
Furthermore, we prove some propositions related to proximal operator and Yosida approximation operator.
Proof. Using the definition of J zψ ρ , linearity of P and ψ, and eorem 1.48 and eorem 1.49 of [31], we have Proof. Using the definition of Y zψ ρ and Proposition 2, we have Journal of Mathematics us, As ψ is strongly convex with modulus λ > 0, then the proximal operator J zψ ρ is strongly monotone with constant σ > 0, where σ � 2λ (see [31]). erefore, Since P is strongly monotone with respect to J zψ ρ with constant μ > 0, we have which implies that at is, J zψ ρ is Lipschitz continuous.

Proposition 5. e Yosida approximation operator is strongly monotone if all the conditions of Proposition 4 hold.
Proof. Using the Lipschitz continuity of proximal operator J zψ ρ , we have □

Description of the Problems and Equivalence Lemmas
Let H be a real ordered positive Hilbert space and C ⊆ H be a closed convex pointed cone. Let A, B, C: C ⟶ CB(H) be the multivalued mappings and N: H × H × H ⟶ H be a single-valued mapping. Suppose ψ: C ⟶ R ∪ +∞ { } is a proper, convex functional and Y zψ ρ : C ⟶ C is the Yosida approximation operator. We consider the following Yosida complementarity problem involving XOR-operation.
In connection with Yosida complementarity problem involving XOR-operation (21), we mention the following Yosida variational inequality problem involving XORoperation. Find In acquaintance with Yosida variational inequality problem involving XOR-operation (22), we mention the following Yosida proximal operator equation involving XOR-operation.
is the proximal operator, P: C ⟶ C is a mapping, and z � P(Y zψ ρ (x)) + ρN (u, v, w). e equivalence between problem (21) and problem (22) and the equivalence between problem (22) and problem (23) are given as follows.
Proof. Let the Yosida complementarity problem involving XOR-operation (21) holds. We have Since By (ii) and (iii) of Proposition 1, we have Using the properties of inner product, we can write Applying (25) and (27), (28) becomes which is the Yosida variational inequality problem involving XOR-operation (22). On the other hand, let the Yosida variational inequality problem (22) holds.
at is,

Journal of Mathematics
As C is a closed convex pointed cone, y � 2x ∈ C as well as y � (1/2)x ∈ C. Putting y � 2x and y � (1/2)x and using linearity of ψ and Proposition 3, we have us, we have Adding (31) and (32), we have Since we have Using (25), from the above inequality, we have it follows that Using (ii) of Proposition 1, we have Combination of (33) and (39) is the required Yosida complementarity problem involving XOR-operation (21). □ e following Lemma guarantees the equivalence between the Yosida variational inequality problem involving XOR-operation (22) and a fixed point equation.

Lemma 2.
Let P: C ⟶ C be a mapping, then the Yosida variational inequality problem involving XOR-operation (22) has a solution x ∈ C, u ∈ A(x), v ∈ B(x), w ∈ C(x), if and only if it satisfies the equation: where ρ > 0 is a constant.
Using the definition of the proximal operator J zψ ρ and from the above equation, we have 6 Journal of Mathematics Applying the definition of subdifferential operator, the above inclusion holds if and only if Using (ii) of Proposition 1, we have It follows that which is the required Yosida variational inequality problem involving XOR-operation (22).  Proof. Let x ∈ C, u ∈ A(x), v ∈ B(x), w ∈ C(x) be the solution of the Yosida variational inequality problem involving XOR-operation (22). en by Lemma 2, it satisfies the equation: Using (ii) of Proposition 1, we have which is the required Yosida proximal operator equation involving XOR-operation (23).
, w ∈ C(x) be the solution of Yosida proximal operator equation involving XOR-operation (23).
at is, we have Using (v) of Proposition 1, definition of R zψ ρ and comparability of N(u, v, w) with R zψ ρ (z), we obtain (51)

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From above, we have

Algorithm and Existence Results
Invoking Lemmas 2 and 3, we suggest the following algorithm for solving Yosida proximal operator equation involving XOR-operation (23).
Take any , by Nadler's theorem [38], there exist u 1 ∈ A(x 1 ), v 0 ∈ B(x 1 ), w 0 ∈ C(x 1 ), using (viii) of Proposition 1 and comparability of u 0 , u 1 ; v 0 , v 1 and w 0 , w 1 , we have where D(·, ·) is the Hausdorff metric on CB(H). v 1 , w 1 ) and take any Continuing the above procedure, we compute the sequences x n , u n , v n and z n by the schemes given below: w n ∈ C x n , w n+1 ∈ C x n+1 such that w n ∝ w n+1 , where ρ > 0 is a constant and n � 0, 1, 2, 3, · · ·.  do m(zψ). Suppose that z n+1 ∝ z n , for n � 0, 1, 2, · · · and if the following condition is satisfied: then there exists x, z ∈ C, u ∈ A(x), v ∈ B(x), w ∈ C(x) satisfying the Yosida proximal operator equation involving XOR-operation (23) and the sequences x n , z n , u n , v n 8 Journal of Mathematics and w n generated by Algorithm 1 converge strongly to x, z, u, v, and w, respectively.
Proof. Using (x) of Algorithm 1 and (ii) of Proposition 1, we have It follows from (60) that Since z n+1 ∝ z n and using (vii) and (viii) of Proposition 1, from (61), we obtain Since N is Lipschitz continuous in all the three arguments with constants λ N 1 , λ N 2 , and λ N 3 , respectively, and A, B, C are D-Lipschitz continuous mappings with constants λ D A , λ D B , λ D C , respectively, and using (vii), (viii), (ix) of Algorithm 1, we have where , it follows that S n (θ) ⟶ S(θ) as n ⟶ ∞. From (59), we have ξ(y) < 1 and S(θ) < 1. Consequently, we conclude from (65) and (67) that x n and z n both are Cauchy sequences. Since H is complete and C ⊆ H is a closed convex subset of H and thus C is also complete, we may assume that x n ⟶ x ∈ C and z n ⟶ z ∈ C. From (vii), (viii), and (ix) of Algorithm 1, it follows that u n , v n , and w n are also Cauchy sequences such that u n ⟶ u, v n ⟶ v and w n ⟶ w, as n ⟶ ∞.
It can be shown easily by using the techniques of [28] that u ∈ A(x), v ∈ B(x), and w ∈ C(x). By Lemma 3, we conclude that x, z ∈ C, u ∈ A(x), v ∈ B(x), and w ∈ C(x) is the solution of Yosida proximal operator equation involving XOR-operation (23).

□
We provide the following numerical example using MATLAB program R2018a along with a computational table and a convergence graphs for different initial values in support of Algorithm 1 and eorem 1.
For ρ � 1, the proximal operator J zψ ρ is given by It is simple to see that P is Lipschitz continuous with constant λ p � (11/10), strongly monotone with respect to J zψ ρ with constant μ � 1/3, and J zψ ρ is Lipschitz continuous with constant θ � � 2 √ /3. In view of proximal operator calculated above, the Yosida approximation operator is given by Also, Hence, Y zψ ρ is strongly monotone with constant δ y � (2/5).
where x ∈ C, u ∈ A(x), v ∈ B(x), and w ∈ C(x). N is Lipschitz continuous in all the three arguments with constants λ N 1 � λ N 2 � λ N 3 � 1. Hence, Below we show that condition (59) is satisfied.
Furthermore, we obtain the sequences x n and z n generated by iterative Algorithm 1 as also, Y zψ ρ x n � J zψ ρ z n 3x n 5 � 2z n 5 x n � 2z n 5 .
Clearly, the sequence z n converges to 0, and consequently, the sequence x n also converges to 0.
It is shown in Figures 1-3 that, for initial values z 0 � − 5, 2.5, and 5, the sequence z n converges to 0. A consolidated graph using Figures 1-3 is provided in Figure 4. In Table 1, comparing different initial values of z n and for different iterations, it is obtained that the sequence z n converges to 0.

Conclusion
In this work, we introduce and study three new problems, that is, a Yosida complementarity problem, a Yosida variational inequality problem, and a Yosida proximal operator equation involving XOR-operation. It is shown that Yosida complementarity problem involving XORoperation is equivalent to a Yosida variational inequality problem involving XOR-operation and Yosida variational inequality problem involving XOR-operation is equivalent to a Yosida proximal operator equation involving XORoperation. An algorithm is established to obtain the solution of Yosida proximal operator equation involving XOR-operation. Finally, an existence and convergence result is proved. A numerical example is given in support of our main result.
It is still an open and interesting problem that how to establish equivalence between Yosida complementarity problem involving XOR-operation and Yosida proximal operator equation problem involving XOR-operation.

Data Availability
No data were used to support this study.   [39]. In this variant form, neither the concept of Yosida approximation operator nor the concept of XOR-operation was used. Moreover, no complementarity problem was considered in the abovementioned thesis.

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