Variational-Like Inequality Problem Involving Generalized Cayley Operator

This article deals with the study of a variational-like inequality problem which involves the generalized Cayley operator. We compare our problem with a fixed point equation, and based on it we construct an iterative algorithm to obtain the solution of our problem. Convergence analysis as well as stability analysis are studied.


Introduction
The mathematical formalism of a classical variational inequality problem is to find y ∈ H such that Ty, x − y ≥ 0, ∀ x ∈ H, (1) where H is a Hilbert space and T : H → H is a nonlinear operator. The concept of variational inequalities was introduced by Stampacchia [1] and Fichera [2], separately. The variational inequality theory has received adequate recognition due to its implementation in a diverse range of problems arising in economics, physics, mathematical finance, structural analysis and in many branches of social, pure and applied sciences, see, for example, in [3][4][5][6][7][8][9][10][11][12][13][14]. Stampacchia [1] proved that the possible problems related with elliptic equations can be analysed through variational inequalities. Combining auxiliary principle technique and projection operator technique, Lions and Stampacchia [15] studied the existence of solution of variational inequalities. The variational-like inequalities are the generalized forms of variational inequalities and provide us cogent tools to study many problems of basic and applied sciences. It is obvious that variational inequalities and variational-like inequalities are analogous of fixed point equations. This flipside equivalent formulation plays a significant role in many aspects of variational inequalities and variational-like inequalities. More precisely, this equivalent formulation is used to develop iterative algorithms and to study numerical methods related to variational inequalities and variational-like inequalities, etc.
It is well known that Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. As far as Hilbert spaces are concerned, Cayley transform is a mapping between linear operators. This transform is a homography having applications in real analysis, complex analysis and quaternionic analysis, etc.
As this subject is application oriented, in this paper, we consider a variational-like inequality problem involving generalized Cayley operator. An iterative algorithm is defined to obtain the solution of variational-like inequality problem involving generalized Cayley operator. For more details and recent developments of the subject, we refer to the works in [16][17][18][19][20][21][22][23][24][25][26][27][28][29] and the references therein. An existence and convergence result is proved. Stability analysis is also discussed.

Preliminaries
Let H be a real Hilbert space with usual norm . and inner product ·, · . Let CB(H) be the family of all nonempty closed and bounded subsets of H and D(·, ·) be the Hausdorff metric on CB(H) defined by The following well known concepts are needed to achieve the goal of this paper.
(iv) g is said to be strongly monotone, if there exists a constant µ g > 0 such that where f * is called a η-subgradient of φ at x. The set of all η-subgradients of φ at x is denoted by ∂ η φ. The mapping ∂ η φ : H → 2 H defined by (iii) there exists a nonempty compact convex subset X 0 of X and a nonempty compact subset K of X such that for each y ∈ X \ K, there is an x ∈ Co(X 0 ∪ {y}) satisfying f (x, y) > 0.
Then there exists y ∈ X such that f (x, y ) ≤ 0, for all x ∈ X. Theorem 1. [30] Let η : H × H → H be continuous and δ-strongly monotone such that η(x, y) = −η(y, x) for all x, y ∈ H and for any given x ∈ H, the functional h(y, u) = x − u, η(y, u) is 0-DQCV in y. Let φ : H → R ∪ {+∞} be a lower semicontinuous, η-subdifferentiable, proper functional. Then, for any given ρ > 0 and x ∈ H, there exist a unique u ∈ H such that Theorem 2. [30] Let η : H × H → H be δ-strongly monotone and τ-Lipschitz continuous such that η(x, y) = −η(y, x) for all x, y ∈ H and for any given x ∈ H, the functional h(y, u) = x − u, η(y, u) is 0-DQCV in y. Let φ : H → R be a lower semicontinuous, η-subdifferentiable, proper functional, and ρ > 0 be a arbitrary constant. Then, the η-proximal mapping J The lim n→∞ ϑ n = 0, implies that u n → x * , and consequently the iterative sequence {x n } is said to be T-stable or stable with respect to T.

Definition 8. [35]
If M is a maximal monotone mapping, then for a fixed ρ > 0, the resolvent operator associated with M is defined as It is well known that the resolvent operator R M I,ρ is single-valued.
As the η-subdifferential operator ∂ η φ of φ is maximal monotone, we define the resolvent operator for a fixed ρ > 0 as In case, if φ : H × H → R ∪ {+∞}, then the resolvent operator is defined for a fixed ρ > 0 as (2) where R is defined by (2).

Lemma 2. The generalized Cayley operator C
Proof. As the resolvent operator R be sequences of nonnegative numbers and 0 ≤ q < 1, so that a n+1 ≤ qa n + b n , f or all n ≥ 0.

Formulation of Problem and Fixed Point Formulation
: H → H is the generalized Cayley operator. We study the following variational-like inequality problem involving generalized Cayley operator.
We list below some special cases of variational-like inequality problem involving generalized Cayley operator (4). Special cases : (4) reduces to a general quasi-variational-like inclusion problem: Problem (5) was introduced and studied by Ding and Luo [37].
, also if K : H → 2 H be a given multi-valued mapping such that each K(x) is a closed convex subset of H(or K(x) = m(x) + K where m : H → H and K is a closed convex subset of H) and if η(x, y) = x − y for all x, y ∈ H, φ : H × H → H is defined by φ(x, y) = I K(y) (x), ∀x, y ∈ H, where I K(y) (x) is the indicator function of K(y), that is, then problem (4) reduces to the following strongly nonlinear quasi-variational inequality problem: Find x ∈ H such that g(x) ∈ K(x) and Problem (7) includes various classes of variational inequalities, quasi-variational inequalities, complementarity and quasi-complementarity problems, studied previously by many authors, see [40,41]. It is shown below that problem (4) is equivalent to a fixed point equation.
Proof. Let x ∈ H, t ∈ F(x) satisfy the Equation (8), that is Using the definition of resolvent operator, the above inequality holds if and only if Applying η-subdifferentiability of ∂ η (., x), the above relation holds if and only if Thus, we have The result follows.

Iterative Algorithm and Convergence Result
Using Lemma 4, we construct an iterative algorithm to obtain the solution of problem (4).
and C Furthermore, if the following condition holds: where . Then, the sequences {x n } and {t n } generated by Algorithm 1 converge strongly to the solution x and t of variational-like inequality problem involving generalized Cayley operator (4), respectively.
Proof. Using (9) of Algorithm 1, we have As g is Lipschitz continuous with constant λ g and strongly monotone with constant δ g , by using technique of [35], we have where λ 2 g > (2δ g − 1). As the resolvent operator R ∂ η φ(.,x) I,ρ is Lipschitz continuous with constant τ δ and using condition (11), we have R We evaluate, Using Lipschitz continuity of G in both the arguments, we get Applying (10) of Algorithm 1 and D-Lipshitz continuity of F, we have Combinings (18) and (19), we have where is Lipschitz continuous with constant 2τ+δ δ and using condition (12), we have (C where θ 2 = 2τ+δ δ + µ ).

Stability Analysis
This component deals with the stability analysis of Iterative Algorithm 1. Proof. Let us consider a sequence {u n } in H such that Also let (24) and Using (24) and (25), we obtain It follows from condition (13) that 0 ≤ ζ(θ) ≤ 1 and applying Lemma 3 it follows that if lim n→∞ χ n = 0, thus lim n→∞ u n → x * . Therefore, the iterative sequences {x n } and {t n } generated by iterative Algorithm 1 are R ∂ η φ(.,x) I,ρ -Stable.

Numerical Example
We provide a numerical example in support of most of the concepts used in Theorem 3.
(v) clearly η is strongly monotone and Lipschitz continuous with constants δ = 1 and τ = 1, respectively, and also η(x, y) = −η(y, x). (vi) As h(y, u) = (x − u)(y − u), using the techniques of Ding [30] it is easy to show that h is 0-DQCV in y. (vii) Under the above observations, we show the convergence of the sequences {x n } and {t n } generated by the Algorithm 1.

Conclusions
We have introduced and study a quite new problem called variational-like inequality problem involving Cayley operator in a real Hilbert space. Applying η-subdifferentiablity, we have shown that variational-like inequality problem involving Cayley operator is equivalent to a fixed point equation. This fixed point formulation is used to construct an iterative algorithm to obtain the solution of variational-like inequality problem involving a Cayley operator. Convergence and stability analysis are discussed, separately.
We remark that our results may be further extended in higher-order dimensional spaces.
Author Contributions: All the authors have contributed equally to this paper. All authors have read and agreed to the published version of the manuscript.