ON WELL-POSEDNESS ASSOCIATED WITH A CLASS OF CONTROLLED VARIATIONAL INEQUALITIES

In this paper, by using the new concepts of monotonicity, pseudomonotonicity and hemicontinuity associated with the considered curvilinear integral functional, we investigate the wellposedness and well-posedness in generalized sense for a class of controlled variational inequality problems. More precisely, by introducing the approximating solution set of the considered class of controlled variational inequality problems, we formulate and prove some characterization results on well-posedness and well-posedness in generalized sense. Also, the theoretical developments presented in the paper are accompanied by illustrative examples. Mathematics Subject Classification. 49K40, 65K10. Received March 15, 2021. Accepted August 31, 2021.

of the sequence of approximate solutions obtained through iterative techniques. Tykhonov [30] first investigated the general concept of well-posedness for unconstrained minimization problems that requires the existence and uniqueness of minimizer and convergence of every minimizing sequence towards the unique minimizer. It is clear that the concept of well-posedness is motivated by the numerical methods producing optimizing sequences for optimization problems. Following the concept of Tykhonov well-posedness, various kinds of well-posedness for optimization problems, such as Levitin-Polyak well-posedness [16] and extended well-posedness, have been introduced and studied by many researchers (see [2,4,14,17,18]).
In the literature, several researchers studied the variational inequality over set valued mappings (see, for example, [31]). Under mild conditions, the variational inequality problems are closely related to the optimization problems, so the concept of Tykhonov well-posedness has also been extended to variational inequalities [5,9,11,34], and thereafter, to some other problems like fixed point problems [7], hemivariational inequality problems [1,21,33,34], equilibrium problems [6], complementary problems [10], and Nash equilibrium problems [19]. Ceng and Yao [3] discussed well-posedness of mixed variational inequality in generalized sense and proved that the well-posed generalized mixed variational inequality is equivalent to that of fixed point problems and inclusion problems. Thereafter, Huang et al. [12] investigated Levitin-Polyak generalized well-posedness for variational inequality problem and established some characterizations for the well-posednesses of constrained variational inequalities. Also, in [20], Lin and Chuang proposed generalized well-posedness for variational disclusion problems, inclusion problems and the minimization problems involving variational disclusion problems, inclusion problems as constraints. Later on, Lalita and Bhatia [15] proposed the well-posedness and generalized well-posedness for parametric type quasi-variational inequality problems and minimization problems, and discussed the cases in which the problem has one or more than one solution. Moreover, Fang et al. [8] extended this notion of well-posedness by applying perturbations on the mixed variational inequality problem over Banach space and proved various results for well-posedness via perturbations. Virmani and Srivastava [32] introduced the necessary and sufficient conditions for α, α − L well-posedness and generalized α, α − L well-posedness for a mixed vector quasi-variational-like inequality using bifunctions. Recently, Jayswal and Shalini [13] derived the well-posedness for generalized mixed vector variational-like inequality and optimization problems including this inequality as a constraint.
Moreover, the multi-time (multidimensional) control theory comes from calculus of variations which is applied to solve various operations research problems involving applied science and technology. In the last few years, it has been intensively studied in applied and theoretical points of view (see [22][23][24][25][26]). Multi-time variational inequality problem is another interesting generalization of variational inequality (see [27,28]). In [29], Treanţȃ has established the weak sharp solutions for a variational-type inequality by defining (ρ, b, d)convex path-independent curvilinear integral functional. Also, he studied an equivalence between the minimum principle sufficiency property and the weak sharpness property of the solution set associated with the considered variational-type inequality.
Inspired and motivated by the above research works, in the current paper, we investigate the well-posedness and well-posedness in generalized sense for a class of controlled variational inequality problems. More precisely, by using the new concepts of monotonicity, pseudomonotonicity and hemicontinuity associated with the considered curvilinear integral functional, and the approximating solution set of the considered class of controlled variational inequality problems, we formulate and prove some characterization results on well-posedness and well-posedness in generalized sense. The presence of the curvilinear integral functionals and of the mathematical framework governed by infinite-dimensional function spaces, represent the main novelty elements of this paper (the former works are studied in the classical finite-dimensional spaces). Furthermore, besides totally new elements mentioned above, due to the physical meaning of the integral functionals used (as is well-known the path-independent curvilinear integrals represent the mechanical work performed by a variable force in order to move its point of application along a given piecewise smooth curve), this paper becomes a fundamental work for researchers in the field of applied mathematics and ingineering.
The paper is organized as follows. The new notions of monotonicity, pseudomonotonicity and hemicontinuity associated with a curvilinear integral functional, and an auxiliary lemma are provided in Section 2. In Section 3, we investigate about well-posedness and well-posedness in generalized sense, by considering the approximating solution set of the considered class of controlled variational inequality problems. Further, under suitable hypotheses, we establish that well-posedness is characterized in the terms of existence and uniqueness of solution. Also, we formulate and prove sufficient conditions for the well-posedness in generalized sense by assuming the boundedness of approximate solutions. In order to illustrate the theoretical results derived in this paper, we also provide some examples. Section 4 concludes the paper.

Notations and preliminaries
In this paper, we consider the following notations and mathematical tools. Denote by Ω a compact domain in R m and consider the point Ω s = (s ν ), ν = 1, m, as a multi-parameter of evolution. Let Ω ⊃ Υ : s = (s(ς)) , ς ∈ [a, b] (or s ∈ s 1 , s 2 ) be a piecewise smooth curve joining two different points s 1 = (s 1 1 , . . . , s m 1 ), s 2 = (s 1 2 , . . . , s m 2 ) in Ω. Consider X as the space of all piecewise smooth state functions a : Ω → R n , equipped with the norm where we used the notation a ν := ∂a ∂s ν , ν = 1, m, and let U be the space of piecewise continuous control functions u : Ω → R k , endowed with the uniform norm · ∞ .
Throughout this paper, we consider that X × U is a nonempty, closed and convex subset of X × U, equipped with the inner product Let J 1 (R m , R n ) be the jet bundle of first-order associated with R m and R n . By using the Lagrange 1-form density f ν : we define the following functional governed by a curvilinear integral (see summation over the repeated indices, Einstein summation): where a κ (s) = ∂a ∂s κ (s), κ ∈ {1, . . . , m}, denotes the partial velocities. By using the above mathematical tools, we introduce the following controlled variational inequality problem (for short, CVIP), formulated as: find (a, u) ∈ X × U such that where D κ := ∂ ∂s κ is the total derivative operator and (ψ a,u (s)) := (s, a(s), a κ (s), u(s)). Let Θ be the set of all feasible solutions of (CVIP), that is, Further, by considering the notion of monotonicity for variational inequality problems, we introduce the concepts of monotonicity and pseudomonotonicity associated with the aforementioned curvilinear integral functional.
Definition 2.1. The curvilinear integral functional F is called monotone on X × U if the following inequality holds: , Ω = [0, 1] 2 and Ω ⊃ Υ be a piecewise smooth curve joining the points (0, 0), (1,1) in Ω. Consider In the following, we present an example of curvilinear integral functional which is pseudomonotone but not monotone.
Now, we show that the curvilinear integral functional Υ f ν (ψ a,u (s))ds ν is pseudomonotone on X × U = But it is not monotone on X × U , because Assumption. In this paper, we assume the following working hypothesis: is an exact total differential satisfying G(s 1 ) = G(s 2 ). Inspired by Usman and Khan [31], we introduce the following definition of hemicontinuity for the aforementioned curvilinear integral functional.
Definition 2.5. The curvilinear integral functional F is said to be hemicontinuous on X × U if, for ∀(a(s), u(s)), (b(s), w(s)) ∈ X × U , the application λ → ((a(s), u(s)) − (b(s), w(s)) , The following lemma is an auxiliary result for proving the main results derived in the present paper.
Lemma 2.6. Consider the curvilinear integral functional F is pseudomonotone and hemicontinuous on X × U . A pair (a, u) ∈ X × U is solution of (CVIP) if and only if it is solution for Proof. Firstly, consider the pair (a, u) ∈ X × U solves (CVIP), that is, By using the definition of pseudomonotonicity, the above inequality implies Conversely, assume that Thus, the above inequality can be rewritten as Taking λ → 0 and using the hemicontinuity property associated with the considered curvilinear integral functional, we have which shows that (a, u) solves (CVIP).

Well-posedness and well-posedness in generalized sense associated with (CVIP)
In this section, by using the notions of monotonicity and hemicontinuity introduced in Section 2, we study the well-posedness and well-posedness in generalized sense of the considered class of controlled variational inequality problems. For this purpose, we introduce the following definitions.
We can see that {(a n , u n )} is bounded in X × U . So, passing to a subsequence if necessary, we may assume that (a n , u n ) → (a, u) weakly in X × U = {(a 0 , u 0 )}.
Further, we provide an illustrative application of Theorem 3.7.

Conclusions
In this paper, based on the new concepts of monotonicity, pseudomonotonicity and hemicontinuity associated with curvilinear integral functionals, we have studied well-posedness and well-posedness in generalized sense for a class of controlled variational inequality problems. More concretely, we have established that, under suitable conditions, the well-posedness is characterized in terms of existence and uniqueness of solution. Moreover, sufficient conditions were presented for the well-posedness in generalized sense by assuming the boundedness of approximating solution set. Also, several illustrative examples have been formulated throughout the paper.