On Well-Posedness of Some Constrained Variational Problems

: By considering the new forms of the notions of lower semicontinuity, pseudomonotonicity, hemicontinuity and monotonicity of the considered scalar multiple integral functional, in this paper we study the well-posedness of a new class of variational problems with variational inequality constraints. More speciﬁcally, by deﬁning the set of approximating solutions for the class of variational problems under study, we establish several results on well-posedness.

In this paper, motivated and inspired by the above research papers, we study the well-posedness property for new constrained variational problems, implying second-order multiple integral functionals and partial derivatives. In this regard, we formulate new forms of monotonicity, lower semicontinuity, hemicontinuity, and pseudomonotonicity for the considered multiple integral-type functional. Further, we introduce the set of approximating solutions for the constrained optimization problem under study and establish several theorems on well-posedness. The previous research works in this scientific area did not take into account the new form of the notions mentioned above. In essence, the results derived here can be considered as dynamic generalizations of the corresponding static results already existing in the literature. In this paper, the framework is based on function spaces of infinite-dimension and multiple integral-type functionals. This element is completely new for the well-posed optimization problems.
The present paper is structured as follows: In Section 2, we formulate the problem under study and introduce the new forms of monotonicity, lower semicontinuity, hemicontinuity, and pseudomonotonicity for the considered multiple integral-type functional. Additionally, an auxiliary lemma is provided. In Section 3, we study the well-posedness for the considered constrained variational problem. More precisely, we prove that wellposedness is equivalent with the existence and uniqueness of a solution in the aforesaid problem. Finally, Section 4 concludes the paper and provides further developments.

Preliminaries and Problem Formulation
In this paper, we consider the following notations and mathematical tools: denote by K a compact domain in R m and consider the point K ζ = (ζ α ), α = 1, m; let E denote the space of state functions of C 4 -class s : K → R n and s α := ∂s ∂ζ α , s βγ := ∂ 2 s ∂ζ β ∂ζ γ denote the partial speed and partial acceleration, respectively; consider E ⊆ E as a nonempty, closed and convex subset, with s| ∂K = given, equipped with the inner product and the induced norm, where dζ = dζ 1 · · · dζ m is the element of volume on R m .
More precisely, the set of all feasible solutions of (VIP) is defined as By direct computation, we obtain which implies that the functional F(s) = K f (π s (ζ))dζ is not monotone on E (in the sense of Definition 1). By considering the work of Usman and Khan [37], we provide the following definition.
Proof. Firstly, let us consider that the function s ∈ E solves (VIP). In consequence, it follows By using the pseudomonotonicity property of F(s) = K f (π s (ζ))dζ, the previous inequality involves Conversely, assume that For z ∈ E and λ ∈ (0, 1], we define Therefore, the above inequality can be rewritten as follows By considering λ → 0 (and the hemicontinuity property of F(s) = K f (π s (ζ))dζ), which shows that s is solution for (VIP). The proof of this lemma is now complete.

Well-Posedness Associated with (CVP)
In this section, we analyze the well-posedness property for the constrained variational problem (CVP). To this aim, we provide the following mathematical tools.

Definition 6.
The problem (CVP) is called well-posed if: (i) It has a unique solution s 0 ; (ii) Each approximating sequence of (CVP) will converge to this unique solution s 0 .
Since diam S(θ n , ϑ n ) → 0 as (θ n , ϑ n ) → (0, 0), {s n } is a Cauchy sequence which converges to somes ∈ E as E is a closed set. By hypothesis, the multiple integral functional K f (π s (ζ))dζ is monotone on E. Therefore, by Definition 1, fors, z ∈ E, we have Taking limit in inequality (3), we have On combining (4) and (5), we obtain Further, taking into account Lemma 1, it follows that which implies thats ∈ Ω.

Conclusions
In this paper, we have studied the well-posedness property of new constrained variational problems governed by second-order partial derivatives. More precisely, by using the concepts of lower semicontinuity, monotonicity, hemicontinuity and pseudomonotonicity of considered multiple integral-type functional, we have proved that the well-posedness property of the problem under study is described in terms of existence and uniqueness of solution.