On weak sharp solutions in (ρ,b,d)$(\rho , \mathbf{b}, \mathbf{d})$-variational inequalities

In this paper, weak sharp solutions are investigated for a variational-type inequality governed by $(\rho , \mathbf{b}, \mathbf{d})$(ρ,b,d)-convex path-independent curvilinear integral functional. Moreover, an equivalence between the minimum principle sufficiency property and the weak sharpness property of the solution set associated with the considered variational-type inequality is established.


Introduction
Based on the works of Burke and Ferris [3], Patriksson [11] and following Marcotte and Zhu [10], the concept of weak sharp solution associated with variational-type inequalities has attracted the attention of many researchers (see, for instance, Hu and Song [7], Liu and Wu [9], Zhu [17] and Jayswal and Singh [8]). Recently, by using gap-type functions, in accordance with Ferris and Mangasarian [5] and following Hiriart-Urruty and Lemaréchal [6], Alshahrani et al. [1] studied the minimum and maximum principle sufficiency properties associated with nonsmooth variational inequalities.
In this paper, motivated and inspired by the ongoing research in this field and by using some variational techniques developed in Ansari [2], Clarke [4] and Treanţă [12][13][14][15][16], we investigate a new class of variational-type inequalities governed by (ρ, b, d)-convex pathindependent curvilinear integral functionals (a new concept introduced in Treanţă [16]). The extended concept of a normal cone (see Treanţă [16]), firstly introduced by Marcotte and Zhu [10], plays a crucial role in our investigations. More precisely, under some working assumptions and using a dual gap-type functional, the weak sharpness property of the solution set for the considered variational-type inequality is studied. In this regard, two characterization results are formulated and proved.
The present paper is organized as follows. Section 2 contains notations, problem description and some auxiliary results. The main results of this paper are included in Sect. 3. Concretely, weak sharp solutions are investigated for an extended variational-type inequality involving (ρ, b, d)-convex path-independent curvilinear integral functional. Finally, Sect. 4 concludes this study.

Preliminaries
In this paper, in order to introduce our study, consider the following notations and mathematical objects: Θ ⊂ R m is a compact domain and the point Θ t = (t β ), β = 1, m, is considered as a multiple parameter of evolution; , a piecewise smooth curve joining the different points t 1 = (t 1 1 , . . . , t m 1 ), t 2 = (t 1 2 , . . . , t m 2 ) in Θ; let L be the space of piecewise smooth functions x : Θ → R n , endowed with the Euclidean inner product x i (t)y i (t) dt m , ∀x, y ∈ L and the induced norm; denote by L a nonempty, closed and convex subset of L, defined as throughout this paper, the summation over the repeated indices is assumed and x, x α are the simplified notations for x(t), x α (t) and x α (t) = ∂x ∂t α (t); consider the real-valued continuously differentiable functions (closed Lagrange 1form densities) (see J 1 (R m , R n ) as the first-order jet bundle associated to R m and R n ) which generate the following path-independent curvilinear integral functionals: Let ρ be a real number, b(x, y) a symmetric positive real-valued functional on L × L and d(x, y) a real-valued functional on L × L.
where D α denotes the total derivative operator.

Definition 2.2 The variational (functional) derivative
and, for any ψ ∈ L with ψ(t 1 ) = ψ(t 2 ) = 0, it satisfies the following relation: Throughout this paper, it is assumed that the inner product between the variational derivative associated with a path-independent curvilinear integral functional and an element ψ ∈ L is accompanied by the condition ψ(t 1 ) = ψ(t 2 ) = 0.
By using the previous mathematical tools, we formulate the following extended variational-type inequality problem: for some given ρ, b, d (introduced as above), find y ∈ L such that for any x ∈ L. The dual extended variational-type inequality problem associated to (EVIP) is formulated as follows: for some given ρ, b, d (introduced as above), find y ∈ L such that for any x ∈ L. Denote by L * and L * the solution set associated with (EVIP) and (DEVIP), respectively, and assume they are nonempty.
Remark 2.1 As can be easily seen, the above extended variational-type inequality problems can be reformulated as follows: for some given ρ, b, d (introduced as above), find y ∈ L such that respectively: for some given ρ, b, d (introduced as above), find y ∈ L such that is an exact total differential and it is satisfied the condition U(t 1 ) = U(t 2 ). Throughout this paper, this working hypothesis is assumed.
Further, in order to investigate the solution set L * , we introduce the following gap functionals.

Definition 2.3
For x ∈ L, the primal gap functional associated to (EVIP) is defined as and, similarly, the dual gap functional associated to (EVIP) is defined as From now onwards, for x ∈ L, consider the following notations: Remark 2.2 By using the previous notations, we can observe the following: denotes the (possibly empty) solution set of In order to formulate and prove the main results of this paper, in accordance with Marcotte and Zhu [10], we introduce the following relevant concepts.

Definition 2.4
The polar set L • associated to L is defined as follows:

Definition 2.5
The projection of a point x ∈ L onto the set L is defined as

Definition 2.6
The normal cone to L at x ∈ L, with respect to ρ, b and d (introduced as above), is defined as and the tangent cone to L at x ∈ L, with respect to ρ, b and d (introduced as above), is T Remark 2.3 Taking into account the definition of normal cone at x ∈ L, we notice that: Further, we establish some working assumptions and auxiliary results.

Working hypotheses
are fulfilled. (ii) For any y ∈ R(x) and x, z ∈ L, the relations are true.
(iii) For any x, v ∈ L and λ > 0, there exists are satisfied.
The continuity property of the variational derivative δ β F δx implies L * ⊂ L * . By Proposition 2.1, we conclude L * = L * . As well, the solution set L * associated to (DEVIP) is convex and, consequently, the solution set L * associated to (EVIP) is a convex set.

Main results
In this section, weak sharp solutions are investigated for the considered extended variational-type inequality governed by (ρ, b, d)-convex path-independent curvilinear integral functional. In accordance with Ferris and Mangasarian [5], following Marcotte and Zhu [10], the weak sharpness property of the solution set L * for (EVIP) is studied. In this regard, two characterization results are formulated and proved.

Definition 3.1 The solution set L * associated to (EVIP) is called weakly sharp
or, equivalently, there exists a positive number γ > 0 such that where int(S) stands for interior of the set S and B denotes the open unit ball in L.

Lemma 3.1
There exists a positive number γ > 0 such that

1)
if and only if Considering B b = z z , z = 0, the previous inequality becomes (3.2). Conversely, if Eq. (3.2) holds, then there exists a positive number γ > 0 such that which implies (3.1) and the proof is complete.

Theorem 3.1 Assume the scalar functional H(x) is differentiable on L * and the scalar functional F(x) is strongly
Proof " ⇒" Consider L * is weakly sharp. Consequently, by Definition 3.1, it follows or, by Lemma 3.1, there exists a positive number γ > 0 such that (3.1) (or (3.2)) is fulfilled. Further, taking into account the convexity property of the solution set L * associated to (EVIP) (see Remark 2.4), it results proj L * (x) =ŷ ∈ L * , ∀x ∈ L and, following Hiriart-Urruty and Lemaréchal [6], we get . By the hypothesis and Lemma 3.1, we get by the strong b-convexity on L of the scalar functional F(x) and the Working hypotheses, it results

Now, by using (3.3), we obtain
"⇐ " Consider there exists a positive number γ > 0 such that Obviously, for any y ∈ L * , the case T ρ,b,d and, consequently, , with u(t 1 ) = u(t 2 ) = 0, u, xy ≤ 0, x ∈ L, involving there exists a sequence u k converging to u with y + t k u k ∈ L (for some sequence of positive numbers {t k } decreasing to zero), such that d y + t k u k , L * ≥ d y + t k u k , H u = t k u, u k u , (3.4) where H u = {x ∈ L : u, xy = 0} is a hyperplane passing through y and orthogonal to u. By the hypothesis and (3.4), it follows or, equivalently (H(y) = 0, ∀y ∈ L * ), Further, by taking the limit for k → ∞ in (3.5) and using a classical result of functional analysis, we get where λ > 0. By Definition 2.2, the inequality (3.6) can be rewritten as Now, taking into account the hypothesis and (3.7), for any b ∈ B, it results and the proof is complete. The second characterization of weak sharpness for L * implies the notion of minimum principle sufficiency property, introduced by Ferris and Mangasarian [5].
Definition 3.2 It is said that (EVIP) satisfies minimum principle sufficiency property if Q(x * ) = L * , for any x * ∈ L * .

Theorem 3.3 Assume the scalar functional H(x)
is differentiable on L * and the scalar functional F(x) is strongly b-convex on L. Also, for any x * ∈ L * , v ∈ L, z ∈ R(x * ), the implication