Weak Sharp Type Solutions for Some Variational Integral Inequalities

Abstract

Weak sharp type solutions are analyzed for a variational integral inequality defined by a convex functional of the multiple integral type. A connection with the sufficiency property associated with the minimum principle is formulated, as well. Also, an illustrative numerical application is provided.

1. Introduction

Functional and variational analysis are increasingly mathematical branches, with various optimization and control theory applications. Burke and Ferris [1], Patriksson [2] and Marcotte and Zhu [3] have introduced and investigated weak sharp type solutions for variational-like inequalities. Thereafter, various researchers, like Hu and Song [4], Liu and Wu [5], and Zhu [6] have continued this study by considering gap functions, extremization problems and an appropriate framework. Also, Alshahrani et al. [7] studied nonsmooth variational inequalities by using gap-type functions.

Knowing the implications of variational analysis in multifarious fields, like optimization or control theory, and taking into account some techniques presented by Clarke [8], Treanţă [9,10,11,12,13,14,15], Jayswal and Singh [16], Kassay and Rădulescu [17], Mititelu and Treanţă [18], in this paper, we investigate weak sharp type solutions for a family of variational integral inequalities defined by a convex functional of the multiple integral type. A connection with the sufficiency property associated with the minimum principle is formulated, as well. Also, an illustrative numerical application is provided. The novelty of this study is the appearance of multiple integrals in the variational-like inequality context and, most importantly, the main results of the current paper are based on the variational (functional) derivative concept, introduced by Treanţă [11]. For other connected ideas on this topic (that is, integral inequalities), the interested reader can consult the papers of Ciurdariu [19] (Bergstrom-type inequality associated with commuting gramian normal operators), Ciurdariu and Grecu [20] (Hermite–Hadamard-type integral inequalities associated with convex functions by using a parameter), and Minculete and Ciurdariu [21] (Grüss type inequality), Khan et al. [22] (fractional calculus associated with convex functions and the related inequalities), Tareq et al. [23] (integral inequalities associated with harmonical $cr-(h_1,h_2)$-Godunova-Levin functions), Tareq and Treanţă [24] (new connections between some interval-valued variational models and the associated inequalities).

The paper continues in the following manner. In Section 2, we introduce the problem under study and formulate some preliminaries. In Section 3, several preliminary findings are established. These results are essential to state the principal theorems of the paper. The main results are included in Section 4. Concretely, we investigate weak sharp-type solutions for a family of variational integral inequalities generated by convex multiple integrals. In addition, a connection with the sufficiency property associated with the minimum principle is provided. The next section illustrates the theoretical findings by a numerical example. The last section provides the conclusions of this study.

2. Preliminaries

In the following, let us consider the notations and working assumptions:

▸

${\displaystyle \Theta }$ is a compact set in ${\displaystyle {\mathbb{R}}^m}$, with a non-empty interior and the smooth boundary ${\displaystyle \partial \Theta }$, and ${\displaystyle t=\left(t^{\iota }\right),\iota =\overline{1,m}}$ is an element of ${\displaystyle \Theta }$;

▸

${\displaystyle dr=d{t}^1\cdots d{t}^m}$ is the element for volume on ${\displaystyle {\mathbb{R}}^m\supset \Theta }$;

▸

${\displaystyle \overline{\mathcal{L}}}$ is the family of piecewise differentiable functions ${\displaystyle a:\Theta \subset {\mathbb{R}}^m\to {\mathbb{R}}^n}$, equipped with (see ${\displaystyle a\left(t\right)·b\left(t\right)}$ as the Euclidean inner product)

$${\displaystyle \langle a,b\rangle ={\int }_{\Theta }a\left(t\right)·b\left(t\right)dr,\forall a,b\in \overline{\mathcal{L}}}$$

and, also, with the induced norm;

▸

define ${\displaystyle \mathcal{L}}$ to be a nonempty, closed and convex subset of ${\displaystyle \overline{\mathcal{L}}}$, defined as

$${\displaystyle \mathcal{L}=\left\{a\in \overline{\mathcal{L}}{:a|}_{\partial \Theta }=a_0=\mathrm{given}\right\};}$$

▸

throughout this paper, we consider the notations ${\displaystyle a,b,a_{\iota }}$ for ${\displaystyle a\left(t\right),b\left(t\right),a_{\iota }\left(t\right)}$, respectively; also, denote ${\displaystyle a_{\iota }:=\frac{\partial a}{\partial t^{\iota }},f_a:=\frac{\partial f}{\partial a},f_{a_{\iota }}:=\frac{\partial f}{\partial a_{\iota }}}$.

For the $C^1$-class functions with real values ${\displaystyle f,g,h:J^1({\mathbb{R}}^m,{\mathbb{R}}^n)\to \mathbb{R}}$ (with ${\displaystyle J^1({\mathbb{R}}^m,{\mathbb{R}}^n)}$ as the jet bundle of first-order for ${\displaystyle {\mathbb{R}}^m}$ and ${\displaystyle {\mathbb{R}}^n}$) and, for ${\displaystyle a\in \overline{\mathcal{L}}}$, we introduce the scalar multiple integral functionals:

$${\displaystyle F:\overline{\mathcal{L}}\to \mathbb{R},F\left(a\right)={\int }_{\Theta }f\left(t,a,a_{\iota }\right)dr,}$$

$${\displaystyle G:\overline{\mathcal{L}}\to \mathbb{R},G\left(a\right)={\int }_{\Theta }g\left(t,a,a_{\iota }\right)dr,}$$

$${\displaystyle K:\overline{\mathcal{L}}\to \mathbb{R},K\left(a\right)={\int }_{\Theta }h\left(t,a,a_{\iota }\right)dr.}$$

Definition 1.

A real-valued multiple integral functional ${\displaystyle F\left(a\right)={\int }_{\Theta }f\left(t,a,a_{\iota }\right)dr,F:\overline{\mathcal{L}}\to \mathbb{R}}$ is called convex on ${\displaystyle \mathcal{L}}$ if for any ${\displaystyle a,b\in \mathcal{L}}$:

$${\displaystyle F\left(a\right)-F\left(b\right)\ge {\int }_{\Theta }\left[f_a\left(t,b,b_{\iota }\right)(a-b)+f_{a_{\iota }}\left(t,b,b_{\iota }\right)D_{\iota }(a-b)\right]dr,}$$

where ${\displaystyle D_{\iota }:=\frac{\partial }{\partial t^{\iota }}}$.

Definition 2.

The functional (variational) derivative associated with the real-valued multiple integral functional ${\displaystyle F\left(a\right)={\int }_{\Theta }f\left(t,a,a_{\iota }\right)dr,F:\overline{\mathcal{L}}\to \mathbb{R}}$ is denoted by ${\displaystyle \frac{\delta F}{\delta a}}$, and it is defined as

$${\displaystyle \frac{\delta F}{\delta a}=f_a\left(t,a,a_{\iota }\right)-D_{\iota }{f}_{a_{\iota }}\left(t,a,a_{\iota }\right)\in \overline{\mathcal{L}},}$$

and satisfies

$${\displaystyle \langle \frac{\delta F}{\delta a},\psi \rangle ={\int }_{\Theta }\frac{\delta F}{\delta a}\left(t\right)·\psi \left(t\right)dr=\underset{\epsilon \to 0}{lim}\frac{F(a+\epsilon \psi )-F\left(a\right)}{\epsilon },\forall \psi \in \overline{\mathcal{L}}{,\psi |}_{\partial \Theta }=0.}$$

Throughout this paper, we assume the condition ${\displaystyle {\psi |}_{\partial \Theta }=0}$ in all scalar products between a functional derivative of a real-valued multiple integral-type functional and ${\displaystyle \psi \in \overline{\mathcal{L}}}$.

Now, by considering the mathematical tools previously defined, we formulate the variational integral inequality problem: let us find ${\displaystyle b\in \mathcal{L}}$ such that

$${\displaystyle \left(MP\right){\int }_{\Theta }\left[f_a\left(t,b,b_{\iota }\right)(a-b)+f_{a_{\iota }}\left(t,b,b_{\iota }\right)D_{\iota }(a-b)\right]dr\ge 0,}$$

for any ${\displaystyle a\in \mathcal{L}}$. The dual integral inequality problem of $\left(MP\right)$ is defined as let us find ${\displaystyle b\in \mathcal{L}}$ such that

$${\displaystyle \left(DMP\right){\int }_{\Theta }\left[f_a\left(t,a,a_{\iota }\right)(a-b)+f_{a_{\iota }}\left(t,a,a_{\iota }\right)D_{\iota }(a-b)\right]dr\ge 0,}$$

for any ${\displaystyle a\in \mathcal{L}}$.

Consider ${\displaystyle {\mathcal{L}}^{*}}$ and ${\displaystyle {\mathcal{L}}_{*}}$ as the nonempty and closed solution set for $\left(MP\right)$ and $\left(DMP\right)$, respectively.

Remark 1.

The variational models $\left(MP\right)$ and $\left(DMP\right)$ are stated as follows: let us find ${\displaystyle b\in \mathcal{L}}$ such that

$${\displaystyle \left(MP\right)\langle \frac{\delta F}{\delta b},a-b\rangle \ge 0,\forall a\in \mathcal{L},}$$

respectively, let us find ${\displaystyle b\in \mathcal{L}}$ such that

$${\displaystyle \left(DMP\right)\langle \frac{\delta F}{\delta a},a-b\rangle \ge 0,\forall a\in \mathcal{L}.}$$

For investigating ${\displaystyle {\mathcal{L}}^{*}}$, we define the gap-type functionals associated with the considered multiple integrals.

Definition 3.

Given ${\displaystyle a\in \overline{\mathcal{L}}}$, the primal gap functional of $\left(MP\right)$ is formulated as follows

$${\displaystyle G\left(a\right)=\underset{b\in \mathcal{L}}{\max }{\int }_{\Theta }\left[f_a\left(t,a,a_{\iota }\right)(a-b)+f_{a_{\iota }}\left(t,a,a_{\iota }\right)D_{\iota }(a-b)\right]dr}$$

and the dual gap functional of $\left(MP\right)$ is formulated as follows

$${\displaystyle K\left(a\right)=\underset{b\in \mathcal{L}}{\max }{\int }_{\Theta }\left[f_a\left(t,b,b_{\iota }\right)(a-b)+f_{a_{\iota }}\left(t,b,b_{\iota }\right)D_{\iota }(a-b)\right]dr.}$$

From now onwards, for ${\displaystyle a\in \overline{\mathcal{L}}}$, we consider:

$${\displaystyle A\left(a\right)=\left\{\beta \in \mathcal{L}:G\left(a\right)={\int }_{\Theta }\left[f_a\left(t,a,a_{\iota }\right)(a-\beta )+f_{a_{\iota }}\left(t,a,a_{\iota }\right)D_{\iota }(a-\beta )\right]dr\right\},}$$

$${\displaystyle Z\left(a\right)=\left\{\beta \in \mathcal{L}:K\left(a\right)={\int }_{\Theta }\left[f_a\left(t,\beta ,{\beta }_{\iota }\right)(a-\beta )+f_{a_{\iota }}\left(t,\beta ,{\beta }_{\iota }\right)D_{\iota }(a-\beta )\right]dr\right\}.}$$

Remark 2.

Taking into account the above tools, we can notice:

(i) 

${\displaystyle G\left(a\right)=\underset{b\in \mathcal{L}}{\max }\langle \frac{\delta F}{\delta a},a-b\rangle ,K\left(a\right)=\underset{b\in \mathcal{L}}{\max }\langle \frac{\delta F}{\delta b},a-b\rangle }$;

(ii) 

${\displaystyle A\left(a\right)=\arg \underset{b\in \mathcal{L}}{\max }\langle \frac{\delta F}{\delta a},a-b\rangle =\arg \underset{b\in \mathcal{L}}{\max }\left\{-\langle \frac{\delta F}{\delta a},y\rangle \right\}}$, where ${\displaystyle \arg \underset{b\in \mathcal{L}}{\max }\langle \frac{\delta F}{\delta a},a-b\rangle }$ is the set of solutions (possibly empty) for ${\displaystyle \underset{b\in \mathcal{L}}{\max }\langle \frac{\delta F}{\delta a},a-b\rangle }$;

(iii) 

${\displaystyle Z\left(a\right)=\arg \underset{b\in \mathcal{L}}{\max }\langle \frac{\delta F}{\delta b},a-b\rangle }$;

(iv) 

if ${\displaystyle A\left(a\right)=Ø}$, then ${\displaystyle G\left(a\right)=\underset{b\in \mathcal{L}}{sup}\langle \frac{\delta F}{\delta a},a-b\rangle }$; similarly, if ${\displaystyle Z\left(a\right)=Ø}$, then ${\displaystyle K\left(a\right)=\underset{b\in \mathcal{L}}{sup}\langle \frac{\delta F}{\delta b},a-b\rangle }$.

Further, in accordance with Marcotte and Zhu [3] and following Matsushita and Xu [25], we formulate the following relevant concepts.

Definition 4.

The polar set ${\displaystyle {\mathcal{L}}^{\circ }}$ for ${\displaystyle \mathcal{L}}$ is

$${\displaystyle {\mathcal{L}}^{\circ }=\left\{b\in \overline{\mathcal{L}}:\langle b,a\rangle \le 0,\forall a\in \mathcal{L}\right\}.}$$

Definition 5.

The projection of ${\displaystyle a\in \overline{\mathcal{L}}}$ onto the set ${\displaystyle \mathcal{L}}$ is

$${\displaystyle {\mathit{proj}}_{\mathcal{L}}a=\arg \underset{b\in \mathcal{L}}{\min }\Vert a-b\Vert .}$$

Remark 3.

If ${\displaystyle \mathcal{L}}$ is a closed convex set, then ${\displaystyle {\mathit{proj}}_{\mathcal{L}}(·)}$ is a nonexpansive mapping and ${\displaystyle {\mathit{proj}}_{\mathcal{L}}a}$ is a singleton set.

Definition 6.

We define the normal cone of ${\displaystyle \mathcal{L}}$ at ${\displaystyle a\in \overline{\mathcal{L}}}$ as below

$${\displaystyle N_{\mathcal{L}}\left(a\right)=\left\{b\in \overline{\mathcal{L}}:\langle b,\beta -a\rangle \le 0,\forall \beta \in \mathcal{L}\right\},a\in \mathcal{L},}$$

$${\displaystyle N_{\mathcal{L}}\left(a\right)=Ø,a\notin \mathcal{L}}$$

and the tangent cone of ${\displaystyle \mathcal{L}}$ at ${\displaystyle a\in \overline{\mathcal{L}}}$ is ${\displaystyle T_{\mathcal{L}}\left(a\right)={\left[N_{\mathcal{L}}\left(a\right)\right]}^{\circ }}$.

Remark 4.

Taking into account the above definition, we notice that: ${\displaystyle a^{*}\in {\mathcal{L}}^{*}⟺-\frac{\delta F}{\delta a^{*}}\in N_{\mathcal{L}}\left(a^{*}\right)}$. This is a characterization of ${\displaystyle {\mathcal{L}}^{*}}$ in terms of variational (functional) derivative.

3. Preliminary Results

This section establishes and proves several results beneficial for establishing the principal results of this study.

Proposition 1.

Assume ${\displaystyle F\left(a\right)={\int }_{\Theta }f\left(t,a,a_{\iota }\right)dr}$ is convex on ${\displaystyle \mathcal{L}}$. Then:

(i) 

for any ${\displaystyle a^1,a^2\in {\mathcal{L}}^{*}}$, it follows

$${\displaystyle {\int }_{\Theta }\left[f_a\left(t,a^2,a_{\iota }^2\right)(a^1-a^2)+f_{a_{\iota }}\left(t,a^2,a_{\iota }^2\right)D_{\iota }(a^1-a^2)\right]dr=0;}$$

(ii) 

${\displaystyle {\mathcal{L}}^{*}\subset {\mathcal{L}}_{*}}$.

Proof.

(i) Since ${\displaystyle a^1\in {\mathcal{L}}^{*}}$, it results

$${\displaystyle {\int }_{\Theta }\left[f_a\left(t,a^1,a_{\iota }^1\right)(a-a^1)+f_{a_{\iota }}\left(t,a^1,a_{\iota }^1\right)D_{\iota }(a-a^1)\right]dr\ge 0,\forall a\in \mathcal{L}.}$$

Since ${\displaystyle a^2\in {\mathcal{L}}^{*}\subset \mathcal{L}}$, the previous inequality becomes

$${\displaystyle {\int }_{\Theta }\left[f_a\left(t,a^1,a_{\iota }^1\right)(a^2-a^1)+f_{a_{\iota }}\left(t,a^1,a_{\iota }^1\right)D_{\iota }(a^2-a^1)\right]dr\ge 0.}$$

(1)

By hypothesis, the functional ${\displaystyle F\left(a\right)={\int }_{\Theta }f\left(t,a,a_{\iota }\right)dr}$ is convex on ${\displaystyle \mathcal{L}}$. Consequently, it results

$${\displaystyle F\left(a^1\right)-F\left(a^2\right)\ge {\int }_{\Theta }\left[f_a\left(t,a^2,a_{\iota }^2\right)(a^1-a^2)+f_{a_{\iota }}\left(t,a^2,a_{\iota }^2\right)D_{\iota }(a^1-a^2)\right]dr}$$

(2)

and

$${\displaystyle F\left(a^2\right)-F\left(a^1\right)\ge {\int }_{\Theta }\left[f_a\left(t,a^1,a_{\iota }^1\right)(a^2-a^1)+f_{a_{\iota }}\left(t,a^1,a_{\iota }^1\right)D_{\iota }(a^2-a^1)\right]dr.}$$

(3)

By $\left(2\right)+\left(3\right)$ and by considering $\left(1\right)$, it follows

$${\displaystyle {\int }_{\Theta }\left[f_a\left(t,a^2,a_{\iota }^2\right)(a^1-a^2)+f_{a_{\iota }}\left(t,a^2,a_{\iota }^2\right)D_{\iota }(a^1-a^2)\right]dr\le 0.}$$

(4)

Similarly as above, by ${\displaystyle a^2\in {\mathcal{L}}^{*}}$, we can write

$${\displaystyle {\int }_{\Theta }\left[f_a\left(t,a^2,a_{\iota }^2\right)(a^1-a^2)+f_{a_{\iota }}\left(t,a^2,a_{\iota }^2\right)D_{\iota }(a^1-a^2)\right]dr\ge 0.}$$

(5)

Now, by using $\left(4\right)$ and $\left(5\right)$, we complete the proof of this implication.

(ii) Since ${\displaystyle a^{*}\in {\mathcal{L}}^{*}}$, we obtain

$${\displaystyle {\int }_{\Theta }\left[f_a\left(t,a^{*},a_{\iota }^{*}\right)(a-a^{*})+f_{a_{\iota }}\left(t,a^{*},a_{\iota }^{*}\right)D_{\iota }(a-a^{*})\right]dr\ge 0,\forall a\in \mathcal{L}.}$$

(6)

Also, the convexity assumption of ${\displaystyle F\left(a\right)}$ (see $\left(2\right)+\left(3\right)$) involves

$${\displaystyle {\int }_{\Theta }\left[f_a\left(t,a^1,a_{\iota }^1\right)(a^1-a^2)+f_{a_{\iota }}\left(t,a^1,a_{\iota }^1\right)D_{\iota }(a^1-a^2)\right]dr}$$

(7)

$${\displaystyle \ge {\int }_{\Theta }\left[f_a\left(t,a^2,a_{\iota }^2\right)(a^1-a^2)+f_{a_{\iota }}\left(t,a^2,a_{\iota }^2\right)D_{\iota }(a^1-a^2)\right]dr,\forall a^1,a^2\in \mathcal{L}.}$$

By using the relations $\left(6\right)$ and $\left(7\right)$, we obtain

$${\displaystyle {\int }_{\Theta }\left[f_a\left(t,a,a_{\iota }\right)(a-a^{*})+f_{a_{\iota }}\left(t,a,a_{\iota }\right)D_{\iota }(a-a^{*})\right]dr\ge 0,\forall a\in \mathcal{L},}$$

and the proof is complete. □

Remark 5.

By using the continuity of ${\displaystyle \frac{\delta F}{\delta a}}$, it follows ${\displaystyle {\mathcal{L}}_{*}\subset {\mathcal{L}}^{*}}$. Proposition 1 concludes ${\displaystyle {\mathcal{L}}^{*}={\mathcal{L}}_{*}}$. Moreover, the solution set ${\displaystyle {\mathcal{L}}_{*}}$ of $\left(DMP\right)$ is convex, and thus the solution set ${\displaystyle {\mathcal{L}}^{*}}$ of $\left(MP\right)$ is a convex set.

Proposition 2.

If the functional ${\displaystyle K\left(a\right)}$ is differentiable on ${\displaystyle \overline{\mathcal{L}}}$, then the inequality

$${\displaystyle \langle \frac{\delta K}{\delta a},v\rangle \ge \langle \frac{\delta F}{\delta b},v\rangle }$$

is true, for any ${\displaystyle a,v\in \overline{\mathcal{L}},b\in Z\left(a\right)}$.

Proof.

For a given ${\displaystyle a\in \overline{\mathcal{L}}}$, it results

$${\displaystyle K\left(a\right)=\underset{b\in \mathcal{L}}{\max }{\int }_{\Theta }\left[f_a\left(t,b,b_{\iota }\right)(a-b)+f_{a_{\iota }}\left(t,b,b_{\iota }\right)D_{\iota }(a-b)\right]dr,}$$

or (see Remark 2), equivalently,

$${\displaystyle K\left(a\right)=\underset{b\in \mathcal{L}}{\max }\langle \frac{\delta F}{\delta b},a-b\rangle ,\forall a\in \overline{\mathcal{L}},}$$

or, obviously,

$${\displaystyle K\left(a\right)=\langle \frac{\delta F}{\delta b},a-b\rangle ,\forall b\in Z\left(a\right).}$$

(8)

Moreover, for any ${\displaystyle b\in \mathcal{L},\beta \in \overline{\mathcal{L}}}$, the inequality

$${\displaystyle K\left(\beta \right)\ge \langle \frac{\delta F}{\delta b},\beta -b\rangle }$$

(9)

holds, and by considering $\left(8\right)$ and $\left(9\right)$ we obtain

$${\displaystyle K\left(\beta \right)-K\left(a\right)\ge \langle \frac{\delta F}{\delta b},\beta -a\rangle ,\forall b\in Z\left(a\right),\forall a,\beta \in \overline{\mathcal{L}}.}$$

For ${\displaystyle \beta =a+\lambda v\in \overline{\mathcal{L}}}$,${\displaystyle \lambda >0}$, we obtain

$${\displaystyle K(a+\lambda v)-K\left(a\right)\ge \langle \frac{\delta F}{\delta b},\lambda v\rangle ,\forall b\in Z\left(a\right),\forall a,v\in \overline{\mathcal{L}},}$$

and, dividing by ${\displaystyle \lambda >0}$, it follows

$${\displaystyle \frac{K(a+\lambda v)-K\left(a\right)}{\lambda }\ge \langle \frac{\delta F}{\delta b},v\rangle ,\forall b\in Z\left(a\right),\forall a,v\in \overline{\mathcal{L}}.}$$

Next, we complete the proof by taking ${\displaystyle \lambda \to 0}$ and using Definition 2. □

Proposition 3.

If ${\displaystyle K}$ is differentiable on ${\displaystyle {\mathcal{L}}^{*}}$ and ${\displaystyle F}$ is convex on ${\displaystyle \mathcal{L}}$, then ${\displaystyle Z\left(a^{*}\right)={\mathcal{L}}^{*},\forall a^{*}\in {\mathcal{L}}^{*}}$ if, for any ${\displaystyle a^{*}\in {\mathcal{L}}^{*},v\in \overline{\mathcal{L}},\beta \in Z\left(a^{*}\right)}$, the following implication

$${\displaystyle \langle \frac{\delta K}{\delta a^{*}},v\rangle \ge \langle \frac{\delta F}{\delta \beta },v\rangle ⟹\frac{\delta K}{\delta a^{*}}=\frac{\delta F}{\delta \beta }}$$

is satisfied.

Proof.

“⊂” Consider ${\displaystyle \beta \in Z\left(a^{*}\right)}$. In consequence, it results

$${\displaystyle K\left(a^{*}\right)={\int }_{\Theta }\left[f_a\left(t,\beta ,{\beta }_{\iota }\right)(a^{*}-\beta )+f_{a_{\iota }}\left(t,\beta ,{\beta }_{\iota }\right)D_{\iota }(a^{*}-\beta )\right]dr,a^{*}\in {\mathcal{L}}^{*}.}$$

(10)

${\displaystyle F\left(a\right)}$ is convex on ${\displaystyle \mathcal{L}}$ and ${\displaystyle a^{*}\in {\mathcal{L}}^{*}}$. By Remark 2 and Proposition 1, we obtain ${\displaystyle a^{*}\in {\mathcal{L}}_{*}}$, implying

$${\displaystyle {\int }_{\Theta }\left[f_a\left(t,a,a_{\iota }\right)(a-a^{*})+f_{a_{\iota }}\left(t,a,a_{\iota }\right)D_{\iota }(a-a^{*})\right]dr\ge 0,}$$

(11)

for ${\displaystyle a\in \mathcal{L}}$. We obtain ${\displaystyle K\left(a^{*}\right)=0,\forall a^{*}\in {\mathcal{L}}^{*}}$, that is

$${\displaystyle {\int }_{\Theta }\left[f_a\left(t,\beta ,{\beta }_{\iota }\right)(a^{*}-\beta )+f_{a_{\iota }}\left(t,\beta ,{\beta }_{\iota }\right)D_{\iota }(a^{*}-\beta )\right]dr=0,a^{*}\in {\mathcal{L}}^{*},}$$

(12)

implying

$${\displaystyle {\int }_{\Theta }\left[f_a\left(t,\beta ,{\beta }_{\iota }\right)(a-\beta )+f_{a_{\iota }}\left(t,\beta ,{\beta }_{\iota }\right)D_{\iota }(a-\beta )\right]dr}$$

(13)

$${\displaystyle ={\int }_{\Theta }\left[f_a\left(t,\beta ,{\beta }_{\iota }\right)(a-a^{*})+f_{a_{\iota }}\left(t,\beta ,{\beta }_{\iota }\right)D_{\iota }(a-a^{*})\right]dr.}$$

Next, by using the definition of ${\displaystyle K\left(a\right)}$ associated to $\left(MP\right)$, for ${\displaystyle \lambda \in [0,1]}$ and ${\displaystyle a\in \mathcal{L}}$, it results

$${\displaystyle \frac{K(a^{*}+\lambda (a-a^{*}))-K\left(a^{*}\right)}{\lambda }}$$

$${\displaystyle \ge {\int }_{\Theta }\left[f_a\left(t,a^{*},a_{\iota }^{*}\right)(a-a^{*})+f_{a_{\iota }}\left(t,a^{*},a_{\iota }^{*}\right)D_{\iota }(a-a^{*})\right]dr.}$$

By taking ${\displaystyle \lambda \to 0}$ and using Definition 2, we obtain

$${\displaystyle \langle \frac{\delta K}{\delta a^{*}},a-a^{*}\rangle \ge {\int }_{\Theta }\left[f_a\left(t,a^{*},a_{\iota }^{*}\right)(a-a^{*})+f_{a_{\iota }}\left(t,a^{*},a_{\iota }^{*}\right)D_{\iota }(a-a^{*})\right]dr.}$$

(14)

We conclude (see Proposition 2) that ${\displaystyle \frac{\delta K}{\delta a^{*}}=\frac{\delta F}{\delta \beta }}$. Therefore, $\left(14\right)$ becomes

$${\displaystyle \langle \frac{\delta F}{\delta \beta },a-a^{*}\rangle \ge {\int }_{\Theta }\left[f_a\left(t,a^{*},a_{\iota }^{*}\right)(a-a^{*})+f_{a_{\iota }}\left(t,a^{*},a_{\iota }^{*}\right)D_{\iota }(a-a^{*})\right]dr,}$$

or, equivalently,

$${\displaystyle {\int }_{\Theta }\left[f_a\left(t,\beta ,{\beta }_{\iota }\right)(a-a^{*})+f_{a_{\iota }}\left(t,\beta ,{\beta }_{\iota }\right)D_{\iota }(a-a^{*})\right]dr}$$

(15)

$${\displaystyle \ge {\int }_{\Theta }\left[f_a\left(t,a^{*},a_{\iota }^{*}\right)(a-a^{*})+f_{a_{\iota }}\left(t,a^{*},a_{\iota }^{*}\right)D_{\iota }(a-a^{*})\right]dr.}$$

The relations $\left(13\right)$ and $\left(15\right)$ imply

$${\displaystyle {\int }_{\Theta }\left[f_a\left(t,\beta ,{\beta }_{\iota }\right)(a-\beta )+f_{a_{\iota }}\left(t,\beta ,{\beta }_{\iota }\right)D_{\iota }(a-\beta )\right]dr}$$

$${\displaystyle \ge {\int }_{\Theta }\left[f_a\left(t,a^{*},a_{\iota }^{*}\right)(a-a^{*})+f_{a_{\iota }}\left(t,a^{*},a_{\iota }^{*}\right)D_{\iota }(a-a^{*})\right]dr.}$$

Since ${\displaystyle a^{*}\in {\mathcal{L}}^{*}}$, the previous inequality implies

$${\displaystyle {\int }_{\Theta }\left[f_a\left(t,\beta ,{\beta }_{\iota }\right)(a-\beta )+f_{a_{\iota }}\left(t,\beta ,{\beta }_{\iota }\right)D_{\iota }(a-\beta )\right]dr\ge 0,\forall a\in \mathcal{L},}$$

involving ${\displaystyle \beta \in {\mathcal{L}}^{*}}$ and, in consequence, ${\displaystyle Z\left(a^{*}\right)\subset {\mathcal{L}}^{*}}$.

“⊃” Consider ${\displaystyle \beta ,a^{*}\in {\mathcal{L}}^{*}}$. By Proposition 1, we obtain

$${\displaystyle {\int }_{\Theta }\left[f_a\left(t,\beta ,{\beta }_{\iota }\right)(a^{*}-\beta )+f_{a_{\iota }}\left(t,\beta ,{\beta }_{\iota }\right)D_{\iota }(a^{*}-\beta )\right]dr=0.}$$

Since ${\displaystyle K\left(a^{*}\right)=0,\forall a^{*}\in {\mathcal{L}}^{*}}$, it results

$${\displaystyle K\left(a^{*}\right)={\int }_{\Theta }\left[f_a\left(t,\beta ,{\beta }_{\iota }\right)(a^{*}-\beta )+f_{a_{\iota }}\left(t,\beta ,{\beta }_{\iota }\right)D_{\iota }(a^{*}-\beta )\right]dr,}$$

involving ${\displaystyle \beta \in Z\left(a^{*}\right)}$. □

4. Main Results

This section investigates the weak sharpness of the solution set of the considered integral variational inequality. Thus, under Ferris and Mangasarian [26], Marcotte and Zhu [3] and following Matsushita and Xu [25], the weak sharpness property of ${\displaystyle {\mathcal{L}}^{*}}$ for $\left(MP\right)$ is studied.

Definition 7.

The set of solutions ${\displaystyle {\mathcal{L}}^{*}}$ of $\left(MP\right)$ is said to be weak sharp if

$${\displaystyle -\frac{\delta F}{\delta a^{*}}\in int\left(\bigcap _{s\in {\mathcal{L}}^{*}}{\left[T_{\mathcal{L}}\left(s\right)\cap N_{{\mathcal{L}}^{*}}\left(s\right)\right]}^{\circ }\right),\forall a^{*}\in {\mathcal{L}}^{*},}$$

is equivalent to the existence of ${\displaystyle \alpha >0}$ such that

$${\displaystyle \alpha B\subset \frac{\delta F}{\delta a^{*}}+{\left[T_{\mathcal{L}}\left(a^{*}\right)\cap N_{{\mathcal{L}}^{*}}\left(a^{*}\right)\right]}^{\circ },\forall a^{*}\in {\mathcal{L}}^{*},}$$

where B is a unit open ball of ${\displaystyle \overline{\mathcal{L}}}$.

Lemma 1.

The existence of ${\displaystyle \alpha >0}$ such that

$${\displaystyle \alpha B\subset \frac{\delta F}{\delta b}+{\left[T_{\mathcal{L}}\left(b\right)\cap N_{{\mathcal{L}}^{*}}\left(b\right)\right]}^{\circ },\forall b\in {\mathcal{L}}^{*}}$$

(16)

is equivalent with

$${\displaystyle \langle \frac{\delta F}{\delta b},\beta \rangle \ge \alpha \Vert \beta \Vert ,\forall \beta \in T_{\mathcal{L}}\left(b\right)\cap N_{{\mathcal{L}}^{*}}\left(b\right).}$$

(17)

Proof.

Relation given in $\left(16\right)$ is written as

$${\displaystyle \alpha b-\frac{\delta F}{\delta b}\in {\left[T_{\mathcal{L}}\left(b\right)\cap N_{{\mathcal{L}}^{*}}\left(b\right)\right]}^{\circ },\forall b\in {\mathcal{L}}^{*},\forall b\in B,}$$

or

$${\displaystyle \langle \alpha b-\frac{\delta F}{\delta b},\beta \rangle \le 0,\forall b\in {\mathcal{L}}^{*},\forall b\in B,\forall \beta \in T_{\mathcal{L}}\left(b\right)\cap N_{{\mathcal{L}}^{*}}\left(b\right).}$$

Considering ${\displaystyle B\ni b=\frac{\beta }{\Vert \beta \Vert },\beta \ne 0}$, we obtain $\left(17\right)$.

Conversely, we assume $\left(17\right)$ holds, that is, the existence of ${\displaystyle \alpha >0}$ such that

$${\displaystyle \langle \alpha b-\frac{\delta F}{\delta b},\beta \rangle =\langle \alpha b,\beta \rangle -\langle \frac{\delta F}{\delta b},\beta \rangle }$$

$${\displaystyle \le \alpha \Vert \beta \Vert -\alpha \Vert \beta \Vert =0,\forall b\in {\mathcal{L}}^{*},\forall b\in B,\forall \beta \in T_{\mathcal{L}}\left(b\right)\cap N_{{\mathcal{L}}^{*}}\left(b\right),}$$

that is

$${\displaystyle \langle \alpha b-\frac{\delta F}{\delta b},\beta \rangle \le 0,\forall b\in {\mathcal{L}}^{*},\forall b\in B,\forall \beta \in T_{\mathcal{L}}\left(b\right)\cap N_{{\mathcal{L}}^{*}}\left(b\right),}$$

or, equivalently,

$${\displaystyle \alpha b-\frac{\delta F}{\delta b}\in {\left[T_{\mathcal{L}}\left(b\right)\cap N_{{\mathcal{L}}^{*}}\left(b\right)\right]}^{\circ },\forall b\in {\mathcal{L}}^{*},\forall b\in B,}$$

involving $\left(16\right)$. □

Theorem 1.

Consider ${\displaystyle K\left(a\right)}$ is differentiable on ${\displaystyle {\mathcal{L}}^{*}}$ and ${\displaystyle F\left(a\right)}$ is convex on ${\displaystyle \mathcal{L}}$. Also, for any ${\displaystyle a^{*}\in {\mathcal{L}}^{*},v\in \overline{\mathcal{L}},\beta \in Z\left(a^{*}\right)}$, the implication

$${\displaystyle \langle \frac{\delta K}{\delta a^{*}},v\rangle \ge \langle \frac{\delta F}{\delta \beta },v\rangle ⟹\frac{\delta K}{\delta a^{*}}=\frac{\delta F}{\delta \beta }}$$

is valid and ${\displaystyle \frac{\delta F}{\delta a^{*}}}$ is constant on ${\displaystyle {\mathcal{L}}^{*}}$. Under these hypotheses, ${\displaystyle {\mathcal{L}}^{*}}$ is weak sharp becomes equivalent with the existence of ${\displaystyle \alpha >0}$ such that

$${\displaystyle K\left(a\right)\ge \alpha d(a,{\mathcal{L}}^{*}),\forall a\in \mathcal{L},}$$

where ${\displaystyle d(a,{\mathcal{L}}^{*})=\underset{b\in {\mathcal{L}}^{*}}{min}\Vert a-b\Vert }$.

Proof.

“⟹” Since ${\displaystyle {\mathcal{L}}^{*}}$ is weak sharp, it follows

$${\displaystyle -\frac{\delta F}{\delta b}\in int\left(\bigcap _{s\in {\mathcal{L}}^{*}}{\left[T_{\mathcal{L}}\left(s\right)\cap N_{{\mathcal{L}}^{*}}\left(s\right)\right]}^{\circ }\right),\forall b\in {\mathcal{L}}^{*}.}$$

Next, by using the convexity of ${\displaystyle {\mathcal{L}}^{*}}$ (see Remark 5), it follows

$${\displaystyle {\mathrm{proj}}_{{\mathcal{L}}^{*}}\left(a\right)=\widehat{b}\in {\mathcal{L}}^{*},\forall a\in \mathcal{L}}$$

and, according to Hiriart–Urruty and Lemaréchal [27], it results in ${\displaystyle a-\widehat{b}\in T_{\mathcal{L}}\left(\widehat{b}\right)\cap N_{{\mathcal{L}}^{*}}\left(\widehat{b}\right)}$. By using the hypothesis and Lemma 1, it results in

$${\displaystyle \langle \frac{\delta F}{\delta \widehat{b}},a-\widehat{b}\rangle \ge \alpha \Vert a-\widehat{b}\Vert =\alpha d(a,{\mathcal{L}}^{*}),}$$

or, equivalently,

$${\displaystyle {\int }_{\Theta }\left[f_a\left(t,\widehat{b},{\widehat{b}}_{\iota }\right)(a-\widehat{b})+f_{a_{\iota }}\left(t,\widehat{b},{\widehat{b}}_{\iota }\right)D_{\iota }(a-\widehat{b})\right]dr\ge \alpha d(a,{\mathcal{L}}^{*}),\forall a\in \mathcal{L}.}$$

(18)

Since

$${\displaystyle K\left(a\right)\ge {\int }_{\Theta }\left[f_a\left(t,\widehat{b},{\widehat{b}}_{\iota }\right)(a-\widehat{b})+f_{a_{\iota }}\left(t,\widehat{b},{\widehat{b}}_{\iota }\right)D_{\iota }(a-\widehat{b})\right]dr,\forall a\in \mathcal{L},}$$

we obtain

$${\displaystyle K\left(a\right)\ge \alpha d(a,{\mathcal{L}}^{*}),\forall a\in \mathcal{L}.}$$

“⟸” Consider there exists ${\displaystyle \alpha >0}$ such that

$${\displaystyle K\left(a\right)\ge \alpha d(a,{\mathcal{L}}^{*}),\forall a\in \mathcal{L}.}$$

Let ${\displaystyle b\in {\mathcal{L}}^{*}}$. Clearly, for ${\displaystyle T_{\mathcal{L}}\left(b\right)\cap N_{{\mathcal{L}}^{*}}\left(b\right)=\left\{0\right\}}$, the result is obvious. Let ${\displaystyle 0\ne s\in T_{\mathcal{L}}\left(b\right)\cap N_{{\mathcal{L}}^{*}}\left(b\right)}$. Now, ${\displaystyle 0\ne s\in T_{\mathcal{L}}\left(b\right)}$ implies that there exists a sequence ${\displaystyle s^k}$ converging to s with ${\displaystyle b+t_k{s}^k\in \mathcal{L}}$ (for ${\displaystyle \left\{t_k\right\}}$ some sequence of positive numbers decreasing to zero). On the other hand, ${\displaystyle 0\ne s\in N_{{\mathcal{L}}^{*}}\left(b\right)}$ implies ${\displaystyle \langle s,\overline{a}-b\rangle \le 0,\forall \overline{a}\in {\mathcal{L}}^{*}}$. Evidently, ${\displaystyle \langle s,s\rangle \ge 0}$. This involves that ${\displaystyle {\mathcal{L}}^{*}}$ is separated from $b+s$ by a hyperplane ${\displaystyle H_s=\left\{a\in \overline{\mathcal{L}}:\langle s,a-b\rangle =0\right\}}$ (passing through b and orthogonal to s). Since s is not on ${\displaystyle H_s}$, without loss of generality, we can say that ${\displaystyle H_s}$ separates ${\displaystyle {\mathcal{L}}^{*}}$ from ${\displaystyle b+t_k{s}^k}$. Therefore,

$${\displaystyle d(b+t_k{s}^k,{\mathcal{L}}^{*})\ge d(b+t_k{s}^k,H_s)=\frac{t_k\langle s,s^k\rangle }{\parallel s\parallel },}$$

(19)

By $\left(19\right)$ and using the hypothesis, we obtain

$${\displaystyle K(b+t_k{s}^k)\ge \alpha \frac{t_k\langle s,s^k\rangle }{\parallel s\parallel },}$$

equivalent with (see ${\displaystyle K\left(b\right)=0,\forall b\in {\mathcal{L}}^{*}}$),

$${\displaystyle \frac{K(b+t_k{s}^k)-K\left(b\right)}{t_k}\ge \alpha \frac{\langle s,s^k\rangle }{\parallel s\parallel }.}$$

(20)

Further, by considering ${\displaystyle k\to \infty }$ in $\left(20\right)$, we obtain

$${\displaystyle \underset{\lambda \to 0}{lim}\frac{K(b+\lambda s)-K\left(b\right)}{\lambda }\ge \alpha \parallel s\parallel ,}$$

(21)

where ${\displaystyle \lambda >0}$. By Definition 2, the relation $\left(21\right)$ becomes

$${\displaystyle \langle \frac{\delta K}{\delta b},s\rangle \ge \alpha \parallel s\parallel .}$$

(22)

In the following, by considering the hypothesis and $\left(22\right)$, for any ${\displaystyle \overline{b}\in B}$, it results in

$${\displaystyle \langle \alpha \overline{b}-\frac{\delta F}{\delta b},s\rangle =\langle \alpha \overline{b},s\rangle -\langle \frac{\delta K}{\delta b},s\rangle \le \alpha \parallel s\parallel -\alpha \parallel s\parallel =0}$$

and, therefore,

$${\displaystyle \alpha B\subset \frac{\delta F}{\delta b}+{\left[T_{\mathcal{L}}\left(b\right)\cap N_{{\mathcal{L}}^{*}}\left(b\right)\right]}^{\circ },\forall b\in {\mathcal{L}}^{*}}$$

and this finishes the proof. □

Remark 6.

Weak sharpness associated with the set of solutions for the following extremization problem

$${\displaystyle \underset{a\in \mathcal{L}}{min}K\left(a\right)}$$

is described by the inequality (${\displaystyle K\left(b\right)=0,\forall b\in {\mathcal{L}}^{*}}$)

$${\displaystyle K\left(a\right)-K\left(a^{*}\right)\ge \alpha d(a,{\mathcal{L}}^{*}),\forall a\in \mathcal{L},a^{*}\in {\mathcal{L}}^{*}.}$$

According to Ferris and Mangasarian [26], we establish the following definition.

Definition 8.

We say that $\left(MP\right)$ fulfills the sufficiency property of minimum principle if ${\displaystyle A\left(a^{*}\right)={\mathcal{L}}^{*}}$, for any ${\displaystyle a^{*}\in {\mathcal{L}}^{*}}$.

Lemma 2.

For any ${\displaystyle a\in int\left(\bigcap _{s\in {\mathcal{L}}^{*}}{\left[T_{\mathcal{L}}\left(s\right)\cap N_{{\mathcal{L}}^{*}}\left(s\right)\right]}^{\circ }\right)\ne Ø}$, the inclusion ${\displaystyle \arg \underset{b\in \mathcal{L}}{\max }\langle a,b\rangle \subset {\mathcal{L}}^{*}}$ is fulfilled.

Proof.

For ${\displaystyle b\in \mathcal{L}\setminus {\mathcal{L}}^{*}}$ and convexity of ${\displaystyle {\mathcal{L}}^{*}}$, it results in

$${\displaystyle {\mathrm{proj}}_{{\mathcal{L}}^{*}}\left(b\right)=\widehat{b}\in {\mathcal{L}}^{*}}$$

and, by taking into account Hiriart–Urruty and Lemaréchal [27], it follows ${\displaystyle b-\widehat{b}\in T_{\mathcal{L}}\left(\widehat{b}\right)\cap N_{{\mathcal{L}}^{*}}\left(\widehat{b}\right)}$. There exists a positive number ${\displaystyle \iota >0}$ such that

$${\displaystyle \langle a+v,b-\widehat{b}\rangle <0,\forall v\in \iota B,}$$

for ${\displaystyle a\in int\left(\bigcap _{s\in {\mathcal{L}}^{*}}{\left[T_{\mathcal{L}}\left(s\right)\cap N_{{\mathcal{L}}^{*}}\left(s\right)\right]}^{\circ }\right)\ne Ø}$, or, equivalently,

$${\displaystyle \langle a,b\rangle <\langle a,\widehat{b}\rangle -\langle v,b-\widehat{b}\rangle ,\forall v\in \iota B,}$$

for ${\displaystyle a\in int\left(\bigcap _{s\in {\mathcal{L}}^{*}}{\left[T_{\mathcal{L}}\left(s\right)\cap N_{{\mathcal{L}}^{*}}\left(s\right)\right]}^{\circ }\right)\ne Ø}$. For ${\displaystyle v=\iota \frac{b-\widehat{b}}{\Vert b-\widehat{b}\Vert }\in \iota B}$, the previous inequality becomes

$${\displaystyle \langle a,b\rangle <\langle a,\widehat{b}\rangle -\iota \Vert b-\widehat{b}\Vert ,}$$

(23)

for ${\displaystyle a\in int\left(\bigcap _{s\in {\mathcal{L}}^{*}}{\left[T_{\mathcal{L}}\left(s\right)\cap N_{{\mathcal{L}}^{*}}\left(s\right)\right]}^{\circ }\right)\ne Ø}$, implying

$${\displaystyle b\notin \arg \underset{b\in \mathcal{L}}{\max }\langle a,b\rangle ,}$$

that is ${\displaystyle \arg \underset{b\in \mathcal{L}}{\max }\langle a,b\rangle \subset {\mathcal{L}}^{*}}$, for any ${\displaystyle a\in int\left(\bigcap _{s\in {\mathcal{L}}^{*}}{\left[T_{\mathcal{L}}\left(s\right)\cap N_{{\mathcal{L}}^{*}}\left(s\right)\right]}^{\circ }\right)\ne Ø}$. □

Theorem 2.

The problem $\left(MP\right)$ satisfies sufficiency property of minimum principle if ${\displaystyle {\mathcal{L}}^{*}}$ is weak sharp and ${\displaystyle F\left(a\right)}$ is convex on ${\displaystyle \mathcal{L}}$.

Proof.

By hypothesis, we have

$${\displaystyle -\frac{\delta F}{\delta a^{*}}\in int\left(\bigcap _{s\in {\mathcal{L}}^{*}}{\left[T_{\mathcal{L}}\left(s\right)\cap N_{{\mathcal{L}}^{*}}\left(s\right)\right]}^{\circ }\right),\forall a^{*}\in {\mathcal{L}}^{*}}$$

or, by Lemma 2,

$${\displaystyle \arg \underset{b\in \mathcal{L}}{\max }\langle -\frac{\delta F}{\delta a^{*}},y\rangle \subset {\mathcal{L}}^{*}⟺A\left(a^{*}\right)\subset {\mathcal{L}}^{*}.}$$

(24)

Further, let ${\displaystyle \beta \in {\mathcal{L}}^{*}}$. For ${\displaystyle a^{*}\in {\mathcal{L}}^{*}}$ (see Proposition 1), we obtain

$${\displaystyle {\int }_{\Theta }\left[f_a\left(t,a^{*},a_{\iota }^{*}\right)(\beta -a^{*})+f_{a_{\iota }}\left(t,a^{*},a_{\iota }^{*}\right)D_{\iota }(\beta -a^{*})\right]dr=0.}$$

(25)

By $\left(25\right)$, for ${\displaystyle b\in \mathcal{L}}$, it results in

$${\displaystyle {\int }_{\Theta }\left[f_a\left(t,a^{*},a_{\iota }^{*}\right)(\beta -b)+f_{a_{\iota }}\left(t,a^{*},a_{\iota }^{*}\right)D_{\iota }(\beta -b)\right]dr}$$

(26)

$${\displaystyle ={\int }_{\Theta }\left[f_a\left(t,a^{*},a_{\iota }^{*}\right)(a^{*}-b)+f_{a_{\iota }}\left(t,a^{*},a_{\iota }^{*}\right)D_{\iota }(a^{*}-b)\right]dr.}$$

Since ${\displaystyle a^{*}\in {\mathcal{L}}^{*}}$, relation $\left(26\right)$ provides

$${\displaystyle {\int }_{\Theta }\left[f_a\left(t,a^{*},a_{\iota }^{*}\right)(\beta -b)+f_{a_{\iota }}\left(t,a^{*},a_{\iota }^{*}\right)D_{\iota }(\beta -b)\right]dr\le 0,\forall b\in \mathcal{L},}$$

that is ${\displaystyle \beta \in A\left(a^{*}\right)}$ and, consequently,

$${\displaystyle {\mathcal{L}}^{*}\subset A\left(a^{*}\right).}$$

(27)

□

Theorem 3.

Consider ${\displaystyle K\left(a\right)}$ is differentiable on ${\displaystyle {\mathcal{L}}^{*}}$ and ${\displaystyle F\left(a\right)}$ is convex on ${\displaystyle \mathcal{L}}$. Also, for any ${\displaystyle a^{*}\in {\mathcal{L}}^{*},v\in \overline{\mathcal{L}},\beta \in Z\left(a^{*}\right)}$, the implication

$${\displaystyle \langle \frac{\delta K}{\delta a^{*}},v\rangle \ge \langle \frac{\delta F}{\delta \beta },v\rangle ⟹\frac{\delta K}{\delta a^{*}}=\frac{\delta F}{\delta \beta }}$$

is valid and ${\displaystyle \frac{\delta F}{\delta a^{*}}}$ is constant on ${\displaystyle {\mathcal{L}}^{*}}$. The problem $\left(MP\right)$ satisfies the sufficiency property of the minimum principle if and only if ${\displaystyle {\mathcal{L}}^{*}}$ is weak sharp.

Proof.

“⟹” By hypothesis, we obtain ${\displaystyle A\left(a^{*}\right)={\mathcal{L}}^{*}}$, for any ${\displaystyle a^{*}\in {\mathcal{L}}^{*}}$. Obviously, for ${\displaystyle a^{*}\in {\mathcal{L}}^{*}}$ and ${\displaystyle a\in \overline{\mathcal{L}}}$, we obtain

$${\displaystyle K\left(a\right)\ge {\int }_{\Theta }\left[f_a\left(t,a^{*},a_{\iota }^{*}\right)(a-a^{*})+f_{a_{\iota }}\left(t,a^{*},a_{\iota }^{*}\right)D_{\iota }(a-a^{*})\right]dr.}$$

(28)

By considering ${\displaystyle P\left(a\right)=\langle \frac{\delta F}{\delta a^{*}},a\rangle ,a\in \mathcal{L}}$, we have ${\displaystyle A\left(a^{*}\right)}$ the set of solutions to ${\displaystyle \underset{a\in \mathcal{L}}{min}P\left(a\right)}$. By Remark 6, we obtain

$${\displaystyle P\left(a\right)-P\left(\overline{a}\right)\ge \alpha d(a,A\left(a^{*}\right)),\forall a\in \mathcal{L},\overline{a}\in A\left(a^{*}\right),}$$

or,

$${\displaystyle \langle \frac{\delta F}{\delta a^{*}},a-a^{*}\rangle \ge \alpha d(a,{\mathcal{L}}^{*}),\forall a\in \mathcal{L},}$$

or, equivalently,

$${\displaystyle {\int }_{\Theta }\left[f_a\left(t,a^{*},a_{\iota }^{*}\right)(a-a^{*})+f_{a_{\iota }}\left(t,a^{*},a_{\iota }^{*}\right)D_{\iota }(a-a^{*})\right]dr\ge \alpha d(a,{\mathcal{L}}^{*}),\forall a\in \mathcal{L}.}$$

(29)

Therefore, $\left(28\right)$, $\left(29\right)$ and Theorem 1, imply ${\displaystyle {\mathcal{L}}^{*}}$ is weak sharp.

“⟸” A simple consequence of Theorem 2. □

5. Numerical Illustrative Example

Let ${\displaystyle {\Theta }_{0,2}}$ be a square determined by the diagonal corners ${\displaystyle 0=\left(0,0\right)}$, ${\displaystyle 2=\left(2,2\right)}$ in ${\displaystyle {\mathbb{R}}^2}$. Define

$${\displaystyle \overline{\mathcal{L}}=\left\{a:{\Theta }_{0,2}\subset {\mathbb{R}}^2\to [-1,4]:a=\mathrm{piecewise}\mathrm{smooth}\mathrm{function}\right\},}$$

$${\displaystyle \mathcal{L}=\left\{a\in \overline{\mathcal{L}}:a\left(t\right)\in [0,1]\subset \mathbb{R},a\left(0\right)=a(0,0)=0,a\left(2\right)=a(2,2)=0\right\}}$$

and the $C^1$-class function

$${\displaystyle f:J^1({\mathbb{R}}^2,\mathbb{R})\to \mathbb{R},f\left(t,a,a_{\iota }\right)=a^2+4a.}$$

Now, we formulate the variational integral inequality problem: let us find ${\displaystyle b\in \mathcal{L}}$ such that

$${\displaystyle \left(BP\right){\int }_{{\Theta }_{0,2}}\left[f_a\left(t,b,b_{\iota }\right)(a-b)+f_{a_{\iota }}\left(t,b,b_{\iota }\right)D_{\iota }(a-b)\right]d{t}^{1}d{t}^2\ge 0,}$$

for any ${\displaystyle a\in \mathcal{L}}$.

The associated dual gap functional

$${\displaystyle K:\overline{\mathcal{L}}\to \mathbb{R},K\left(a\right)={\int }_{{\Theta }_{0,2}}h\left(t,a,a_{\iota }\right)d{t}^{1}d{t}^2}$$

is, by direct computations, the following

$${\displaystyle K\left(a\right)=\underset{b\in \mathcal{L}}{\max }{\int }_{{\Theta }_{0,2}}\left[f_a\left(t,b,b_{\iota }\right)(a-b)+f_{a_{\iota }}\left(t,b,b_{\iota }\right)D_{\iota }(a-b)\right]d{t}^{1}d{t}^2}$$

$${\displaystyle =\underset{b\in \mathcal{L}}{\max }{\int }_{{\Theta }_{0,2}}(2b+4)(a-b)d{t}^{1}d{t}^2=\left\{\begin{array}{cc}\hfill {\displaystyle {\int }_{{\Theta }_{0,2}}4ad{t}^{1}d{t}^2,}& -1\le a<2\hfill \\ \hfill {\displaystyle {\int }_{{\Theta }_{0,2}}\frac{{(a+2)}^2}{2}d{t}^{1}d{t}^2,}& 2\le a\le 4.\hfill \end{array}\right.}$$

Also, the scalar functional ${\displaystyle F:\overline{\mathcal{L}}\to \mathbb{R},F\left(a\right)={\int }_{{\Theta }_{0,2}}f\left(t,a,a_{\iota }\right)d{t}^{1}d{t}^2}$ is convex on ${\displaystyle \mathcal{L}}$:

$${\displaystyle F\left(a\right)-F\left(b\right)-{\int }_{{\Theta }_{0,2}}\left[f_a\left(t,b,b_{\iota }\right)(a-b)+f_{a_{\iota }}\left(t,b,b_{\iota }\right)D_{\iota }(a-b)\right]d{t}^{1}d{t}^2}$$

$${\displaystyle ={\int }_{{\Theta }_{0,2}}{(a-b)}^{2}d{t}^{1}d{t}^2\ge 0,\forall a,b\in \mathcal{L}.}$$

We obtain

$${\displaystyle {\mathcal{L}}^{*}=\left\{b:{\Theta }_{0,2}\subset {\mathbb{R}}^2\to [0,1]:b\left(t\right)=0,\forall t\in {\Theta }_{0,2}\right\};}$$

$${\displaystyle \hspace{1em}Z\left(a^{*}\right)={\mathcal{L}}^{*},\forall a^{*}\in {\mathcal{L}}^{*};\frac{\delta F}{\delta a}=2a+4,}$$

and, obviously, ${\displaystyle K\left(a\right)}$ is differentiable on ${\displaystyle {\mathcal{L}}^{*}}$. Moreover, for ${\displaystyle a\in \mathcal{L}}$, there exists ${\displaystyle \alpha >0}$ satisfying

$${\displaystyle K\left(a\right)={\int }_{{\Theta }_{0,2}}4ad{t}^{1}d{t}^2\ge \alpha d(a,{\mathcal{L}}^{*}).}$$

Thus, by Theorem 1, we obtain ${\displaystyle {\mathcal{L}}^{*}}$ is weak sharp. In addition, in accordance with Theorems 2 and 3, we obtain that the considered problem $\left(BP\right)$ fulfills ${\displaystyle A\left(a^{*}\right)={\mathcal{L}}^{*}}$, for ${\displaystyle a^{*}\in {\mathcal{L}}^{*}}$.

Remark 7.

Based on the authors’ knowledge, the weak sharp study outcomes associated with such kinds of variational models are new in the field. Related to some future research directions of the current works, let us consider the situations where the partial derivatives of second-order are included, and the functionals are not under convexity assumptions (here, we can use the ideas formulated in Treanţă [28], by considering concepts of monotonicity, pseudomonotonicity and hemicontinuity for curvilinear integral-type functionals).

6. Conclusions

In this paper, weak sharp type solutions are studied for a variational integral inequality defined by a convex functional of multiple integral types. A connection with the sufficiency property associated with the minimum principle has been formulated, as well. Also, an illustrative numerical application was provided.