On Some Variational Inequalities Involving Second-Order Partial Derivatives

Abstract

By using the monotonicity, hemicontinuity, and pseudomonotonicity of the considered integral functionals, we studied the well-posedness of some variational inequality problems governed by partial derivatives of the second-order. To this aim, we introduce the approximating solution set and the concept of approximating sequences for the considered controlled variational inequality problem. Further, by using the aforementioned new mathematical tools, we established some theorems on well-posedness. Moreover, the theoretical tools and results included in the paper are accompanied by some examples.

1. Introduction

At times, it is not easy to figure out the solutions associated with optimization problems, using certain methods. In this regard, the well-posedness of an optimization problem is important since this condition provides the convergence of the approximate solution sequence. Many researchers have studied this concept for unconstrained optimization problems (see Tykhonov [1]) and, moreover, they have introduced many types of well-posedness, namely: well-posedness of the Levitin–Polyak type [2] and extended well-posedness (Chen et al. [3]). Well-posedness of the Tykhonov-type was also extended to variational inequalities (see Huang et al. [4], Fang et al. [5], Virmani and Srivastava [6], Hu et al. [7]), and to other connected problems, such as hemivariational inequality problems (see Ceng et al. [8], Wang et al. [9], Shu et al. [10]), complementary problems, equilibrium problems, and Nash equilibrium problems (see Lignola and Morgan [11]). A generalized well-posedness for variational disclusion problems and inclusion problems has been proposed by Lin and Chuang [12]. Moreover, Lalita and Bhatia [13] analyzed the well-posedness for minimization problems. In the last few years, multi-time variational inequality appeared as a necessary generalization of variational inequality (Treanţă [14,15]). Moreover, multidimensional optimization problems have been investigated, with remarkable results, by Treanţă [16,17,18]. For other (different but connected) ideas on well-posedness and differential inclusions of the second-order, the reader is directed to Antonelli et al. [19], Sawano [20], Lakzian and Munn [21], Vijayakumar et al. [22].

In this paper, motivated by previous research works, we studied the well-posedness associated with a class of controlled variational inequality problems. More specifically, by considering the concepts of hemicontinuity, monotonicity, and pseudomonotonicity for functionals of path-independent curvilinear integral types, and by defining the approximating solution set of the considered problem, we established some results on well-posedness. The main novelty elements of this paper are given by the presence of the curvilinear integral functionals governed by second-order partial derivatives, and of the mathematical context determined by function spaces of infinite-dimensions. Moreover, due to the new elements highlighted above and thanks to the physical significance of the involved functionals (the path-independent curvilinear integrals compute the mechanical work done by a variable force, for moving its application point along a piecewise differentiable curve), this paper represents important work for researchers in the fields of abstract and applied mathematics.

The paper is structured as follows. In Section 2, we introduce the preliminary mathematical tools, namely, the notions of pseudomonotonicity, monotonicity, and hemicontinuity of an integral functional of curvilinear type, and an auxiliary lemma. The well-posedness of the problem under study is investigated in Section 3 by considering the approximating solution set. Concretely, we establish that well-posedness is characterized in terms of existence and uniqueness of the solution. Moreover, to validate the theoretical results, we also provide some illustrative examples. In Section 4, we conclude the paper.

2. Preliminaries

In the following, in accordance with Treanţă [15,16], we consider: ${\displaystyle \mathsf{\Omega }}$ is a compact set in ${\displaystyle {\mathbb{R}}^m}$, ${\displaystyle \mathsf{\Omega }\ni s=\left(s^{\nu }\right),\nu =\overline{1,m}}$, is a multi-variate evolution parameter, ${\displaystyle \mathsf{\Omega }\supset \mathsf{{\rm Y}}}$ is a piecewise differentiable curve that links the points ${\displaystyle s_1=(s_1^1,\dots ,s_1^m),s_2=(s_2^1,\dots ,s_2^m)}$ in ${\displaystyle \mathsf{\Omega }}$; consider ${\displaystyle \mathcal{P}}$ is the space of $C^4$-class state functions ${\displaystyle p:\mathsf{\Omega }\to {\mathbb{R}}^n}$ and ${\displaystyle p_{\kappa }:=\frac{\partial p}{\partial s^{\kappa }},p_{\alpha \beta }:=\frac{{\partial }^{2}p}{\partial s^{\alpha }\partial s^{\beta }}}$ denote the partial speed and partial acceleration, respectively; also, let ${\displaystyle \mathsf{\Xi }}$ be the space of $C^1$-class control functions ${\displaystyle y:\mathsf{\Omega }\to {\mathbb{R}}^k}$, and assume that ${\displaystyle P\times \mathsf{\Xi }}$ is a (nonempty) convex and closed subset of ${\displaystyle \mathcal{P}\times \mathsf{\Xi }}$, equipped with

$${\displaystyle ⟨(p,y),(q,z)⟩={\int }_{\mathsf{{\rm Y}}}[p\left(s\right)·q\left(s\right)+y\left(s\right)·z\left(s\right)]d{s}^{\nu }}$$

$${\displaystyle ={\int }_{\mathsf{{\rm Y}}}[\sum _{i=1}^n{p}^i\left(s\right)q^i\left(s\right)+\sum _{j=1}^k{y}^j\left(s\right)z^j\left(s\right)]d{s}^{\nu }}$$

$${\displaystyle ={\int }_{\mathsf{{\rm Y}}}[\sum _{i=1}^n{p}^i\left(s\right)q^i\left(s\right)+\sum _{j=1}^k{y}^j\left(s\right)z^j\left(s\right)]d{s}^1+\cdots +[\sum _{i=1}^n{p}^i\left(s\right)q^i\left(s\right)+\sum _{j=1}^k{y}^j\left(s\right)z^j\left(s\right)]d{s}^m,}$$

$${\displaystyle \forall (p,y),(q,z)\in \mathcal{P}\times \mathsf{\Xi }}$$

and the norm induced by it.

Let ${\displaystyle J^2({\mathbb{R}}^m,{\mathbb{R}}^n)}$ be the jet bundle of the second-order of ${\displaystyle {\mathbb{R}}^m}$ and ${\displaystyle {\mathbb{R}}^n}$. Assume that the following second-order Lagrangians ${\displaystyle f_{\nu }:J^2({\mathbb{R}}^m,{\mathbb{R}}^n)\times {\mathbb{R}}^k\to \mathbb{R},\nu =\overline{1,m}}$ provide a controlled closed Lagrange 1-form (see Einstein summation)

$${\displaystyle f_{\nu }(s,p\left(s\right),p_{\kappa }\left(s\right),p_{\alpha \beta }\left(s\right),y\left(s\right))d{s}^{\nu },}$$

which gives the following integral functional

$${\displaystyle F:\mathcal{P}\times \mathsf{\Xi }\to \mathbb{R},F(p,y)={\int }_{\mathsf{{\rm Y}}}{f}_{\nu }\left(s,p\left(s\right),p_{\kappa }\left(s\right),p_{\alpha \beta }\left(s\right),y\left(s\right)\right)d{s}^{\nu }}$$

$${\displaystyle ={\int }_{\mathsf{{\rm Y}}}{f}_1\left(s,p\left(s\right),p_{\kappa }\left(s\right),p_{\alpha \beta }\left(s\right),y\left(s\right)\right)d{s}^1+\cdots +f_m\left(s,p\left(s\right),p_{\kappa }\left(s\right),p_{\alpha \beta }\left(s\right),y\left(s\right)\right)d{s}^m.}$$

To formulate the problem under study, we shall introduce the Saunders’s multi-index notation (see Saunders [23]).

Now, we introduce the variational problem: find $(p,y)\in P\times \mathsf{\Xi }$ such that

$${\displaystyle {\int }_{\mathsf{{\rm Y}}}\left[{\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{p,y}\left(s\right)\right)(q\left(s\right)-p\left(s\right))+{\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{p,y}\left(s\right)\right)D_{\kappa }(q\left(s\right)-p\left(s\right))\right]d{s}^{\nu }}$$

$${\displaystyle +{\int }_{\mathsf{{\rm Y}}}\left[\frac{1}{x(\alpha ,\beta )}{\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{p,y}\left(s\right)\right)D_{\alpha \beta }^2(q\left(s\right)-p\left(s\right))\right]d{s}^{\nu }}$$

$${\displaystyle +{\int }_{\mathsf{{\rm Y}}}\left[{\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{p,y}\left(s\right)\right)(z\left(s\right)-y\left(s\right))\right]d{s}^{\nu }\ge 0,\forall (q,z)\in P\times \mathsf{\Xi },}$$

where ${\displaystyle D_{\kappa }:=\frac{\partial }{\partial s^{\kappa }}}$ is the total derivative operator, ${\displaystyle D_{\alpha \beta }^2:=D_{\alpha }\left(D_{\beta }\right)}$, and ${\displaystyle \left({\psi }_{p,y}\left(s\right)\right):=(s,p\left(s\right),p_{\kappa }\left(s\right),p_{\alpha \beta }\left(s\right),y\left(s\right))}$.

Let K be the feasible solution set of (variational problem equation),

$$\begin{array}{cc}\hfill & K=\{(p,y)\in P\times \mathsf{\Xi }:{\int }_{\mathsf{{\rm Y}}}[(q\left(s\right)-p\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{p,y}\left(s\right)\right)\hfill \\ \hfill & +D_{\kappa }(q\left(s\right)-p\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{p,y}\left(s\right)\right)\hfill \\ \hfill & +\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(q\left(s\right)-p\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{p,y}\left(s\right)\right)\hfill \\ \hfill & +(z\left(s\right)-y\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{p,y}\left(s\right)\right)]d{s}^{\nu }\ge 0,\hfill \\ \hfill & \forall (q,z)\in P\times \mathsf{\Xi }\}.\hfill \end{array}$$

Assumption 1.

The working hypothesis is assumed:

$${\displaystyle dG:=D_{\kappa }\left[\frac{\partial f_{\nu }}{\partial p_{\kappa }}(p-q)\right]d{s}^{\nu }}$$

is a total exact differential, with ${\displaystyle G\left(s_1\right)=G\left(s_2\right)}$.

Further, in accordance to assumption (hypothesis equation) and by considering the notion of monotonicity associated with variational inequalities, we introduce the monotonicity and pseudomonotonicity of the above functional ${\displaystyle F}$.

Definition 1.

The functional ${\displaystyle F}$ is monotone on $P\times \mathsf{\Xi }$ if:

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{{\rm Y}}}[(p\left(s\right)-q\left(s\right))\left({\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{p,y}\left(s\right)\right)-{\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{q,z}\left(s\right)\right)\right)\hfill \\ \hfill & +(y\left(s\right)-z\left(s\right))\left({\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{p,y}\left(s\right)\right)-{\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{q,z}\left(s\right)\right)\right)\hfill \\ \hfill & +D_{\kappa }(p\left(s\right)-q\left(s\right))\left({\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{p,y}\left(s\right)\right)-{\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{q,z}\left(s\right)\right)\right)\hfill \\ \hfill & +\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(p\left(s\right)-q\left(s\right))\left({\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{p,y}\left(s\right)\right)-{\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{q,z}\left(s\right)\right)\right)]d{s}^{\nu }\ge 0,\hfill \end{array}$$

$$\forall (p,y),(q,z)\in P\times \mathsf{\Xi }$$

is satisfied.

Example 1.

For $\mathsf{\Omega }={[0,1]}^2,\nu \in \{1,2\}$ and ${\displaystyle \mathsf{\Omega }\supset \mathsf{{\rm Y}}}$ a piecewise differentiable curve that links ${\displaystyle (0,0),(1,1)}$ in ${\displaystyle \mathsf{\Omega }}$, we define

$$f_{\nu }\left({\psi }_{p,y}\left(s\right)\right)d{s}^{\nu }=f_1\left({\psi }_{p,y}\left(s\right)\right)d{s}^1+f_2\left({\psi }_{p,y}\left(s\right)\right)d{s}^2$$

$$=\left[\frac{\partial p}{\partial s^1}+y\left(s\right)\right]d{s}^1+(e^{p\left(s\right)}-1)d{s}^2.$$

The functional ${\displaystyle {\int }_{\mathsf{{\rm Y}}}{f}_{\nu }\left({\psi }_{p,y}\left(s\right)\right)d{s}^{\nu }}$ is monotone on $P\times \mathsf{\Xi }=C^4(\mathsf{\Omega },\mathbb{R})\times C^1(\mathsf{\Omega },\mathbb{R})$ since

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{{\rm Y}}}[(p\left(s\right)-q\left(s\right))\left({\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{p,y}\left(s\right)\right)-{\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{q,z}\left(s\right)\right)\right)\hfill \\ \hfill & +(y\left(s\right)-z\left(s\right))\left({\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{p,y}\left(s\right)\right)-{\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{q,z}\left(s\right)\right)\right)\hfill \\ \hfill & +D_{\kappa }(p\left(s\right)-q\left(s\right))\left({\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{p,y}\left(s\right)\right)-{\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{q,z}\left(s\right)\right)\right)\hfill \\ \hfill & +\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(p\left(s\right)-q\left(s\right))\left({\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{p,y}\left(s\right)\right)-{\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{q,z}\left(s\right)\right)\right)]d{s}^{\nu }\hfill \\ \hfill & ={\int }_{\mathsf{{\rm Y}}}(p\left(s\right)-q\left(s\right))(e^{p\left(s\right)}-e^{q\left(s\right)})d{s}^2\ge 0,\forall (p,y),(q,z)\in P\times \mathsf{\Xi }.\hfill \end{array}$$

Definition 2.

The functional ${\displaystyle F}$ is pseudo-monotone on $P\times \mathsf{\Xi }$ if:

$${\int }_{\mathsf{{\rm Y}}}[(p\left(s\right)-q\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{q,z}\left(s\right)\right)+(y\left(s\right)-z\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{q,z}\left(s\right)\right)$$

$$+D_{\kappa }(p\left(s\right)-q\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{q,z}\left(s\right)\right)$$

$$+\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(p\left(s\right)-q\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{q,z}\left(s\right)\right)]d{s}^{\nu }\ge 0$$

$$\Rightarrow {\int }_{\mathsf{{\rm Y}}}[(p\left(s\right)-q\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{p,y}\left(s\right)\right)+(y\left(s\right)-z\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{p,y}\left(s\right)\right)$$

$$+D_{\kappa }(p\left(s\right)-q\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{p,y}\left(s\right)\right)$$

$$+\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(p\left(s\right)-q\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{p,y}\left(s\right)\right)]d{s}^{\nu }\ge 0,$$

$$\forall (p,y),(q,z)\in P\times \mathsf{\Xi }$$

is valid.

Example 2.

For $\mathsf{\Omega }={[0,1]}^2,\nu \in \{1,2\}$ and ${\displaystyle \mathsf{\Omega }\supset \mathsf{{\rm Y}}}$ a piecewise differentiable curve that links ${\displaystyle (0,0),(1,1)}$ in ${\displaystyle \mathsf{\Omega }}$, we define

$$f_{\nu }\left({\psi }_{p,y}\left(s\right)\right)d{s}^{\nu }=f_1\left({\psi }_{p,y}\left(s\right)\right)d{s}^1+f_2\left({\psi }_{p,y}\left(s\right)\right)d{s}^2$$

$$=\left[\frac{\partial p}{\partial s^1}+siny\left(s\right)\right]d{s}^1+p\left(s\right)e^{p\left(s\right)}d{s}^2.$$

The functional ${\displaystyle {\int }_{\mathsf{{\rm Y}}}{f}_{\nu }\left({\psi }_{p,y}\left(s\right)\right)d{s}^{\nu }}$ is pseudo-monotone on

$$P\times \mathsf{\Xi }=C^4(\mathsf{\Omega },[-1,1])\times C^1(\mathsf{\Omega },[-1,1])$$

since

$${\int }_{\mathsf{{\rm Y}}}[(p\left(s\right)-q\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{q,z}\left(s\right)\right)+(y\left(s\right)-z\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{q,z}\left(s\right)\right)$$

$$+D_{\kappa }(p\left(s\right)-q\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{q,z}\left(s\right)\right)$$

$$+\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(p\left(s\right)-q\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{q,z}\left(s\right)\right)]d{s}^{\nu }$$

$$\begin{array}{cc}\hfill & ={\int }_{\mathsf{{\rm Y}}}\left[(y\left(s\right)-z\left(s\right))cosz\left(s\right)+D_1(p\left(s\right)-q\left(s\right))\right]d{s}^1\hfill \\ \hfill & +(p\left(s\right)-q\left(s\right))(e^{q\left(s\right)}+q\left(s\right)e^{q\left(s\right)})d{s}^2\ge 0,\forall (p,y),(q,z)\in P\times \mathsf{\Xi }\hfill \end{array}$$

$$\Rightarrow {\int }_{\mathsf{{\rm Y}}}[(p\left(s\right)-q\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{p,y}\left(s\right)\right)+(y\left(s\right)-z\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{p,y}\left(s\right)\right)$$

$$+D_{\kappa }(p\left(s\right)-q\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{p,y}\left(s\right)\right)$$

$$+\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(p\left(s\right)-q\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{p,y}\left(s\right)\right)]d{s}^{\nu }$$

$$\begin{array}{cc}\hfill & ={\int }_{\mathsf{{\rm Y}}}\left[(y\left(s\right)-z\left(s\right))cosy\left(s\right)+D_1(p\left(s\right)-q\left(s\right))\right]d{s}^1\hfill \\ \hfill & +(p\left(s\right)-q\left(s\right))(e^{p\left(s\right)}+p\left(s\right)e^{p\left(s\right)})d{s}^2\ge 0,\forall (p,y),(q,z)\in P\times \mathsf{\Xi }.\hfill \end{array}$$

On the other hand, it is not monotone on $P\times \mathsf{\Xi }$,

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{{\rm Y}}}[(p\left(s\right)-q\left(s\right))\left({\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{p,y}\left(s\right)\right)-{\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{q,z}\left(s\right)\right)\right)\hfill \\ \hfill & +(y\left(s\right)-z\left(s\right))\left({\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{p,y}\left(s\right)\right)-{\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{q,z}\left(s\right)\right)\right)\hfill \\ \hfill & +D_{\kappa }(p\left(s\right)-q\left(s\right))\left({\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{p,y}\left(s\right)\right)-{\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{q,z}\left(s\right)\right)\right)\hfill \\ \hfill & +\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(p\left(s\right)-q\left(s\right))\left({\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{p,y}\left(s\right)\right)-{\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{q,z}\left(s\right)\right)\right)]d{s}^{\nu }\hfill \\ \hfill & ={\int }_{\mathsf{{\rm Y}}}(y\left(s\right)-z\left(s\right))(cosy\left(s\right)-cosz\left(s\right))d{s}^1\hfill \\ \hfill & +(p\left(s\right)-q\left(s\right))(p\left(s\right)e^{p\left(s\right)}+e^{p\left(s\right)}-q\left(s\right)e^{q\left(s\right)}-e^{q\left(s\right)})d{s}^2\ngeqq 0,\hfill \\ \hfill & \forall (p,y),(q,z)\in P\times \mathsf{\Xi }.\hfill \end{array}$$

By using Usman and Khan [24], we introduce the following definition:

Definition 3.

The functional ${\displaystyle F}$ is hemicontinuous on $P\times \mathsf{\Xi }$ if

$$\begin{array}{c}\hfill \lambda \to ⟨\left(\left(p\right(s),y(s\left)\right)-\left(q\right(s),z(s)\right),\left(\frac{{\delta }_{\nu }F}{\delta p_{\lambda }},\frac{{\delta }_{\nu }F}{\delta y_{\lambda }}\right)⟩,0\le \lambda \le 1\end{array}$$

is continuous at $0^{+}$, for $\forall (p,y),(q,z)\in P\times \mathsf{\Xi }$, where

$$\frac{{\delta }_{\nu }F}{\delta p_{\lambda }}:=\frac{\partial f_{\nu }}{\partial p}\left({\psi }_{p_{\lambda },y_{\lambda }}\left(s\right)\right)-D_{\kappa }{\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{p_{\lambda },y_{\lambda }}\left(s\right)\right)+\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2{\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{p_{\lambda },y_{\lambda }}\left(s\right)\right)\in P,$$

$$\frac{{\delta }_{\nu }F}{\delta y_{\lambda }}:={\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{p_{\lambda },y_{\lambda }}\left(s\right)\right)\in \mathsf{\Xi },$$

$$p_{\lambda }:=\lambda p+(1-\lambda )q,y_{\lambda }:=\lambda y+(1-\lambda )z.$$

Lemma 1.

Let the functional ${\displaystyle F}$ be hemicontinuous and pseudo-monotone on $P\times \mathsf{\Xi }$. A point $(p,y)\in P\times \mathsf{\Xi }$ solves (variational problem equation) if and only if $(p,y)\in P\times \mathsf{\Xi }$ solves

$${\int }_{\mathsf{{\rm Y}}}[(q\left(s\right)-p\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{q,z}\left(s\right)\right)+(z\left(s\right)-y\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{q,z}\left(s\right)\right)$$

$$+D_{\kappa }(q\left(s\right)-p\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{q,z}\left(s\right)\right)$$

$$+\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(q\left(s\right)-p\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{q,z}\left(s\right)\right)]d{s}^{\nu }\ge 0,\forall (q,z)\in P\times \mathsf{\Xi }.$$

Proof. 

Firstly, we consider the pair $(p,y)\in P\times \mathsf{\Xi }$ solves (variational problem equation); that is,

$${\int }_{\mathsf{{\rm Y}}}[(q\left(s\right)-p\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{p,y}\left(s\right)\right)+(z\left(s\right)-y\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{p,y}\left(s\right)\right)$$

$$+D_{\kappa }(q\left(s\right)-p\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{p,y}\left(s\right)\right)$$

$$+\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(q\left(s\right)-p\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{p,y}\left(s\right)\right)]d{s}^{\nu }\ge 0,\forall (q,z)\in P\times \mathsf{\Xi }.$$

By pseudomonotonicity, we have

$${\int }_{\mathsf{{\rm Y}}}[(q\left(s\right)-p\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{q,z}\left(s\right)\right)+(z\left(s\right)-y\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{q,z}\left(s\right)\right)$$

$$+D_{\kappa }(q\left(s\right)-p\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{q,z}\left(s\right)\right)$$

$$+\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(q\left(s\right)-p\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{q,z}\left(s\right)\right)]d{s}^{\nu }\ge 0,\forall (q,z)\in P\times \mathsf{\Xi }.$$

Conversely, assume that

$${\int }_{\mathsf{{\rm Y}}}[(q\left(s\right)-p\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{q,z}\left(s\right)\right)+(z\left(s\right)-y\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{q,z}\left(s\right)\right)$$

$$+D_{\kappa }(q\left(s\right)-p\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{q,z}\left(s\right)\right)$$

$$+\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(q\left(s\right)-p\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{q,z}\left(s\right)\right)]d{s}^{\nu }\ge 0,\forall (q,z)\in P\times \mathsf{\Xi }.$$

Now, for $(q,z)\in P\times \mathsf{\Xi }$ and $\lambda \in (0,1]$, we consider

$${\displaystyle (q_{\lambda },z_{\lambda })=((1-\lambda )p+\lambda q,(1-\lambda )y+\lambda z)\in P\times \mathsf{\Xi }.}$$

The above inequality is reformulated as

$${\int }_{\mathsf{{\rm Y}}}[(q_{\lambda }\left(s\right)-p\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{q_{\lambda },z_{\lambda }}\left(s\right)\right)+(z_{\lambda }\left(s\right)-y\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{q_{\lambda },z_{\lambda }}\left(s\right)\right)$$

$$+D_{\kappa }(q_{\lambda }\left(s\right)-p\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{q_{\lambda },z_{\lambda }}\left(s\right)\right)$$

$$+\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(q_{\lambda }\left(s\right)-p\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{q_{\lambda },z_{\lambda }}\left(s\right)\right)]d{s}^{\nu }\ge 0,(q,z)\in P\times \mathsf{\Xi }.$$

Taking $\lambda \to 0$ and considering the hemicontinuity of the integral functional, we obtain

$${\int }_{\mathsf{{\rm Y}}}[(q\left(s\right)-p\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{p,y}\left(s\right)\right)+(z\left(s\right)-y\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{p,y}\left(s\right)\right)$$

$$+D_{\kappa }(q\left(s\right)-p\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{p,y}\left(s\right)\right)$$

$$+\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(q\left(s\right)-p\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{p,y}\left(s\right)\right)]d{s}^{\nu }\ge 0,\forall (q,z)\in P\times \mathsf{\Xi },$$

showing that $(p,y)$ solves (variational problem equation). □

3. Main Results

In this section, we investigate the well-posedness of the considered variational inequality problem involving second-order PDEs.

Definition 4.

The sequence $\left\{(p_n,y_n)\right\}\in P\times \mathsf{\Xi }$ is called an approximating sequence of (variational problem equation) if there exists a sequence of positive real numbers ${\sigma }_n\to 0$ as $n\to \infty$, such that the following inequality holds:

$${\int }_{\mathsf{{\rm Y}}}[(q\left(s\right)-p_n\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{p_n,y_n}\left(s\right)\right)+(z\left(s\right)-y_n\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{p_n,y_n}\left(s\right)\right)$$

$$+D_{\kappa }(q\left(s\right)-p_n\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{p_n,y_n}\left(s\right)\right)$$

$$+\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(q\left(s\right)-p_n\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{p_n,y_n}\left(s\right)\right)]d{s}^{\nu }+{\sigma }_n\ge 0,\forall (q,z)\in P\times \mathsf{\Xi }.$$

Definition 5.

The problem (variational problem equation) is called well-posed if:

(i) 

The problem (variational problem equation) has one solution $(p_0,y_0)$;

(ii) 

Each approximating sequence of (variational problem equation) converges to $(p_0,y_0)$.

The approximating solution set of (variational problem equation) is given as follows:

$$\begin{array}{cc}\hfill & K_{\sigma }=\{(p,y)\in P\times \mathsf{\Xi }:{\int }_{\mathsf{{\rm Y}}}[(q\left(s\right)-p\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{p,y}\left(s\right)\right)+(z\left(s\right)-y\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{p,y}\left(s\right)\right)\hfill \\ \hfill & +D_{\kappa }(q\left(s\right)-p\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{p,y}\left(s\right)\right)\hfill \\ \hfill & +\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(q\left(s\right)-p\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{p,y}\left(s\right)\right)]d{s}^{\nu }+\sigma \ge 0,\forall (q,z)\in P\times \mathsf{\Xi }\}.\hfill \end{array}$$

Remark 1.

We have: $K=K_{\sigma }$, when $\sigma =0$ and $K\subseteq K_{\sigma },\forall \sigma >0.$

Further, for a set B, the diameter of B is defined as follows

$$\mathrm{diam}B=\underset{\varphi ,\eta \in B}{sup}\parallel \varphi -\eta \parallel .$$

Theorem 1.

Let the functional ${\displaystyle F}$ be hemicontinuous and monotone on $P\times \mathsf{\Xi }$. The problem (variational problem equation) is well-posed if and only if

$$K_{\sigma }\ne \varnothing ,\forall \sigma >0\mathit{and}\mathit{diam}{K}_{\sigma }\to 0\mathit{as}\sigma \to 0.$$

Proof. 

Let us consider the problem (variational problem equation) is well-posed. By Definition 5, it has one solution $(\overline{p},\overline{y})\in K$. Since $K\subseteq K_{\sigma },\forall \sigma >0$; therefore, $K_{\sigma }\ne \varnothing ,\forall \sigma >0$. Contrary, we consider that $\mathrm{diam}{K}_{\sigma }\nrightarrow 0\mathrm{as}\sigma \to 0.$ Then there exist $r>0$, a positive integer m, a sequence of real numbers ${\sigma }_n>0$ with ${\sigma }_n\to 0$, and two elements $(p_n,y_n)$ and $(p_n^{\prime },y_n^{\prime })\in K_{{\sigma }_n}$, so that

$$\parallel (p_n\left(s\right),y_n\left(s\right))-(p_n^{\prime }\left(s\right),y_n^{\prime }\left(s\right))\parallel >r,\forall n\ge m.$$

(1)

Since $(p_n,y_n),(p_n^{\prime },y_n^{\prime })\in K_{{\sigma }_n}$, we have

$${\int }_{\mathsf{{\rm Y}}}[(q\left(s\right)-p_n\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{p_n,y_n}\left(s\right)\right)+(z\left(s\right)-y_n\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{p_n,y_n}\left(s\right)\right)$$

$$+D_{\kappa }(q\left(s\right)-p_n\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{p_n,y_n}\left(s\right)\right)$$

$$+\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(q\left(s\right)-p_n\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{p_n,y_n}\left(s\right)\right)]d{s}^{\nu }+{\sigma }_n\ge 0,\forall (q,z)\in P\times \mathsf{\Xi }$$

and

$${\int }_{\mathsf{{\rm Y}}}[(q\left(s\right)-p_n^{\prime }\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{p_n^{\prime },y_n^{\prime }}\left(s\right)\right)+(z\left(s\right)-y_n^{\prime }\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{p_n^{\prime },y_n^{\prime }}\left(s\right)\right)$$

$$+D_{\kappa }(q\left(s\right)-p_n^{\prime }\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{p_n^{\prime },y_n^{\prime }}\left(s\right)\right)$$

$$+\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(q\left(s\right)-p_n^{\prime }\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{alpha\beta }}}\left({\psi }_{p_n^{\prime },y_n^{\prime }}\left(s\right)\right)]d{s}^{\nu }+{\sigma }_n\ge 0,\forall (q,z)\in P\times \mathsf{\Xi }.$$

It results that $\left\{(p_n,y_n)\right\}$ and $\left\{(p_n^{\prime },y_n^{\prime })\right\}$ are approximating sequences of (variational problem equation) converging to $(\overline{p},\overline{y})$. By computation, we obtain

$$\parallel (p_n\left(s\right),y_n\left(s\right))-(p_n^{\prime }\left(s\right),y_n^{\prime }\left(s\right))\parallel$$

$$=\parallel (p_n\left(s\right),y_n\left(s\right))-(\overline{p}\left(s\right),\overline{y}\left(s\right))+(\overline{p}\left(s\right),\overline{y}\left(s\right))-(p_n^{\prime }\left(s\right),y_n^{\prime }\left(s\right))\parallel$$

$$\le \parallel (p_n\left(s\right),y_n\left(s\right))-(\overline{p}\left(s\right),\overline{y}\left(s\right))\parallel +\parallel (\overline{p}\left(s\right),\overline{y}\left(s\right))-(p_n^{\prime }\left(s\right),y_n^{\prime }\left(s\right))\parallel \le \sigma ,$$

which enters in contradiction with (1), for some $\sigma =r$.

Now, let us consider $\left\{(p_n,y_n)\right\}$ is an approximating sequence of (variational problem equation). Thus, there exists a sequence of positive real numbers ${\sigma }_n\to 0$ as $n\to \infty$ so that

$${\int }_{\mathsf{{\rm Y}}}[(q\left(s\right)-p_n\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{p_n,y_n}\left(s\right)\right)+(z\left(s\right)-y_n\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{p_n,y_n}\left(s\right)\right)$$

$$+D_{\kappa }(q\left(s\right)-p_n\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{p_n,y_n}\left(s\right)\right)$$

$$+\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2\left((q\left(s\right)-p_n\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{p_n,y_n}\left(s\right)\right)\right]d{s}^{\nu }+{\sigma }_n\ge 0,\forall (q,z)\in P\times \mathsf{\Xi }$$

(2)

is true, involving $(p_n,y_n)\in K_{{\sigma }_n}$. Since $\mathrm{diam}{K}_{{\sigma }_n}\to 0\mathrm{as}{\sigma }_n\to 0$, we have $\left\{(p_n,y_n)\right\}$ is a Cauchy sequence converging to some $(\overline{p},\overline{y})\in P\times \mathsf{\Xi }$.

Since the functional ${\displaystyle {\int }_{\mathsf{{\rm Y}}}{f}_{\nu }\left({\psi }_{p,y}\left(s\right)\right)d{s}^{\nu }}$ is monotone on $P\times \mathsf{\Xi }$, for $(\overline{p},\overline{y}),(q,z)\in P\times \mathsf{\Xi }$, we have

$${\int }_{\mathsf{{\rm Y}}}[(\overline{p}\left(s\right)-q\left(s\right))\left({\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{\overline{p},\overline{y}}\left(s\right)\right)-{\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{q,z}\left(s\right)\right)\right)$$

$$+(\overline{y}\left(s\right)-z\left(s\right))\left({\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{\overline{p},\overline{y}}\left(s\right)\right)-{\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{q,z}\left(s\right)\right)\right)$$

$$+D_{\kappa }(\overline{p}\left(s\right)-q\left(s\right))\left({\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{\overline{p},\overline{y}}\left(s\right)\right)-{\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{q,z}\left(s\right)\right)\right)$$

$$+\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(\overline{p}\left(s\right)-q\left(s\right))\left({\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{\overline{p},\overline{y}}\left(s\right)\right)-{\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{q,z}\left(s\right)\right)\right)]d{s}^{\nu }\ge 0$$

or, equivalently,

$${\int }_{\mathsf{{\rm Y}}}[(\overline{p}\left(s\right)-q\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{\overline{p},\overline{y}}\left(s\right)\right)+(\overline{y}\left(s\right)-z\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{\overline{p},\overline{y}}\left(s\right)\right)$$

$$+D_{\kappa }(\overline{p}\left(s\right)-q\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{\overline{p},\overline{y}}\left(s\right)\right)$$

$$+\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(\overline{p}\left(s\right)-q\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{\overline{p},\overline{y}}\left(s\right)\right)]d{s}^{\nu }$$

$$\ge {\int }_{\mathsf{{\rm Y}}}[(\overline{p}\left(s\right)-q\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{q,z}\left(s\right)\right)+(\overline{y}\left(s\right)-z\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{q,z}\left(s\right)\right)$$

$$+D_{\kappa }(\overline{p}\left(s\right)-q\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{q,z}\left(s\right)\right)$$

$$+\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(\overline{p}\left(s\right)-q\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{q,z}\left(s\right)\right)]d{s}^{\nu }.$$

(3)

Taking the limit in (2), we obtain

$${\int }_{\mathsf{{\rm Y}}}[(\overline{p}\left(s\right)-q\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{\overline{p},\overline{y}}\left(s\right)\right)+(\overline{y}\left(s\right)-z\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{\overline{p},\overline{y}}\left(s\right)\right)$$

$$+D_{\kappa }(\overline{p}\left(s\right)-q\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{\overline{p},\overline{y}}\left(s\right)\right)$$

$$+\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(\overline{p}\left(s\right)-q\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{\overline{p},\overline{y}}\left(s\right)\right)]d{s}^{\nu }\le 0,\forall (q,z)\in P\times \mathsf{\Xi }.$$

(4)

By (3) and (4), we have

$${\int }_{\mathsf{{\rm Y}}}[(q\left(s\right)-\overline{p}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{q,z}\left(s\right)\right)+(z\left(s\right)-\overline{y}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{q,z}\left(s\right)\right)$$

$$+D_{\kappa }(q\left(s\right)-\overline{p}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{q,z}\left(s\right)\right)$$

$$+\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(q\left(s\right)-\overline{p}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{q,z}\left(s\right)\right)]d{s}^{\nu }\ge 0,\forall (q,z)\in P\times \mathsf{\Xi }.$$

Further, by Lemma 1, we have

$${\int }_{\mathsf{{\rm Y}}}[(q\left(s\right)-\overline{p}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{\overline{p},\overline{y}}\left(s\right)\right)+(z\left(s\right)-\overline{y}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{\overline{p},\overline{y}}\left(s\right)\right)$$

$$+D_{\kappa }(q\left(s\right)-\overline{p}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{\overline{p},\overline{y}}\left(s\right)\right)$$

$$+\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(q\left(s\right)-\overline{p}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{\overline{p},\overline{y}}\left(s\right)\right)]d{s}^{\nu }\ge 0,\forall (q,z)\in P\times \mathsf{\Xi },$$

which implies that $(\overline{p},\overline{y})\in K$. Now, we prove that $(\overline{p},\overline{y})$ is the single solution of (variational problem equation). We suppose that $(p_1,y_1),(p_2,y_2)$ are two different solutions of (variational problem equation). Then

$$\begin{array}{c}\hfill 0<\parallel (p_1\left(s\right),y_1\left(s\right))-(p_2\left(s\right),y_2\left(s\right))\parallel \le \mathrm{diam}{K}_{\sigma }\to 0\mathrm{as}\sigma \to 0,\end{array}$$

which is a contradiction. □

Theorem 2.

Let the functional ${\displaystyle F}$ be hemicontinuous and monotone on $P\times \mathsf{\Xi }$. Then (variational problem equation) is well-posed if and only if it has one solution.

Proof. 

Consider the problem (variational problem equation) is well-posed. Consequently, by Definition 5, it has one solution $(p_0,y_0)$. Conversely, suppose that (variational problem equation) has a unique solution $(p_0,y_0)$ but it is not well-posed. Thus, there exists an approximating sequence $\left\{(p_n,y_n)\right\}$ of (variational problem equation), which does not converge to $(p_0,y_0)$. Since $\left\{(p_n,y_n)\right\}$ is an approximating sequence of (variational problem equation), there must exist a sequence of positive real numbers ${\sigma }_n\to 0$ as $n\to \infty$ such that

$${\int }_{\mathsf{{\rm Y}}}[(q\left(s\right)-p_n\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{p_n,y_n}\left(s\right)\right)+(z\left(s\right)-y_n\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{p_n,y_n}\left(s\right)\right)$$

$$+D_{\kappa }(q\left(s\right)-p_n\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{p_n,y_n}\left(s\right)\right)$$

$$+\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(q\left(s\right)-p_n\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{p_n,y_n}\left(s\right)\right)]d{s}^{\nu }+{\sigma }_n\ge 0,\forall (q,z)\in P\times \mathsf{\Xi }.$$

(5)

To prove the boundedness of $\left\{(p_n,y_n)\right\}$, suppose $\left\{(p_n,y_n)\right\}$ is not bounded; that is, $\parallel (p_n\left(s\right),y_n\left(s\right))\parallel \to +\infty$ as $n\to +\infty$. Consider ${\delta }_n\left(s\right)={\displaystyle \frac{1}{\parallel (p_n\left(s\right),y_n\left(s\right))-(p_0\left(s\right),y_0\left(s\right))\parallel }}$ and $({\mathsf{p}}_n,{\mathsf{y}}_n)=(p_0,y_0)+{\delta }_n[(p_n,y_n)-(p_0,y_0)]$.

We obtain that $\left\{({\mathsf{p}}_n,{\mathsf{y}}_n)\right\}$ is bounded in $P\times \mathsf{\Xi }$. So, we may assume that

$$({\mathsf{p}}_n,{\mathsf{y}}_n)\to (\mathsf{p},\mathsf{y})\mathrm{weakly} \mathrm{in} P\times \mathsf{\Xi }\ne (p_0,y_0).$$

It is not difficult to verify that $(\mathsf{p},\mathsf{y})\ne (p_0,y_0)$, thanks to

$$\parallel {\delta }_n\left(s\right)[(p_n\left(s\right),y_n\left(s\right))-(p_0\left(s\right),y_0\left(s\right))]\parallel =1,$$

for all $n\in \mathbb{N}$. Since $(p_0,y_0)$ is a solution of (variational problem equation); therefore,

$${\int }_{\mathsf{{\rm Y}}}[(q\left(s\right)-p_0\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{p_0,y_0}\left(s\right)\right)+(z\left(s\right)-y_0\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{p_0,y_0}\left(s\right)\right)$$

$$+D_{\kappa }(q\left(s\right)-p_0\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{p_0,y_0}\left(s\right)\right)$$

$$+\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(q\left(s\right)-p_0\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{p_0,y_0}\left(s\right)\right)]d{s}^{\nu }\ge 0,\forall (q,z)\in P\times \mathsf{\Xi }.$$

Thus, by considering Lemma 1, we have

$${\int }_{\mathsf{{\rm Y}}}[(q\left(s\right)-p_0\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{q,z}\left(s\right)\right)+(z\left(s\right)-y_0\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{q,z}\left(s\right)\right)$$

$$+D_{\kappa }(q\left(s\right)-p_0\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{q,z}\left(s\right)\right)$$

$$+\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(q\left(s\right)-p_0\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{q,z}\left(s\right)\right)]d{s}^{\nu }\ge 0,\forall (q,z)\in P\times \mathsf{\Xi }.$$

(6)

The integral functional ${\displaystyle F}$ is monotone on $P\times \mathsf{\Xi }$. Consequently, for $(p_n,y_n),(q,z)\in P\times \mathsf{\Xi }$, we have

$${\int }_{\mathsf{{\rm Y}}}[(p_n\left(s\right)-q\left(s\right))\left({\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{p_n,y_n}\left(s\right)\right)-{\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{q,z}\left(s\right)\right)\right)$$

$$+(y_n\left(s\right)-z\left(s\right))\left({\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{p_n,y_n}\left(s\right)\right)-{\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{q,z}\left(s\right)\right)\right)$$

$$+D_{\kappa }(p_n\left(s\right)-q\left(s\right))\left({\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{p_n,y_n}\left(s\right)\right)-{\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{q,z}\left(s\right)\right)\right)$$

$$+\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(p_n\left(s\right)-q\left(s\right))\left({\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{p_n,y_n}\left(s\right)\right)-{\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{q,z}\left(s\right)\right)\right)]d{s}^{\nu }\ge 0,$$

or, equivalently,

$${\int }_{\mathsf{{\rm Y}}}[(q\left(s\right)-p_n\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{p_n,y_n}\left(s\right)\right)+(z\left(s\right)-y_n\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{p_n,y_n}\left(s\right)\right)$$

$$+D_{\kappa }(q\left(s\right)-p_n\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{p_n,y_n}\left(s\right)\right)$$

$$+\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(q\left(s\right)-p_n\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{p_n,y_n}\left(s\right)\right)]d{s}^{\nu }$$

$$\le {\int }_{\mathsf{{\rm Y}}}[(q\left(s\right)-p_n\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{q,z}\left(s\right)\right)+(z\left(s\right)-y_n\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{q,z}\left(s\right)\right)$$

$$+D_{\kappa }(q\left(s\right)-p_n\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{q,z}\left(s\right)\right)$$

$$+\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(q\left(s\right)-p_n\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{q,z}\left(s\right)\right)]d{s}^{\nu }.$$

(7)

Combining with (5) and (7), we have

$${\int }_{\mathsf{{\rm Y}}}[(q\left(s\right)-p_n\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{q,z}\left(s\right)\right)+(z\left(s\right)-y_n\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{q,z}\left(s\right)\right)$$

$$+D_{\kappa }(q\left(s\right)-p_n\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{q,z}\left(s\right)\right)$$

$$+\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(q\left(s\right)-p_n\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{q,z}\left(s\right)\right)]d{s}^{\nu }\ge -{\sigma }_n,\forall (q,z)\in P\times \mathsf{\Xi }.$$

Since ${\delta }_n\to 0$ as $n\to \infty$, we can consider $n_0\in \mathbb{N}$ be large enough so that ${\delta }_n<1$ for all $n\ge n_0$. By multiplying the above inequality and (6) by ${\delta }_n>0$ and $1-{\delta }_n>0$, respectively, we sum the resulting inequalities to obtain

$${\int }_{\mathsf{{\rm Y}}}[(q\left(s\right)-{\mathsf{p}}_n\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{q,z}\left(s\right)\right)+(z\left(s\right)-{\mathsf{y}}_n\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{q,z}\left(s\right)\right)$$

$$+D_{\kappa }(q\left(s\right)-{\mathsf{p}}_n\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{q,z}\left(s\right)\right)$$

$$+\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(q\left(s\right)-{\mathsf{p}}_n\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{q,z}\left(s\right)\right)]d{s}^{\nu }\ge -{\sigma }_n,\forall (q,z)\in P\times \mathsf{\Xi },\forall n\ge n_0.$$

Since $({\mathsf{p}}_n,{\mathsf{y}}_n)\to (\mathsf{p},\mathsf{y})\ne (p_0,y_0)$ and $({\mathsf{p}}_n,{\mathsf{y}}_n)=(p_0,y_0)+{\delta }_n[(p_n,y_n)-(p_0,y_0)]$, we have

$${\int }_{\mathsf{{\rm Y}}}[(q\left(s\right)-\mathsf{p}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{q,z}\left(s\right)\right)+(z\left(s\right)-\mathsf{y}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{q,z}\left(s\right)\right)$$

$$+D_{\kappa }(q\left(s\right)-\mathsf{p}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{q,z}\left(s\right)\right)$$

$$+\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(q\left(s\right)-\mathsf{p}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{q,z}\left(s\right)\right)]d{s}^{\nu }$$

$$=\underset{n\to \infty }{lim}{\int }_{\mathsf{{\rm Y}}}[(q\left(s\right)-{\mathsf{p}}_n\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{q,z}\left(s\right)\right)+(z\left(s\right)-{\mathsf{y}}_n\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{q,z}\left(s\right)\right)$$

$$+D_{\kappa }(q\left(s\right)-{\mathsf{p}}_n\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{q,z}\left(s\right)\right)$$

$$+\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(q\left(s\right)-{\mathsf{p}}_n\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{q,z}\left(s\right)\right)]d{s}^{\nu }$$

$$\ge -\underset{n\to \infty }{lim}{\sigma }_n=0,\forall (q,z)\in P\times \mathsf{\Xi }.$$

Thus, by Lemma 1, we have

$${\int }_{\mathsf{{\rm Y}}}[(q\left(s\right)-\mathsf{p}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{\mathsf{p},\mathsf{y}}\left(s\right)\right)+(z\left(s\right)-\mathsf{y}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{\mathsf{p},\mathsf{y}}\left(s\right)\right)$$

$$+D_{\kappa }(q\left(s\right)-\mathsf{p}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{\mathsf{p},\mathsf{y}}\left(s\right)\right)$$

$$+\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(q\left(s\right)-\mathsf{p}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{\mathsf{p},\mathsf{y}}\left(s\right)\right)]d{s}^{\nu }\ge 0,\forall (q,z)\in P\times \mathsf{\Xi }.$$

(8)

This implies that $(\mathsf{p},\mathsf{y})$ is a solution of (variational problem equation), which contradicts the uniqueness of $(p_0,y_0)$. Therefore, $\left\{(p_n,y_n)\right\}$ is a bounded sequence having a convergent subsequence $\left\{(p_{n_k},y_{n_k})\right\}$, which converges to $(\overline{p},\overline{y})\in P\times \mathsf{\Xi }$ as $k\to \infty$. From monotonicity, for $(p_{n_k},y_{n_k}),(q,z)\in P\times \mathsf{\Xi }$, we obtain (see (7))

$${\int }_{\mathsf{{\rm Y}}}[(q\left(s\right)-p_{n_k}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{p_{n_k},y_{n_k}}\left(s\right)\right)+(z\left(s\right)-y_{n_k}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{p_{n_k},y_{n_k}}\left(s\right)\right)$$

$$+D_{\kappa }(q\left(s\right)-p_{n_k}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{p_{n_k},y_{n_k}}\left(s\right)\right)$$

$$+\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(q\left(s\right)-p_{n_k}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{p_{n_k},y_{n_k}}\left(s\right)\right)]d{s}^{\nu }$$

$$\le {\int }_{\mathsf{{\rm Y}}}[(q\left(s\right)-p_{n_k}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{q,z}\left(s\right)\right)+(z\left(s\right)-y_{n_k}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{q,z}\left(s\right)\right)$$

$$+D_{\kappa }(q\left(s\right)-p_{n_k}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{q,z}\left(s\right)\right)$$

$$+\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(q\left(s\right)-p_{n_k}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{q,z}\left(s\right)\right)]d{s}^{\nu }.$$

(9)

Moreover, on behalf of (5), we can write

$$\underset{k\to \infty }{lim}{\int }_{\mathsf{{\rm Y}}}[(q\left(s\right)-p_{n_k}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{p_{n_k},y_{n_k}}\left(s\right)\right)+(z\left(s\right)-y_{n_k}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{p_{n_k},y_{n_k}}\left(s\right)\right)$$

$$+D_{\kappa }(q\left(s\right)-p_{n_k}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{p_{n_k},y_{n_k}}\left(s\right)\right)$$

$$+\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(q\left(s\right)-p_{n_k}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{p_{n_k},y_{n_k}}\left(s\right)\right)]d{s}^{\nu }\ge 0.$$

(10)

Combining (9) and (10), we have

$$\underset{k\to \infty }{lim}{\int }_{\mathsf{{\rm Y}}}[(q\left(s\right)-p_{n_k}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{q,z}\left(s\right)\right)+(z\left(s\right)-y_{n_k}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{q,z}\left(s\right)\right)$$

$$+D_{\kappa }(q\left(s\right)-p_{n_k}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{q,z}\left(s\right)\right)$$

$$+\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(q\left(s\right)-p_{n_k}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{q,z}\left(s\right)\right)]d{s}^{\nu }\ge 0$$

$$\Rightarrow {\int }_{\mathsf{{\rm Y}}}[(q\left(s\right)-\overline{p}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{q,z}\left(s\right)\right)+(z\left(s\right)-\overline{y}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{q,z}\left(s\right)\right)$$

$$+D_{\kappa }(q\left(s\right)-\overline{p}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{q,z}\left(s\right)\right)$$

$$+\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(q\left(s\right)-\overline{p}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{q,z}\left(s\right)\right)]d{s}^{\nu }\ge 0$$

Thus, by Lemma 1, the above inequality implies that

$${\int }_{\mathsf{{\rm Y}}}[(q\left(s\right)-\overline{p}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p}}\left({\psi }_{\overline{p},\overline{y}}\left(s\right)\right)+(z\left(s\right)-\overline{y}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial y}}\left({\psi }_{\overline{p},\overline{y}}\left(s\right)\right)$$

$$+D_{\kappa }(q\left(s\right)-\overline{p}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\kappa }}}\left({\psi }_{\overline{p},\overline{y}}\left(s\right)\right)$$

$$+\frac{1}{x(\alpha ,\beta )}{D}_{\alpha \beta }^2(q\left(s\right)-\overline{p}\left(s\right)){\displaystyle \frac{\partial f_{\nu }}{\partial p_{\alpha \beta }}}\left({\psi }_{\overline{p},\overline{y}}\left(s\right)\right)]d{s}^{\nu }\ge 0,$$

which implies that $(\overline{p},\overline{y})$ solves (variational problem equation). So, $(p_{n_k},y_{n_k})\to (\overline{p},\overline{y})$; that is, $(p_{n_k},y_{n_k})\to (p_0,y_0)$, involving $(p_n,y_n)\to (p_0,y_0)$ and the proof is complete. □

Further, we provide an illustrative application of the previous theoretical results.

Example 3.

As in the previous section, consider $\mathsf{\Omega }={[0,1]}^2$, $\nu \in \{1,2\}$, ${\displaystyle \mathsf{\Omega }\supset \mathsf{{\rm Y}}}$ is a piecewise differentiable curve that links ${\displaystyle (0,0),(1,1)}$ in ${\displaystyle \mathsf{\Omega }}$. Moreover, we consider

$$f_{\nu }\left({\psi }_{p,y}\left(s\right)\right)d{s}^{\nu }=f_1\left({\psi }_{p,y}\left(s\right)\right)d{s}^1+f_2\left({\psi }_{p,y}\left(s\right)\right)d{s}^2=y^2\left(s\right)d{s}^1+(e^{p\left(s\right)}-p\left(s\right))d{s}^2.$$

(VP-1): Find $(p,y)\in P\times \mathsf{\Xi }=C^4(\mathsf{\Omega },[-10,10])\times C^1(\mathsf{\Omega },[-10,10])$ so that

$${\int }_{\mathsf{{\rm Y}}}2(z\left(s\right)-y\left(s\right))y\left(s\right)d{s}^1+(q\left(s\right)-p\left(s\right))(e^{p\left(s\right)}-1)d{s}^2\ge 0,\forall (q,z)\in P\times \mathsf{\Xi }.$$

We have $K=\left\{\right(0,0\left)\right\}$ and the functional ${\displaystyle {\int }_{\mathsf{{\rm Y}}}{f}_{\nu }\left({\psi }_{p,y}\left(s\right)\right)d{s}^{\nu }}$ is hemicontinuous and monotone on the set $P\times \mathsf{\Xi }=C^4(\mathsf{\Omega },[-10,10])\times C^1(\mathsf{\Omega },[-10,10])$. Since Theorem 2 holds, the variational problem (VP-1) is well-posed. In addition, $K_{\sigma }=\left\{(0,0)\right\}$ and consequently, $K_{\sigma }\ne \varnothing$ and $diam{K}_{\sigma }\to 0$ as $\sigma \to 0$. By Theorem 1, we also obtain that the variational problem (VP-1) is well-posed.

4. Conclusions

In this paper, by using some properties associated with path-independent curvilinear integral functionals governed by partial derivatives of second-order, we investigated the well-posedness of some variational problems. Moreover, in order to support the mathematical development, some illustrative examples were formulated. Possible lines of research that this study can open, among other aspects, are the following: formulating the derived results by using the variational/functional derivative (see Treanţă [16]); establishing the associated duality results (Wolfe, Mond-Weir, mixed dual).