Reciprocal Theorems for Multi-Cost Problems with S-Type I Functionals

Abstract

In this paper, for the considered multi-cost variational problem (P), we associate a dual model (D) in order to study and state the connections between the solution sets of these control problems. Thus, under S-type I assumptions associated with the integral functionals involved, we formulate and prove various reciprocal results, such as weak, strong, and converse-type dualities.

1. Introduction

Over time, many researchers have been interested in studying and solving an optimization problem by transforming it into a simpler one, both from a computational point of view and from the point of view of the equivalence of the solution sets associated with the two models. In this regard, duality theory has gained momentum among scientists. Recently, Abdulaleem and Treanţă [1] formulated optimality conditions and a duality theory for E-differentiable multiobjective programming problems involving V-E-type I functions. Aghezzaf and Hachimi [2] used generalized invexity and duality in some classes of multi-objective programming problems. Antczak [3,4] studied the G-invex and G-type I variational optimization problems that provide sufficient optimality criteria and duality results. Also, considering various types of invexity, several authors (see Bhatia and Mehra [5], Hachimi and Aghezzaf [6], Khazafi and Rueda [7,8], Kim and Kim [9]) have investigated some families of optimization problems and provided optimality conditions and duality connections. Very recently, Marghescu and Treanţă [10] investigated a class of control problems driven by approximately pseudo-convex multiple integral functionals. Mishra et al. [11,12] involved vanishing constraints and tangential subdifferentials in the considered multiobjective semi-infinite programs. In addition, approximate solutions associated with non-smooth infinite optimization problems have been studied by Pham [13]. Infinite interval constraints via integral-type penalty function have been included by Qian et al. [14] in the analysis of some vector interval-valued optimization problems. Sadeghieh et al. [15] investigated stationarity in non-smooth multiple-objective problems involving vanishing constraints. Saeed et al. [16] presented optimality criteria in multiobjective models with semi-infinite constraints and some characterization theorems for new classes of multi-objective control models. Very recently, Treanţă et al. [17] formulated necessary and sufficient efficiency criteria for multi-cost models with H-type I functionals. Tung [18] presented Karush–Kuhn–Tucker-type optimality criteria and duality outcomes for some multiobjective semi-infinite models with vanishing constraints. Yadav and Gupta [19,20] formulated an optimality and duality analysis in conic and interval-valued semi-infinite programs having vanishing constraints. In 2012, Zhang et al. [21] established sufficiency and duality theorems for multi- objective control models implying G-invexity.

In this study, based on the above research papers, keeping in mind the results established in Treanţă et al. [17], we associate a dual model (D) for the considered multi-cost variational problem (P) in order to study and state the connections between the solution sets of these control problems. Thus, under S-type I assumptions associated with the integral functionals involved, we formulate and prove various reciprocal results, such as weak, strong, and converse-type dualities. The novelty elements of our study are given by the use of S-type I functionals for the analysis of the reciprocal outcomes established in this article. Compared with other papers in the specialized area, this is the principal new tool for the duality study of the considered family of constrained optimization problems. The tool used in this paper is more effective or different from previous methods in terms of efficiency or coverage. More precisely, the S-type I assumptions associated with the multiple integral functionals involved cover broader and more general classes of problems, where the convexity of the functionals is not fulfilled or the functionals considered are not of simple integral type.

2. Preliminary Elements

In the following, according to Treanţă et al. [17], we state various definitions and the following convention for equalities and inequalities: for any two vectors, $\gamma ={\left({\gamma }_1,{\gamma }_2,\dots ,{\gamma }_o\right)}^T,\delta ={\left({\delta }_1,{\delta }_2,\dots ,{\delta }_o\right)}^T$, we assume

(i)  

] $\gamma =\delta ⟺{\gamma }_{\varsigma }={\delta }_{\varsigma }$ for $\varsigma =1,\dots ,o$;

(ii) 

$\gamma <\delta ⟺{\gamma }_{\varsigma }<{\delta }_{\varsigma }$ for $\varsigma =1,\dots ,o$;

(iii)

$\gamma \leqq \delta ⟺{\gamma }_{\varsigma }\le {\delta }_{\varsigma }$ for $\varsigma =1,\dots ,o$;

(iv)

$\gamma \le \delta ⟺\gamma \leqq \delta$ and $\gamma \ne \delta$.

Consider $A=A_{a,b}\subset {\mathbb{R}}^s$ to be compact and $V:=\{1,\dots ,\eta \},Z:=\{1,\dots ,\pi \}$. Suppose $ϵ\left(w\right)$ is a smooth (piecewise) mapping of $w\in A$, and ${ϵ}_{\alpha }\left(w\right):={\displaystyle \frac{\mathsf{\partial }ϵ}{\mathsf{\partial }{w}^{\alpha }}}\left(w\right)$. Also, consider $q\left(w\right)$ to be a smooth (piecewise) mapping of $w\in A$, and $q_{\beta }\left(w\right):={\displaystyle \frac{\mathsf{\partial }q}{\mathsf{\partial }{w}^{\beta }}}\left(w\right)$. Consider R to be the family of functions $ϵ:A\to {\mathbb{R}}^o$ (state variables) with $\parallel ϵ\parallel =$${\parallel ϵ\parallel }_{\infty }+{\parallel {ϵ}_{\alpha }\parallel }_{\infty }$, and Q is the family of functions $q:A\to {\mathbb{R}}^t$ (control variables) with the same norm, as well.

In the following, we investigate the multi-cost model defined as:

$$\begin{array}{l}\left(\mathrm{P}\right)\underset{(ϵ,q)}{min}{\mathit{\int }}_A\mathsf{\Phi }(w,ϵ\left(w\right),{ϵ}_{\alpha }\left(w\right),q\left(w\right),q_{\beta }\left(w\right))d\theta \hfill \\ =\left({\mathit{\int }}_A{\mathsf{\Phi }}^1(w,ϵ\left(w\right),{ϵ}_{\alpha }\left(w\right),q\left(w\right),q_{\beta }\left(w\right))d\theta ,\dots ,{\mathit{\int }}_A{\mathsf{\Phi }}^{\eta }(w,ϵ\left(w\right),{ϵ}_{\alpha }\left(w\right),q\left(w\right),q_{\beta }\left(w\right))d\theta \right)\hfill \\ \mathrm{constrained}\mathrm{by}\hfill \\ \mathsf{\Theta }(w,ϵ\left(w\right),{ϵ}_{\alpha }\left(w\right),q\left(w\right),q_{\beta }\left(w\right)):={ϵ}_{\alpha }\left(w\right)-K(w,ϵ\left(w\right),q\left(w\right),q_{\beta }\left(w\right))\leqq 0,w\in A,\hfill \\ ϵ\left(a\right)=f=\mathrm{given},ϵ\left(b\right)=g=\mathrm{given},\hfill \end{array}$$

with $\mathsf{\Phi }=\left({\mathsf{\Phi }}^1,\dots ,{\mathsf{\Phi }}^{\eta }\right):A\times {\mathbb{R}}^o\times {\mathbb{R}}^{os}\times {\mathbb{R}}^t\times {\mathbb{R}}^{ts}\to {\mathbb{R}}^{\eta }$, $\mathsf{\Theta }:A\times {\mathbb{R}}^o\times$${\mathbb{R}}^{os}\times {\mathbb{R}}^t\times {\mathbb{R}}^{ts}\to {\mathbb{R}}^{\pi }$ as $C^1$-class functionals, and $d\theta$ as the volume element in ${\mathbb{R}}^s$.

Also, we consider the notations $ϵ\left(w\right)$ and ${ϵ}_{\alpha }\left(w\right)$ for $ϵ$ and ${ϵ}_{\alpha }$, respectively, $q\left(w\right)$ and $q_{\beta }\left(w\right)$ for q and $q_{\beta }$, respectively, and

$$\begin{array}{c}{\tau }_{ϵq}\left(w\right):=(w,ϵ\left(w\right),{ϵ}_{\alpha }\left(w\right),q\left(w\right),q_{\beta }\left(w\right)),{\tau }_{\overline{ϵ}\overline{q}}\left(w\right):=(w,\overline{ϵ}\left(w\right),{\overline{ϵ}}_{\alpha }\left(w\right),\overline{q}\left(w\right),{\overline{q}}_{\beta }\left(w\right)),\\ {\tau }_{ϵq\overline{ϵq}}\left(w\right):=(w,ϵ\left(w\right),\overline{ϵ}\left(w\right),q\left(w\right),\overline{q}\left(w\right)).\end{array}$$

Consider U, defined as

$$U=\left\{\right(ϵ,q):ϵ\in R,q\in Q\mathrm{fulfilling}\mathrm{the}\mathrm{restrictions}\mathrm{given}\mathrm{in}\left(\mathrm{P}\right)\},$$

is the feasible set of points for (P).

Definition 1.

The feasible point $(\overline{ϵ},\overline{q})\in U$ is weakly efficient point for (P) if there exists no $(ϵ,q)\in U$ satisfying

$${\mathit{\int }}_A\mathsf{\Phi }\left({\tau }_{ϵq}\left(w\right)\right)d\theta <{\mathit{\int }}_A\mathsf{\Phi }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta .$$

Definition 2.

The feasible point $(\overline{ϵ},\overline{q})\in U$ is efficient point for (P) if there exists no $(ϵ,q)\in U$ satisfying

$${\mathit{\int }}_A\mathsf{\Phi }\left({\tau }_{ϵq}\left(w\right)\right)d\theta \le {\mathit{\int }}_A\mathsf{\Phi }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta .$$

Definition 3.

The feasible point $(\overline{ϵ},\overline{q})\in U$ is properly efficient point for (P) if there exists $F>0$ such that

$${\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta -{\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{ϵq}\left(w\right)\right)d\theta$$

$$\le F\left({\mathit{\int }}_A{\mathsf{\Phi }}^k\left({\tau }_{ϵq}\left(w\right)\right)d\theta -{\mathit{\int }}_A{\mathsf{\Phi }}^k\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right)$$

is satisfied, for every $\varsigma \in V$, and for k, fulfilling

$${\displaystyle {\mathit{\int }}_A{\mathsf{\Phi }}^k\left({\tau }_{ϵq}\left(w\right)\right)d\theta >{\mathit{\int }}_A{\mathsf{\Phi }}^k\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta ,}$$

whenever $(ϵ,q)\in U$ and ${\displaystyle {\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{ϵq}\left(w\right)\right)d\theta <{\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta }$.

Definition 4.

A mapping $h:\mathbb{R}\to \mathbb{R}$ is said to be strictly increasing if

$$\forall x,y\in \mathbb{R},x<y⟹h\left(x\right)<h\left(y\right).$$

Consider $Im\left(J^{\varsigma }\right),\varsigma \in V$, as the range for

$${\displaystyle J^{\varsigma }:R\times Q\to \mathbb{R},J^{\varsigma }(ϵ,q):={\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }(w,ϵ\left(w\right),{ϵ}_{\alpha }\left(w\right),q\left(w\right),q_{\beta }\left(w\right))d\theta ,}$$

where $ϵ\in R,q\in Q$, and $Im\left(G^{\sigma }\right),\sigma \in Z$, be the range of

$${\displaystyle G^{\sigma }:R\times Q\to \mathbb{R},G^{\sigma }(ϵ,q):={\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }(w,ϵ\left(w\right),{ϵ}_{\alpha }\left(w\right),q\left(w\right),q_{\beta }\left(w\right))d\theta ,}$$

where $ϵ\in R,q\in Q$.

Definition 5

(Treanţă et al. [17]). For $(\overline{ϵ},\overline{q})\in R\times Q$, if there exists a differentiable functional $S_{\mathsf{\Phi }}=\left(S_{{\mathsf{\Phi }}^1},\dots ,S_{{\mathsf{\Phi }}^{\eta }}\right):\mathbb{R}\to {\mathbb{R}}^{\eta }$ with $S_{{\mathsf{\Phi }}^{\varsigma }}:Im\left(J^{\varsigma }\right)\to \mathbb{R}$ as a real-valued strictly increasing mapping, a differentiable functional $S_{\mathsf{\Theta }}=\left(S_{{\mathsf{\Theta }}^1},\dots ,S_{{\mathsf{\Theta }}^{\pi }}\right):\mathbb{R}\to {\mathbb{R}}^{\pi }$ with $S_{{\mathsf{\Theta }}^{\sigma }}:Im\left(G^{\sigma }\right)\to \mathbb{R}$ as a real-valued strictly increasing mapping, $\varphi :A\times {\mathbb{R}}^{2\sigma }\times {\mathbb{R}}^{2\varsigma }\to {\mathbb{R}}^o$ with $\varphi (w,\overline{ϵ}\left(w\right),\overline{ϵ}\left(w\right),\overline{q}\left(w\right),\overline{q}\left(w\right)){=\varphi |}_{w=a,b}=0$, and $\psi :A\times {\mathbb{R}}^{2\sigma }\times {\mathbb{R}}^{2\varsigma }\to {\mathbb{R}}^t$ with $\psi (w,\overline{ϵ}\left(w\right),\overline{ϵ}\left(w\right),\overline{q}\left(w\right),\overline{q}\left(w\right)){=\psi |}_{w=a,b}=0$, such that [see $D_{\alpha }:={\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }{w}^{\alpha }}}$]

$$\begin{array}{c}{S}_{{\mathsf{\Phi }}^{\varsigma }}\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{ϵq}\left(w\right)\right)d\theta \right)-S_{{\mathsf{\Phi }}^{\varsigma }}\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right)\hfill \\ \ge S_{{\mathsf{\Phi }}^{\varsigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right){\mathit{\int }}_A\left\{{\left[\varphi \left({\tau }_{ϵq\overline{ϵq}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Phi }}_{ϵ}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)-D_{\alpha }{\mathsf{\Phi }}_{{ϵ}_{\alpha }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]\right.\hfill \\ \hfill \mathrm{(1)}& \left.+{\left[\psi \left({\tau }_{ϵq\overline{ϵq}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Phi }}_q^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]-D_{\beta }{\mathsf{\Phi }}_{q_{\beta }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right\}d\theta ,\varsigma \in V,\hfill \end{array}$$

and

$$\begin{array}{c}-S_{{\mathsf{\Theta }}^{\sigma }}\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right)\hfill \\ \ge S_{{\mathsf{\Theta }}^{\sigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right){\mathit{\int }}_A\left\{{\left[\varphi \left({\tau }_{ϵq\overline{ϵq}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Theta }}_{ϵ}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)-D_{\alpha }{\mathsf{\Theta }}_{{ϵ}_{\alpha }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]\right.\hfill \\ \hfill \mathrm{(2)}& \left.+{\left[\psi \left({\tau }_{ϵq\overline{ϵq}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Theta }}_q^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)-D_{\beta }{\mathsf{\Theta }}_{q_{\beta }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]\right\}d\theta ,\sigma \in Z,\hfill \end{array}$$

hold, for all $(ϵ,q)\in R\times Q$, then $(\mathsf{\Phi },\mathsf{\Theta })$ is called S-type I at $(\overline{ϵ},\overline{q})\in R\times Q$ on $R\times Q$ (related to $S_{\mathsf{\Phi }},S_{\mathsf{\Theta }},\varphi$ and ψ).

If (1) and (2) are valid for each $(\overline{ϵ},\overline{q})\in R\times Q$, then $(\mathsf{\Phi },\mathsf{\Theta })$ is called S-type I on $R\times Q$ related to $S_{\mathsf{\Phi }},S_{\mathsf{\Theta }}$, ϕ and ψ.

Definition 6

(Treanţă et al. [17]). For $(\overline{ϵ},\overline{q})\in R\times Q$, if there exists a differentiable functional $S_{\mathsf{\Phi }}=\left(S_{{\mathsf{\Phi }}^1},\dots ,S_{{\mathsf{\Phi }}^{\eta }}\right):\mathbb{R}\to {\mathbb{R}}^{\eta }$ with $S_{{\mathsf{\Phi }}^{\varsigma }}:Im\left(J^{\varsigma }\right)\to \mathbb{R}$ as a real-valued strictly increasing mapping, a differentiable functional $S_{\mathsf{\Theta }}=\left(S_{{\mathsf{\Theta }}^1},\dots ,S_{{\mathsf{\Theta }}^{\pi }}\right):\mathbb{R}\to {\mathbb{R}}^{\pi }$ with $S_{{\mathsf{\Theta }}^{\sigma }}:Im\left(G^{\sigma }\right)\to \mathbb{R}$ as a real-valued strictly increasing mapping, $\varphi :A\times {\mathbb{R}}^{2\sigma }\times {\mathbb{R}}^{2\varsigma }\to {\mathbb{R}}^o$ with $\varphi (w,\overline{ϵ}\left(w\right),\overline{ϵ}\left(w\right),\overline{q}\left(w\right),\overline{q}\left(w\right)){=\varphi |}_{w=a,b}=0$, and $\psi :A\times {\mathbb{R}}^{2\sigma }\times {\mathbb{R}}^{2\varsigma }\to {\mathbb{R}}^t$ with $\psi (w,\overline{ϵ}\left(w\right),\overline{ϵ}\left(w\right),\overline{q}\left(w\right),\overline{q}\left(w\right)){=\psi |}_{w=a,b}=0$, such that the inequalities

$$\begin{array}{c}{S}_{{\mathsf{\Phi }}^{\varsigma }}\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{ϵq}\left(w\right)\right)d\theta \right)-S_{{\mathsf{\Phi }}^{\varsigma }}\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right)\hfill \\ >S_{{\mathsf{\Phi }}^{\varsigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right){\mathit{\int }}_A\left\{{\left[\varphi \left({\tau }_{ϵq\overline{ϵq}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Phi }}_{ϵ}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)-D_{\alpha }{\mathsf{\Phi }}_{{ϵ}_{\alpha }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]\right.\hfill \\ \hfill \mathrm{(3)}& \left.+{\left[\psi \left({\tau }_{ϵq\overline{ϵq}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Phi }}_q^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]-D_{\beta }{\mathsf{\Phi }}_{q_{\beta }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right\}d\theta ,\varsigma \in V,\hfill \end{array}$$

and

$$\begin{array}{c}-S_{{\mathsf{\Theta }}^{\sigma }}\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right)\hfill \\ \ge S_{{\mathsf{\Theta }}^{\sigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right){\mathit{\int }}_A\left\{{\left[\varphi \left({\tau }_{ϵq\overline{ϵq}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Theta }}_{ϵ}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)-D_{\alpha }{\mathsf{\Theta }}_{{ϵ}_{\alpha }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]\right.\hfill \\ \hfill \mathrm{(4)}& \left.+{\left[\psi \left({\tau }_{ϵq\overline{ϵq}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Theta }}_q^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)-D_{\beta }{\mathsf{\Theta }}_{q_{\beta }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]\right\}d\theta ,\sigma \in Z,\hfill \end{array}$$

hold, for all $(ϵ,q)\in R\times Q,(ϵ,q)\ne (\overline{ϵ},\overline{q})$, then $(\mathsf{\Phi },\mathsf{\Theta })$ is called strictly-S-type I at $(\overline{ϵ},\overline{q})\in R\times Q$ on $R\times Q$ related to $S_{\mathsf{\Phi }},S_{\mathsf{\Theta }},\varphi$ and ψ.

If (3) and (4) are valid for each $(\overline{ϵ},\overline{q})\in R\times Q$, then $(\mathsf{\Phi },\mathsf{\Theta })$ is called strictly-S-type I on $R\times Q$ related to $S_{\mathsf{\Phi }},S_{\mathsf{\Theta }},\varphi$ and ψ.

Definition 7

(Treanţă et al. [17]). For $(\overline{ϵ},\overline{q})\in R\times Q$, if there exists a differentiable functional $S_{\mathsf{\Phi }}=\left(S_{{\mathsf{\Phi }}^1},\dots ,S_{{\mathsf{\Phi }}^{\eta }}\right):\mathbb{R}\to {\mathbb{R}}^{\eta }$ with $S_{{\mathsf{\Phi }}^{\varsigma }}:Im\left(J^{\varsigma }\right)\to \mathbb{R}$ as a real-valued strictly increasing mapping, a differentiable functional $S_{\mathsf{\Theta }}=\left(S_{{\mathsf{\Theta }}^1},\dots ,S_{{\mathsf{\Theta }}^{\pi }}\right):\mathbb{R}\to {\mathbb{R}}^{\pi }$ with $S_{{\mathsf{\Theta }}^{\sigma }}:Im\left(G^{\sigma }\right)\to \mathbb{R}$ as a real-valued strictly increasing mapping, $\varphi :A\times {\mathbb{R}}^{2\sigma }\times {\mathbb{R}}^{2\varsigma }\to {\mathbb{R}}^o$ with $\varphi (w,\overline{ϵ}\left(w\right),\overline{ϵ}\left(w\right),\overline{q}\left(w\right),\overline{q}\left(w\right)){=\varphi |}_{w=a,b}=0$, and $\psi :A\times {\mathbb{R}}^{2\sigma }\times {\mathbb{R}}^{2\varsigma }\to {\mathbb{R}}^t$ with $\psi (w,\overline{ϵ}\left(w\right),\overline{ϵ}\left(w\right),\overline{q}\left(w\right),\overline{q}\left(w\right)){=\psi |}_{w=a,b}=0$, such that

$$\begin{array}{c}{S}_{{\mathsf{\Phi }}^{\varsigma }}\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{ϵq}\left(w\right)\right)d\theta \right)<S_{{\mathsf{\Phi }}^{\varsigma }}\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right)\hfill \\ ⟹S_{{\mathsf{\Phi }}^{\varsigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right){\mathit{\int }}_A\left\{{\left[\varphi \left({\tau }_{ϵq\overline{ϵq}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Phi }}_{ϵ}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)-D_{\alpha }{\mathsf{\Phi }}_{{ϵ}_{\alpha }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]\right.\hfill \\ \hfill \mathrm{(5)}& \left.+{\left[\psi \left({\tau }_{ϵq\overline{ϵq}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Phi }}_q^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]-D_{\beta }{\mathsf{\Phi }}_{q_{\beta }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right\}d\theta <0,\varsigma \in V,\hfill \end{array}$$

and

$$\begin{array}{c}-S_{{\mathsf{\Theta }}^{\sigma }}\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right)\le 0\hfill \\ ⟹S_{{\mathsf{\Theta }}^{\sigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right){\mathit{\int }}_A\left\{{\left[\varphi \left({\tau }_{ϵq\overline{ϵq}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Theta }}_{ϵ}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)-D_{\alpha }{\mathsf{\Theta }}_{{ϵ}_{\alpha }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]\right.\hfill \\ \hfill \mathrm{(6)}& \left.+{\left[\psi \left({\tau }_{ϵq\overline{ϵq}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Theta }}_q^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)-D_{\beta }{\mathsf{\Theta }}_{q_{\beta }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]\right\}d\theta \le 0,\sigma \in Z,\hfill \end{array}$$

hold, for all $(ϵ,q)\in R\times Q$, then $(\mathsf{\Phi },\mathsf{\Theta })$ is called pseudo-quasi-S-type I at $(\overline{ϵ},\overline{q})\in R\times Q$ on $R\times Q$ (related to $S_{\mathsf{\Phi }},S_{\mathsf{\Theta }},\varphi$ and ψ).

If (5) and (6) are valid for each $(\overline{ϵ},\overline{q})\in R\times Q$, then $(\mathsf{\Phi },\mathsf{\Theta })$ is called pseudo-quasi-S-type I on $R\times Q$ related to $S_{\mathsf{\Phi }},S_{\mathsf{\Theta }},\varphi$ and ψ.

Definition 8

(Treanţă et al. [17]). For $(\overline{ϵ},\overline{q})\in R\times Q$, if there exists a differentiable functional $S_{\mathsf{\Phi }}=\left(S_{{\mathsf{\Phi }}^1},\dots ,S_{{\mathsf{\Phi }}^{\eta }}\right):\mathbb{R}\to {\mathbb{R}}^{\eta }$ with $S_{{\mathsf{\Phi }}^{\varsigma }}:Im\left(J^{\varsigma }\right)\to \mathbb{R}$ as a real-valued strictly increasing mapping, a differentiable functional $S_{\mathsf{\Theta }}=\left(S_{{\mathsf{\Theta }}^1},\dots ,S_{{\mathsf{\Theta }}^{\pi }}\right):\mathbb{R}\to {\mathbb{R}}^{\pi }$ with $S_{{\mathsf{\Theta }}^{\sigma }}:Im\left(G^{\sigma }\right)\to \mathbb{R}$ as a real-valued strictly increasing mapping, $\varphi :A\times {\mathbb{R}}^{2\sigma }\times {\mathbb{R}}^{2\varsigma }\to {\mathbb{R}}^o$ with $\varphi (w,\overline{ϵ}\left(w\right),\overline{ϵ}\left(w\right),\overline{q}\left(w\right),\overline{q}\left(w\right)){=\varphi |}_{w=a,b}=0$, and $\psi :A\times {\mathbb{R}}^{2\sigma }\times {\mathbb{R}}^{2\varsigma }\to {\mathbb{R}}^t$ with $\psi (w,\overline{ϵ}\left(w\right),\overline{ϵ}\left(w\right),\overline{q}\left(w\right),\overline{q}\left(w\right)){=\psi |}_{w=a,b}=0$, such that

$$\begin{array}{c}{S}_{{\mathsf{\Phi }}^{\varsigma }}\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{ϵq}\left(w\right)\right)d\theta \right)\le S_{{\mathsf{\Phi }}^{\varsigma }}\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right)\hfill \\ ⟹S_{{\mathsf{\Phi }}^{\varsigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right){\mathit{\int }}_A\left\{{\left[\varphi \left({\tau }_{ϵq\overline{ϵq}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Phi }}_{ϵ}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)-D_{\alpha }{\mathsf{\Phi }}_{{ϵ}_{\alpha }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]\right.\hfill \\ \hfill \mathrm{(7)}& \left.+{\left[\psi \left({\tau }_{ϵq\overline{ϵq}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Phi }}_q^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]-D_{\beta }{\mathsf{\Phi }}_{q_{\beta }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right\}d\theta <0,\varsigma \in V,\hfill \end{array}$$

and

$$\begin{array}{c}-S_{{\mathsf{\Theta }}^{\sigma }}\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right)\le 0\hfill \\ ⟹S_{{\mathsf{\Theta }}^{\sigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right){\mathit{\int }}_A\left\{{\left[\varphi \left({\tau }_{ϵq\overline{ϵq}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Theta }}_{ϵ}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)-D_{\alpha }{\mathsf{\Theta }}_{{ϵ}_{\alpha }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]\right.\hfill \\ \hfill \mathrm{(8)}& \left.+{\left[\psi \left({\tau }_{ϵq\overline{ϵq}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Theta }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)-D_{\beta }{\mathsf{\Theta }}_{q_{\beta }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]\right\}d\theta \le 0,\sigma \in Z,\hfill \end{array}$$

hold, for all $(ϵ,q)\in R\times Q,(ϵ,q)\ne (\overline{ϵ},\overline{q})$, then $(\mathsf{\Phi },\mathsf{\Theta })$ is called strictly-pseudo-quasi-S-type I at $(\overline{ϵ},\overline{q})\in R\times Q$ on $R\times Q$ related to $S_{\mathsf{\Phi }},S_{\mathsf{\Theta }},\varphi$ and ψ.

If (7) and (8) are valid for each $(\overline{ϵ},\overline{q})\in R\times Q$, then $(\mathsf{\Phi },\mathsf{\Theta })$ is called strictly-pseudo-quasi-S-type I on $R\times Q$ related to $S_{\mathsf{\Phi }},S_{\mathsf{\Theta }},\varphi$ and ψ.

Definition 9

(Treanţă et al. [17]). For $(\overline{ϵ},\overline{q})\in R\times Q$, if there exists a differentiable functional $S_{\mathsf{\Phi }}=\left(S_{{\mathsf{\Phi }}^1},\dots ,S_{{\mathsf{\Phi }}^{\eta }}\right):\mathbb{R}\to {\mathbb{R}}^{\eta }$ with $S_{{\mathsf{\Phi }}^{\varsigma }}:Im\left(J^{\varsigma }\right)\to \mathbb{R}$ as a real-valued strictly increasing mapping, a differentiable functional $S_{\mathsf{\Theta }}=\left(S_{{\mathsf{\Theta }}^1},\dots ,S_{{\mathsf{\Theta }}^{\pi }}\right):\mathbb{R}\to {\mathbb{R}}^{\pi }$ with $S_{{\mathsf{\Theta }}^{\sigma }}:Im\left(G^{\sigma }\right)\to \mathbb{R}$ as a real-valued strictly increasing mapping, $\varphi :A\times {\mathbb{R}}^{2\sigma }\times {\mathbb{R}}^{2\varsigma }\to {\mathbb{R}}^o$ with $\varphi (w,\overline{ϵ}\left(w\right),\overline{ϵ}\left(w\right),\overline{q}\left(w\right),\overline{q}\left(w\right)){=\varphi |}_{w=a,b}=0$, and $\psi :A\times {\mathbb{R}}^{2\sigma }\times {\mathbb{R}}^{2\varsigma }\to {\mathbb{R}}^t$ with $\psi (w,\overline{ϵ}\left(w\right),\overline{ϵ}\left(w\right),\overline{q}\left(w\right),\overline{q}\left(w\right)){=\psi |}_{w=a,b}=0$, such that

$$\begin{array}{c}{S}_{{\mathsf{\Phi }}^{\varsigma }}\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{ϵq}\left(w\right)\right)d\theta \right)<S_{{\mathsf{\Phi }}^{\varsigma }}\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right)\hfill \\ ⟹\left[\underset{\varsigma \in V}{\forall }{S}_{{\mathsf{\Phi }}^{\varsigma }}^{\prime }\left({\mathit{\int }}_a^b{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{\overline{ϵq}}\left(w\right)\right)d\theta \right){\mathit{\int }}_a^b\left\{{\left[\varphi \left({\tau }_{ϵq\overline{ϵq}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Phi }}_{ϵ}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)-D_{\alpha }{\mathsf{\Phi }}_{{ϵ}_{\alpha }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]\right.\right.\hfill \end{array}$$

$$\begin{array}{c}\left.+{\left[\psi \left({\tau }_{ϵq\overline{ϵq}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Phi }}_q^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]-D_{\beta }{\mathsf{\Phi }}_{q_{\beta }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right\}d\theta \le 0\hfill \\ \wedge \underset{\varsigma \in V}{\exists }{S}_{{\mathsf{\Phi }}^{\varsigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right){\mathit{\int }}_A\left\{{\left[\varphi \left({\tau }_{ϵq\overline{ϵq}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Phi }}_{ϵ}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)-D_{\alpha }{\mathsf{\Phi }}_{{ϵ}_{\alpha }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]\right.\hfill \\ \hfill \mathrm{(9)}& \left.\left.+{\left[\psi \left({\tau }_{ϵq\overline{ϵq}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Phi }}_q^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]-D_{\beta }{\mathsf{\Phi }}_{q_{\beta }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right\}d\theta <0\right]\hfill \end{array}$$

and

$$\begin{array}{c}-S_{{\mathsf{\Theta }}^{\sigma }}\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right)\le 0\hfill \\ ⟹S_{\mathsf{\Theta }\sigma }^{\prime }\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right){\mathit{\int }}_A\left\{{\left[\varphi \left({\tau }_{ϵq\overline{ϵq}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Theta }}_{ϵ}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)-D_{\alpha }{\mathsf{\Theta }}_{{ϵ}_{\alpha }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]\right.\hfill \\ \hfill \mathrm{(10)}& \left.+{\left[\psi \left({\tau }_{ϵq\overline{ϵq}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Theta }}_q^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)-D_{\beta }{\mathsf{\Theta }}_{q_{\beta }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]\right\}d\theta \le 0,\sigma \in Z,\hfill \end{array}$$

hold, for all $(ϵ,q)\in R\times Q,(ϵ,q)\ne (\overline{ϵ},\overline{q})$, then $(\mathsf{\Phi },\mathsf{\Theta })$ is called weak-pseudo-quasi-S-type I at $(\overline{ϵ},\overline{q})\in R\times Q$ on $R\times Q$ related to $S_{\mathsf{\Phi }},S_{\mathsf{\Theta }},\varphi$ and ψ.

If (9) and (10) are valid for each $(\overline{ϵ},\overline{q})\in R\times Q$, then $(\mathsf{\Phi },\mathsf{\Theta })$ is called weak-pseudo-quasi-S-type I on $R\times Q$ related to $S_{\mathsf{\Phi }},S_{\mathsf{\Theta }},\varphi$ and ψ.

Definition 10

(Treanţă et al. [17]). For $(\overline{ϵ},\overline{q})\in R\times Q$, if there exists a differentiable functional $S_{\mathsf{\Phi }}=\left(S_{{\mathsf{\Phi }}^1},\dots ,S_{{\mathsf{\Phi }}^{\eta }}\right):\mathbb{R}\to {\mathbb{R}}^{\eta }$ with $S_{{\mathsf{\Phi }}^{\varsigma }}:Im\left(J^{\varsigma }\right)\to \mathbb{R}$ as a real-valued strictly increasing mapping, a differentiable functional $S_{\mathsf{\Theta }}=\left(S_{{\mathsf{\Theta }}^1},\dots ,S_{{\mathsf{\Theta }}^{\pi }}\right):\mathbb{R}\to {\mathbb{R}}^{\pi }$ with $S_{{\mathsf{\Theta }}^{\sigma }}:Im\left(G^{\sigma }\right)\to \mathbb{R}$ as a real-valued strictly increasing mapping, $\varphi :A\times {\mathbb{R}}^{2\sigma }\times {\mathbb{R}}^{2\varsigma }\to {\mathbb{R}}^o$ with $\varphi (w,\overline{ϵ}\left(w\right),\overline{ϵ}\left(w\right),\overline{q}\left(w\right),\overline{q}\left(w\right)){=\varphi |}_{w=a,b}=0$, and $\psi :A\times {\mathbb{R}}^{2\sigma }\times {\mathbb{R}}^{2\varsigma }\to {\mathbb{R}}^t$ with $\psi (w,\overline{ϵ}\left(w\right),\overline{ϵ}\left(w\right),\overline{q}\left(w\right),\overline{q}\left(w\right)){=\psi |}_{w=a,b}=0$, such that

$$\begin{array}{c}\left[\underset{\varsigma \in V}{\forall }{S}_{{\mathsf{\Phi }}^{\varsigma }}\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{ϵq}\left(w\right)\right)d\theta \right)\le S_{{\mathsf{\Phi }}^{\varsigma }}\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right)\right.\hfill \\ \left.\wedge \underset{\varsigma \in V}{\exists }{S}_{{\mathsf{\Phi }}^{\varsigma }}\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{ϵq}\left(w\right)\right)d\theta \right)<S_{{\mathsf{\Phi }}^{\varsigma }}\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right)\right]\hfill \\ ⟹S_{{\mathsf{\Phi }}^{\varsigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right){\mathit{\int }}_A\left\{{\left[\varphi \left({\tau }_{ϵq\overline{ϵq}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Phi }}_{ϵ}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)-D_{\alpha }{\mathsf{\Phi }}_{{ϵ}_{\alpha }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]\right.\hfill \\ \hfill \mathrm{(11)}& \left.+{\left[\psi \left({\tau }_{ϵq\overline{ϵq}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Phi }}_q^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]-D_{\beta }{\mathsf{\Phi }}_{q_{\beta }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right\}d\theta \le 0,\varsigma \in V,\hfill \end{array}$$

$$\begin{array}{c}\left[\underset{\varsigma \in V}{\forall }{S}_{{\mathsf{\Phi }}^{\varsigma }}\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{ϵq}\left(w\right)\right)d\theta \right)\le S_{{\mathsf{\Phi }}^{\varsigma }}\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right)\right.\hfill \\ \left.\wedge \underset{\varsigma \in V}{\exists }{S}_{{\mathsf{\Phi }}^{\varsigma }}\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{ϵq}\left(w\right)\right)d\theta \right)<S_{{\mathsf{\Phi }}^{\varsigma }}\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right)\right]\hfill \\ ⟹S_{{\mathsf{\Phi }}^{\varsigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right){\mathit{\int }}_A\left\{{\left[\varphi \left({\tau }_{ϵq\overline{ϵq}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Phi }}_{ϵ}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)-D_{\alpha }{\mathsf{\Phi }}_{{ϵ}_{\alpha }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]\right.\hfill \\ \hfill \mathrm{(12)}& \left.+{\left[\psi \left({\tau }_{ϵq\overline{ϵq}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Phi }}_q^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]-D_{\beta }{\mathsf{\Phi }}_{q_{\beta }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right\}d\theta <0,\mathrm{for}\mathrm{at}\mathrm{least}\mathrm{one}\varsigma \in V,\hfill \end{array}$$

and

$$\begin{array}{c}-S_{{\mathsf{\Theta }}^{\sigma }}\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right)\le 0\hfill \\ ⟹S_{\mathsf{\Theta }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right){\mathit{\int }}_A\left\{{\left[\varphi \left({\tau }_{ϵq\overline{ϵq}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Theta }}_{ϵ}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)-D_{\alpha }{\mathsf{\Theta }}_{{ϵ}_{\alpha }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]\right.\hfill \\ \hfill \mathrm{(13)}& \left.+{\left[\psi \left({\tau }_{ϵq\overline{ϵq}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Theta }}_q^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)-D_{\beta }{\mathsf{\Theta }}_{q_{\beta }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]\right\}d\theta \le 0,\sigma \in Z,\hfill \end{array}$$

hold, for all $(ϵ,q)\in R\times Q$, then $(\mathsf{\Phi },\mathsf{\Theta })$ is called strong-pseudo-quasi-S-type I at $(\overline{ϵ},\overline{q})\in R\times Q$ on $R\times Q$ related to $S_{\mathsf{\Phi }},S_{\mathsf{\Theta }},\varphi$ and ψ.

If (11), (12) and (13) are valid for each $(\overline{ϵ},\overline{q})\in R\times Q$, then $(\mathsf{\Phi },\mathsf{\Theta })$ is called strong-pseudo-quasi-S-type I on $R\times Q$ related to $S_{\mathsf{\Phi }},S_{\mathsf{\Theta }},\varphi$ and ψ.

Definition 11

(Treanţă et al. [17]). For $(\overline{ϵ},\overline{q})\in R\times Q$, if there exists a differentiable functional $S_{\mathsf{\Phi }}=\left(S_{{\mathsf{\Phi }}^1},\dots ,S_{{\mathsf{\Phi }}^{\eta }}\right):\mathbb{R}\to {\mathbb{R}}^{\eta }$ with $S_{{\mathsf{\Phi }}^{\varsigma }}:Im\left(J^{\varsigma }\right)\to \mathbb{R}$ as a real-valued strictly increasing mapping, a differentiable functional $S_{\mathsf{\Theta }}=\left(S_{{\mathsf{\Theta }}^1},\dots ,S_{{\mathsf{\Theta }}^{\pi }}\right):\mathbb{R}\to {\mathbb{R}}^{\pi }$ with $S_{{\mathsf{\Theta }}^{\sigma }}:Im\left(G^{\sigma }\right)\to \mathbb{R}$ as a real-valued strictly increasing mapping, $\varphi :A\times {\mathbb{R}}^{2\sigma }\times {\mathbb{R}}^{2\varsigma }\to {\mathbb{R}}^o$ with $\varphi (w,\overline{ϵ}\left(w\right),\overline{ϵ}\left(w\right),\overline{q}\left(w\right),\overline{q}\left(w\right)){=\varphi |}_{w=a,b}=0$, and $\psi :A\times {\mathbb{R}}^{2\sigma }\times {\mathbb{R}}^{2\varsigma }\to {\mathbb{R}}^t$ with $\psi (w,\overline{ϵ}\left(w\right),\overline{ϵ}\left(w\right),\overline{q}\left(w\right),\overline{q}\left(w\right)){=\psi |}_{w=a,b}=0$, such that

$$\begin{array}{c}\left[\underset{\varsigma \in V}{\forall }{S}_{{\mathsf{\Phi }}^{\varsigma }}\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{ϵq}\left(w\right)\right)d\theta \right)\le S_{{\mathsf{\Phi }}^{\varsigma }}\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right)\right.\hfill \\ \left.\wedge \underset{\varsigma \in V}{\exists }{S}_{{\mathsf{\Phi }}^{\varsigma }}\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{ϵq}\left(w\right)\right)d\theta \right)<S_{{\mathsf{\Phi }}^{\varsigma }}\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right)\right]\hfill \\ ⟹S_{{\mathsf{\Phi }}^{\varsigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right){\mathit{\int }}_A\left\{{\left[\varphi \left({\tau }_{ϵq\overline{ϵq}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Phi }}_{ϵ}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)-D_{\alpha }{\mathsf{\Phi }}_{{ϵ}_{\alpha }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]\right.\hfill \\ \hfill \mathrm{(14)}& \left.+{\left[\psi \left({\tau }_{ϵq\overline{ϵq}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Phi }}_q^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]-D_{\beta }{\mathsf{\Phi }}_{q_{\beta }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right\}d\theta <0,\varsigma \in V,\hfill \end{array}$$

and

$$\begin{array}{c}-S_{{\mathsf{\Theta }}^{\sigma }}\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right)\le 0\hfill \\ ⟹S_{{\mathsf{\Theta }}^{\sigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right){\mathit{\int }}_A\left\{{\left[\varphi \left({\tau }_{ϵq\overline{ϵq}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Theta }}_{ϵ}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)-D_{\alpha }{\mathsf{\Theta }}_{{ϵ}_{\alpha }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]\right.\hfill \\ \hfill \mathrm{(15)}& \left.+{\left[\psi \left({\tau }_{ϵq\overline{ϵq}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Theta }}_q^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)-D_{\beta }{\mathsf{\Theta }}_{q_{\beta }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]\right\}d\theta \le 0,\sigma \in Z,\hfill \end{array}$$

hold, for all $(ϵ,q)\in R\times Q$, then $(\mathsf{\Phi },\mathsf{\Theta })$ is called weak-strictly-pseudo-quasi-S-type I at $(\overline{ϵ},\overline{q})\in R\times Q$ on $R\times Q$ related to $S_{\mathsf{\Phi }},S_{\mathsf{\Theta }},\varphi$ and ψ.

If (14) and (15) are valid for each $(\overline{ϵ},\overline{q})\in R\times Q$, then $(\mathsf{\Phi },\mathsf{\Theta })$ is called weak-strictly-pseudo-quasi-S-type I on $R\times Q$ related to $S_{\mathsf{\Phi }},S_{\mathsf{\Theta }},\varphi$ and ψ.

3. Reciprocal Theorems

In this section, for the considered multi-cost variational problem (P), we associate a dual model (D) in order to study and investigate the connections between the solution sets of these two variational control problems. Thus, we introduce the following multiple objective variational control problem:

$$\begin{array}{c}\left(\mathrm{D}\right)\underset{(y,v)}{max}{\mathit{\int }}_A\mathsf{\Phi }\left({\tau }_{yv}\left(w\right)\right)d\theta =\left({\mathit{\int }}_A{\mathsf{\Phi }}^1\left({\tau }_{yv}\left(w\right)\right)d\theta ,\dots ,{\mathit{\int }}_A{\mathsf{\Phi }}^{\eta }\left({\tau }_{yv}\left(w\right)\right)d\theta \right)\hfill \\ \mathrm{subject}\mathrm{to}\hfill \\ \sum _{\varsigma =1}^{\eta }{\varrho }_{\varsigma }{S}_{{\mathsf{\Phi }}^{\varsigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{yv}\left(w\right)\right)d\theta \right)\left[{\mathsf{\Phi }}_y^{\varsigma }\left({\tau }_{yv}\left(w\right)\right)-D_{\alpha }{\mathsf{\Phi }}_{y_{\alpha }}^{\varsigma }\left({\tau }_{yv}\left(w\right)\right)\right]\hfill \\ +\sum _{\sigma =1}^{\pi }{\xi }_{\sigma }\left(w\right)S_{{\mathsf{\Theta }}^{\sigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{yv}\left(w\right)\right)d\theta \right)\left[{\mathsf{\Theta }}_y^{\sigma }\left({\tau }_{yv}\left(w\right)\right)-D_{\alpha }{\mathsf{\Theta }}_{y_{\alpha }}^{\sigma }\left({\tau }_{yv}\left(w\right)\right)\right]=0,w\in A,\hfill \\ \sum _{\varsigma =1}^{\eta }{\varrho }_{\varsigma }{S}_{{\mathsf{\Phi }}^{\varsigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{yv}\left(w\right)\right)d\theta \right)\left[{\mathsf{\Phi }}_v^{\varsigma }\left({\tau }_{yv}\left(w\right)\right)-D_{\beta }{\mathsf{\Phi }}_{v_{\beta }}^{\varsigma }\left({\tau }_{yv}\left(w\right)\right)\right]\hfill \\ +\sum _{\sigma =1}^{\pi }{\xi }_{\sigma }\left(w\right)S_{{\mathsf{\Theta }}^{\sigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{yv}\left(w\right)\right)d\theta \right)\left[{\mathsf{\Theta }}_v^{\sigma }\left({\tau }_{yv}\left(w\right)\right)-D_{\beta }{\mathsf{\Theta }}_{v_{\beta }}^{\sigma }\left({\tau }_{yv}\left(w\right)\right)\right]=0,w\in A,\hfill \\ {\xi }_{\sigma }\left(w\right)S_{{\mathsf{\Theta }}^{\sigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{yv}\left(w\right)\right)d\theta \right)\ge 0,w\in A,\hfill \\ \varrho \in R^{\eta },\varrho \ge 0,\xi \left(w\right)\in R^{\pi },\xi \left(w\right)\geqq 0,y\left(a\right)=f,y\left(b\right)=g,\hfill \end{array}$$

with $\mathsf{\Phi }$ and $\mathsf{\Theta }$ defined as above.

Consider B is the set of all feasible solutions for (D), that is, the set

$$B=\left\{(y,v,\varrho ,\xi ):(y,v)\in R\times Q,\mathrm{satisfying}\mathrm{the}\mathrm{restrictions}\mathrm{in}\left(\mathrm{D}\right)\right\}.$$

The following result formulates a weak-type reciprocal outcome associated with the considered variational problems.

Theorem 1

(Weak duality). Consider $(ϵ,q)\in U$ and $(y,v,\varrho ,\xi )\in B$ and suppose that one of the following assumptions is valid:

(a)

$(\mathsf{\Phi },\mathsf{\Theta })$ is strictly S-type I at $(y,v)$ with respect to $S_{\mathsf{\Phi }},S_{\mathsf{\Theta }},\varphi$ and ψ;

(b)

$(\mathsf{\Phi },\mathsf{\Theta })$ is strictly-pseudo-quasi-S-type I at $(y,v)$ with respect to $S_{\mathsf{\Phi }},S_{\mathsf{\Theta }},\varphi$ and ψ;

(c)

$(\mathsf{\Phi },\mathsf{\Theta })$ is strong-pseudo-quasi-S-type I at $(y,v)$ with respect to $S_{\mathsf{\Phi }},S_{\mathsf{\Theta }},\varphi$ and ψ.

Then, the relations

$${\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{ϵq}\left(w\right)\right)d\theta \le {\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{yv}\left(w\right)\right)d\theta ,\mathrm{for}\mathrm{each}\varsigma \in V$$

(16)

and

$${\mathit{\int }}_A{\mathsf{\Phi }}^{{\varsigma }^{*}}\left({\tau }_{ϵq}\left(w\right)\right)d\theta <{\mathit{\int }}_A{\mathsf{\Phi }}^{{\varsigma }^{*}}\left({\tau }_{yv}\left(w\right)\right)d\theta ,\mathrm{for}\mathrm{some}{\varsigma }^{*}\in V$$

(17)

cannot hold.

Proof. 

Let $(ϵ,q)\in U$ and $(y,v,\varrho ,\xi )\in B$. We proceed by contradiction and consider that we are in hypothesis a) (for instance). It follows

$$\begin{array}{c}{S}_{{\mathsf{\Phi }}^{\varsigma }}\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{ϵq}\left(w\right)\right)d\theta \right)-S_{{\mathsf{\Phi }}^{\varsigma }}\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{yv}\left(w\right)\right)d\theta \right)\hfill \\ >S_{{\mathsf{\Phi }}^{\varsigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{yv}\left(w\right)\right)d\theta \right){\mathit{\int }}_A\left\{{\left[\varphi \left({\tau }_{ϵqyv}\left(w\right)\right)\right]}^T\left[{\mathsf{\Phi }}_y^{\varsigma }\left({\tau }_{yv}\left(w\right)\right)-D_{\alpha }{\mathsf{\Phi }}_{y_{\alpha }}^{\varsigma }\left({\tau }_{yv}\left(w\right)\right)\right]\right.\hfill \\ \hfill \mathrm{(18)}& \left.+{\left[\psi \left({\tau }_{ϵqyv}\left(w\right)\right)\right]}^T\left[{\mathsf{\Phi }}_v^{\varsigma }\left({\tau }_{yv}\left(w\right)\right)\right]-D_{\beta }{\mathsf{\Phi }}_{v_{\beta }}^{\varsigma }\left({\tau }_{yv}\left(w\right)\right)\right\}d\theta ,\varsigma \in V\hfill \end{array}$$

and

$$\begin{array}{c}-S_{{\mathsf{\Theta }}^{\sigma }}\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{yv}\left(w\right)\right)d\theta \right)\hfill \\ \ge S_{{\mathsf{\Theta }}^{\sigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{yv}\left(w\right)\right)d\theta \right){\mathit{\int }}_A\left\{{\left[\varphi \left({\tau }_{ϵqyv}\left(w\right)\right)\right]}^T\left[{\mathsf{\Theta }}_y^{\sigma }\left({\tau }_{yv}\left(w\right)\right)-D_{\alpha }{\mathsf{\Theta }}_{y_{\alpha }}^{\sigma }\left({\tau }_{yv}\left(w\right)\right)\right]\right.\hfill \\ \hfill \mathrm{(19)}& \left.+{\left[\psi \left({\tau }_{ϵqyv}\left(w\right)\right)\right]}^T\left[{\mathsf{\Theta }}_v^{\sigma }\left({\tau }_{yv}\left(w\right)\right)-D_{\beta }{\mathsf{\Theta }}_{v_{\beta }}^{\sigma }\left({\tau }_{yv}\left(w\right)\right)\right]\right\}d\theta ,\sigma \in Z.\hfill \end{array}$$

Since every $S_{{\mathsf{\Phi }}^{\varsigma }},\varsigma \in V$, is a strictly increasing functional on its domain, the inequalities (16) and (17) yield

$$S_{{\mathsf{\Phi }}^{\varsigma }}\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{ϵq}\left(w\right)\right)d\theta \right)\le S_{{\mathsf{\Phi }}^{\varsigma }}\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{yv}\left(w\right)\right)d\theta \right),\varsigma \in V$$

(20)

and

$$S_{{\mathsf{\Phi }}^{{\varsigma }^{*}}}\left({\mathit{\int }}_A{\mathsf{\Phi }}^{{\varsigma }^{*}}\left({\tau }_{ϵq}\left(w\right)\right)d\theta \right)<S_{{\mathsf{\Phi }}^{{\varsigma }^{*}}}\left({\mathit{\int }}_A{\mathsf{\Phi }}^{{\varsigma }^{*}}\left({\tau }_{yv}\left(w\right)\right)d\theta \right),\mathrm{for}\mathrm{some}{\varsigma }^{*}\in V.$$

(21)

By (18), (20) and (21), it follows that

$$\begin{array}{c}{S}_{{\mathsf{\Phi }}^{\varsigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{yv}\left(w\right)\right)d\theta \right){\mathit{\int }}_A\left[{\left[\varphi \left({\tau }_{ϵqyv}\left(w\right)\right)\right]}^T\left[{\mathsf{\Phi }}_y^{\varsigma }\left({\tau }_{yv}\left(w\right)\right)-D_{\alpha }{\mathsf{\Phi }}_{y_{\alpha }}^{\varsigma }\left({\tau }_{yv}\left(w\right)\right)\right]\right.\hfill \\ \hfill \mathrm{(22)}& \left.+{\left[\psi \left({\tau }_{ϵqyv}\left(w\right)\right)\right]}^T\left[{\mathsf{\Phi }}_v^{\varsigma }\left({\tau }_{yv}\left(w\right)\right)\right]-D_{\beta }{\mathsf{\Phi }}_{v_{\beta }}^{\varsigma }\left({\tau }_{yv}\left(w\right)\right)\right\}d\theta <0,\varsigma \in V.\hfill \end{array}$$

Multiplying by ${\varrho }_{\varsigma },\varsigma \in V$, and then adding, it results

$$\begin{array}{c}\sum _{\varsigma =1}^{\eta }{\varrho }_{\varsigma }{S}_{{\mathsf{\Phi }}^{\varsigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{yv}\left(w\right)\right)d\theta \right){\mathit{\int }}_A\left\{{\left[\varphi \left({\tau }_{ϵqyv}\left(w\right)\right)\right]}^T\left[{\mathsf{\Phi }}_y^{\varsigma }\left({\tau }_{yv}\left(w\right)\right)-D_{\alpha }{\mathsf{\Phi }}_{y_{\alpha }}^{\varsigma }\left({\tau }_{yv}\left(w\right)\right)\right]\right.\hfill \\ \hfill \mathrm{(23)}& \left.+{\left[\psi \left({\tau }_{ϵqyv}\left(w\right)\right)\right]}^T\left[{\mathsf{\Phi }}_v^{\varsigma }\left({\tau }_{yv}\left(w\right)\right)\right]-D_{\beta }{\mathsf{\Phi }}_{v_{\beta }}^{\varsigma }\left({\tau }_{yv}\left(w\right)\right)\right\}d\theta <0.\hfill \end{array}$$

Multiplying each inequality

$$\begin{array}{c}-S_{{\mathsf{\Theta }}^{\sigma }}\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{yv}\left(w\right)\right)d\theta \right)\hfill \\ \ge S_{{\mathsf{\Theta }}^{\sigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{yv}\left(w\right)\right)d\theta \right){\mathit{\int }}_A\left\{{\left[\varphi \left({\tau }_{ϵqyv}\left(w\right)\right)\right]}^T\left[{\mathsf{\Theta }}_y^{\sigma }\left({\tau }_{yv}\left(w\right)\right)-D_{\alpha }{\mathsf{\Theta }}_{y_{\alpha }}^{\sigma }\left({\tau }_{yv}\left(w\right)\right)\right]\right.\hfill \\ \left.+{\left[\psi \left({\tau }_{ϵqyv}\left(w\right)\right)\right]}^T\left[{\mathsf{\Theta }}_v^{\sigma }\left({\tau }_{yv}\left(w\right)\right)-D_{\beta }{\mathsf{\Theta }}_{v_{\beta }}^{\sigma }\left({\tau }_{yv}\left(w\right)\right)\right]\right\}d\theta ,\sigma \in Z,\hfill \end{array}$$

by ${\xi }_{\sigma }\left(w\right)\ge 0,\sigma \in Z$, and then adding, we obtain

$$\begin{array}{c}-\sum _{\sigma =1}^{\pi }{\xi }_{\sigma }\left(w\right)S_{{\mathsf{\Theta }}^{\sigma }}\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{yv}\left(w\right)\right)d\theta \right)\hfill \\ \ge \sum _{\sigma =1}^{\pi }{\xi }_{\sigma }\left(w\right)S_{{\mathsf{\Theta }}^{\sigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{yv}\left(w\right)\right)d\theta \right){\mathit{\int }}_A\left\{{\left[\varphi \left({\tau }_{ϵqyv}\left(w\right)\right)\right]}^T\left[{\mathsf{\Theta }}_y^{\sigma }\left({\tau }_{yv}\left(w\right)\right)-D_{\alpha }{\mathsf{\Theta }}_{y_{\alpha }}^{\sigma }\left({\tau }_{yv}\left(w\right)\right)\right]\right.\hfill \\ \hfill \mathrm{(24)}& \left.+{\left[\psi \left({\tau }_{ϵqyv}\left(w\right)\right)\right]}^T\left[{\mathsf{\Theta }}_v^{\sigma }\left({\tau }_{yv}\left(w\right)\right)-D_{\beta }{\mathsf{\Theta }}_{v_{\beta }}^{\sigma }\left({\tau }_{yv}\left(w\right)\right)\right]\right\}d\theta .\hfill \end{array}$$

Using the feasibility of $(y,v,\varrho ,\xi )$ in (D) together with (24), we get

$$\begin{array}{c}\sum _{\sigma =1}^{\pi }{\xi }_{\sigma }\left(w\right)S_{\mathsf{\Theta }\sigma }^{\prime }\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{yv}\left(w\right)\right)d\theta \right){\mathit{\int }}_A\left\{{\left[\varphi \left({\tau }_{ϵqyv}\left(w\right)\right)\right]}^T\left[{\mathsf{\Theta }}_y^{\sigma }\left({\tau }_{yv}\left(w\right)\right)-D_{\alpha }{\mathsf{\Theta }}_{y_{\alpha }}^{\sigma }\left({\tau }_{yv}\left(w\right)\right)\right]\right.\hfill \\ \hfill \mathrm{(25)}& \left.{\left[\psi \left({\tau }_{ϵqyv}\left(w\right)\right)\right]}^T\left[{\mathsf{\Theta }}_v^{\sigma }\left({\tau }_{yv}\left(w\right)\right)-D_{\beta }{\mathsf{\Theta }}_{v_{\beta }}^{\sigma }\left({\tau }_{yv}\left(w\right)\right)\right]\right\}d\theta \le 0.\hfill \end{array}$$

Adding (23) and (25), we obtain that

$$\begin{array}{c}{\mathit{\int }}_A{\left[\varphi \left({\tau }_{ϵqyv}\left(w\right)\right)\right]}^T\left\{\sum _{\varsigma =1}^{\eta }{\varrho }_{\varsigma }{S}_{{\mathsf{\Phi }}^{\varsigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{yv}\left(w\right)\right)d\theta \right)\left[{\mathsf{\Phi }}_y^{\varsigma }\left({\tau }_{yv}\left(w\right)\right)-D_{\alpha }{\mathsf{\Phi }}_{y_{\alpha }}^{\varsigma }\left({\tau }_{yv}\left(w\right)\right)\right]\right.\hfill \\ \left.+\sum _{\sigma =1}^{\pi }{\xi }_{\sigma }\left(w\right)S_{{\mathsf{\Theta }}^{\sigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{yv}\left(w\right)\right)d\theta \right)\left[{\mathsf{\Theta }}_y^{\sigma }\left({\tau }_{yv}\left(w\right)\right)-D_{\alpha }{\mathsf{\Theta }}_{y_{\alpha }}^{\sigma }\left({\tau }_{yv}\left(w\right)\right)\right]\right\}d\theta \hfill \\ +{\mathit{\int }}_A{\left[\psi \left({\tau }_{ϵqyv}\left(w\right)\right)\right]}^T\left\{\sum _{\varsigma =1}^{\eta }{\varrho }_{\varsigma }{S}_{{\mathsf{\Phi }}^{\varsigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{yv}\left(w\right)\right)d\theta \right)\left[{\mathsf{\Phi }}_v^{\varsigma }\left({\tau }_{yv}\left(w\right)\right)-D_{\beta }{\mathsf{\Phi }}_{v_{\beta }}^{\varsigma }\left({\tau }_{yv}\left(w\right)\right)\right]\right.\hfill \\ \left.+\sum _{\sigma =1}^{\pi }{\xi }_{\sigma }\left(w\right)S_{{\mathsf{\Theta }}^{\sigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{yv}\left(w\right)\right)d\theta \right)\left[{\mathsf{\Theta }}_v^{\sigma }\left({\tau }_{yv}\left(w\right)\right)-D_{\beta }{\mathsf{\Theta }}_v^{\sigma }\left({\tau }_{yv}\left(w\right)\right)\right]\right\}d\theta <0\hfill \end{array}$$

holds, contradicting the feasibility of $(y,v,\varrho ,\xi )$ in (D). This completes the proof. □

Under weaker generalized invexity assumptions, the next result is true.

Theorem 2

(Weak duality). Consider $(ϵ,q)\in U$ and $(y,v,\varrho ,\xi )\in B$ and suppose that one of the following assumptions is valid:

(a)

$(\mathsf{\Phi },\mathsf{\Theta })$ is S-type I at $(y,v)$ with respect to $S_{\mathsf{\Phi }}$, $S_{\mathsf{\Theta }},\varphi$ and ψ;

(b)

$(\mathsf{\Phi },\mathsf{\Theta })$ is pseudo-quasi-S-type I at $(y,v)$ with respect to $S_{\mathsf{\Phi }},S_{\mathsf{\Theta }},\varphi$ and ψ;

(c)

$(\mathsf{\Phi },\mathsf{\Theta })$ is weak-strictly-pseudo-quasi-S-type I at $(y,v)$ with respect to $S_{\mathsf{\Phi }},S_{\mathsf{\Theta }},\varphi$ and ψ.

Then, the relation

$${\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{ϵq}\left(w\right)\right)d\theta <{\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{yv}\left(w\right)\right)d\theta ,\mathrm{for}\mathrm{each}\varsigma \in V$$

cannot hold.

The next two results formulate strong-type reciprocal outcomes associated with the considered variational problems.

Theorem 3

(Strong duality). Consider $(\overline{ϵ},\overline{q})$ is an (weakly efficient) efficient point for (P), implying

$$\begin{array}{cc}\hfill & \sum _{\varsigma =1}^{\eta }{\overline{\varrho }}_{\varsigma }{S}_{{\mathsf{\Phi }}^{\varsigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right)\left[{\mathsf{\Phi }}_{ϵ}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)-D_{\alpha }{\mathsf{\Phi }}_{{ϵ}_{\alpha }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]\hfill \\ \hfill & +\sum _{\sigma =1}^{\pi }{\overline{\xi }}_{\sigma }{S}_{{\mathsf{\Theta }}^{\sigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right)\left[{\mathsf{\Theta }}_{ϵ}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)-D_{\alpha }{\mathsf{\Theta }}_{{ϵ}_{\alpha }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]=0,w\in A,\hfill \\ \hfill & \sum _{\varsigma =1}^{\eta }{\overline{\varrho }}_{\varsigma }{S}_{{\mathsf{\Phi }}^{\varsigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right)\left[{\mathsf{\Phi }}_q^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)-D_{\beta }{\mathsf{\Phi }}_{q_{\beta }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]\hfill \\ \hfill & +\sum _{\sigma =1}^{\pi }{\overline{\xi }}_{\sigma }{S}_{{\mathsf{\Theta }}^{\sigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right)\left[{\mathsf{\Theta }}_q^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)-D_{\beta }{\mathsf{\Theta }}_{q_{\beta }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]=0,w\in A,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\overline{\xi }}_{\sigma }\left(w\right)S_{{\mathsf{\Theta }}^{\sigma }}\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right)=0,w\in A,\sigma \in Z,\hfill \\ \hfill & \overline{\varrho }\ge 0,{\overline{\varrho }}^{T}e=1,\overline{\xi }\left(w\right)\geqq 0,\hfill \end{array}$$

are satisfied at this point. Then, $(\overline{ϵ},\overline{q},\overline{\varrho },\overline{\xi })$ is a feasible point for (D) and, in addition, if Theorem 1 (or Theorem 2) is true, then $(\overline{ϵ},\overline{q},\overline{\varrho },\overline{\xi })$ is an (weakly efficient) efficient solution for (D).

Theorem 4

(Strong duality). Consider $(\overline{ϵ},\overline{q})$ is a properly efficient point for (P), involving

$$\begin{array}{cc}\hfill & \sum _{\varsigma =1}^{\eta }{\overline{\varrho }}_{\varsigma }{S}_{{\mathsf{\Phi }}^{\varsigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right)\left[{\mathsf{\Phi }}_{ϵ}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)-D_{\alpha }{\mathsf{\Phi }}_{{ϵ}_{\alpha }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]\hfill \\ \hfill & +\sum _{\sigma =1}^{\pi }{\overline{\xi }}_{\sigma }{S}_{{\mathsf{\Theta }}^{\sigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right)\left[{\mathsf{\Theta }}_{ϵ}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)-D_{\alpha }{\mathsf{\Theta }}_{{ϵ}_{\alpha }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]=0,w\in A,\hfill \\ \hfill & \sum _{\varsigma =1}^{\eta }{\overline{\varrho }}_{\varsigma }{S}_{{\mathsf{\Phi }}^{\varsigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right)\left[{\mathsf{\Phi }}_q^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)-D_{\beta }{\mathsf{\Phi }}_{q_{\beta }}^{\varsigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]\hfill \\ \hfill & +\sum _{\sigma =1}^{\pi }{\overline{\xi }}_{\sigma }{S}_{{\mathsf{\Theta }}^{\sigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right)\left[{\mathsf{\Theta }}_q^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)-D_{\beta }{\mathsf{\Theta }}_{q_{\beta }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)\right]=0,w\in A,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\overline{\xi }}_{\sigma }\left(w\right)S_{{\mathsf{\Theta }}^{\sigma }}\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta \right)=0,w\in A,\sigma \in Z,\hfill \\ \hfill & \overline{\varrho }\ge 0,{\overline{\varrho }}^{T}e=1,\overline{\xi }\left(w\right)\geqq 0,\hfill \end{array}$$

are valid, and assume the assumptions in Theorem 1 (or Theorem 2) hold. Then, $(\overline{ϵ},\overline{q},\overline{\varrho },\overline{\xi })$ is a feasible point for (D) and, in addition, $(\overline{ϵ},\overline{q},\overline{\varrho },\overline{\xi })$ is a properly efficient solution for (D) and the cost values are equal at these points.

Proof. 

As $(\overline{ϵ},\overline{q})$ is a properly efficient solution for (P), there exist $\overline{\varrho }\in R^{\eta },\overline{\varrho }\ge 0$ and $\overline{\xi }(·):A\to R^{\pi }$ such that the conditions mentioned abobe are satisfied. Thus, $(\overline{ϵ},\overline{q},\overline{\varrho },\overline{\xi })$ is feasible point for (D). By using Theorem 1 (or Theorem 2), it results that $(\overline{ϵ},\overline{q},\overline{\varrho },\overline{\xi })$ is an efficient solution for (D). Now, by considering the method of contradiction, we prove that $(\overline{ϵ},\overline{q},\overline{\varrho },\overline{\xi })$ is a properly efficient solution for (D). Therefore, there exists $(\tilde{y},\tilde{v},\tilde{\varrho },\tilde{\xi })\in B$ and ${\varsigma }^{*}\in V$ such that

$$\begin{array}{c}{\mathit{\int }}_A{\mathsf{\Phi }}^{{\varsigma }^{*}}\left({\tau }_{\tilde{y}\tilde{v}}\left(w\right)\right)d\theta -{\mathit{\int }}_A{\mathsf{\Phi }}^{{\varsigma }^{*}}\left({\tau }_{\overline{ϵq}}\left(w\right)\right)d\theta \\ \hfill \mathrm{(26)}& >F\left({\mathit{\int }}_A{\mathsf{\Phi }}^k\left({\tau }_{\overline{ϵq}}\left(w\right)\right)d\theta -{\mathit{\int }}_A{\mathsf{\Phi }}^k\left({\tau }_{\tilde{y}\tilde{v}}\left(w\right)\right)d\theta \right)\end{array}$$

is valid for every scalar $F>0$ and all k, satisfying

$${\mathit{\int }}_A{\mathsf{\Phi }}^k\left({\tau }_{\overline{ϵq}}\left(w\right)\right)d\theta >{\mathit{\int }}_A{\mathsf{\Phi }}^k\left({\tau }_{\tilde{y}\tilde{v}}\left(w\right)\right)d\theta .$$

(27)

We consider the set of indexes $V=V_1\cup V_2$, with $V_1$ as the set of indexes for the cost functionals that satisfy (27). Define $V_2$ as follows $V_2:=V\setminus \left(V_1\cup \left\{{\varsigma }^{*}\right\}\right)$. Relation in (26) is valid for all $F>0$. We set $F>{\displaystyle \frac{{\overline{\varrho }}_k}{{\overline{\varrho }}_{{\varsigma }^{*}}}}\left|V_1\right|$, where $\left|V_1\right|$ denotes the number of elements in the set $V_1$. Thus, (26) and (27) yield

$$\begin{array}{c}{\overline{\varrho }}_{{\varsigma }^{*}}\left({\mathit{\int }}_A{\mathsf{\Phi }}^{{\varsigma }^{*}}\left({\tau }_{\overline{ϵq}}\left(w\right)\right)d\theta -{\mathit{\int }}_A{\mathsf{\Phi }}^{{\varsigma }^{*}}\left({\tau }_{\tilde{y}\tilde{v}}\left(w\right)\right)d\theta \right)\hfill \\ \hfill \mathrm{(28)}& >\sum _{k\in V_1}{\overline{\varrho }}_k\left({\mathit{\int }}_A{\mathsf{\Phi }}^k\left({\tau }_{\overline{ϵq}}\left(w\right)\right)d\theta -{\mathit{\int }}_A{\mathsf{\Phi }}^k\left({\tau }_{\tilde{y}\tilde{v}}\left(w\right)\right)d\theta \right).\hfill \end{array}$$

By the definition of the set $V_2$, (26)–(28), it follows that

$$\begin{array}{cc}& \sum _{\varsigma =1}^{\eta }{\overline{\varrho }}_{\varsigma }{\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{\overline{ϵq}}\left(w\right)\right)d\theta ={\overline{\varrho }}_{{\varsigma }^{*}}{\mathit{\int }}_A{\mathsf{\Phi }}^{{\varsigma }^{*}}\left({\tau }_{\overline{ϵq}}\left(w\right)\right)d\theta +\sum _{k\in V_1}{\overline{\varrho }}_k{\mathit{\int }}_A{\mathsf{\Phi }}^k\left({\tau }_{\overline{ϵq}}\left(w\right)\right)d\theta \hfill \\ & +\sum _{k\in V_2}{\overline{\varrho }}_k{\mathit{\int }}_A{\mathsf{\Phi }}^k\left({\tau }_{\overline{ϵq}}\left(w\right)\right)d\theta <{\overline{\varrho }}_{{\varsigma }^{*}}{\mathit{\int }}_A{\mathsf{\Phi }}^{{\varsigma }^{*}}\left({\tau }_{\tilde{y}\tilde{v}}\left(w\right)\right)d\theta \hfill \\ & +\sum _{k\in V_1}{\overline{\varrho }}_k{\mathit{\int }}_A{\mathsf{\Phi }}^k\left({\tau }_{\tilde{y}\tilde{\eta }}\left(w\right)\right)d\theta +\sum _{k\in V_2}{\overline{\varrho }}_k{\mathit{\int }}_A{\mathsf{\Phi }}^k\left({\tau }_{\tilde{y}\tilde{\eta }}\left(w\right)\right)d\theta =\sum _{\varsigma =1}^{\eta }{\overline{\varrho }}_{\varsigma }{\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{\tilde{y}\tilde{v}}\left(w\right)\right)d\theta .\hfill \end{array}$$

This is a contradiction to the weak duality theorem (see Theorem 1). Hence, $(\overline{ϵ},\overline{q},\overline{\varrho },\overline{\xi })$ is a properly efficient solution for (D), and the cost values are equal at these points. □

The last result derived in the current paper is provided by the following theorem. It establishes a strict converse duality associated with the two considered variational models.

Theorem 5

(Strict converse duality). Let $(\overline{ϵ},\overline{q})\in U$ and $(\overline{y},\overline{v},\overline{\varrho },\overline{\xi })\in B$ such that

$${\overline{\varrho }}_{\varsigma }{S}_{{\mathsf{\Phi }}^{\varsigma }}\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{\overline{ϵq}}\left(w\right)\right)d\theta \right)={\overline{\varrho }}_{\varsigma }{S}_{{\mathsf{\Phi }}^{\varsigma }}\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{\overline{yv}}\left(w\right)\right)d\theta \right).$$

(29)

Further, assume that $(\mathsf{\Phi },\mathsf{\Theta })$ is strictly-S-type I at $(\overline{y},\overline{v})$ with respect to $S_{\mathsf{\Phi }},S_{\mathsf{\Theta }},\varphi$ and ψ. Then $(\overline{ϵ},\overline{q})=(\overline{y},\overline{v})$.

Proof. 

Contrary to the result, we assume that $(\overline{ϵ},\overline{q})\ne (\overline{y},\overline{v})$. By assumption, $(\mathsf{\Phi },\mathsf{\Theta })$ is strictly-S-type I at $(\overline{y},\overline{v})$ with respect to $S_{\mathsf{\Phi }},S_{\mathsf{\Theta }},\varphi$ and $\psi$. Then, we have

$$\begin{array}{c}{S}_{{\mathsf{\Phi }}^{\varsigma }}\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{\overline{ϵq}}\left(w\right)\right)d\theta \right)-S_{{\mathsf{\Phi }}^{\varsigma }}\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{\overline{yv}}\left(w\right)\right)d\theta \right)\hfill \\ >S_{{\mathsf{\Phi }}^{\varsigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{\overline{yv}}\left(w\right)\right)d\theta \right){\mathit{\int }}_A\left\{{\left[\varphi \left({\tau }_{\overline{ϵqyv}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Phi }}_y^{\varsigma }\left({\tau }_{\overline{yv}}\left(w\right)\right)-D_{\alpha }{\mathsf{\Phi }}_{y_{\alpha }}^{\varsigma }\left({\tau }_{\overline{yv}}\left(w\right)\right)\right]\right.\hfill \\ \hfill \mathrm{(30)}& \left.+{\left[\psi \left({\tau }_{\overline{ϵqy\overline{v}}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Phi }}_v^{\varsigma }\left({\tau }_{\overline{yv}}\left(w\right)\right)\right]-D_{\beta }{\mathsf{\Phi }}_{v_{\beta }}^{\varsigma }\left({\tau }_{\overline{y}\overline{v}}\left(w\right)\right)\right\}d\theta ,\varsigma \in V\hfill -S_{{\mathsf{\Theta }}^{\sigma }}\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{\overline{y}\overline{v}}\left(w\right)\right)d\theta \right)\hfill \\ \ge S_{{\mathsf{\Theta }}^{\sigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{\overline{y}\overline{v}}\left(w\right)\right)d\theta \right){\mathit{\int }}_A\left\{{\left[\varphi \left({\tau }_{\overline{ϵqyv}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Theta }}_y^{\sigma }\left({\tau }_{\overline{y}\overline{v}}\left(w\right)\right)-D_{\alpha }{\mathsf{\Theta }}_{y_{\alpha }}^{\sigma }\left({\tau }_{\overline{yv}}\left(w\right)\right)\right]\right.\hfill \\ \hfill \mathrm{(31)}& \left.+{\left[\psi \left({\tau }_{\overline{ϵqy\overline{v}}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Theta }}_v^{\sigma }\left({\tau }_{\overline{yv}}\left(w\right)\right)-D_{\beta }{\mathsf{\Theta }}_{v_{\beta }}^{\sigma }\left({\tau }_{\overline{yv}}\left(w\right)\right)\right]\right\}d\theta ,\sigma \in Z.\hfill \end{array}$$

Multiplying each inequality (30) by ${\overline{\varrho }}_{\varsigma },\varsigma \in V$, then (29) gives

$$\begin{array}{c}{\overline{\varrho }}_{\varsigma }{S}_{{\mathsf{\Phi }}^{\varsigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{\overline{yv}}\left(w\right)\right)d\theta \right){\mathit{\int }}_A\left\{{\left[\varphi \left({\tau }_{\overline{ϵqyv}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Phi }}_y^{\varsigma }\left({\tau }_{\overline{yv}}\left(w\right)\right)-D_{\alpha }{\mathsf{\Phi }}_{y_{\alpha }}^{\varsigma }\left({\tau }_{\overline{yv}}\left(w\right)\right)\right]\right.\hfill \\ \left.+{\left[\psi \left({\tau }_{\overline{ϵqyv}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Phi }}_v^{\varsigma }\left({\tau }_{\overline{yv}}\left(w\right)\right)\right]-D_{\beta }{\mathsf{\Phi }}_{v_{\beta }}^{\varsigma }\left({\tau }_{\overline{yv}}\left(w\right)\right)\right\}d\theta <0,\varsigma \in V.\hfill \end{array}$$

Adding both sides of the above inequalities, we get

$$\begin{array}{c}\sum _{\varsigma =1}^{\eta }{\overline{\varrho }}_{\varsigma }{S}_{{\mathsf{\Phi }}^{\varsigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{\overline{yv}}\left(w\right)\right)d\theta \right){\mathit{\int }}_A\left\{{\left[\varphi \left({\tau }_{\overline{ϵqyv}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Phi }}_y^{\varsigma }\left({\tau }_{\overline{yv}}\left(w\right)\right)-D_{\alpha }{\mathsf{\Phi }}_{y_{\alpha }}^{\varsigma }\left({\tau }_{\overline{yv}}\left(w\right)\right)\right]\right.\hfill \\ \hfill \mathrm{(32)}& \left.+{\left[\psi \left({\tau }_{\overline{ϵqy\overline{v}}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Phi }}_v^{\varsigma }\left({\tau }_{\overline{y}\overline{v}}\left(w\right)\right)\right]-D_{\beta }{\mathsf{\Phi }}_{v_{\beta }}^{\varsigma }\left({\tau }_{\overline{y}\overline{v}}\left(w\right)\right)\right\}d\theta <0.\hfill \end{array}$$

Multiplying each inequality (31) by ${\overline{\xi }}_{\sigma }\left(w\right)\ge 0,\sigma \in Z$, and then adding both sides of the obtained inequalities, we obtain

$$\begin{array}{c}-\sum _{\sigma =1}^{\pi }{\overline{\xi }}_{\sigma }\left(w\right)S_{{\mathsf{\Theta }}^{\sigma }}\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{\overline{yv}}\left(w\right)\right)d\theta \right)\hfill \\ \ge \sum _{\sigma =1}^{\pi }{S}_{{\mathsf{\Theta }}^{\sigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{\overline{yv}}\left(w\right)\right)d\theta \right){\mathit{\int }}_A\left\{{\overline{\xi }}_{\sigma }\left(w\right){\left[\varphi \left({\tau }_{\overline{ϵqyv}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Theta }}_y^{\sigma }\left({\tau }_{\overline{yv}}\left(w\right)\right)-D_{\alpha }{\mathsf{\Theta }}_{y_{\alpha }}^{\sigma }\left({\tau }_{\overline{yv}}\left(w\right)\right)\right]\right.\hfill \\ \hfill \mathrm{(33)}& \left.+{\left[\psi \left({\tau }_{\overline{ϵqy\overline{v}}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Theta }}_v^{\sigma }\left({\tau }_{\overline{yv}}\left(w\right)\right)-D_{\beta }{\mathsf{\Theta }}_{v_{\beta }}^{\sigma }\left({\tau }_{\overline{yv}}\left(w\right)\right)\right]\right\}d\theta ,\sigma \in Z.\hfill \end{array}$$

Hence, the feasibility of $(\overline{y},\overline{v},\overline{\varrho },\overline{\xi })$ in (D) implies

$$\begin{array}{c}\sum _{\sigma =1}^{\pi }{S}_{{\mathsf{\Theta }}^{\sigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{\overline{y}\overline{v}}\left(w\right)\right)d\theta \right){\mathit{\int }}_A\left\{{\overline{\xi }}_{\sigma }\left(w\right){\left[\varphi \left({\tau }_{\overline{ϵqy\overline{v}}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Theta }}_y^{\sigma }\left({\tau }_{\overline{y}\overline{v}}\left(w\right)\right)-D_{\alpha }{\mathsf{\Theta }}_{y_{\alpha }}^{\sigma }\left({\tau }_{\overline{y}\overline{v}}\left(w\right)\right)\right]\right.\hfill \\ \hfill \mathrm{(34)}& \left.+{\left[\psi \left({\tau }_{\overline{ϵqyv}}\left(w\right)\right)\right]}^T\left[{\mathsf{\Theta }}_v^{\sigma }\left({\tau }_{\overline{yv}}\left(w\right)\right)-D_{\beta }{\mathsf{\Theta }}_{v_{\beta }}^{\sigma }\left({\tau }_{\overline{yv}}\left(w\right)\right)\right]\right\}d\theta \le 0.\hfill \end{array}$$

Adding both sides of (32) and (34), we get that the following inequality

$$\begin{array}{cc}& {\mathit{\int }}_A{\left[\varphi \left({\tau }_{\overline{ϵqyv}}\left(w\right)\right)\right]}^T\left\{\sum _{\varsigma =1}^{\eta }{\varrho }_{\varsigma }{S}_{{\mathsf{\Phi }}^{\varsigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{\overline{yv}}\left(w\right)\right)d\theta \right)\left[{\mathsf{\Phi }}_y^{\varsigma }\left({\tau }_{\overline{yv}}\left(w\right)\right)-D_{\alpha }{\mathsf{\Phi }}_{y_{\alpha }}^{\varsigma }\left({\tau }_{\overline{yv}}\left(w\right)\right)\right]\right.\hfill \\ & \left.+\sum _{\sigma =1}^{\pi }{\xi }_{\sigma }\left(w\right)S_{{\mathsf{\Theta }}^{\sigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{\overline{yv}}\left(w\right)\right)d\theta \right)\left[{\mathsf{\Theta }}_y^{\sigma }\left({\tau }_{\overline{yv}}\left(w\right)\right)-D_{\alpha }{\mathsf{\Theta }}_{y_{\alpha }}^{\sigma }\left({\tau }_{\overline{yv}}\left(w\right)\right)\right]\right\}d\theta \hfill \\ & +{\mathit{\int }}_A{\left[\psi \left({\tau }_{\overline{ϵqyv}}\left(w\right)\right)\right]}^T\left\{\sum _{\varsigma =1}^{\eta }{\varrho }_{\varsigma }{S}_{{\mathsf{\Phi }}^{\varsigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Phi }}^{\varsigma }\left({\tau }_{\overline{yv}}\left(w\right)\right)d\theta \right)\left[{\mathsf{\Phi }}_v^{\varsigma }\left({\tau }_{\overline{y}\overline{v}}\left(w\right)\right)-D_{\beta }{\mathsf{\Phi }}_{v_{\beta }}^{\varsigma }\left({\tau }_{\overline{yv}}\left(w\right)\right)\right]\right.\hfill \\ & \left.+\sum _{\sigma =1}^{\pi }{\xi }_{\sigma }\left(w\right)S_{{\mathsf{\Theta }}^{\sigma }}^{\prime }\left({\mathit{\int }}_A{\mathsf{\Theta }}^{\sigma }\left({\tau }_{\overline{yv}}\left(w\right)\right)d\theta \right)\left[{\mathsf{\Theta }}_v^{\sigma }\left({\tau }_{\overline{y}\overline{v}}\left(w\right)\right)-D_{\beta }{\mathsf{\Theta }}_{v_{\beta }}^{\sigma }\left({\tau }_{\overline{yv}}\left(w\right)\right)\right]\right\}d\theta <0\hfill \end{array}$$

holds, which is a contradiction to the feasibility of $(\overline{y},\overline{v},\overline{\varrho },\overline{\xi })$ in (D). □

Illustrative Example.

We formulate the following constrained multi-objective nonlinear control problem:

$$\begin{array}{cc}\hfill \left(P\right)\underset{\left(ϵ\right(·),q(·\left)\right)}{min}& \left({\mathit{\int }}_{\Lambda }({ϵ}^2+{\displaystyle \frac{5}{4}}q)d{t}^{1}d{t}^2,{\mathit{\int }}_{\Lambda }{q}^{2}d{t}^{1}d{t}^2\right)\hfill \\ \hfill \mathrm{subject} \mathrm{to}& ϵ(ϵ-3)\le 0,\hfill \\ & {\displaystyle \frac{\mathsf{\partial }ϵ}{\mathsf{\partial }{t}^1}}=2-q,\hfill \\ & {\displaystyle \frac{\mathsf{\partial }ϵ}{\mathsf{\partial }{t}^2}}=2-q,\hfill \\ & ϵ(0,0)=0,ϵ\left({\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}}\right)={\displaystyle \frac{1}{2}},\hfill \end{array}$$

where  $t=(t^1,t^2)\in \Lambda ={\left[0,{\displaystyle \frac{1}{4}}\right]}^2.$

Let $(\overline{ϵ},\overline{q})=(t^1+t^2,1)$ be a feasible solution to the problem (P). The dual problem associated with (P) is defined as follows:

$$\begin{array}{c}\hfill \left(D\right)\underset{\left(y\right(·),v(·\left)\right)}{max}\left({\mathit{\int }}_{\Lambda }(y^2+{\displaystyle \frac{5}{4}}v)d{t}^{1}d{t}^2,{\mathit{\int }}_{\Lambda }{v}^{2}d{t}^{1}d{t}^2\right)\end{array}$$

subject to

$$2{\mu }_{1}y+\nu (2y-3)-{\displaystyle \frac{\mathsf{\partial }{\gamma }_1}{\mathsf{\partial }{t}^1}}-{\displaystyle \frac{\mathsf{\partial }{\gamma }_2}{\mathsf{\partial }{t}^2}}=0,$$

$${\displaystyle \frac{5}{4}}{\mu }_1+2v{\mu }_2+{\gamma }_1+{\gamma }_2=0,$$

$$y(0,0)=0,y\left({\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}}\right)={\displaystyle \frac{1}{2}},$$

$${\mu }^T>0,e^T\mu =1,e=(1,1)\in {\mathbb{R}}^2.$$

We note that $B=\left\{\right(y,v,\mu ,\nu ,\gamma \left)\mathit{satisfying}\mathit{constraints}\mathit{in}D\right\}$ is the feasible solution set to (D). Let us consider $\overline{y}=t^1+t^2,\overline{v}=-t^1{t}^2-{\displaystyle \frac{5}{16}},\overline{\mu }=({\overline{\mu }}_1,{\overline{\mu }}_2)=({\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}),\overline{\nu }=0,\overline{\gamma }=({\overline{\gamma }}_1,{\overline{\gamma }}_2)=(t^1{t}^2,t^1{t}^2)$. Then $(\overline{y},\overline{v},\overline{\mu },\overline{\nu },\overline{\gamma })$ is a feasible solution to (D). Further, it can be easily verified that all the involved functionals are strictly S-type I at $(\overline{y},\overline{v})$ with respect to $S_{\mathsf{\Phi }}=\mathit{identity}\mathit{map},S_{\mathsf{\Theta }}=\mathit{identity}\mathit{map},\varphi =ϵ-\overline{ϵ}$ and $\psi =q-\overline{q}$. Also, the following inequality

$${\mathit{\int }}_A\mathsf{\Phi }\left({\tau }_{\overline{ϵ}\overline{q}}\left(w\right)\right)d\theta >{\mathit{\int }}_A\mathsf{\Phi }\left({\tau }_{\overline{y}\overline{v}}\left(w\right)\right)d\theta$$

shows that the duality gap is positive. In consequence, Theorem 1 (Weak duality) is verified.

4. Conclusions and Further Developments

In this study, new results and characterizations of the set of solutions associated with a family of constrained optimization problems have been stated. Namely, a new family of multi-cost models has been introduced and, under S-type I hypotheses of the involved multiple integral-type functionals, various connections were highlighted between the solution sets of these control problems. We have formulated and proved various reciprocal results, such as weak, strong, and converse-type dualities. More specifically, firstly, under suitable hypotheses, we have established that the value of the cost functional associated with the original (primal) model can not be greater than the value of the cost functional associated with the dual model. Then, the results presented a strong and converse-type duality between the considered variational models. The limitations of the presented results could be, for instance, the appearance of (path-dependent or path-independent) curvilinear integrals as functionals instead of multiple integrals. This situation would add additional conditions to the study of this class of problems. In addition, as for issues that remain unsolved, we can mention the analysis of well-posedness associated with this type of extremization model and, also, establish new efficiency criteria based on saddle-points or a modified cost functional approach. Such research directions are still unexplored and, due to their importance and efficiency, deserve to be studied as soon as possible. Consequently, we note the applicability of the proposed technique to larger or more complex optimization problems.