Duality for Multitime Multiobjective Ratio Variational Problems on First Order Jet Bundle

Abstract

We consider a new class of multitime multiobjective variational problems of minimizing a vector of quotients of functionals of curvilinear integral type. Based on the efficiency conditions for multitime multiobjective ratio variational problems, we introduce a ratio dual of generalized Mond-Weir-Zalmai type, and under some assumptions of generalized convexity, duality theorems are stated. We prove our weak duality theorem for efficient solutions, showing that the value of the objective function of the primal cannot exceed the value of the dual. Direct and converse duality theorems are stated, underlying the connections between the values of the objective functions of the primal and dual programs. As special cases, duality results of Mond-Weir-Zalmai type for a multitime multiobjective variational problem are obtained. This work further develops our studies in (Pitea and Postolache (2011)).

1. Introduction

The duality for scalar variational problems involving convex functions has been formulated by Mond and Hanson in [1]. This study was further developed to various classes of convex and generalized convex functions. In [2], Hanson extended the duality results (in the sense of Wolfe) to a class of functions subsequently called invex. In order to weaken the convexity conditions, Bector et al. [3], introduced a dual to a variational problem (in the sense of Mond and Weir), different from that formulated by Mond and Hanson in [1]. Mond et al. [4], extended the concept of invexity to continuous functions and used it to generalize earlier Wolfe duality results for a class of variational problems. Duality theory for scalar optimization can be found in many other works and see Mond and Husain [5] and Preda [6].

In [7], using a vector-valued Lagrangian function, Tanino and Sawaragi introduced a duality theory for multiobjective optimization. This idea is employed by Bitran [8], which associated a matrix of dual variables with the constraints in the primal problem. In [9], Mond and Weir considered a pair of symmetric dual nonlinear programs and developed a duality theory under assumptions of pseudoconvexity. Under various types of generalized convex functions, Mukherjee and Purnachandra [10], Preda [11] and Zalmai [12] established several weak efficiency conditions and developed different types of dualities for multiobjective variational problems. D. S. Kim and A. L. Kim [13] used the efficiency property of nondifferentiable multiobjective variational problems in duality theory. In a recent study [14], Pitea and Postolache considered a new class of multitime multiobjective variational problems of minimizing a vector of functionals of curvilinear integral type. Based on appropriate normal efficiency conditions, they studied duals of Mond-Weir type, generalized Mond-Weir-Zalmai type, and under appropriate assumptions of generalized convexity, stated duality theorems.

In time, several authors have been interested in the study of (vector) ratio programs in connection with generalized convexity. This study is motivated by many practical optimization problems whose objective functions are quotients of two functions. In [15], Jagannathan introduced a duality study using some results connecting solutions of a nonlinear ratio program with those of suitably defined parametric convex program. Concerning advances on single-objective ratio programs, see [16] by Khan and Hanson and [17] by Reddy and Mukherjee, which utilized invexity assumptions in the sense of Hanson [18] to obtain optimality conditions and duality results. As concerns vector ratio problems, Singh and Hanson [19] applied invex functions to derive duality results, while Jeyakumar and Mond [20] generalized these results to the class of 𝑉-invex functions. Later, Liang et al. [21] introduced a unified formulation of the generalized convexity to derive optimality conditions and duality results for vector ratio problems.

In this paper, we are motivated by previous published research articles (please refer to [22] by Antczak, [23] by Nahak and Mohapatra, and [14] by Pitea and Postolache) to consider a new class of multitime multiobjective variational problems of minimizing a vector of quotients of functionals of curvilinear integral type and, as a particular case, a vector of functionals of curvilinear integral type. We state and prove new duality results, of Mond-Weir-Zalmai type, for feasible solutions of our multitime multiobjective ratio variational problems, under assumptions of (𝜌,𝑏)-quasi-invexity. This study is encouraged by its possible application in mechanical engineering, where curvilinear integral objectives are extensively used due to their physical meaning as mechanical work. The objective vector function is of curvilinear integral type, the integrand depending on velocities; that is why we consider it adequate to introduce our results within the framework offered by the first order jet bundle, [24–26] by Pitea et al., [27] by Sounders, and [28] by Udriste et al.

2. Preliminaries

Before presenting our results, we need the following background, which is necessary for the completeness of the exposition. For more details, we address the reader to [27].

2.1. The First Order Jet Bundle

To make our presentation self-contained and reader-friendly, we recall the notion of jet bundle of the first order, 𝐽¹(𝑇,𝑀).

Let be given the smooth manifolds (𝑀,𝑔) and (𝑇,ℎ), of dimensions 𝑛, and 𝑝, respectively. The corresponding local coordinates are 𝑥=(𝑥^𝑖), 𝑖=1,𝑛, and 𝑡=(𝑡^𝛼), 𝛼=1,𝑝, respectively.

In the following, the set {1,2,…,𝑛} will be indexed by Latin characters, while the set {1,2,…,𝑝} will be indexed by Greek characters.

Denote by 𝐶^∞(𝑇,𝑀) the set of mappings of class 𝐶^∞, from 𝑇 to 𝑀.

Consider (𝑡₀,𝑥₀) an arbitrary point of the product manifold 𝑇×𝑀. On the set 𝐶^∞(𝑇,𝑀), define the equivalence relation 𝑓∼_{(𝑡0,𝑥0)}𝑓𝑡𝑔⟺⁰=𝑔𝑡₀=𝑥₀𝑑𝑓_𝑡0=𝑑𝑔_𝑡0. (2.1)

If 𝑓 and 𝑔 are arbitrary mappings from 𝐶^∞(𝑇,𝑀), we denote 𝑡^𝛽𝑡₀=𝑡^𝛽₀,𝛽=𝑥1,𝑝,_𝑖=𝑥^𝑖𝑦∘𝑓,_𝑖=𝑥^𝑖𝑥∘𝑔,_𝑖𝑥₀=𝑥^𝑖₀,𝑖=1,𝑛. (2.2)

Thus, the equivalence relation 𝑓∼_{(𝑡0,𝑥0)}𝑔 has the local expression 𝑥^𝑖𝑡^𝛽₀=𝑦^𝑖𝑡^𝛽₀=𝑥^𝑖₀,𝜕𝑥^𝑖𝜕𝑡^𝛼𝑡^𝛽₀=𝜕𝑦^𝑖𝜕𝑡^𝛼𝑡^𝛽₀,𝑖=1,𝑛,𝛼=1,𝑝. (2.3)

The equivalence class of a mapping 𝑓∈𝐶^∞(𝑇,𝑀) will be denoted by 𝑓_{(𝑡0,𝑥0)}=𝑔∈𝐶^∞(𝑇,𝑀)∣𝑔∼_{(𝑡0,𝑥0)}𝑓. (2.4) The quotient space obtained by the factorization of the space 𝐶^∞(𝑇,𝑀) by the equivalence relation “∼_{(𝑡0,𝑥0)}”, 𝐽¹_𝑡0,𝑥0()=𝐶𝑇,𝑀^∞(𝑇,𝑀)∼_{(𝑡0,𝑥0)}, (2.5) is called 1-jet at the point (𝑡₀,𝑥₀).

The total space of the set of 1-jets, 𝐽¹(𝑇,𝑀)=_{(𝑡0,𝑥0)∈𝑇×𝑀}𝐽¹_{(𝑡0,𝑥0)}(𝑇,𝑀), (2.6) can be organized as a vector bundle over the base space 𝑇×𝑀, endowed with the differentiable structure of the product space.

Let 𝜍 be the canonical projection, defined as 𝜍∶𝐽¹(𝑇,𝑀)→𝑇×𝑀,𝜍𝑓_{(𝑡0,𝑥0)}=𝑡₀,𝑓𝑡₀. (2.7)

The mapping 𝜍 is well defined, having the property to be onto.

For every local chart 𝑈×𝑉 of the product manifold 𝑇×𝑀, we define the bijection 𝜙_𝑈×𝑉∶𝜍⁻¹(𝑈×𝑉)→𝑈×𝑉×ℝ^𝑛𝑝,𝜙_𝑈×𝑉𝑓_{(𝑡0,𝑥0)}=𝑡₀,𝑥₀,𝜕𝑥^𝑖𝜕𝑡^𝛼𝑡^𝛽₀,𝑥₀=𝑓𝑡₀. (2.8)

Therefore, the 1-jet space is endowed with a differentiable structure of dimension 𝑛+𝑝+𝑛𝑝, such that the mappings 𝜙_𝑈×𝑉 are diffeomorphisms.

The local coordinates on the space 𝐽¹(𝑇,𝑀) are (𝑡^𝛼,𝑥^𝑖,𝑥^𝑖_𝛼), where 𝑡^𝛼𝑓_{(𝑡0,𝑥0)}=𝑡^𝛼𝑡₀,𝑥^𝑖𝑓_{(𝑡0,𝑥0)}=𝑥^𝑖𝑥₀,𝑥^𝑖_𝛼𝑓_{(𝑡0,𝑥0)}=𝜕𝑥^𝑖𝜕𝑡^𝛼𝑡^𝛽₀,𝛼=1,𝑝,𝑖=1,𝑛. (2.9)

Remark 2.1. From physical viewpoint, the differentiable manifold 𝑇 should be thought as a “temporal manifold" or a “multitime", while 𝑀 is a “space manifold."

Remark 2.2. From geometrical viewpoint, an element [𝑓] of a fiber 1-jets 𝐽¹(𝑇,𝑀) should be thought as a class of parametrized sheet. The sections of fiber 1-jets are “physical fields."
To simplify the notations, denote by 𝑥_𝛾(𝑡)=(𝜕𝑥/𝜕𝑡^𝛾)(𝑡), 𝛾=1,𝑝, the partial velocities. Also, in our subsequent theory, we will set 𝜋_𝑥(𝑡)=(𝑡,𝑥(𝑡),𝑥_𝛾(𝑡)).

2.2. On Lagrange 1-Forms

A Lagrange 1-form of the first order on the jets space 𝐽¹(𝑇,𝑀) has the form 𝜔=𝐿_𝛼𝜋_𝑥(𝑡)𝑑𝑡^𝛼+𝑀_𝑖𝜋_𝑥(𝑡)𝑑𝑥^𝑖+𝑁^𝛽_𝑖𝜋_𝑥(𝑡)𝑑𝑥^𝑖_𝛽, (2.10) where 𝐿_𝛼, 𝑀_𝑖 and 𝑁^𝛽_𝑖 are Lagrangians of the first order. The pullback 𝑥^∗𝜔=𝐿_𝛼+𝑀_𝑖𝑥^𝑖_𝛼+𝑁^𝛽_𝑖𝑥^𝑖_𝛽𝛼𝑑𝑡^𝛼 (2.11) is a Lagrange 1-form of the second order on 𝑀. The coefficients 𝐿_𝛼+𝑀_𝑖𝑥^𝑖_𝛼+𝑁^𝛽_𝑖𝑥^𝑖_𝛽𝛼 are second order Lagrangians, which are linear in the partial accelerations.

A smooth Lagrangian 𝐿(𝜋_𝑥(𝑡)), 𝑡∈ℝ^𝑚₊ produces two smooth closed (completely integrable) 1-forms:

 (i) the differential 𝑑𝐿=𝜕𝐿𝜕𝑥^𝑖𝑑𝑥^𝑖+𝜕𝐿𝜕𝑥^𝑖_𝛾𝑑𝑥^𝑖_𝛾+𝜕𝐿𝜕𝑡^𝛾𝑑𝑡^𝛾 (2.12)  of components (𝜕𝐿/𝜕𝑡^𝛾,𝜕𝐿/𝜕𝑥^𝑖,𝜕𝐿/𝜕𝑥^𝑖_𝛾), with respect to the basis (𝑑𝑡^𝛾,𝑑𝑥^𝑖,𝑑𝑥^𝑖_𝛾);

  (ii) the restriction of 𝑑𝐿 to 𝜋_𝑥(𝑡), that is, the pullback 𝑑𝐿|_{𝜋𝑥(𝑡)}=𝜕𝐿𝜕𝑥^𝑖𝜕𝑥^𝑖𝜕𝑡^𝛽+𝜕𝐿𝜕𝑥^𝑖_𝛾𝜕𝑥^𝑖_𝛾𝜕𝑡^𝛽+𝜕𝐿𝜕𝑡^𝛽𝑑𝑡^𝛽 (2.13) of components containing partial accelerations 𝐷_𝛽𝐿=𝜕𝐿𝜕𝑥^𝑖𝜋_𝑥(𝑡)𝜕𝑥^𝑖𝜕𝑡^𝛽(𝑡)+𝜕𝐿𝜕𝑥^𝑖_𝛾𝜋_𝑥(𝑡)𝜕𝑥^𝑖_𝛾𝜕𝑡^𝛽(𝑡)+𝜕𝐿𝜕𝑡^𝛽𝜋_𝑥(𝑡), (2.14) with respect to the basis 𝑑𝑡^𝛽 (for other ideas, see [28]).

3. Problem Description

Let (𝑇,ℎ) and (𝑀,𝑔) be Riemannian manifolds of dimensions 𝑝 and 𝑛, respectively. Denote by 𝑡=(𝑡^𝛼), 𝛼=1,𝑝, and 𝑥=(𝑥^𝑖), 𝑖=1,𝑛, the local coordinates on 𝑇 and 𝑀, respectively. Consider 𝐽¹(𝑇,𝑀) the first order jet bundle associated to 𝑇 and 𝑀.

To develop our theory, we recall the following relations between two vectors 𝑣=(𝑣^𝑗) and 𝑤=(𝑤^𝑗), 𝑗=1,𝛿: 𝑣=𝑤⟺𝑣^𝑗=𝑤^𝑗,𝑗=1,𝛿,𝑣<𝑤⟺𝑣^𝑗<𝑤^𝑗,𝑗=1,𝛿,𝑣≦𝑤⟺𝑣^𝑗≤𝑤^𝑗,𝑗=1,𝛿productorderrelation,𝑣≤𝑤⟺𝑣≦𝑤,𝑣≠𝑤. (3.1)

Using the product order relation on ℝ^𝑝, the hyperparallelepiped Ω_𝑡0,𝑡1, in ℝ^𝑝, with diagonal opposite points 𝑡₀=(𝑡¹₀,…,𝑡^𝑝₀) and 𝑡₁=(𝑡¹₁,…,𝑡^𝑝₁), can be written as being the interval [𝑡₀,𝑡₁]. Suppose 𝛾_𝑡0,𝑡1 is a piecewise 𝐶¹-class curve joining the points 𝑡₀ and 𝑡₁.

The closed Lagrange 1-forms densities of 𝐶^∞-class 𝑓_𝛼=𝑓^ℓ_𝛼∶𝐽¹(𝑇,𝑀)→ℝ^𝑟,𝑘_𝛼=𝑘^ℓ_𝛼∶𝐽¹(𝑇,𝑀)→ℝ^𝑟,ℓ=1,𝑟,𝛼=1,𝑝, (3.2) determine the following path independent curvilinear functionals (actions, mechanical work): 𝐹^ℓ(𝑥(⋅))=_{𝛾𝑡01,𝑡}𝑓^ℓ_𝛼𝜋_𝑥(𝑡)𝑑𝑡^𝛼,𝐾^ℓ(𝑥(⋅))=_{𝛾𝑡01,𝑡}𝑘^ℓ_𝛼𝜋_𝑥(𝑡)𝑑𝑡^𝛼. (3.3)

The closeness conditions (complete integrability conditions) are 𝐷_𝛽𝑓^ℓ_𝛼=𝐷_𝛼𝑓^ℓ_𝛽,𝐷_𝛽𝑘^ℓ_𝛼=𝐷_𝛼𝑘^ℓ_𝛽,𝛼,𝛽=1,𝑝,𝛼≠𝛽,ℓ=1,𝑟, (3.4) where 𝐷_𝛽 is the total derivative.

Suppose 𝐾^ℓ(𝑥(⋅))>0, for all ℓ=1,𝑟, and accept that the Lagrange matrix densities 𝑔=𝑔^𝑏_𝑎∶𝐽¹(𝑇,𝑀)→ℝ^𝑞𝑠,𝑎=1,𝑠,𝑏=1,𝑞,ℎ𝑞<𝑛,ℎ=^𝑏_𝑎∶𝐽¹(𝑇,𝑀)→ℝ^𝑞𝑠,𝑎=1,𝑠,𝑏=1,𝑞,𝑞<𝑛, (3.5) of 𝐶^∞-class define the partial differential inequations (PDIs) (of evolution) 𝑔𝜋_𝑥(𝑡)≦0,𝑡∈Ω_𝑡0,𝑡1, (3.6) and the partial differential equations (PDE) (of evolution) ℎ𝜋_𝑥(𝑡)=0,𝑡∈Ω_𝑡0,𝑡1. (3.7)

On the set 𝐶^∞(Ω_𝑡0,𝑡1,𝑀) of all functions 𝑥∶Ω_𝑡0,𝑡1→𝑀 of 𝐶^∞-class, we set the norm ‖𝑥‖=‖𝑥‖_∞+∑^𝑝_𝛼=1‖x_𝛼‖_∞.

For each ℓ=1,𝑟, suppose 𝐾^ℓ(𝑥(⋅))>0, and consider 𝐹(𝑥(⋅))𝐾(𝑥(⋅=𝐹))¹(𝑥(⋅))𝐾¹(𝑥(⋅𝐹)),…,^𝑟(𝑥(⋅))𝐾^𝑟(𝑥(⋅.)) (3.8)

The aim of this work is to introduce and study the variational problem of minimizing a vector of quotients of functionals of curvilinear integral type: min_𝑥(⋅)𝐹(𝑥(⋅))𝐾(𝑥(⋅(⋅)Ω))subjectto𝑥∈ℱ_𝑡0,𝑡1, (MFP) where ℱ(Ω_𝑡0,𝑡1) denotes the set of all feasible solutions of problem (MFP), defined as ℱΩ_𝑡0,𝑡1=𝑥∈𝐶^∞Ω_𝑡0,𝑡1,𝑀𝑥𝑡₀=𝑥₀,𝑥𝑡₁=𝑥₁,o𝑟𝑥(𝑡)_{𝜕Ω𝑡01,𝑡}=𝜒=given,𝑔𝜋_𝑥(𝑡)≦0,ℎ𝜋_𝑥(𝑡)=0,𝑡∈Ω_𝑡0,𝑡1. (3.9)

This kind of problems, of considerable interest, arises in various branches of mathematical, engineering, and economical sciences. We especially have in mind mechanical engineering, where curvilinear integral objectives are extensively used due to their physical meaning as mechanical work. These objectives play an essential role in mathematical modeling of certain processes in relation with robotics, tribology, engines, and much more.

4. Main Results

The following two definitions are crucial in developing our results. For more details, see [26] by Pitea et al., and [14], by Pitea and Postolache.

Definition 4.1. A feasible solution 𝑥^∘(⋅)∈ℱ(Ω_𝑡0,𝑡1) is called an efficient solution of (MFP) if there is no 𝑥(⋅)∈ℱ(Ω_𝑡0,𝑡1), 𝑥(⋅)≠𝑥^∘(⋅), such that 𝐹(𝑥(⋅))/𝐾(𝑥(⋅))≤𝐹(𝑥^∘(⋅))/𝐾(𝑥^∘(⋅)).

Definition 4.2. Let 𝜌 be a real number and let 𝑏∶𝐶^∞(Ω_𝑡0,𝑡1,𝑀)×𝐶^∞(Ω_𝑡0,𝑡1,𝑀)→[0,∞) be a functional. To any closed 1-form 𝑎=(𝑎_𝛼) we associate the path independent curvilinear functional 𝐴(𝑥(⋅))=_{𝛾𝑡01,𝑡}𝑎_𝛼𝜋_𝑥(𝑡)𝑑𝑡^𝛼. (4.1)
The functional 𝐴 is called [strictly] (𝜌,𝑏)-quasi-invex at the point 𝑥^∘(⋅) if there is a vector function 𝜂∶𝐽¹(Ω_𝑡0,𝑡1,𝑀)×𝐽¹(Ω_𝑡0,𝑡1,𝑀)→ℝ^𝑛, vanishing at the point (𝜋_𝑥∘(𝑡),𝜋_𝑥∘(𝑡)), and 𝜃∶𝐶^∞(Ω_𝑡0,𝑡1,𝑀)×𝐶^∞(Ω_𝑡0,𝑡1,𝑀)→ℝ^𝑛, such that for any 𝑥(⋅) in 𝐶^∞(Ω_𝑡0,𝑡1,𝑀), [𝑥(⋅)≠𝑥^∘(⋅)], the following implication holds: 𝐴(𝑥(⋅))≤𝐴𝑥^∘(⋅)⇒𝑏𝑥(⋅),𝑥^∘(⋅)_{𝛾𝑡01,𝑡}𝜂𝜋_𝑥(𝑡),𝜋_𝑥∘(𝑡),𝜕𝑎_𝛼𝜋𝜕𝑥^𝑥∘(𝑡)+𝐷_𝛾𝜂𝜋_𝑥(𝑡),𝜋_𝑥∘(𝑡),𝜕𝑎_𝛼𝜕𝑥_𝛾𝜋_𝑥∘(𝑡)𝑑𝑡^𝛼[<]≤−𝜌𝑏𝑥(⋅),𝑥^∘(⋅)𝜃𝑥(⋅),𝑥^∘(⋅)². (4.2)
Several examples which illustrate our concept could be found in [14]. However, the following example is interesting. It is a generalization of Example 1 in [14].

Example 4.3. Let 𝑎∶[0,1]×𝐶^∞([0,1])×𝐶^∞([0,1])→ℝ, 𝑥(⋅)=(𝑥¹(⋅),𝑥²(⋅),…,𝑥^𝑛(⋅)). With 𝑥^∘(⋅)∈𝐶^∞([0,1]), denote 𝐺(𝑡)=𝜕𝑎𝜕𝑥𝑡,𝑥^∘(𝑡),𝑥̇^∘(𝑡)−𝐷𝜕𝑎𝜕𝑥̇𝑡,𝑥^∘(𝑡),𝑥̇^∘(𝑡),𝜕𝑎𝜕𝑥̇𝑡,𝑥^∘(𝑡),𝑥̇^∘(𝑡),𝑡∈[0,1], (4.3) where by 𝐷 we denoted the total derivative operator.
Suppose that 𝐺(1)−𝐺(0)=0. Then, the functional 𝐴(𝑥(⋅))=¹₀𝑎(𝑡,𝑥(𝑡),𝑥̇(𝑡))𝑑𝑡 (4.4) is (𝜌,1)-quasi-invex, for 𝜌≤0 and any 𝜃, at the point 𝑥^∘(⋅), with respect to 𝜂𝜋_𝑥(𝑡),𝜋_𝑥∘(𝑡)=𝐴(𝑥(⋅))−𝐴𝑥^∘(⋅)𝜕𝑎𝜕𝑥𝑡,𝑥^∘(𝑡),𝑥̇^∘(𝑡)−𝐷𝜕𝑎𝜕𝑥̇𝑡,𝑥^∘(𝑡),𝑥̇^∘(𝑡). (4.5)
In their recent work [26], Pitea et al. established necessary efficiency conditions for problem (MFP). More accurately, they proved that if 𝑥^∘(⋅) is an efficient solution of problem (MFP), then there are two vectors Λ¹⁰, Λ²⁰ in ℝ^𝑟 and the smooth functions 𝜇^∘ and 𝜈^∘, the first from Ω_𝑡0,𝑡1 to ℝ^𝑞𝑠𝑝, and the second from Ω_𝑡0,𝑡1 to ℝ^𝑞𝑠𝑝, such that Λ¹⁰,𝜕𝑓_𝛼𝜋𝜕𝑥^𝑥∘(𝑡)−Λ²⁰,𝜕𝑘_𝛼𝜋𝜕𝑥^𝑥∘(𝑡)+𝜇^∘_𝛼(𝑡),𝜕𝑔𝜋𝜕𝑥^𝑥∘(𝑡)+𝜈^∘_𝛼(𝑡),𝜕ℎ𝜋𝜕𝑥^𝑥∘(𝑡)−𝐷_𝛾Λ¹⁰,𝜕𝑓_𝛼𝜕𝑥_𝛾𝜋_𝑥∘(𝑡)−Λ²⁰,𝜕𝑘_𝛼𝜕𝑥_𝛾𝜋_𝑥∘(𝑡)+𝜇^∘_𝛼𝜋_𝑥∘(𝑡),𝜕𝑔𝜕𝑥_𝛾𝜋_𝑥∘(𝑡)+𝜈^∘_𝛼(𝑡),𝜕ℎ𝜕𝑥_𝛾𝜋_𝑥∘(𝑡)=0,𝑡∈Ω_𝑡0,𝑡1,𝛼=1,𝑝Euler-LagrangePDEs𝜇^∘_𝛼(𝑡),𝑔𝜋_𝑥∘(𝑡)=0,𝜇^∘_𝛼(𝑡)≧0,𝑡∈Ω_𝑡0,𝑡1,𝛼=1,𝑝. (4.6)(4.7)
If Λ¹⁰≥0 and Λ²⁰≥0, then 𝑥^∘(⋅), from conditions (4.6), is called normal efficient solution.
Let 𝑥^∘(⋅) be an efficient solution of primal (MFP), the scalars Λ¹⁰, Λ²⁰ in ℝ^𝑟, and the smooth functions 𝜇^∘∶Ω_𝑡0,𝑡1→ℝ^𝑞𝑠𝑝, 𝜈^∘∶Ω_𝑡0,𝑡1→ℝ^𝑞𝑠𝑝, given previously.
In order to use the idea of "grouping the resources," consider {𝑃₀,𝑃₁,…,𝑃_𝑞} and {𝑄₀,𝑄₁,…,𝑄_𝑞} partitions of the sets {1,…,𝑞} and {1,…,𝑞}, respectively.
For each ℓ=1,𝑟 and 𝛼=1,𝑝, we denote 𝑓^ℓ_𝛼𝜋_𝑦(𝑡)=𝑓^ℓ_𝛼𝜋_𝑦(𝑡)+𝜇_𝛼P0(𝑡),𝑔^𝑃0𝜋_𝑦(𝑡)+𝜈_𝛼𝑄0(𝑡),ℎ^𝑄0𝜋_𝑦(𝑡),𝐹^ℓ𝑦(⋅)=_{𝛾𝑡01,𝑡}𝑓^ℓ_𝛼𝜋_𝑦(𝑡)𝑑𝑡^𝛼. (4.8)
Consider a function 𝑦(⋅)∈𝐶^∞(Ω_𝑡0,𝑡1,𝑀) and associate to (MFP) the multiobjective ratio variational problem max_𝑦(⋅)𝐹¹𝑦(⋅)𝐾¹𝑦(⋅),…,𝐹^𝑟𝑦(⋅)𝐾^𝑟𝑦(⋅Λ)subjectto¹⁰,𝜕𝑓_𝛼𝜋𝜕𝑦^𝑦(𝑡)−Λ²⁰,𝜕𝑘_𝛼𝜋𝜕𝑦^𝑦(𝑡+𝜇_𝛼)(𝑡),𝜕𝑔𝜋𝜕𝑦^𝑦(𝑡)+𝜈_𝛼(𝑡),𝜕ℎ𝜋𝜕𝑦^𝑦(𝑡)−𝐷_𝛾Λ¹⁰,𝜕𝑓_𝛼𝜕𝑦_𝛾𝜋_𝑦(𝑡)−Λ²⁰𝜕𝑘_𝛼𝜕𝑦_𝛾𝜋_𝑦(𝑡)+𝜇_𝛼(𝑡),𝜕𝑔𝜕𝑦_𝛾𝜋_𝑦(𝑡)+𝜈_𝛼(𝑡),𝜕ℎ𝜕𝑦_𝛾𝜋_𝑦(𝑡)=0,𝛼=1,𝑝,𝑡∈Ω_𝑡0,𝑡1,𝜇_𝛼𝑃𝜅(𝑡),𝑔^𝑃𝜅𝜋_𝑦(𝑡)+𝜈_𝛼𝑄𝜅(𝑡),ℎ^𝑄𝜅𝜋_𝑦(𝑡)≥0,𝜅=1,𝑞,𝛼=1,𝑝,𝑡∈Ω_𝑡0,𝑡1Λ¹⁰≥0, (MFZD)(4.9) taking into account that the function 𝑦(𝑡) has to satisfy the boundary conditions 𝑦(𝑡₀)=𝑥₀, 𝑦(𝑡₁)=𝑥₁, or 𝑦(𝑡)|_{𝜕Ω𝑡01,𝑡}=𝜒=given.
𝜛(𝑥^∘(⋅)) is the value of the objective function of problem (MFP) at 𝑥^∘(⋅)∈ℱ(Ω_𝑡0,𝑡1), and 𝛿(𝑦(⋅),𝑦_𝛾(⋅),Λ¹⁰,Λ²⁰,𝜇(⋅),𝜈(⋅)) is the maximizing functional vector of dual problem (MFZD) at the point (𝑦(⋅),𝑦_𝛾(⋅),Λ¹⁰,Λ²⁰,𝜇(⋅),𝜈(⋅))∈Δ, where Δ is the domain of problem (MFZD).
We prove three duality results in a generalized sense of Mond-Weir-Zalmai.

Theorem 4.4 (weak duality). Let 𝑥^∘(⋅) be a feasible solution of problem (MFP) and let (𝑦(⋅),𝑦_𝛾(⋅),Λ¹⁰,Λ²⁰,𝜇(⋅),𝜈(⋅)) be a feasible point of problem (MFZD). Assume that the following conditions are satisfied: (a)⟨𝜇_𝑃0(𝑡),𝑔^𝑃0(𝜋_𝑦(𝑡))⟩+⟨𝜈_𝑄0(𝑡),ℎ^𝑄0(𝜋_𝑦(𝑡))⟩≤0, for all 𝑡∈Ω_𝑡0,𝑡1; (b)Λ_ℓ²⁰>0 and Λ_ℓ¹⁰𝐹^ℓ(𝑦(⋅))−Λ_ℓ²⁰𝐾^ℓ(𝑦(⋅))=0, for each ℓ=1,𝑟; (c)for each ℓ=1,𝑟, the functional Λ_ℓ¹⁰𝐹^ℓ(𝑥(⋅))−Λ_ℓ²⁰𝐾^ℓ(𝑥(⋅)) is (𝜌^′ℓ,𝑏)-quasi-invex at the point 𝑦(⋅) with respect to 𝜂 and 𝜃, on ℱ(Ω_𝑡0,𝑡1); (d)∫_{𝛾𝑡01,𝑡}[⟨𝜇_𝛼𝑃𝜅(𝑡),𝑔^𝑃𝜅(𝜋_𝑥(𝑡))⟩+⟨𝜈_𝛼𝑄𝜅(𝑡),ℎ^𝑄𝜅(𝜋_𝑥(𝑡))⟩]𝑑𝑡^𝛼 is (𝜌_𝜅^′′,𝑏)-quasi-invex at 𝑦(⋅) with respect to 𝜂 and 𝜃, for each 𝜅=1,𝑞, on 𝐹(Ω_𝑡0,𝑡1); (e)at least one of the functionals of (c), (d) is strictly quasi-invex; (f)∑^𝑟_ℓ=1𝜌^′ℓ+∑^𝑞_𝜅=1𝜌_𝜅^′′≥0.
Then, the inequality 𝜛(𝑥^∘(⋅))≤𝛿(𝑦(⋅),𝑦_𝛾(⋅),Λ¹⁰,Λ²⁰,𝜇(⋅),𝜈(⋅)) is false.

Proof. By reductio ad absurdum, suppose we have 𝐹^ℓ𝑥^∘(⋅)𝐾^ℓ(𝑥^∘(⋅≤))𝐹^ℓ𝑦(⋅)𝐾^ℓ𝑦(⋅),ℓ=1,𝑟. (4.10) From these inequalities, it follows Λ_ℓ¹⁰𝐹^ℓ𝑥^∘(⋅)Λ_ℓ²⁰𝐾^ℓ(𝑥^∘(⋅1))≤1+𝐹^ℓ𝑦(⋅)_{𝛾𝑡01,𝑡}𝜇_𝛼𝑃0(𝑡),𝑔^𝑃0𝜋_𝑦(𝑡)+𝜈_𝛼𝑄0(𝑡),ℎ^𝑄0𝜋_𝑦(𝑡)𝑑𝑡^𝛼, (4.11) and taking into account the hypotheses (a) and (b), we get Λ_ℓ¹⁰𝐹^ℓ𝑥^∘(⋅)≤Λ_ℓ²⁰𝐾^ℓ𝑥^∘(⋅),∀ℓ=1,𝑟. (4.12)
We obtain the following implications: Λ¹⁰,𝐹𝑥^∘(⋅)−Λ²⁰,𝐾𝑥^∘(⋅⇒𝑏𝑥)≤0_∘(⋅)(⋅)×,𝑦_{𝛾𝑡01,𝑡}𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),Λ¹⁰,𝜕𝑓_𝛼𝜋𝜕𝑦^𝑦(𝑡)−Λ²⁰,𝜕𝑘_𝛼𝜋𝜕𝑦^𝑦(𝑡)+𝐷_𝛾𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),Λ¹⁰,𝜕𝑓_𝛼𝜕𝑦_𝛾𝜋_𝑦(𝑡)−Λ²⁰,𝜕𝑘_𝛼𝜕𝑦_𝛾𝜋_𝑦(𝑡)𝑑𝑡^𝛼≤−𝑏𝑥^∘(⋅),𝑦(⋅)𝜃𝑥^∘(⋅),𝑦(⋅)^2𝑟_ℓ=1𝜌^′ℓ. (4.13)
Hypothesis (d) regarding the (𝜌_𝜅^′′,𝑏)-quasi-invexity property of each functional implies (𝜅=1,𝑞)∶_{𝛾𝑡01,𝑡}𝜇_𝛼𝑃𝜅(𝑡),𝑔^𝑃𝜅𝜋_𝑥∘(𝑡)+𝜈_𝛼𝑄𝜅(𝑡),ℎ^𝑄𝜅𝜋_𝑥∘(𝑡)𝑑𝑡^𝛼≤_{𝛾𝑡01,𝑡}𝜇_𝛼𝑃𝜅(𝑡),𝑔^𝑃𝜅𝜋_𝑦(𝑡)+𝜈_𝛼𝑄𝜅(𝑡),ℎ^𝑄𝜅𝜋_𝑦(𝑡)𝑑𝑡^𝛼⇒𝑏𝑥^∘(⋅),𝑦(⋅)×_{𝛾𝑡01,𝑡}𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),𝜇_𝛼𝑃𝜅(𝑡),𝜕𝑔^𝑃𝜅𝜋𝜕𝑦^𝑦(𝑡)+𝜈_𝛼𝑄𝜅(𝑡),𝜕ℎ^𝑄𝜅𝜋𝜕𝑦^𝑦(𝑡)+𝐷_𝛾𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),𝜇_𝛼𝑃𝜅(𝑡),𝜕𝑔^𝑃𝜅𝜕𝑦_𝛾𝜋_𝑦(𝑡)+𝜈_𝛼𝑄𝜅(𝑡),𝜕ℎ^𝑄𝜅𝜕𝑦_𝛾𝜋_𝑦(𝑡)𝑑𝑡^𝛼≤−𝜌_𝜅^′′𝑏𝑥^∘(⋅),𝑦(⋅)𝜃𝑥^∘(⋅),𝑦(⋅)². (4.14)
Now, we make the sum of implications (4.13) and (4.14) side by side and from 𝜅=1 to 𝜅=𝑞. It follows Λ¹⁰,𝐹𝑥^∘(⋅)−Λ²⁰,𝐾𝑥^∘(⋅)+_{𝛾𝑡01,𝑡}𝜇_𝛼(𝑡),𝑔𝜋_𝑥∘(𝑡)+𝜈_𝛼(𝑡),ℎ𝜋_𝑥∘(𝑡)𝑑𝑡^𝛼−_{𝛾𝑡01,𝑡}𝜇_𝛼(𝑡),𝑔𝜋_𝑦(𝑡)+𝜈_𝛼(𝑡),ℎ𝜋_𝑦(𝑡)𝑑𝑡^𝛼⇒𝑏𝑥≤0_∘(⋅)(⋅)×,𝑦_{𝛾𝑡01,𝑡}𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),Λ¹⁰,𝜕𝑓_𝛼𝜋𝜕𝑦^𝑦(𝑡)−Λ²⁰,𝜕𝑘_𝛼𝜋𝜕𝑦^𝑦(𝑡)+𝜇_𝛼(𝑡),𝜕𝑔𝜋𝜕𝑦^𝑦(𝑡)+𝜈_𝛼(𝑡),𝜕ℎ𝜋𝜕𝑦^𝑦(𝑡)+𝐷_𝛾𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),Λ¹⁰,𝜕𝑓_𝛼𝜕𝑦_𝛾𝜋_𝑦(𝑡)−Λ²⁰,𝜕𝑘_𝛼𝜕𝑦_𝛾𝜋_𝑦(𝑡)+𝜇_𝛼(𝑡),𝜕𝑔𝜕𝑦_𝛾𝜋_𝑦(𝑡)+𝜈_𝛼(𝑡),𝜕ℎ𝜕𝑦_𝛾𝜋_𝑦(𝑡)𝑑𝑡^𝛼−<𝑏𝑥^∘(⋅),𝑦(⋅)𝜃𝑥^∘(⋅),𝑦(⋅)^2𝑟_ℓ=1𝜌^′ℓ+^𝑞_𝜅=1𝜌_𝜅^′′. (4.15)
Since 𝑏(𝑥^∘(⋅),𝑦(⋅))>0, we obtain _{𝛾𝑡01,𝑡}𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),𝜕𝑉_𝛼𝜋𝜕𝑦^𝑦(𝑡),Λ¹⁰,Λ²⁰(⋅)(⋅)+𝐷,𝜇,𝜈_𝛾𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),𝜕𝑉_𝛼𝜕𝑦_𝛾𝜋_𝑦(𝑡),Λ¹⁰,Λ²⁰,𝜇(⋅),𝜈(⋅)𝑑𝑡^𝛼<−𝜃𝑥^∘(⋅),𝑦(⋅)^2𝑟_ℓ=1𝜌^′ℓ+^𝑞_𝜅=1𝜌_𝜅^′′, (4.16) where 𝑉_𝛼𝜋_𝑦(⋅),Λ¹⁰,Λ²⁰,𝜇(⋅),𝜈(⋅)=Λ¹⁰,𝑓_𝛼𝜋_𝑦(𝑡)−Λ²⁰,𝑘_𝛼𝜋_𝑦(𝑡+𝜇_𝛼)𝑦(𝑡),𝑔𝜋_𝑦(𝑡)+𝜈_𝛼(𝑡),ℎ𝜋_𝑦(𝑡),𝑡∈Ω_𝑡0,𝑡1,𝛼=1,𝑝. (4.17)
The following relation holds: 𝐷_𝛾𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),𝜕𝑉_𝛼𝜕𝑦_𝛾𝜋_𝑦(𝑡),Λ¹⁰,Λ²⁰,𝜇(𝑡),𝜈(𝑡)=𝐷_𝛾𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),𝜕𝑉_𝛼𝜕𝑦_𝛾𝜋_𝑦(𝑡),Λ¹⁰,Λ²⁰(𝑡)(𝑡)−𝜂𝜋,𝜇,𝜈_𝑥∘(𝑡),𝜋_𝑦(𝑡),𝐷_𝛾𝜕𝑉_𝛼𝜕𝑦_𝛾𝜋_𝑦(𝑡),Λ¹⁰,Λ²⁰,𝜇(𝑡),𝜈(𝑡). (4.18)
By replacing relations (4.18) and by using Euler-Lagrange PDE, relation (4.16) becomes _{𝛾𝑡01,𝑡}𝐷_𝛾𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),𝜕𝑉_𝛼𝜕𝑦_𝛾𝜋_𝑦(𝑡),Λ¹⁰,Λ²⁰,𝜇(𝑡),𝜈(𝑡)𝑑𝑡^𝛼<−𝜃𝑥^∘(⋅),𝑦(⋅)^2𝑟_ℓ=1𝜌^′ℓ+^𝑞_𝜅=1𝜌_𝜅^′′. (4.19)

For 𝛼,𝛾=1,𝑝, let us denote by 𝑄^𝛾_𝛼(𝑡)=𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),𝜕𝑉_𝛼𝜕𝑦_𝛾𝜋_𝑦(𝑡),Λ¹⁰,Λ²⁰,𝜇(𝑡),𝜈(𝑡),𝐼=_{𝛾𝑡01,𝑡}𝐷_𝛾𝑄^𝛾_𝛼(𝑡)𝑑𝑡^𝛼. (4.20)

According to [28], § 9, a total divergence is equal to a total derivative. Consequently, there exists 𝑄(𝑡), with 𝑄(𝑡₀)=0 and 𝑄(𝑡₁)=0 such that 𝐷_𝛾𝑄^𝛾_𝛼(𝑡)=𝐷_𝛼𝑄(𝑡) and 𝐼=_{𝛾𝑡01,𝑡}𝐷_𝛼𝑄(𝑡)𝑑𝑡^𝛼=𝑄𝑡₁−𝑄𝑡₀=0. (4.21)

Replacing into inequality (4.19), it follows that 0<−𝜃𝑥^∘(⋅),𝑦(⋅)^2𝑟_ℓ=1𝜌^′ℓ+^𝑞_𝑎=1𝜌_𝑎^′′, (4.22) contradicting hypothesis (f).

From relation (4.15), it follows 0≤Λ¹⁰,𝐹𝑥^∘(⋅)−Λ²⁰,𝐾𝑥^∘(⋅)+_{𝛾𝑡01,𝑡}𝜇_𝛼(𝑡),𝑔𝜋_𝑥∘(𝑡)+𝜈_𝛼(𝑡),ℎ𝜋_𝑥∘(𝑡)𝑑𝑡^𝛼−_{𝛾𝑡01,𝑡}𝜇_𝛼(𝑡),𝑔𝜋_𝑦(𝑡)+𝜈_𝛼(𝑡),ℎ𝜋_𝑦(𝑡)𝑑𝑡^𝛼. (4.23) According to the constraints of problems (MFP) and (MFZD), the previously mentioned relation becomes Λ_ℓ¹⁰𝐹^ℓ(𝑥^∘(⋅))−Λ_ℓ²⁰𝐾^ℓ(𝑥^∘(⋅))>0, that is: 𝐾^ℓ𝑥^∘(⋅)𝐾^ℓ𝑦(⋅)𝐹^ℓ𝑥^∘(⋅)𝐾^ℓ(𝑥^∘(⋅−𝐹))^ℓ𝑦(⋅)𝐾^ℓ𝑦(⋅)>0. (4.24)

Because 𝐾^ℓ(𝑥^∘(⋅))𝐾^ℓ(𝑦(⋅))>0, ℓ=1,𝑟, we conclude that 𝐹¹𝑥^∘(⋅)𝐾¹(𝑥^∘(⋅−𝐹))¹𝑦(⋅)𝐾¹𝑦(⋅)𝐹,…,^𝑟𝑥^∘(⋅)𝐾^𝑟(𝑥^∘(⋅−𝐹))^𝑟𝑦(⋅)𝐾^𝑟𝑦(⋅)≰(0,…,0), (4.25) or 𝐹¹𝑥^∘(⋅)𝐾¹(𝑥^∘(⋅𝐹)),…,^𝑟𝑥^∘(⋅)𝐾^𝑟(𝑥^∘(⋅≰𝐹))¹𝑦(⋅)𝐾¹𝑦(⋅)𝐹,…,^𝑟𝑦(⋅)𝐾^𝑟𝑦(⋅). (4.26)

Therefore, the inequality 𝜛(𝑥^∘(⋅))≤𝛿(𝑦(⋅),𝑦_𝛾(⋅),Λ¹⁰,Λ²⁰,𝜇(⋅),𝜈(⋅)) contradicts relations (4.12), and this completes the proof.

Theorem 4.5 (direct duality). Let 𝑥^∘(⋅) be an efficient solution of primal (MFP). Suppose the hypotheses of Theorem 4.4 hold. Then there are Λ¹⁰ and Λ²⁰ in ℝ^𝑟, and the smooth functions 𝜇^∘∶Ω_𝑡0,𝑡1→ℝ^𝑞𝑠𝑝 and 𝜈^∘∶Ω_𝑡0,𝑡1→ℝ^𝑞𝑠𝑝, such that (𝑥^∘(⋅),𝑥^∘_𝛾(⋅),Λ¹⁰,Λ²⁰,𝜇^∘(⋅),𝜈^∘(⋅)) is an efficient solution of dual program (MFZD). Moreover 𝜛𝑥^∘(⋅)=𝛿𝑥^∘(⋅),𝑥^∘_𝛾(⋅),Λ¹⁰,Λ²⁰,𝜇^∘(⋅),𝜈^∘(⋅). (4.27)

Proof. 𝑥^∘(⋅) being efficient solution of primal (MFP), there are Λ¹⁰ and Λ²⁰ in ℝ^𝑟 and the smooth functions 𝜇^∘∶Ω_𝑡0,𝑡1→ℝ^𝑞𝑠𝑝 and 𝜈^∘∶Ω_𝑡0,𝑡1→ℝ^𝑞𝑠𝑝, such that relations (4.6) are verified. Therefore, ⟨𝜇_𝛼(𝑡),𝑔(𝜋_𝑥∘(𝑡))⟩=0, 𝛼=1,𝑝. It follows (𝑥^∘(⋅),𝑥^∘_𝛾(⋅),Λ¹⁰,Λ²⁰,𝜇^∘(⋅),𝜈^∘(⋅)) is a feasible solution for program (MFZD). Obviously, 𝜛𝑥^∘(⋅)=𝛿𝑥^∘(⋅),𝑥^∘_𝛾(⋅),Λ¹⁰,Λ²⁰,𝜇^∘(⋅),𝜈^∘(⋅). (4.28) The weak duality theorem assures the efficiency of 𝑥^∘(⋅).

If we remark that the notion of efficient solution of problem (MFZD) is similar to those in Definition 4.1, we can state the result in the following.

Theorem 4.6 (converse duality). Let (𝑥^∘(⋅),𝑥^∘_𝛾(⋅),Λ¹⁰,Λ²⁰,𝜇^∘(⋅),𝜈^∘(⋅)) be an efficient solution to dual (MFZD) and suppose the following conditions are satisfied: (i)𝑥(⋅) is a normal efficient solution of primal (MFP); (ii)the hypotheses of Theorem 4.4 hold at (𝑥^∘(⋅),𝑥^∘_𝛾(⋅),Λ¹⁰,Λ²⁰,𝜇^∘(⋅),𝜈^∘(⋅)).
Then 𝑥^∘(⋅) is an efficient solution to (MFP). Moreover, one has the equality 𝜛𝑥^∘(⋅)=𝛿𝑥^∘(⋅),𝑥^∘_𝛾(⋅),Λ¹⁰,Λ²⁰,𝜇^∘(⋅),𝜈^∘(⋅). (4.29)

Proof. Suppose 𝑥(⋅)≠𝑥^∘(⋅). The efficiency of 𝑥(⋅) of primal (MFP) implies the existence of Λ¹⁰ and Λ²⁰ in ℝ^𝑟 and the smooth functions 𝜇^∘∶Ω_𝑡0,𝑡1→ℝ^𝑞𝑠𝑝 and 𝜈^∘∶Ω_𝑡0,𝑡1→ℝ^𝑞𝑠𝑝, such that relations (4.6) are satisfied. It follows that (𝑥(⋅),𝑥_𝛾(⋅),Λ¹⁰,Λ²⁰,𝜇^∘(⋅),𝜈^∘(⋅)) is a feasible solution of (MFZD) and 𝜛(𝑥(⋅))=𝛿𝑥(⋅),𝑥_𝛾(⋅),Λ¹⁰,Λ²⁰,𝜇^∘(⋅),𝜈^∘(⋅). (4.30)
On the other hand, from the weak duality theorem, 𝜛(𝑥(⋅))≰𝛿𝑥^∘(⋅),𝑥^∘_𝛾(⋅),Λ¹⁰,Λ²⁰,𝜇^∘(⋅),𝜈^∘(⋅) (4.31) holds, that is: 𝛿𝑥(⋅),𝑥_𝛾(⋅),Λ¹⁰,Λ²⁰,𝜇^∘(⋅),𝜈^∘(⋅)≰𝛿𝑥^∘(⋅),𝑥^∘_𝛾(⋅),Λ¹⁰,Λ²⁰,𝜇^∘(⋅),𝜈^∘(⋅). (4.32)
This last relation contradicts the efficiency of (𝑥^∘(⋅),𝑥^∘_𝛾(⋅),Λ¹⁰,Λ²⁰,𝜇^∘(⋅),𝜈^∘(⋅)) for program (MFZD). Therefore 𝑥(⋅)=𝑥^∘(⋅), and the theorem is proved.
We would like to conclude the study in this section with the following important particular case.
In this respect, suppose that 𝑘^ℓ_𝛼=positiveconstant, ℓ=1,𝑟, 𝛼=1,𝑝. Denote 𝐹(𝑥(⋅))=(𝐹¹(𝑥(⋅)),…,𝐹^𝑟(𝑥(⋅))). Then, program (MFP) becomes (𝑥)(⋅)Ωmin𝐹subjectto𝑥∈ℱ_𝑡0,𝑡1, (MP) and its dual is max^𝑏_𝑎𝐿_𝑒𝜋_𝑦(𝑡)Λ𝑑𝑡subjectto_ℓ10𝜕𝑓^ℓ_𝛼𝜋𝜕𝑦^𝑦(𝑡)+𝜇_𝛼(𝑡),𝜕𝑔𝜋𝜕𝑦^𝑦(𝑡)+𝜈_𝛼(𝑡),𝜕ℎ𝜋𝜕𝑦^𝑦(𝑡)−𝐷_𝛾Λ_ℓ¹⁰𝜕𝑓^ℓ_𝛼𝜕𝑦_𝛾𝜋_𝑦(𝑡)+𝜇_𝛼(𝑡),𝜕𝑔𝜕𝑦_𝛾𝜋_𝑦(𝑡)+𝜈_𝛼(𝑡),𝜕ℎ𝜕𝑦_𝛾𝜋_𝑦(𝑡)=0,𝛼=1,𝑝,𝑡∈Ω_𝑡0,𝑡1,𝜇_𝛼𝑃𝜅(𝑡),𝑔^𝑃𝜅𝜋_𝑦(𝑡)+𝜈_𝛼𝑄𝜅(𝑡),ℎ^𝑄𝜅𝜋_𝑦(𝑡)≥0,𝜅=1,𝑞,𝛼=1,𝑝,𝑡∈Ω_𝑡0,𝑡1,Λ¹⁰≥0, (GMWD) where 𝐿_𝑒𝜋_𝑦(𝑡)=𝑓𝜋_𝑦(𝑡)+𝜇_𝑃0(𝑡),𝑔^𝑃0𝜋_𝑥(𝑡)+𝜈_𝑄0(𝑡),ℎ^𝑄0𝜋_𝑥(𝑡)𝑒, (4.33) with 𝐿_𝑒=(𝐿₁,…,𝐿_𝑟).
It can be seen that we have obtained precisely the programs studied in the work [14]. Therefore, the results in this paper are stronger than the ones mentioned before.

5. Conclusion

In our recent study [14], we initiated an optimization theory on the first order jet bundle. As natural continuation, in this paper we considered a new class of multitime multiobjective variational problems of minimizing a vector of quotients of functionals of curvilinear integral type. We derived duality results for efficient solutions of multitime multiobjective ratio variational problems under the assumptions of (𝜌,𝑏)-quasi-invexity. We proved our weak duality theorem for efficient solutions, showing that the value of the objective function of the primal cannot exceed the value of the dual. Direct and converse duality theorems are stated, underlying the connections between the values of the objective functions of the primal and dual programs. As special cases, duality results of Mond-Weir-Zalmai type for a multitime multiobjective variational problem are obtained.

Having in mind the physical significance of the objective function, this study is strongly motivated by its possible applications of nonlinear optimization to mechanical engineering and economics [29].