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)).

The duality for scalar variational problems involving convex functions has been formulated by Mond and Hanson in [

In [

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 [

In this paper, we are motivated by previous published research articles (please refer to [

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 [

To make our presentation self-contained and reader-friendly, we recall the notion of jet bundle of the first order,

Let be given the smooth manifolds

In the following, the set

Denote by

Consider

If

Thus, the equivalence relation

The equivalence class of a mapping

The total space of the set of 1-jets,

Let

The mapping

For every local chart

Therefore, the 1-jet space is endowed with a differentiable structure of dimension

The local coordinates on the space

From physical viewpoint, the differentiable manifold

From geometrical viewpoint, an element

To simplify the notations, denote by

A Lagrange 1-form of the first order on the jets space

A smooth Lagrangian

(i) the differential

(ii) the restriction of

Let

To develop our theory, we recall the following relations between two vectors

Using the product order relation on

The closed Lagrange 1-forms densities of

The closeness conditions (complete integrability conditions) are

Suppose

On the set

For each

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:

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.

The following two definitions are crucial in developing our results. For more details, see [

A feasible solution

Let

The functional

Several examples which illustrate our concept could be found in [

Let

Suppose that

In their recent work [

If

Let

In order to use the idea of "grouping the resources," consider

For each

Consider a function

We prove three duality results in a generalized sense of Mond-Weir-Zalmai.

Let

for each

at least one of the functionals of (c), (d) is strictly quasi-invex;

Then, the inequality

By reductio ad absurdum, suppose we have

We obtain the following implications:

Hypothesis (d) regarding the

Now, we make the sum of implications (

Since

The following relation holds:

By replacing relations (

For

According to [

Replacing into inequality (

From relation (

Because

Therefore, the inequality

Let

If we remark that the notion of efficient solution of problem

Let

the hypotheses of Theorem

Then

Suppose

On the other hand, from the weak duality theorem,

This last relation contradicts the efficiency of

We would like to conclude the study in this section with the following important particular case.

In this respect, suppose that

It can be seen that we have obtained precisely the programs studied in the work [

In our recent study [

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 [