Efﬁciency for Vector Variational Quotient Problems with Curvilinear Integrals on Riemannian Manifolds via Geodesic Quasiinvexity

: In the paper, we analyze the necessary efﬁciency conditions for scalar, vectorial and vector fractional variational problems using curvilinear integrals as objectives and we establish sufﬁcient conditions of efﬁciency to the above variational problems. The efﬁciency sufﬁcient conditions use of notions of the geodesic invex set and of (strictly, monotonic) ( ρ , b)-geodesic quasiinvex

Throughout this paper, for two vectors v = (v 1 , . . . , v n ) and w = (w 1 , . . . , w n ) the relations of the The purpose of this paper is to give the optimality conditions for multitime variational problems with objectives of the following forms: where C κ r α (j 1 t x)dt α > 0 for r = 1, . . . , p.
As usual, the functionals of mechanical work type, due to their physical meaning, or similarly, the cost functionals in economics, are very important in applications. Thus, in our opinion, the centrality of the present work is supported by both theoretical and practical reasoning. As well, the ideas and techniques of this paper may stimulate further research in this dynamic field. The paper was organized as follows. Section 1 (structured in two sub-sections) is an introduction presenting the aim of the study and the technical tools useful for the sequel. Section 2 presents a scalar multi-time variational problem with constraints. An efficiency solution is defined and efficiency conditions for the program (SPV) are given. Section 3, contains necessary conditions for a vector curvilinear program establishing a Pareto minimum point. In the Section 4, necessary conditions for the quotient variational curvilinear problem are presented. This case study is particularly strategical for potential applications in the frame of multiobjective programming. Sections 5 and 6 condense sufficient efficiency conditions related to the classes of problems precedently introduced. The paper, in the last section, contains the conclusions and potential further developments.

Geodesic Invex Set and (ρ, b)-Geodesic Quasiinvex Functionals
In [24], Barani and Pouryayaei introduced the notions of the invex set and invex function in the following ways.

Definition 1 ([24]
). Let (M, g) be a complete Riemannian manifold. Let η : M × M → TM, η(u, x) ∈ T x M, u, x ∈ M be a vector function and S ⊂ M be a nonempty set.
(i) The set S is called η-geodesic invex if, for every u, x ∈ S, there exists exactly one geodesic γ u,x : . For examples of geodesic invex sets, see [24].
We say that x(·) is a geodesic deformation of x 0 (·).
In order to get our sufficient conditions of efficiency, we shall introduce the notion of (monotonic) (ρ, b)-geodesic quasiinvex functionals.

Remark 1. This paper uses functionals of the form
dt α is a closed differential, to establish the sufficient optimality conditions (in Sections 5 and 6).
Theorem 1 ([16,22,25]). If x 0 (·) ∈ D minimizes the functionalS(x) then x 0 is an optimal solution of the multitime system of equations whereL α is completely integrable and satisfies the limit conditions on the boundary.

Theorem 2.
(Fritz-John conditions) If x 0 (·) ∈ D is an optimal solution of the variational problem (SVP) then there exist the real scalar τ ∈ R and the piecewise smooth functions λ = (λ β (t)) ∈ R m and µ = (µθ(t)) ∈ R q defined on Ω, satisfying the following conditions: Theorem 2 is used forL.

Necessary Efficiency Conditions for Vector Variational Curvilinear Problem
Consider the vector curvilinear functional . . , The domain of (VVCP) is also the set D.

Necessary Efficiency Conditions for Quotient Variational Curvilinear Problem
Consider the vector functional of ratios of integrals and the vector curvilinear integral variational problem The domain of (QVCP) is also the set D.
We now give the necessary efficiency conditions for (QVCP). We define and consider the problem We also consider the following two problems:

Definition 9.
A point x 0 (·) ∈ D is called a normal efficient solution to (QVCP) if it is an efficient solution to this problem and if it is an optimal point to at least one of the scalar problems (SRP) r , r = 1 . . . , p. Now, we prove the main problem of the paper. Theorem 5. (Necessary efficiency in (QVCP)) Let x 0 (·) ∈ D be a normal efficient solution of problem (QVCP). Then there exist a vector τ = (τ r ) ∈ R p and the piecewise smooth functions λ = (λ β (t)) ∈ R m and µ = (µ κ (t)) ∈ R q defined on Ω that satisfy the conditions: Proof of Theorem 5. According to Definitions 5 and 7, there exists r ∈ {1, . . . , p} such that x 0 (·) is an optimal point to the scalar problem (SRP) r . Then, the proof of the theorem is similar to the one of Theorem 4, where for r = 1, . . . , p, f r Theorem 6. (Necessary efficiency in (QVCP)) Let x 0 (·) ∈ D be a normal efficient solution to problem (QVCP). Then, there exist a vector τ = (τ r ) ∈ R p and piecewise smooth functions λ = (λ β (t)) ∈ R m and µ = (µ κ (t)) ∈ R q defined on Ω that satisfy the conditions: Proof of Theorem 6. In Theorem 5 we replaced R r (x 0 ) = F r (x 0 (·))/K r (x 0 (·)), r = 1, . . . , p, where F r (x 0 (·)) = C f r α (j 1 t x)dt α and K r (x 0 (·)) = C k r α (j 1 t x)dt α ; then, we denote the multipliers λ β (t) and µ κ (t).

Sufficient Efficiency Conditions for (VVCP) and (SCP)
In the following, we establish the sufficient efficiency conditions for variational problems with curvilinear integrals.
main condition: Suppose that the subset G ⊂ F(Ω, M) is an η-geodesic invex set, where the C 1 vector function η(t) is as in Definition 3 and G ⊃ D. Furthermore, suppose that the differential is a closed Lagrange 1-form whose primitive U(t) satisfies the condition U(t 1 ) ≤ U(t 2 ). For a fixed x 0 (·) ∈ G, we denote x(·) ∈ G as its geodesic perturbation.
Proof of Theorem 7. Let us suppose a contradiction. If x 0 (·) is not an efficient solution for (VVCP); then, for each r = 1 . . . , p, there exists x(·) ∈ D (x = x 0 ), a feasible solution to (VVCP), such that According to (a) it follows that: (according to Definition 3). By multiplying this inequality by τ r ≥ 0 and summing over r = 1, . . . , p, we obtain From the continuity of the functions we choose Then, taking into account condition (b) and Definition 3, this inequality implies where α = 1, . . . , p.
Taking into account condition (c) and Definition 4, for each α = 1, . . . , p, from By summing relations (5), (6) and (7) and taking into account (d), for x ∈ D, we obtain From (8), it results that b(x, x 0 ) and We denote and then relation (9) becomes where we denoted ∂L α ∂x Integration by parts in the second integral of (10) gives Using relation (11), relation (10) becomes Taking into account the first relation of (VFJ), (12) becomes According to [18], the total divergence is equal to the total derivative dU. Moreover, according to the main condition, dU is a closed 1-form. Then, there exists the primitive U(T), with U(t 1 ) ≤ U(t 2 ), and the integral of (13) becomes Consequently, relation (13) becomes With d(x, x 0 ) ≥ 0 and hypothesis (e), we obtain the inequality 0 < 0, which is a false. Therefore, x 0 (·) is an efficient solution to (VVP).

If in Theorems 7 the integrals from hypotheses (b) and (c) are replaced by the integral
]dt α , then the following results are obtained: Corollary 1. (Sufficient efficiency conditions for (VVCP)) Let x 0 (·) ∈ G, τ, λ and µ satisfy the relations (VFJ) from Theorem 4, let arbitrary x(·) ∈ D and let the main condition be satisfied. We consider a vector function η as in Definition 3 and assume that the following conditions are satisfied: (a ) For each r = 1, . . . , p, C f r α (j 1 t x)dt α is (ρ r 1 , b)-geodesic quasiinvex at x 0 (·) with respect to η and d.
Then x 0 (·) is an efficient solution to (SCP). (c ) One of the integrals of (a ) and (b ) is strictly (ρ, b)-geodesic quasiinvex at x 0 (·) with respect to η and d.

Conclusions and Further Developments
In this paper, new classes of variational control problems of minimizing a vector of path-independent curvilinear integral (mechanical or cost) functionals ratios, were considered. Starting from scalar variational problems (SVP) elaborated by Udriste et al. [16,17,20,22,23] by which optimality conditions of variational problems in the multitime approach (so called multitime variational problems) with constraints were introduced in literature, in this paper by using curvilinear integrals and generalized invex functionals, new necessary and sufficient conditions of efficiency were obtained. In particular, we have formulated and proved necessary geodesic efficiency conditions in the considered scalar, vector and vector quotient variational control problems, by using the notation of normal geodesic efficient solution and new notions of geodesic efficient solution. As well, by using the original concept of (ρ, b)-geodesic quasiinvexity associated with path-independent curvilinear integral functionals, sufficient conditions of geodesic efficiency for a feasible solution in the considered vector and vector quotient variational control problems have been derived. The proposed framework could be depth considering the "theory of functionals," taking into account the variational methodology so useful for the study of regularity properties of integral functionals. In this direction of ongoing research, see [26,27] for more.