相关论文: Time scales: from Nabla calculus to Delta calculus…

The discrete-time, the quantum, and the continuous calculus of variations have been recently unified and extended. Two approaches are followed in the literature: one dealing with minimization of delta integrals; the other dealing with…

The discrete, the quantum, and the continuous calculus of variations, have been recently unified and extended by using the theory of time scales. Such unification and extension is, however, not unique, and two approaches are followed in the…

In this work we propose a new and more general approach to the calculus of variations on time scales that allows to obtain, as particular cases, both delta and nabla results. More precisely, we pose the problem of minimizing or maximizing…

We introduce a fractional calculus on time scales using the theory of delta (or nabla) dynamic equations. The basic notions of fractional order integral and fractional order derivative on an arbitrary time scale are proposed, using the…

We consider a general problem of the calculus of variations on time scales with a cost functional that is the composition of a certain scalar function with delta and nabla integrals of a vector valued field. Euler-Lagrange delta-nabla…

We develop the calculus of variations on time scales for a functional that is the composition of a certain scalar function with the delta and nabla integrals of a vector valued field. Euler-Lagrange equations, transversality conditions, and…

We formulate a coherent approach to signals and systems theory on time scales. The two derivatives from the time-scale calculus are used, i.e., nabla (forward) and delta (backward), and the corresponding eigenfunctions, the so-called nabla…

We study more general variational problems on time scales. Previous results are generalized by proving necessary optimality conditions for (i) variational problems involving delta derivatives of more than the first order, and (ii) problems…

We prove a version of the Euler-Lagrange equations for certain problems of the calculus of variations on time scales with higher-order delta derivatives.

The theory of the calculus of variations was recently extended to the more general time scales setting, both for delta and nabla integrals. The primary purpose of this paper is to further extend the theory on time scales, by establishing…

We prove a necessary optimality condition of Euler-Lagrange type for variational problems on time scales involving nabla derivatives of higher-order. The proof is done using a new and more general fundamental lemma of the calculus of…

The problem of existence of solutions to nabla differential equations and nabla differential inclusions on time scales is considered. Under a special form of the set-valued constraint map, sufficient conditions for the existence of at least…

We establish necessary optimality conditions for variational problems with an action depending on the free endpoints. New transversality conditions are also obtained. The results are formulated and proved using the recent and general theory…

The calculus of variations on time scales is considered. We propose a new approach to the subject that consists in applying a differentiation tool called the contingent epiderivative. It is shown that the contingent epiderivative applied to…

The objective of this paper is twofold: (i) to survey existing results of generalized polynomials on time scales, covering definitions and properties for both delta and nabla derivatives; (ii) to extend previous results by using the more…

In this paper, we introduce the nabla fractional derivative and fractional integral on time scales in the Riemann-Liouville sense. We also introduce the nabla fractional derivative in Gr\"unwald-Letnikov sense. Some of the basic properties…

We deal with direct and inverse problems of the calculus of variations on arbitrary time scales. Firstly, using the Euler-Lagrange equation and the strengthened Legendre condition, we give a general form for a variational functional to…

We prove Euler-Lagrange and natural boundary necessary optimality conditions for problems of the calculus of variations which are given by a composition of nabla integrals on an arbitrary time scale. As an application, we get optimality…

Recently, some authors have proved monotonicity results for delta and nabla fractional differences separately. In this article, we use dual identities relating delta and nabla fractional difference operators to prove shortly the…

We prove the Euler-Lagrange delta-differential equations for problems of the calculus of variations on arbitrary time scales with delta-integral functionals depending on higher-order delta derivatives.