Rui A. C. Ferreira

NECESSARY OPTIMALITY CONDITIONS FOR FRACTIONAL DIFFERENCE PROBLEMS OF THE CALCULUS OF VARIATIONS

We introduce a discrete-time fractional calculus of variations. First and second order necessary optimality conditions are established. Examples illustrating the use of the new Euler--Lagrange and Legendre type conditions are given. They show that the solutions of the fractional problems coincide with the solutions of the corresponding non-fractional variational problems when the order of the discrete derivatives is an integer value.

Higher-Order Calculus of Variations on Time Scales

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.

A discrete fractional Gronwall inequality

One of the most important inequalities in the theory of differential equations is known as the Gronwall inequality. It was published in the year 1919 in the work by Gronwall [14]. Since then many generalizations and extensions of this inequality became part of the literature [19]. The discrete Gronwall inequality seems to have appeared first in the work of Mikeladze [16] in the year 1935, and now it is used for example in proving convergence of the discrete variable methods for ordinary, partial as well as integral equations [1]. Difference equations arise as mathematical models describing many real life situations, e.g., queueing problems, electrical networks, economics, etc., and this is reason enough to explore such a theory. Nevertheless, perhaps due to the advent of computers, only relatively recently difference equations have started receiving the attention they deserve [1] and nowadays it is a discipline of mathematics by itself. Much more recently, maybe due to the explosion in research within the fractional differential calculus (see for example the books [15, 18]), the theory of discrete fractional calculus is being developed. Usually there are many definitions of derivatives within a certain fractional setting, and researchers use the ones that are more suitable in a determined context. Indeed, perhaps the first time that a definition of a discrete fractional derivative appeared in the literature was in the work by Dı́az et al. [10] in the year 1974. This definition was given through an infinite sum, while later, Gray et al. [13], in 1988, presented a definition based on a finite sum. This latter definition uses what is called in the literature the nabla difference operator [we refer the reader to the recent work by Anastassiou [3] obtaining some inequalities (e.g., Opial, Ostrowski) using the fractional nabla operator]. In 1989, Miller et al. [17] defined a fractional sum of order α > 0 via the solution of a linear difference equation (cf. (2.1) below) and proved some basic properties of this operator. The calculus emerging from this definition has become appealing to many authors and now it is a matter of strong research, in various

Existence and uniqueness of solution to some discrete fractional boundary value problems of order less than one

We provide criteria for the existence and uniqueness of solutions to a class of discrete fractional boundary value problems of order . An example illustrating our results is presented at the end of the paper.

Diamond- Jensen's Inequality on Time Scales

The theory and applications of dynamic derivatives on time scales have recently received considerable attention. The primary purpose of this paper is to give basic properties of diamond-Open image in new window derivatives which are a linear combination of delta and nabla dynamic derivatives on time scales. We prove a generalized version of Jensens inequality on time scales via the diamond-Open image in new window integral and present some corollaries, including Holders and Minkowskis diamond-Open image in new window integral inequalities.The theory and applications of dynamic derivatives on time scales have recently received considerable attention. The primary purpose of this paper is to give basic properties of diamond- derivatives which are a linear combination of delta and nabla dynamic derivatives on time scales. We prove a generalized version of Jensens inequality on time scales via the diamond- integral and present some corollaries, including Hölders and Minkowskis diamond- integral inequalities.

Generalized retarded integral inequalities

Abstract We prove some new retarded integral inequalities. The results generalize those in [Y.G. Sun, On retarded integral inequalities and their applications, J. Math. Anal. Appl. 301 (2) (2005) 265–275].

Generalizations of Gronwall–Bihari inequalities on time scales

We establish some nonlinear integral inequalities for functions defined on a time scale. The results extend some previous Gronwall and Bihari type inequalities on time scales. Some examples of time scales for which our results can be applied are provided. An application to the qualitative analysis of a nonlinear dynamic equation is discussed.

Integral inequalities and their applications to the calculus of variations on Time Scales

We discuss the use of inequalities to obtain the solution of certain variational problems on time scales.

Optimality conditions for the calculus of variations with higher-order delta derivatives

Abstract 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.

ISOPERIMETRIC PROBLEMS OF THE CALCULUS OF VARIATIONS WITH FRACTIONAL DERIVATIVES

Abstract In this article, we study isoperimetric problems of the calculus of variations with left and right Riemann-Liouville fractional derivatives. Both situations when the lower bound of the variational integrals coincide and do not coincide with the lower bound of the fractional derivatives are considered.