Parameter-dependent integral on time scales

In this paper, we focus on the parameter-dependent integral on time scales. By combining multiple integration theory on time scales and usual parameter-dependent integration theory in classical mathematical analysis, both normal parameter-dependent integral on time scales and improper parameter-dependent integral on time scales are investigated efficiently. Many criteria for uniform convergence and important analytical properties are developed, respectively. Finally, a lot of examples are also presented to substantiate the theoretical results.     

Noether's theorem on time scales

We show that for any variational symmetry of the problem of the calculus of variations on time scales there exists a conserved quantity along the respective Euler-Lagrange extremals.     

Partial dynamic equations on time scales

In this work, we generalize existing ideas of the univariate case of the time scales calculus to the bivariate case. Formal definitions of partial derivatives and iterated integrals are offered, and bivariate partial differential operators are examined. In particular, solutions of the homogeneous and nonhomogeneous heat and wave operators are found when initial distributions given are in terms of elementary functions by means of the generalized Laplace Transform for the time scale setting. Finally, the so-termed mixed time scale setting is discussed. Examples are given and solutions are provided in tabular form.     

Differential equations on variable time scales

We introduce a class of differential equations on variable   time scales with a transition condition between two consecutive parts of the scale. Conditions for existence and uniqueness of solutions are obtained. Periodicity, boundedness and stability of solutions are considered. The method of investigation is by means of two successive reductions: B$B$-equivalence of the system [E. Akalín, M.U. Akhmet, The principles of B-smooth discontinuous flows, Computers and Mathematics with Applications 49 (2005) 981–995; M.U. Akhmet, Perturbations and Hopf bifurcation of the planar discontinuous dynamical system, Nonlinear Analysis 60 (2005) 163–178; M.U. Akhmet, N.A. Perestyuk, The comparison method for differential equations with impulse action, Differential Equations 26 (9) (1990) 1079–1086] on a variable time scale to a system on a time scale, a reduction to an impulsive differential equation [M.U. Akhmet, Perturbations and Hopf bifurcation of the planar discontinuous dynamical system, Nonlinear Analysis 60 (2005) 163–178; M.U. Akhmet, M. Turan, The differential equations on time scales through impulsive differential equations, Nonlinear Analysis 65 (2006) 2043–2060]. Appropriate examples are constructed to illustrate the theory.     

Constantin integral inequalities on time scales

We establish an integral inequality, on a so called time scale, related to those appearing in Constantin (J. Math. Anal. Appl. 197 (1996), 855–863) and Yang and Tan (JIPAM J. Inequal. Pure Appl. Math. 8 (2007), No. 2, Art 57). Our result can be used to obtain estimates for solutions of certain dynamic equations. Moreover, the bounds obtained in this paper are sharper than those known previously in the literature. This work was completed with the support of the Portuguese Foundation for Science and Technology (FCT) through the PhD fellowship SFRH/BD/39816/2007.     

Averaging dynamic equations on time scales

The aim of this paper is to generalize the classical theorems on averaging of differential equations. We focus on dynamic equations on time scales and prove both periodic and nonperiodic version of the averaging theorem, as well as a related theorem on the existence of periodic solutions.     

Some nonlinear dynamic inequalities on time scales

The aim of this paper is to investigate some nonlinear dynamic inequalities on time scales, which provide explicit bounds on unknown functions. The inequalities given here unify and extend some inequalities in (B G Pachpatte, On some new inequalities related to a certain inequality arising in the theory of differential equation, J. Math. Anal. Appl. 251 (2000) 736–751).     

Frequency measures on time scales with applications

The concept of frequency measures of subsets of a time scale is introduced and the relevant properties are discussed. Then, frequent oscillation is defined to strengthen the classical concept of oscillation. Applications are shown by deriving oscillation criteria for first-order dynamic equations on time scales.     

Oscillation of integro-dynamic equations on time scales

In this paper, the authors initiate the study of oscillation theory for integro-dynamic equations on time-scales. They present some new sufficient conditions guaranteeing that the oscillatory character of the forcing term is inherited by the solutions.     

Integration of functions rd-continuous on time scales

Doklady Mathematics -     

The Laplace transform on time scales revisited

In this work, we reexamine the time scale Laplace transform as defined by Bohner and Peterson [M. Bohner, A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, 2001; M. Bohner, A. Peterson, Laplace transform and Z-transform: Unification and extension, Methods Appl. Anal. 9 (1) (2002) 155-162]. In particular, we give conditions on the class of functions which have a transform, develop an inversion formula for the transform, and further, we provide a convolution for the transform. The notion of convolution leads to considering its algebraic structure—in particular the existence of an identity element—motivating the development of the Dirac delta functional on time scales. Applications and examples of these concepts are given.     

First order dynamic inclusions on time scales

In this paper, we study existence of solutions of first order dynamic inclusions on time scales with general boundary conditions. Both the ∇-derivative and Δ-derivative cases are considered. Examples are presented to illustrate that the Δ-derivative case needs more restrictive assumptions.     

Some new dynamic inequalities on time scales

In this paper, based on some known dynamic inequalities, we investigate certain new dynamic inequalities on time scales, which provide explicit bounds on unknown functions. Our results unify and extend some continuous inequalities and their corresponding discrete analogues.     

Halanay type inequalities on time scales with applications

This paper aims to introduce Halanay type inequalities on time scales. By means of these inequalities we derive new global stability conditions for nonlinear dynamic equations on time scales. Giving several examples we show that besides generalization and extension to q-difference case, our results also provide improvements for the existing theory regarding differential and difference inequalities, which are the most important particular cases of dynamic inequalities on time scales.     

Higher order boundary value problems on time scales

In this paper we study the existence of solutions for Lidstone boundary value problems on time scale. Firstly, by using Schauder fixed point theorem in a cone, we obtain the existence of solutions to a Lidstone boundary value problem (LBVP). Secondly, existence result for this problem is also given by the monotone method. Finally, by using Krasnosel'skii fixed point theorem, it is proved that the LBVP has a positive solution.     

Oscillation of delay differential equations on time scales

Consider the following equation: , where t is in a measure chain. We apply the theory of measure chains to investigate the oscillation and nonoscillation of the above equation on the basis of some well-known results. And in some sense, we show a method to unify the delay differential equation and delay difference equation.