In this study, linear second-order self-adjoint delta-nabla matrix systems on time scales are considered with the motivation of extending the analysis of dominant and recessive solutions from the differential and discrete cases to any arbitrary dynamic equations on time scales. These results emphasize the case when the system is non-oscillatory.

∧ second-order half-linear dynamic equation (r(t)(x(t))) + p(t)x(σ (t)) = 0, where r(t) > 0, p(t) are continuous, ∫ ∞ t0 (r(t)) 1 α ∆t = ∞, α is a quotient of odd positive integers. In particular, no explicit sign assumptions aremadewith respect to the coefficient p(t). We give conditions under which every positive solution of the equations is strictly increasing. For α = 1, T = R, the result i...

Some new Gronwall-Bellman-type delay integral inequalities in two independent variables on time scales are established, which provide a handy tool in the research of qualitative and quantitative properties of solutions of delay dynamic equations on time scales. The established inequalities generalize some of the results in the work of Zhang and Meng 2008, Pachpatte 2002, and Ma 2010.

i=1 pi(t)Φβi (x(τi(t))) = e(t), t ∈ [t0,∞)T where T is a time scale, t0 ∈ T a fixed number; [t0,∞)T is a time scale interval; Φ∗(u) = |u|∗−1u; the functions r, pi, e : [t0,∞)T → R are right-dense continuous with r > 0 nondecreasing; τk : T → T are nondecreasing right-dense continuous with τk(t) ≤ t , limt→∞ τk(t) = ∞; and the exponents satisfy β1 > · · · > βm > α > βm+1 > · · ·βn > 0. All resul...

In the study of dynamic equations on time scales we deal with certain dynamic inequalities which provide explicit bounds on the unknown functions and their derivatives. Most of the inequalities presented are of comparison or Gronwall type and, more specifically, of Pachpatte type.

[Article] Connection between kinetic methods for fluid-dynamic equations and macroscopic finite-difference schemes Connection between kinetic methods for fluid-dynamic equations and macroscopic finite-difference schemes. Porto, the institutional repository of the Politecnico di Torino, is provided by the University Library and the IT-Services. The aim is to enable open access to all the world. ...

In [1], the authors study the stability of the delay dynamic equation x (t)+ p (t) x (t − τ (t)) = 0 for all t ∈ T, (1) where T (i.e., a nonempty subset of reals) is a time scale and p ∈ Crd ( T,R ) . The statements of their theorems and representations include some mistakes as follows: τ : T → (0, r] and t ∈ T with t − τ (t) ≥ minT, which implies t − τ (t) ∈ T. The forward jump operator σ (t) ...

We investigate the asymptotic behavior of solutions of the following higher-order dynamic equation xΔ n t f t, x t , xΔ t , . . . , xΔ n−1 t 0, on an arbitrary time scale T, where the function f is defined on T × R. We give sufficient conditions under which every solution x of this equation satisfies one of the following conditions: 1 limt→∞x n−1 t 0; 2 there exist constants ai 0 ≤ i ≤ n − 1 wi...

Using a known correspondence between the solutions of impulsive measure functional differential equations and the solutions of impulsive functional dynamic equations on time scales, we prove that the limit of solutions of impulsive functional dynamic equations over a convergent sequence of time scales converges to a solution of an impulsive functional dynamic equation over the limiting time sca...