where T is a time scale unbounded above, the variable delays k, : [t0,∞)T → T are nondecreasing with k(t), (t) < t for all t ∈ [t0,∞)T such that limt→∞ k(t), (t)=∞. The coefficient functions p,q : T→R are right-dense continuous with p bounded and q ≥ 0. To clarify some notation, take −1(t) := sup{s : (s) ≤ t}, −(n+1)(t)= −1( −n(t)) for t ∈ [ (t0),∞)T, and n+1(t) = ( n(t)) for t ∈ [ −3(t0),∞)T. ...

New oscillation and nonoscillation criteria are established for second order linear equations with damping and forcing terms. Examples are given to illustrate the results.

We give conditions under which the trivial solution of a first-order nonlinear variable-delay dynamic equation is asymptotically stable, for arbitrary time scales that are unbounded above. In an example, we apply our techniques to a logistic dynamic equation on isolated, unbounded time scales. c © 2006 Elsevier Ltd. All rights reserved. MSC: 39A10; 34B10

Oscillation criteria are established for a second-order forced dynamic equation on time scales containing both delay and advance arguments. Moreover, the potentials are allowed to change sign. Several nontrivial examples from difference equations are provided to illustrate the easy application of the results. The theory can be applied to second-order dynamic equations regardless of the choice o...

We establish some new oscillation criteria for the second-order quasilinear neutral delay dynamic equations r t zΔ t γ Δ q1 t x τ1 t q2 t x τ2 t 0 on a time scale T, where z t x t p t x τ0 t , 0 < α < γ < β. Our results generalize and improve some known results for oscillation of second-order nonlinear delay dynamic equations on time scales. Some examples are considered to illustrate our main r...

*Correspondence: [email protected] School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, P.R. China Abstract In this paper, we study some new retarded Volterra-Fredholm type integral inequalities on time scales, which provide explicit bounds on unknown functions. These inequalities generalize and extend some known inequalities and can be used as tools in the qualitative theory ...