On a new class of dynamic Hardy-type inequalities and some related generalizations

Aequationes mathematicae - In this paper, we establish a new class of dynamic inequalities of Hardy’s type which generalize and improve some recent results given in the literature. More...     

NEW DYNAMIC INEQUALITIES FOR DECREASING FUNCTIONS AND THEOREMS OF HIGHER INTEGRABILITY

In this paper we establish some new dynamic inequalities on time scales which contain in particular generalizations of integral and discrete inequalities due to Hardy, Littlewood, P′olya, D'Apuzzo, Sbordone and Popoli. We also apply these inequalities to prove a higher integrability theorem for decreasing functions on time scales.     

Oscillation of second order neutral delay differential equations of Emden-Fowler type

We present new oscillation criteria for the second order nonlinear neutral delay differential equation [y(t)-py(t-τ)]'+ q(t)y ^λ (g(t)) sgn y(g(t)) = 0, t ≧ t ₀. Our results solve an open problem posed by James S.W . Wong [24]. The relevance of our results becomes clear due to a carefully selected example. This revised version was published online in June 2006 with corrections to the Cover Date.     

Hille and Nehari type criteria for third-order dynamic equations

In this paper, we extend the oscillation criteria that have been established by Hille [E. Hille, Non-oscillation theorems, Trans. Amer. Math. Soc. 64 (1948) 234-252] and Nehari [Z. Nehari, Oscillation criteria for second-order linear differential equations, Trans. Amer. Math. Soc. 85 (1957) 428-445] for second-order differential equations to third-order dynamic equations on an arbitrary time scale T, which is unbounded above. Our results are essentially new even for third-order differential and difference equations, i.e., when T=R and T=N. We consider several examples to illustrate our results.     

Oscillation of second-order damped dynamic equations on time scales

The study of dynamic equations on time scales has been created in order to unify the study of differential and difference equations. The general idea is to prove a result for a dynamic equation where the domain of the unknown function is a so-called time scale, which may be an arbitrary closed subset of the reals. This way results not only related to the set of real numbers or set of integers but those pertaining to more general time scales are obtained. In this paper, by employing the Riccati transformation technique we will establish some oscillation criteria for second-order linear and nonlinear dynamic equations with damping terms on a time scale . Our results in the special case when and extend and improve some well-known oscillation results for second-order linear and nonlinear differential and difference equations and are essentially new on the time scales , h>0, for q>1, , etc. Some examples are considered to illustrate our main results.     

Oscillation and stability of nonlinear discrete models exhibiting the Allee effect

In this paper, we consider the discrete nonlinear delay population model exhibiting the Allee effect

Periodic solutions and oscillation of discrete non-linear delay population dynamics model with external force

^** Email: emelabbasy{at}mans.edu.eg^*** Email: shsaker{at}mans.edu.eg In this paper, we consider the discrete non-linear delay populationdynamics model [graphic: see PDF] where m is a positive integer, p(n), Q(n) and (n) are positiveperiodic sequences of period . By the method that involves theapplication of the Gaines and Mawhins coincidence degree theory,we prove that there exists a positive -periodic solution (n). We prove that every positive solutionof (*) which does not oscillate about (n)satisfies lim_t[y(n)–(n)]=0.We establish some necessary and sufficient conditions for theoscillation of every positive solution about (n), and finally, we establish the lower and upperbounds of the oscillatory solutions.     

Oscillation criteria for quasi-linear functional dynamic equations on time scales

This paper is concerned with oscillation of the second-order quasilinear functional dynamic equation

$$(r(t)(x^\Delta (t))^\gamma )^\Delta + p(t)x^\beta (\tau (t)) = 0,$$

on a time scale \(\mathbb{T}\) where γ and β are quotient of odd positive integers, r, p, and τ are positive rd-continuous functions defined on \(\mathbb{T},\tau :\mathbb{T} \to \mathbb{T}\) and \(\mathop {\lim }\limits_{t \to \infty } \tau (t) = \infty \). We establish some new sufficient conditions which ensure that every solution oscillates or converges to zero. Our results improve the oscillation results in the literature when γ = β, and τ(t) ≤ t and when τ(t) > t the results are essentially new. Some examples are considered to illustrate the main results.