Samir H. Saker

OSCILLATION CRITERIA FOR SECOND-ORDER NONLINEAR DYNAMIC EQUATIONS ON TIME SCALES

By means of generalized Riccati transformation techniques and generalized exponential functions, some oscillation criteria are given for the nonlinear dynamic equation \[ (p(t)x^{\Delta} (t))^{\Delta}+q(t)(f\circ x^{\sigma})=0 \] on time scales. The results are also applied to linear and nonlinear dynamic equations with damping, and some sufficient conditions are obtained for the oscillation of all solutions.

Oscillation criteria for perturbed nonlinear dynamic equations

In this paper, we discuss the oscillatory behavior of a certain nonlinear perturbed dynamic equation on time scales. We establish some new oscillation criteria for such dynamic equations and supply examples.

Oscillation results for second-order nonlinear neutral delay dynamic equations on time scales

In this article, we consider the second-order nonlinear neutral delay dynamic equation on a time scale and establish some new oscillation and nonoscillation criteria. Also from these we deduce the Leighton–Wintner, Hille–Kneser, Kamenev, and Philos types oscillation criteria. Our results are different and complement the existence oscillation results for neutral delay dynamic equations on time scales in (Agarwal et al. 2004, Oscillation criteria for second-order nonlinear neutral dynamic equations. Journal of Mathematical Analysis and Applications, 300, 203–217) and (S.H. Saker, 2006, Oscillation of second-order nonlinear neutral delay dynamic equations on time scales. Journal of Computational and Applied Mathematics, 187, 123–141). Our results can be applied on the time scales , , , for h > 0, , , , and when where {t n } is the set of harmonic numbers, etc. Some examples are considered to illustrate the main results.

Oscillation and global attractivity in a nonlinear delay periodic model of respiratory dynamics

In this paper, we shall consider the nonlinear delay differential equation where m and n are positive integers, and V(t) and λ(t) are positive periodic functions of period ω. In the nondelay case, we shall show that (∗) has a unique positive periodic solution x(t) and provides sufficient conditions for the global attractivity of x(t). In the delay case, we shall present sufficient conditions for the oscillation of all positive solutions of (∗) about x(t) and establish sufficient conditions for the global attractivity of x(t).

Positive periodic solutions of discrete three-level food-chain model of Holling type II

With the help of differential equations with piecewise constant arguments, we first derive a discrete analogy of continuous three level food-chain model of Holling type II, which is governed by difference equations with periodic coefficients. A set of sufficient conditions is derived for the existence of positive periodic solutions with strictly positive components by using the continuation theorem in coincidence degree theory. Particularly, the upper and lower bounds of the periodic solutions are also established.

On the Impulsive Delay Hematopoiesis Model with Periodic Coefficients

In this paper we shall consider the nonlinear impulsive delay hematopoiesis model p0(t) = β(t) 1 + pn(t−mω) − γ(t)p(t), t 6= tk, p(t+k ) = (1 + bk)p(tk), k ∈ N = {1, 2, . . .}, where n,m ∈ N, β(t), γ(t) and 0 0. We prove that the solutions are bounded and persist. The persistence implies the survival of the mature cells for long term. By means of the continuation theorem of coincidence degree, we prove the existence of a positive periodic solution p(t). We also establish some sufficient conditions for the global attractivity of p(t). These conditions imply the absence of any dynamic diseases in the mammal. Moreover, we obtain some sufficient conditions for the oscillation of all positive solutions about the positive periodic solution p(t). These conditions lead to the prevalence of the mature cells around the periodic solution. Our results extend and improve some well known theorems in the literature for the autonomous case without impulse. An example is considered to illustrate the main results.

OSCILLATION CRITERIA FOR NONLINEAR PERTURBED DYNAMIC EQUATIONS OF SECOND-ORDER ON TIME SCALES

In this paper, by using the Riccati transformation technique we establish some new oscillation criteria for second-order nonlinear perturbed dynamic equation on time scales. An example illustrating our main results is also given.

Dynamic inequalities on time scales

Preliminaries.- Basic Inequalities.- Opial Inequalities.- Lyapunov Inequalities.- Halanay Inequalities.- Wirtinger Inequalities.

Hille-Kneser-type criteria for second-order dynamic equations on time scales

We consider the pair of second-order dynamic equations, (r(t)(xΔ)γ)Δ + p(t)xγ(t) = 0 and (r(t)(xΔ)γ)Δ + p(t)xγσ(t) = 0, on a time scale , where γ > 0 is a quotient of odd positive integers. We establish some necessary and sufficient conditions for nonoscillation of Hille-Kneser type. Our results in the special case when involve the well-known Hille-Kneser-type criteria of second-order linear differential equations established by Hille. For the case of the second-order half-linear differential equation, our results extend and improve some earlier results of Li and Yeh and are related to some work of Došlý and Řehák and some results of Řehák for half-linear equations on time scales. Several examples are considered to illustrate the main results.

Oscillation criteria for a forced second-order nonlinear dynamic equation

In this paper, we will establish some new interval oscillation criteria for the forced second-order nonlinear dynamic equation on a time scale 𝕋 where γ ≥ 1. As a special case when 𝕋 = ℝ our results not only include the oscillation results for second-order differential equations established by Wong (Oscillation criteria for a forced second-order linear differential equation, J. Math. Anal. Appl. 231 (1999), pp. 233–240) and Nasr (Sufficient condition for the oscillation of forced super-linear second order differential equations with oscillatory potential, Proc. Amer. math. Soc. 126 (1998), pp. 123–125) but also improve these results. When 𝕋 = ℕ, 𝕋 = hℕ or 𝕋 = q N , i.e. for difference equations, generalized difference equations or q-difference equations our results are essentially new and also can be applied on different types of time scales, as illustrated in several examples.