Taher S. Hassan

Oscillation criteria for nonlinear damped dynamic equations on time scales

Abstract We present new oscillation criteria for the second order nonlinear damped delay dynamic equation ( r ( t ) ( x Δ ( t ) ) γ ) Δ + p ( t ) ( x Δ σ ( t ) ) γ + q ( t ) f x ( τ ( t ) ) = 0 . Our results generalize and improve some known results for oscillation of second order nonlinear delay dynamic equation. Our results are illustrated with examples.

Oscillation criteria for nonlinear functional neutral dynamic equations on time scales

This paper is concerned with the oscillation of the second-order nonlinear functional dynamic equations on a time scale where γ is the quotient of odd positive integers, r(t), p(t), and are positive rd-continuous functions on , , and . We establish some new sufficient conditions for oscillation for the above equation. Several examples illustrating our results will be given.

Kamenev-type oscillation criteria for second order nonlinear dynamic equations on time scales

The purpose of this paper to establish oscillation criteria for second order nonlinear dynamic equation (r(t)(xΔ(t))γ)Δ+f(t,x(g(t)))=0, on an arbitrary time scale T, where γ is a quotient of odd positive integers and r is a positive rd-continuous function on T. The function g:T→T satisfies g(t) ⩾ t and limt→∞g(t) = ∞ and f∈C(T×R,R). We establish some new sufficient conditions such that the above equation is oscillatory by using generalized Riccati transformation. Our results in the special cases when T=R and T=N involve and improve some oscillation results for second-order differential and difference equations; and when T=hN,T=qN0 and T=N2 our oscillation results are essentially new. Some examples illustrating the importance of our results are included.

Forced oscillation of second order differential equations with mixed nonlinearities

This article is concerned with the oscillation of the forced second order differential equation with mixed nonlinearities (a(t)(x′(t))γ)′+po(t)xγ(go(t))+∑i=1npi(t)|x(gi(t))|αisgnx(gi(t))=e(t) where γ is a quotient of odd positive integers, αi > 0, i = 1, 2, …, n, a, e, and pi∈C([0,∞),ℝ),a(t)>o,gi:ℝ→ℝ are positive continuous functions on ℝ with limt→∞gi(t)=∞,i=0,1,…,n Our results generalize and improve the results in a recent article by Sun and Wong [29].

Some nonlinear dynamic integral inequalities on time scales

In this paper, we study some dynamic integral inequalities with mixed nonlinearities on time scales, which provide explicit bounds on unknown functions. Our results include many existing ones in the literature as special cases and can be used as tools in the qualitative theory of certain classes of dynamic equations with mixed nonlinearities on time scales.

Interval oscillation for second order sublinear differential equations with a damping term

We present new oscillation criteria for the second order nonlinear differential equation with a damping term (a(t)y′(t))′ + p(t)y′(t) + q(t)|y(t)|α−1y(t) = 0, t ≥ t0 where 0 < α ≤ 1. Our results here are different, generalise and improve some known results for oscillation of second order nonlinear differential equations that are different from most known ones in the sencse they are based on the information only on a sequence of subintervals of [t0, ∞), rather than on the whole half-line and can be applied to extreme cases such as ∫t0∞ q(t)dt = −∞. Our results are illustrated with examples.

Oscillation criteria for second-order nonlinear dynamic equations

AbstractThis paper concerns the oscillation of solutions to the second-order dynamic equation (r(t)xΔ(t))Δ+p(t)xΔ(t)+q(t)f(xσ(t))=0, on a time scale T which is unbounded above. No sign conditions are imposed on r(t), p(t), and q(t). The function f∈C(R,R) is assumed to satisfy xf(x)>0 and f′(x)>0 for x≠0. In addition, there is no need to assume certain restrictive conditions and also the both cases ∫t0∞Δtr(t)=∞and∫t0∞Δtr(t)<∞ are considered. Our results will improve and extend results in (Baoguo et al. in Can. Math. Bull. 54:580-592, 2011; Bohner et al. in J. Math. Anal. Appl. 301:491-507, 2005; Hassan et al. in Comput. Math. Anal. 59:550-558, 2010; Hassan et al. in J. Differ. Equ. Appl. 17:505-523, 2011) and many known results on nonlinear oscillation. These results have significant importance to the study of oscillation criteria on discrete time scales such as T=Z, T=hZ, h>0, or T={t:t=qk,k∈N0,q>1} and the space of harmonic numbers T=Hn. Some examples illustrating the importance of our results are also included.MSC:34K11, 39A10, 39A99.

Oscillation criteria for second order sublinear dynamic equations with damping term

This paper concerns the oscillation of solutions to the second order sublinear dynamic equations with damping. No sign conditions are imposed on coefficients. We illustrate the results by several examples.

Comparison criteria for odd order forced nonlinear functional neutral dynamic equations

The purpose of this paper is to establish comparison criteria for forced odd order neutral dynamic equation x ( t ) + p ( t ) x ( ? ( t ) ) n + q t ? γ x ? t = g ( t ) , on an above-unbounded time scale T , where n ? 3 ; ? γ ( u ) ? u γ - 1 u , γ 0 ; p , q ? C rd t 0 , ∞ T , R + on t 0 , ∞ T ; g ? C rd ( t 0 , ∞ ) T , R ) ; and ? , ? : T ? T are rd-continuous functions such that lim t ? ∞ ? ( t ) = lim t ? ∞ ? ( t ) = ∞ . Comparison criteria have been established without assuming certain restrictive conditions on the time scale T which improve some results in a number of recent papers.

Comparison Criteria for Nonlinear Functional Dynamic Equations of Higher Order

We will consider the higher order functional dynamic equations with mixed nonlinearities of the form , on an above-unbounded time scale , where , , with , and . The function is a rd-continuous function such that for . The results extend and improve some known results in the literature on higher order nonlinear dynamic equations.