Oscillation criteria for first-order neutral nonlinear difference equations with variable coefficients

AbstractConsider the first-order neutral nonlinear difference equation of the form δ (yn−pnyn−r)+ qn ∏i=lm |yn−σi|ai sgn yn−σi = 0, n=0,1,…, where τ > 0, σi ≥ 0 (i = 1, 2,…, m) are integers, {pn} and {qn} are nonnegative sequences. We obtain new criteria for the oscillation of the above equation without the restrictions Σn=0∞ qn = ∞ or Σn=0∞ nqn Σj=n∞ qj = ∞ commonly used in the literature

Oscillation of first order neutral differential equations with positive and negative coefficients

We obtain some new sharp sufficient conditions for the oscillation of all solutions of the first order neutral differential equation with positive and negative coefficients of the form $$\frac{d}{dt}\bigl(x(t) - R(t)x(t - r)\bigr)+ P(t)x(t - \tau ) - Q(t)x(t -\delta) = 0$$ where P,Q,R\in C([t_0,\infty),R^^), r\in (0,\infty) and $\tau,\delta\in R^+$. In particular, the conditions are necessary and sufficient when the coefficients are constants. As corollaries, many known results are extended and improved in the literature

Existence results for dynamic inclusions on time scales with nonlocal initial conditions

AbstractThis paper is mainly concerned with the existence of solutions for first order dynamic inclusions on time scales with nonlocal initial conditions. By using Bohnenblust–Karlin’s fixed point theorem and Leray–Schauder nonlinear alternative for multivalued maps, some sufficient conditions are established. An example is also included to illustrate our results

Global attractivity of positive periodic solutions for an impulsive delay periodic model of respiratory dynamics

AbstractIn this paper we shall consider the following nonlinear impulsive delay differential equation x′(t)+αV(t)x(t)xn(t−mω)θn+xn(t−mω)=λ(t),a.e.t>0,t≠tk,x(tk+)=1(1+bk)x(tk),k=1,2,…,where m and n are positive integers, V(t) and λ(t) are positive periodic continuous functions with period ω>0. In the nondelay case (m=0), we show that the above equation has a unique positive periodic solution x∗(t) which is globally asymptotically stable. In the delay case, we present sufficient conditions for the global attractivity of x∗(t). Our results imply that under the appropriate periodic impulsive perturbations, the impulsive delay equation shown above preserves the original periodic property of the nonimpulsive delay equation. In particular, our work extends and improves some known results