Comparison theorems and necessary/sufficient conditions for existence of nonoscillatory solutions of forced impulsive delay differential equations

  • Sui Sun Cheng
  • Yuan Huang Shao Tsing Hua Univ., Taiwan

Abstract

In 1997, A. H. Nasr provided necessary and sufficient conditions for the oscillation of the equation $$x''(t) + p(t) |x(g(t))|^{\eta} \text{sgn} (x(g(t))) = e(t),$$ where $\eta > 0$, $p$, and $g$ are continuous functions on $[0, \infty)$ such that $p(t) \geq 0,\;\; g(t) \leq t,\;\; g'(t) \geq \alpha > 0$, and $\lim_{t \rightarrow \infty} g(t) = \infty$ It is important to note that the condition $g'(t) \geq \alpha > 0$ is required. In this paper, we remove this restriction under the superlinear assumption $\eta > 0$. Infact, we can do even better by considering impulsive differential equations with delay and obtain necessary and sufficient conditions for the existence of nonoscillatory solutions and also a comparison theorem that enables us to apply known oscillation results for impulsive equations without forcing terms to yield oscillation criteria for our equations.
Published
25.09.2012
How to Cite
ChengS. S., and ShaoY. H. “Comparison Theorems and necessary/Sufficient Conditions for Existence of Nonoscillatory Solutions of Forced Impulsive Delay Differential Equations”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, no. 9, Sept. 2012, pp. 1233-48, http://umj.imath.kiev.ua/index.php/umj/article/view/2654.
Section
Research articles