Oscillation criteria for impulsive differential equations of first order with retarded argument
The author deals with impulsive differential equations of the forms: \[ x'(t) + p(t)x(t-h) = 0,\quad t \neq \tau_k,\tag{1} \] \[ x'(t) + q(t)x(t) + p(t)x(t-h) = 0,\quad t \neq \tau_k,\tag{2} \] \[ \bigtriangleup x(\tau_k) =x(\tau_k+0)-x(\tau_k-h)= I_k(x(\tau_k),x(\tau_k -h)), k\in\mathbb{N}, \] \[ 0 < h < \tau_1 < \tau_2 <\dots, \quad\lim_{k\to \infty } \tau_k = \infty , \] with the initial condition: \[ x(t) =\varphi (t)\text{ for }t\in[-h,0], \quad \varphi (t) \in C([-h,0];\mathbb{R}). \] The author gives sufficient conditions under which all solutions to (1) and (2) oscillate, respectively.
scientific article; zbMATH DE number 1406292