On oscillations of neutral equations with mixed arguments
Consider the neutral differential equation \[ (1)\quad d/dt[y(t)+py(t- \tau)]+qy(t-\sigma)=0, \] where p, \(\tau\), q and \(\sigma\) are real numbers. The main results in this paper are the following: Theorem 1. The following statements are equivalent: (a) Every bounded solution of equation (1) oscillates. (b) The characteristic equation associated with equation (1) \[ (2)\quad F(\lambda)=\lambda +\lambda pe^{-\lambda \tau}+qe^{-\lambda \sigma}=0 \] has no roots in (- \(\infty,0].\) Theorem 2. The following statements are equivalent: (a) Every unbounded solution of equation (1) oscillates. (b) The characteristic equation (2) associated with equation (1) has no roots in (0,\(\infty)\) and 0 is not a double root of equation (2). An immediate corollary of the above theorems is the following result: Corollary. Every solution of (1) oscillates if and only if its characteristic equation (2) has no real roots.
scientific article; zbMATH DE number 4142516