In this paper we investigate the asymptotic properties of all solutions of the delay differential equation y'(x)=a(x)y(\tau(x))+b(x)y(x),\qquad x\in I=[x_0,\infty). We set up conditions under which every solution of this equation can be represented in terms of a solution of the differential equation z'(x)=b(x)z(x),\qquad x\in I and a solution of the functional equation |a(x)|\varphi(\tau(x))=|b(x)|\varphi(x),\qquad x\in I.
The aim of this paper is to present sufficient conditions for all bounded solutions of the second order neutral differential equation \[ \big (x(t)-px(t-\tau )\big )^{\prime \prime }- q(t)x\big (\sigma (t)\big )=0 \] to be oscillatory and to improve some existing results. The main results are based on the comparison principles.
We study oscillatory behavior of a class of fourth-order quasilinear differential equations without imposing restrictive conditions on the deviated argument. This allows applications to functional differential equations with delayed and advanced arguments, and not only these. New theorems are based on a thorough analysis of possible behavior of nonoscillatory solutions; they complement and improve a number of results reported in the literature. Three illustrative examples are presented.
In this paper we present some new oscillatory criteria for the $n$-th order neutral differential equations of the form \[ (x(t)\pm p(t)x[\tau (t)])^{(n)} +q(t)x[\sigma (t)] =0. \] The results obtained extend and improve a number of existing criteria.