Analytical and numerical treatment of oscillatory mixed differential equations with differentiable delays and advances (Q633966)

The oscillatory behaviour of the following mixed type differential equation is studied \[ x'(t)=\int\limits_{-1}^{0}x(t-r(\theta))d\nu(\theta)+\int\limits_{-1}^{0}x(t+\tau(\theta))d\eta(\theta), \] where \(x(t)\in \mathbb{R}\), \(\nu(\theta)\) and \(\eta(\theta)\) are real functions of bounded variation on \([-1,0]\), normalized so that \(\nu(-1)=\eta(-1)=0\). Delays \(r(\theta)\) and advances \(\tau(\theta)\) are nonnegative real differentiable functions on \([-1,0]\). Some analytical and numerical criteria are obtained in order to guarantee that all solutions of this equation are oscillatory. The paper also contains a large number of numerical examples, which show how numerical approximations can be used to derive information about oscillation or non-oscillation of solutions.