Oscillation Properties of Some Functional Fourth Order Ordinary Differential Equations

2010 Mathematics Subject Classification: 34A30, 34A40, 34C10.

In this paper are considered oscillation properties of some classes of functional ordinary differential equations, namely equations of the type ziv(t) + mz′′(t) + g(z(t), z′(t), z′′(t), z′′′(t)) +nXi=1_i(t)z(t − i) = f(t), where m > 0 is constant, f(t) 2 C([T,1);R), T _ 0 is a large enough constant, g(z, _, _, _) 2 C(R4;R), _i(t) 2 C([0,1); [0,1)), 8 i = 1, n, n 2 N and {i}n i=1 are nonnegative constants. As a main result of this work we derive a sufficient condition for the distribution of the zeros of the above equations. Furthermore we discuss the complexity of the oscillation behavior of such equations and its relation to some properties of the corresponding solutions. Finally, we comment the oscillation behavior of a neutral fourth order ordinary differential equation, which appears in two papers of Ladas and Stavroulakis, as well as in a paper of Grammatikopoulos et al.