Existence of asymptotic solutions of second order neutral differential equation with multiple delays.
For a certain class of functional differential equations with perturbations conditions are given such that there exist solutions which converge to solutions of the equations without perturbation.
The paper deals with the quasi-linear ordinary differential equation with . We treat the case when is not necessarily monotone in its second argument and assume usual conditions on and . We find necessary and sufficient conditions for the existence of unbounded non-oscillatory solutions. By means of a fixed point technique we investigate their growth, proving the coexistence of solutions with different asymptotic behaviors. The results generalize previous ones due to Elbert–Kusano, [Acta...
Consider the third order nonlinear dynamic equation , (*) on a time scale which is unbounded above. The function f ∈ C(,) is assumed to satisfy xf(x) > 0 for x ≠ 0 and be nondecreasing. We study the oscillatory behaviour of solutions of (*). As an application, we find that the nonlinear difference equation , where α ≥ -1, γ > 0, c > 3, is oscillatory.
In this paper we consider a linear operator on an unbounded interval associated with a matrix linear Hamiltonian system. We characterize its Friedrichs extension in terms of the recessive system of solutions at infinity. This generalizes a similar result obtained by Marletta and Zettl for linear operators defined by even order Sturm-Liouville differential equations.
Suppose that the function in the differential equation (1) is decreasing on where . We give conditions on which ensure that (1) has a pair of solutions such that the -th derivative () of the function has the sign for sufficiently large and that the higher differences of a sequence related to the zeros of solutions of (1) are ultimately regular in sign.