Abstract
Differential equations of second order appear in numerous applications such as fluid dynamics, electromagnetism, quantum mechanics, neural networks and the field of time symmetric electrodynamics. The aim of this work is to establish necessary and sufficient conditions for the oscillation of the solutions to a second-order neutral differential equation. First, we have taken a single delay and later the results are generalized for multiple delays. Some examples are given and open problems are presented.
Highlights
Consider the class of nonlinear neutral delay differential equations of the form a w μ (y) + c(y)g u ς (y) = 0, (1)where w(y) = u(y) + b(y)u(θ(y)) and μ is the ratio of two odd positive integers
5 Open problem This work leads to some open problems: 1. Can we find necessary and sufficient conditions for the oscillation of solutions to second-order differential equation (1) for the other ranges of the neutral coefficient b?
Μ = 3/5, a(y) = e–y, b(y) = –e–y, ς1(y) = u – 2, ς2(y) = u – 1, A(y) =
Summary
Consider the class of nonlinear neutral delay differential equations of the form a w μ (y) + c(y)g u ς (y) = 0, (1). Where w(y) = u(y) + b(y)u(θ(y)) and μ is the ratio of two odd positive integers. (A2) g ∈ C(R, R) is non-decreasing and odd with ug(u) > 0 for u = 0. Baculikova et al [2] considered (1) and studied the oscillatory behavior of (1) for g(u) = u, Santra et al Journal of Inequalities and Applications. They obtained sufficient conditions for the oscillation of the solutions of the linear counterpart of (1), using comparison techniques. Chatzarakis et al [3] considered the equation a u μ2 (y) + c(y)uμ[2] ς (y) = 0
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