Oscillation Criteria in Higher Order Neutral Equations

A proof of characterisation of oscillation for higher-order neutral differential equations of mixed type by the Laplace transform

“…In the oscillation theory of linear delay differential equations, one of the most important objectives is to give a necessary and sufficient condition for oscillation via the characteristic equation. Such a result for scalar delay differential equations was proved in [1,4,7] using various methods.…”

“…The characterisation of oscillation for the solutions of a linear functional equation in terms of the characteristic equation was recently extended by S. J. Bilchev, M. K. Grammatikopoulos and I. P. Stavroulakis [4] to scalar higher-order neutral differential equations of mixed type.…”

Exact regions of oscillation for a neutral differential equation

“…In [1], it is shown that a necessary and su¯cient condition for the oscillation of all solutions of the neutral di¬erential equation…”

Oscillatory and nonoscillatory properties of solutions of functional differential equations and difference equations

“…This is established utilizing the method of steps. The method may also be utilized to obtain a global existence theorem for the NFDE of the form ^[x(^) + g(i,a:(ri(i)),...,a:(ri(0))] + /(i,®(<7'i(i)).-)«(o-n(0)) = 0, (1.9) where for some T G R and some positive integer m, gGC{ [T,oo)xR"'x---xR"',R"') (1.10) f GC{ [T,oo)x R"" x---xR"',R"') (1.11) and for i = 1,2,..., / and j = 1,2, ...,n nEC([T,(X))), o'X()eC([r,oo)) ( (1.10), (1.11), (1.12), and (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13) are satisfied.…”