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Abstract

The aim of our paper is to study oscillatory and asymptotic properties of solutions of nonlinear differential equations of the third order with deviating argument. In particular, we prove a comparison theorem for properties A and B as well as a comparison result on property A between nonlinear equations with and without deviating arguments. Our assumptions on nonlinearity f are related to its behavior only in a neighbourhood of zero and/or of infinity.

Abstract

Consider the first-order linear differential equation with several retarded arguments

x(t)+k=1npk(t)x(τk(t))=0,tt0,

where the functions pk, τkC([t 0, ∞), ℝ+), τk(t) < t for tt 0 and limt→∞ τk(t) = ∞, for every k = 1, 2, …, n. Oscillation conditions which essentially improve known results in the literature are established. An example illustrating the results is given.

Abstract

In this paper, we establish some new oscillation criteria for the higher-order half-linear neutral differential equation
[a(t)([x(t)+p(t)x(σ(t))](n1))γ]+q(t)xγ(τ(t))=0.
An example is given to illustrate the main results.

Abstract

The objective of this paper is to study asymptotic properties of the third-order neutral differential equation $$ \left[ {a\left( t \right)\left( {\left[ {x\left( t \right) + p\left( t \right)x\left( {\sigma \left( t \right)} \right)} \right]^{\prime \prime } } \right)^\gamma } \right]^\prime + q\left( t \right)f\left( {x\left[ {\tau \left( t \right)} \right]} \right) = 0, t \geqslant t_0 . \left( E \right) $$. We will establish two kinds of sufficient conditions which ensure that either all nonoscillatory solutions of (E) converge to zero or all solutions of (E) are oscillatory. Some examples are considered to illustrate the main results.

Abstract

It is shown that the strongly noncanonical fourth order operator

Ly=r3(t)r2(t)r1(t)y(t)

can be written in essentially unique canonical form as

Ly=q4(t)q3(t)q2(t)q1(t)q0(t)y(t).

The canonical representation essentially simplifies examination of the fourth order strongly noncanonical equations

r3(t)r2(t)r1(t)y(t)+p(t)y(τ(t))=0.

Abstract

In this paper, some new sufficient conditions are established for the oscillation of all solutions of the second-order neutral differential equation with mixed neutral terms of the form

(r(t)(x(t)+ax(t-τ)+bx(t+σ)))+p(t)xα(t-k)+q(t)xβ(t+m)=0

for all tt0, under the condition t01r(t)𝑑t<. The results obtained in this paper are new, and they generalize and complement some known results. Examples are provided to illustrate the main results.

Abstract

This paper deals with asymptotic behavior of nonoscillatory solutions of certain third-order forced dynamic equations on time scales. The main goal is to investigate when all solutions behave at infinity like certain nontrivial nonlinear functions.

Abstract

The purpose of this paper is to investigate the oscillation of the second-order neutral differential equations of the form (E)$$ (r(t)|z'(t)|^{\alpha - 1} z'(t))' + q(t)|x(\sigma (t))|^{\alpha - 1} x(\sigma (t)) = 0, $$ where z(t) = x(t) + p(t)x(τ(t)). The obtained comparison principles essentially simplify the examination of the studied equations. Further, our results extend and improve the results in the literature.

Abstract

This article concerns the oscillatory behavior of solutions to second-order half-linear delay differential equations with mixed neutral terms. The authors present new oscillation criteria that improve, extend, and simplify existing ones in the literature. The results are illustrated with examples.

Abstract

In this paper, we provide a test under which every solution of a first-order delay differential equation oscillates. An example is given to illustrate the significance of the result.