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
where the functions pk, τk ∈ C([t 0, ∞), ℝ+), τk(t) < t for t ≥ t 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
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
can be written in essentially unique canonical form as
The canonical representation essentially simplifies examination of the fourth order strongly noncanonical equations
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
for all
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.
