In this paper we obtain sufficient conditions under which every solution of the retarded differential equation <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 1 right-parenthesis x prime left-parenthesis t plus p minus tau equals 0 comma greater-than-or-slanted-equals comma"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> stretchy="false">)</mml:mo> <mml:mspace width="1em"/>...

This paper is concerned with the oscillatory behavior of first-order delay differential equations form (1) x′(t) + p(t)x(τ(t)) = 0, t ≥ T, where p, τ ∈ C([T,∞),R+), R+ [0,∞), τ(t) nondecreasing, < for T and limt→∞ ∞. Let numbers k L be defined by

(1983). Necessary and Sufficient Conditions for Oscillations. The American Mathematical Monthly: Vol. 90, No. 9, pp. 637-640.

Sufficient oscillation conditions involving $\limsup $ and $\liminf for first-order differential equations with several non-monotone deviating arguments nonnegative coefficients are obtained. The results based on the iterative application of Gr\"{o}nwall inequality. Examples illustrating significance also given.

This paper is concerned with the oscillatory behavior of first-order delay differential equations form \begin{eqnarray} x' (t)+p(t)x({\tau }(t))=0, \quad t\geq t_{0}, \end{eqnarray} where $p, {\tau } \in C([t_{0}, \infty ), \mathbb {R}^+), {R}^+=[0, \tau (t)$ non-decreasing, $\tau (t) <t$ for $t \geq t_{0}$ and $\lim _{t{\rightarrow }{\infty }} = \infty$. Let numbers $k$ $L$ be defined by \[ k=\liminf \int _{\tau (t)}^{t}p(s)ds \mbox {and} L=\limsup (t)}^{t}p(s)ds. \] It proved here that...

Oscillation and nonoscillation criteria for the first-order delay differential equation <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x prime left-parenthesis t right-parenthesis plus p x tau equals 0 comma greater-than-or-equal-to greater-than comma"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>x</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> stretchy="false">)</mml:mo>...

Consider the first-order delay differential equation in critical case where Sufficient condltions are established under which all solutions spite of fact that corrrsponding "limiting" admits a non-oscillatory solution .It should be emphasized conditions obtained unimproveble is some sense. Esitimates for intervals length between successive zeroes also obtained.

Consider the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n th"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>th</mml:mtext> </mml:mrow> <mml:annotation encoding="application/x-tex">n{\text {th}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> order delay differential equation (1) <disp-formula \[ alttext="x Superscript left-parenthesis n right-parenthesis Baseline...

Consider the first order linear difference equation2 and lim k→+∞ τ (k) = +∞.Optimal conditions for oscillation of all proper solutions this equation are established.The results lead to a sharp condition, when kτ → +∞ as k +∞.Examples illustrating given.

We consider difference equations with several non-monotone deviating arguments and non-negative coefficients. The deviations (delays advances) are, generally, unbounded. Sufficient oscillation conditions are obtained in an explicit iterative form. Additional results terms of lim inf for bounded deviations. Examples illustrating the tests presented.

This paper is concerned with the oscillatory behaviour of first-order delay differential equations form x ′ ( t ) + p τ = 0 , ⩾ (1) where ∈ C [ ∞ non-decreasing, τ(t) < for ⩾t0 and lim → . Let numbers k andL be defined by inf ∫ s d a n L sup It proved here that when 1 ⩽ 1/e all solutions equation oscillate in several cases which condition > ln λ − 5 2 holds, λ1 smaller root ekλ. 2000 Mathematics Subject Classification 34K11 (primary); 34C10 (secondary).

Abstract Consider the nth order ( n ≥ 1) delay differential inequalities and equation , where q t ) 0 is a continuous function p τ are positive constants. Under condition pτe 1 we prove that when odd (1) has no eventually solutions, (2) negative (3) only oscillatory solutions even bounded every solution of oscillatory. The &gt; sharp. above results, which generalize previous results by Ladas Stavroulakis for first inequalities, caused retarded argument do not hold = 0.

Our aim in this paper is to obtain sufficient conditions under which certain functional differential equations have a large number of nonoscillatory solutions. Using the characteristic equation majorant delay with constant coefficients and Schauder's fixed point theorem, we question has solution. Then known comparison theorem employed as tool demonstrate that if solution, then it really such

This paper presents a new sufficient condition for the oscillation of all solutions linear difference equations with general delay argument. The significance this is demonstrated by comparing known conditions. An example illustrating results also given.

Consider the first-order linear differential equation with several retarded arguments x′(t) + Σmi=1 pi(t)x(τi(t)) = 0, t ≥ t0, where functions pi, τi ∈ C([t0, ∞), R+), for every i 1,2, ..., m, τi(t) ≤ t0 and limt→∞ ∞. In this paper state-of-the-art on oscillation of all solutions to these equations is reviewed new sufficient conditions are established, especially in case nonmonotone arguments. Examples illustrating results given.