Sufficient Conditions For Oscillation Research Articles

Note on oscillation of neutral differential equations with multiple delays

This note is a reaction on a recently published sufficient condition for oscillation of all solutions of a neutral delay differential equation. It is shown by a counterexample that the result is not correct and the problem is explained in details. Several, more general, classes of neutral differential equations with time-dependent discrete, distributed as well as mixed delays are considered. New sufficient conditions for oscillation of all their solutions are proved. Applications are given for illustration.

Oscillation Theorems for Nonlinear Second-Order Delay Differential Equations with Some Sublinear Neutral Terms via Canonical Transform

Abstract New sufficient conditions for oscillation of a second-order nonlinear differential equation with some sublinear neutral terms are established via canonical transform and integral averaging method. Examples are provided to illustrate the significance and novelty of the presented results.

"Necessary and sufficient conditions for oscillation of second-order differential equation with several delays"

"In this paper, necessary and sufficient conditions are establish of the solutions to second-order delay differential equations of the form \begin{equation} \Big(r(t)\big(x'(t)\big)^\gamma\Big)' +\sum_{i=1}^m q_i(t)f_i\big(x(\sigma_i(t))\big)=0 \text{ for } t \geq t_0,\notag \end{equation} We consider two cases when $f_i(u)/u^\beta$ is non-increasing for $\beta<\gamma$, and non-decreasing for $\beta>\gamma$ where $\beta$ and $\gamma$ are the quotient of two positive odd integers. Our main tool is Lebesgue's Dominated Convergence theorem. Examples illustrating the applicability of the results are also given, and state an open problem."

Sufficient Conditions for Oscillations in Competitive Linear-Threshold Brain Networks

Oscillatory activity is highly prevalent in the brain. Oscillations with specific characteristics are associated with a variety of healthy and diseased brain functions. This paper considers two mesoscale brain network models described by linear-threshold and threshold-linear dynamics and takes on the analytical characterization of the emergence of oscillations. The synaptic connectivity is described by an arbitrary network interconnection topology that allows for self-excitatory nodes. We provide a structural characterization for the existence of stable node sets that support asymptotically stable equilibria and identify sufficient conditions for oscillatory behavior in competitive linear-threshold and threshold-linear dynamics. Simulations illustrate our results.

Necessary and Sufficient Conditions for Oscillations to a Second-Order Neutral Differential Equations with Impulses

Necessary and Sufficient Conditions for Oscillations to a Second-Order Neutral Differential Equations with Impulses

Necessary and Sufficient Conditions for Oscillation of Nonlinear Neutral Difference Systems of dim-2

Abstract This work is concerned about the necessary and sufficient conditions for oscillation of solutions of 2-dimensional nonlinear neutral delay difference systems of the form: Δ [ x ( n ) + p ( n ) x ( n − m ) y ( n ) + p ( n ) y ( n − m ) ] = [ a ( n ) b ( n ) c ( n ) d ( n ) ] [ f ( x ( n − α ) ) g ( y ( n − β ) ) ] , \Delta \left[ {\matrix{ {x\left( n \right) + p\left( n \right)x\left( {n - m} \right)} \hfill \cr {y\left( n \right) + p\left( n \right)y\left( {n - m} \right)} \hfill \cr } } \right] = \left[ {\matrix{ {a\left( n \right)} \hfill & {b\left( n \right)} \hfill \cr {c\left( n \right)} \hfill & {d\left( n \right)} \hfill \cr } } \right]\,\,\left[ {\matrix{ {f\left( {x\left( {n - \alpha } \right)} \right)} \hfill \cr {g\left( {y\left( {n - \beta } \right)} \right)} \hfill \cr } } \right], where m > 0, α ≥ 0, β ≥ 0 are integers, a(n), b(n), c(n), d(n), p(n) are real sequences and f, g ∈ 𝒞(ℝ, ℝ).

Asymptotic properties of solutions of third-order nonlinear dynamic equations on time scales

This paper is concerned with the asymptotic properties of solutions of third-order nonlinear dynamic equations on time scales. Some sufficient conditions for oscillation and nonoscillation of solutions as well as the boundedness of the solutions are established.

New Sufficient Conditions for Oscillation of Second-Order Neutral Delay Differential Equations

In this work, new sufficient conditions for the oscillation of all solutions of the second-order neutral delay differential equations with the non-canonical operator are established. Using a generalized Riccati substitution, we obtained criteria that complement and extend some previous results in the literature.

First-order impulsive differential systems: sufficient and necessary conditions for oscillatory or asymptotic behavior

In this paper, we study the oscillatory and asymptotic behavior of a class of first-order neutral delay impulsive differential systems and establish some new sufficient conditions for oscillation and sufficient and necessary conditions for the asymptotic behavior of the same impulsive differential system. To prove the necessary part of the theorem for asymptotic behavior, we use the Banach fixed point theorem and the Knaster–Tarski fixed point theorem. In the conclusion section, we mention the future scope of this study. Finally, two examples are provided to show the defectiveness and feasibility of the main results.

New Conditions for the Oscillation of Second-Order Differential Equations with Sublinear Neutral Terms

In continuous applications in electrodynamics, neural networks, quantum mechanics, electromagnetism, and the field of time symmetric, fluid dynamics, neutral differential equations appear when modeling many problems and phenomena. Therefore, it is interesting to study the qualitative behavior of solutions of such equations. In this study, we obtained some new sufficient conditions for oscillations to the solutions of a second-order delay differential equations with sub-linear neutral terms. The results obtained improve and complement the relevant results in the literature. Finally, we show an example to validate the main results, and an open problem is included.

Some properties of the Osvillatory and nonoscillatory solutions of second order

in this paper sufficient conditions of oscillation of all of nonlinear second order neutral differential eqiation and sifficient conditions for nonoscillatory soloitions to onverage to zero are obtained

Second-order Nonlinear Differential Equations: Oscillation Tests and Applications

The 2nd order Differential Equations (DEs) are applied to analyze various phenomena in physics and it is extended to engineering. Specifically 2nd order linear DEs are having a big role in this field. We already observed this working rule in moving systems like a vertical spring attached with a mass. Another example is an electric circuit with a resistor, an inductor, and a capacitor in series connection. These models are applicable to approximate more complex situations like bond structure between atoms or molecules. They are generally patterned as vibrating springs which is explained by the same DEs. Qualitative theory analysis of DEs is an active area of investigation in theory and application point of view. For instance, oscillatory behavior of 2nd order DEs have many applications in the research related to distributed networks where in high-speed computations lossless transmission lines evolve. These lossless transmission lines are employed to connect switching circuits. Recently the study the functional DEs of neutral type is an active area of research. Many researchers are working in the field of oscillation theory and studying only sufficient conditions. Merely few are considering the necessary and sufficient conditions. We have attempted to set up the necessary and sufficient conditions for oscillation of solutions of 2nd order nonlinear neutral DEs. Mathematics Subject Classification (2010): 34K.

Oscillation of a Class of Third-Order Neutral Differential Equations with Noncanonical Operators

The aim of this paper is to complement existing oscillation results for third-order neutral advanced differential equations under the condition of $$\gamma >0$$ ; in particular, the sufficient conditions are given in different way when $$\gamma =1$$ . Our main idea is by establishing sufficient conditions for nonexistence of so-called Kneser solutions. Then, combining with the results which guarantee the equation almost oscillation, we establish sufficient condition for oscillation of all solutions.

On stability and oscillation of fractional differential equations with a distributed delay

In this paper, we study the stability and oscillation of fractional differential equations \begin{equation*} ^cD^\alpha x(t)+ax(t)+\int_0^1x(s+[t-1])dR(s)=0. \end{equation*} We discretize the fractional differential equation by variation of constant formula and semigroup property of Mittag-Leffler function, and get the difference equation corresponding to the integer points. From the equivalence analogy of qualitative properties between the difference equations and the original fractional differential equations, the necessary and sufficient conditions of oscillation, stability and exponential stability of the equations are obtained.

Nonlinear oscillation and second order impulsive neutral difference equations

In this work, the authors have discussed the necessary and sufficient conditions for oscillation and asymptotic behaviour of solutions of second order nonlinear (sublinear/superlinear) neutral impulsive difference equations. The results are illustrated with examples under suitable fixed moments of impulsive effect.

On oscillatory second order nonlinear impulsive systems of neutral type

In this work, the necessary and sufficient conditions for oscillation of a class of second order neutral impulsive systems are established and our impulse satisfies a discrete neutral nonlinear equation of similar type. Further, one illustrative example showing applicability of the new result is included.

Explicit criteria for the oscillation of second-order differential equations with several sub-linear neutral coefficients

In this work, we present sufficient conditions for oscillation of all solutions of a second-order functional differential equation. We consider two special cases when gamma >beta and gamma <beta . This new theorem complements and improves a number of results reported in the literature. Finally, we provide examples illustrating our results and state an open problem.

Necessary and Sufficient Conditions for Oscillation of Solutions to Second-Order Neutral Differential Equations with Impulses

Abstract In this work, necessary and sufficient conditions for oscillation of solutions of second-order neutral impulsive differential system { ( r ( t ) ( z ′ ( t ) ) γ ) ′ + q ( t ) x α ( σ ( t ) ) = 0 , t ≥ t 0 , t ≠ λ k , Δ ( r ( λ k ) ( z ′ ( λ k ) ) γ ) + h ( λ k ) x α ( σ ( λ k ) ) = 0 , k ∈ 𝕅 \left\{ {\matrix{{{{\left( {r\left( t \right){{\left( {z'\left( t \right)} \right)}^\gamma }} \right)}^\prime } + q\left( t \right){x^\alpha }\left( {\sigma \left( t \right)} \right) = 0,} \hfill &amp; {t \ge {t_0},\,\,\,t \ne {\lambda _k},} \hfill \cr {\Delta \left( {r\left( {{\lambda _k}} \right){{\left( {z'\left( {{\lambda _k}} \right)} \right)}^\gamma }} \right) + h\left( {{\lambda _k}} \right){x^\alpha }\left( {\sigma \left( {{\lambda _k}} \right)} \right) = 0,} \hfill &amp; {k \in \mathbb{N}} \hfill \cr } } \right. are established, where z ( t ) = x ( t ) + p ( t ) x ( τ ( t ) ) z\left( t \right) = x\left( t \right) + p\left( t \right)x\left( {\tau \left( t \right)} \right) Under the assumption ∫ ∞ ( r ( η ) ) - 1 / α d η = ∞ \int {^\infty {{\left( {r\left( \eta \right)} \right)}^{ - 1/\alpha }}d\eta = \infty } two cases when γ&gt;α and γ&lt;α are considered. The main tool is Lebesgue’s Dominated Convergence theorem. Examples are given to illustrate the main results, and state an open problem.

Nonlinear Second Order Impulsive Difference Equations and their Oscillation Properties

Abstract In this work, we state and discuss the sufficient conditions for oscillation and nonoscillation of a class of nonlinear second order neutral impulsive difference equations with fixed moments of impulsive effect for various ranges of the neutral coefficient p(n).

A Survey on Sharp Oscillation Conditions of Differential Equations with Several Delays

This paper deals with the oscillation of the first-order differential equation with several delay arguments x′t+∑i=1mpitxτit=0,t≥t0, where the functions pi,τi∈Ct0,∞,R+, for every i=1,2,…,m,τit≤t for t≥t0 and limt→∞τit=∞. In this paper, the state-of-the-art on the sharp oscillation conditions are presented. In particular, several sufficient oscillation conditions are presented and it is shown that, under additional hypotheses dealing with slowly varying at infinity functions, some of the “liminf” oscillation conditions can be essentially improved replacing “liminf” by “limsup”. The importance of the slowly varying hypothesis and the essential improvement of the sufficient oscillation conditions are illustrated by examples.