Oscillatory Behavior Of Solutions Research Articles

Simplified and improved criteria for oscillation of delay differential equations of fourth order

An interesting point in studying the oscillatory behavior of solutions of delay differential equations is the abbreviation of the conditions that ensure the oscillation of all solutions, especially when studying the noncanonical case. Therefore, this study aims to reduce the oscillation conditions of the fourth-order delay differential equations with a noncanonical operator. Moreover, the approach used gives more accurate results when applied to some special cases, as we explained in the examples.

New Results on Qualitative Behavior of Second Order Nonlinear Neutral Impulsive Differential Systems with Canonical and Non-Canonical Conditions

The oscillation of impulsive differential equations plays an important role in many applications in physics, biology and engineering. The symmetry helps to deciding the right way to study oscillatory behavior of solutions of impulsive differential equations. In this work, several sufficient conditions are established for oscillatory or asymptotic behavior of second-order neutral impulsive differential systems for various ranges of the bounded neutral coefficient under the canonical and non-canonical conditions. Here, one can see that if the differential equations is oscillatory (or converges to zero asymptotically), then the discrete equation of similar type do not disturb the oscillatory or asymptotic behavior of the impulsive system, when impulse satisfies the discrete equation. Further, some illustrative examples showing applicability of the new results are included.

More Effective Results for Testing Oscillation of Non-Canonical Neutral Delay Differential Equations

In this work, we address an interesting problem in studying the oscillatory behavior of solutions of fourth-order neutral delay differential equations with a non-canonical operator. We obtained new criteria that improve upon previous results in the literature, concerning more than one aspect. Some examples are presented to illustrate the importance of the new results.

Oscillatory behavior of solutions of certain third-order neutral differential equation with continuously distributed delay

This paper concerns the oscillatory behavior of the solutions of the third-order neutral differential equation with continuously distributed delay. We prove many new sufficient conditions which ensure that every solution to be either oscillatory or converges to zero asymptotically. Examples dwelling upon the importance of our theorems are also included.

New Theorems for Oscillations to Differential Equations with Mixed Delays

The oscillation of differential equations plays an important role in many applications in physics, biology and engineering. The symmetry helps to deciding the right way to study oscillatory behavior of solutions of this equations. The purpose of this article is to establish new oscillatory properties which describe both the necessary and sufficient conditions for a class of nonlinear second-order differential equations with neutral term and mixed delays of the form p(ι)w′(ι)α′+r(ι)uβ(ν(ι))=0,ι≥ι0 where w(ι)=u(ι)+q(ι)u(ζ(ι)). Furthermore, examining the validity of the proposed criteria has been demonstrated via particular examples.

Different techniques for studying oscillatory behavior of solution of differential equations

The aim of this work is to study oscillatory behavior of solutions for a fourth-order neutral nonlinear differential equation (b(x)(wm−1(x))γ)′+ ∑i=1jqi(x)f(w(gi(x)))=0, x≥x0. The results obtained are based on the Riccati transformation, integral averaging technique and the theory of comparison with second-order delay equations. The obtained results complements and generalize the earlier ones. Some examples are illustrated to show the applicability of the obtained results.

Neutral differential equations with noncanonical operator: Oscillation behavior of solutions

The objective of this work is to study the oscillatory behavior of neutral differential equations with several delays. By using both Riccati substitution technique and comparison with delay equations of first-order, we establish new oscillation criteria. Our new criteria are simplifying and complementing some related results that have been published in the literature. Moreover, some examples are given to show the applicability of our results.

New oscillation criteria for third-order differential equations with bounded and unbounded neutral coefficients

This paper examines the oscillatory behavior of solutions to a class of thirdorder differential equations with bounded and unbounded neutral coefficients. Sufficient conditions for all solutions to be oscillatory are given. Some examples are considered to illustrate the main results and suggestions for future research are also included.

Some New Oscillation Results for Fourth-Order Neutral Differential Equations with a Canonical Operator

By this work, our aim is to study oscillatory behaviour of solutions to 4th-order differential equation of neutral typeLy′+∑j=1kqjyzβgjy=0whereLy=ξyw‴yα,wy:=zy+ryzg˜y. By using the comparison method with first-order differential inequality, we find new oscillation conditions for this equation.

English

The Riccati equation method is used for study the oscillatory and non oscillatory behavior of solutions of linear four dimensional hamiltonian systems. An oscillatory and two non oscillatory criteria are proved. On an example the obtained oscillatory criterion is compared with some well known results.

Oscillation of damped second-order linear mixed neutral differential equations

This article concerns the oscillatory behavior of solutions to damped second-order linear functional differential equations with a mixed neutral term. The authors present new oscillation criteria that improve and extend some existing ones in the literature. Examples to illustrate the main results are included.

Oscillation of higher order functional differential equations with an advanced argument

This paper is concerned with the oscillatory behavior of solutions of the nth-order nonlinear differential equations with an advanced argument x(n)(t)+δq(t)xλ(g(t))=0,t≥t0>0, where n≥2 is even or odd and δ=±1. Here q(t)>0, g(t)≥t, and λ is the ratio of odd positive integers. The results are illustrated with an example.

Asymptotic Properties of Neutral Differential Equations with Variable Coefficients

The aim of this work is to study oscillatory behavior of solutions for even-order neutral nonlinear differential equations. By using the Riccati substitution, a new oscillation conditions is obtained which insures that all solutions to the studied equation are oscillatory. The obtained results complement the well-known oscillation results present in the literature. Some example are illustrated to show the applicability of the obtained results.

Oscillatory behavior of solutions of odd-order nonlinear delay differential equations

The objective of this study is to establish new sufficient criteria for oscillation of solutions of odd-order nonlinear delay differential equations. Based on creating comparison theorems that compare the odd-order equation with a couple of first-order equations, we improve and complement a number of related ones in the literature. To show the importance of our results, we provide an example.

Oscillatory Behavior of a Type of Generalized Proportional Fractional Differential Equations with Forcing and Damping Terms

In this paper, we study the oscillatory behavior of solutions for a type of generalized proportional fractional differential equations with forcing and damping terms. Several oscillation criteria are established for the proposed equations in terms of Riemann-Liouville and Caputo settings. The results of this paper generalize some existing theorems in the literature. Indeed, it is shown that for particular choices of parameters, the obtained conditions in this paper reduce our theorems to some known results. Numerical examples are constructed to demonstrate the effectiveness of the our main theorems. Furthermore, we present and illustrate an example which does not satisfy the assumptions of our theorem and whose solution demonstrates nonoscillatory behavior.

Kamenev and Philos-types oscillation criteria for fourth-order neutral differential equations

This work is concerned with the oscillatory behavior of solutions of fourth-order neutral differential equations. By using the Riccati transformation and integral averaging techniques we obtain some new Kamenev-type and Philos-type oscillation criteria. Our results extend and improve some known results in the literature. An example is given to illustrate our main results.

Oscillatory behavior of solutions to third-order nonlinear differential equations with a superlinear neutral term

This article studies the oscillatory and asymptotic behavior of solutions to a class of third-order nonlinear differential equations with superlinear neutral term. The results are obtained by a comparison with first-order delay differential equations whose oscillatory behavior is known, and by using integral criteria. Two examples are provided to illustrate the results.
 For more information see https://ejde.math.txstate.edu/Volumes/2020/32/abstr.html

New results for oscillation of fractional partial differential equations with damping term

<p style='text-indent:20px;'>In this paper, we study the oscillatory behavior of solutions of a class of damped fractional partial differential equations subject to Robin and Dirichlet boundary value conditions. By using integral averaging technique and Riccati type transformations, we obtain some new sufficient conditions for oscillation of all solutions of this kind of fractional differential equations with damping term. Our results essentially enrich the ones in the existing literature. Finally, we also give two specific examples to illustrate our main results.</p>

Oscillatory behavior of solutions of dynamic equations of higher order on time scales

We study the n -th-order nonlinear dynamic equations x [ n ] ( t ) + p ( t ) ϕ α n − 1 [ ( x [ n − 2 ] ( t ) ) Δ σ ] + q ( t ) ϕ γ ( x ( g ( t ) ) ) = 0 on an unbounded time scale 𝕋 , where n ≥ 2 and for i = 1 , … , n − 1 x [ i ] ( t ) : = r i ( t ) ϕ α i [ ( x [ i − 1 ] ( t ) ) Δ ] , with r n = α n = 1 and x [ 0 ] = x ; here the constants α i and the functions r i , i = 1 , … , n − 1 , are positive and p , q are nonnegative functions. Criteria are established for the oscillation of solutions for both even- and odd-order cases. The results improve several known results in the literature on second-order, third-order, and higher-order linear and nonlinear dynamic equations. In particular our results can be applied when g is not (delta) differentiable and the forward jump operator σ and g do not commute.

Some New Oscillation Criteria for Fourth-Order Nonlinear Delay Difference Equations

In this paper, the authors studied oscillatory behavior of solutions of fourth-order delay difference equation Δc3nΔc2nΔc1nΔun+pnfun−k=0 under the conditions ∑n=n0∞cin<∞, i=1, 2, 3. New oscillation criteria have been obtained which greatly reduce the number of conditions required for the studied equation. Some examples are presented to show the strength and applicability of the main results.