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We present two sufficient conditions for the oscillatory behaviour of a class of noncanonical fourth-order dynamic equations on arbitrary time scales. An illustrative example is included to show that new criteria improve related results reported in the literature.

In this work, we study the oscillation and asymptotic behaviour of third-order nonlinear dynamic equations on time scales. The findings are obtained using an integral criterion as well as a comparison theorem with the oscillatory properties of a first-order dynamic equation. As a consequence, we give conditions which guarantee that all solutions to the aforementioned problem are only oscillatory, different from any other result in the literature. We propose novel oscillation criteria that improve, extend, and simplify existing ones in the literature. The results are associated with a numerical example. We point out that the results are new even for the case \(\mathbb{T}=\mathbb{R}\) or \(\mathbb{T}=\mathbb{Z}\).

Abstract In this paper, we present a single-condition sharp criterion for the oscillation of the fourth-order linear delay differential equation x ( 4 ) ( t ) + p ( t ) x ( τ ( t ) ) = 0 $$ x^{(4)}(t) + p(t)x\bigl(\tau (t)\bigr) = 0 $$ by employing a novel method of iteratively improved monotonicity properties of nonoscillatory solutions. The result obtained improves a large number of existing ones in the literature.

Abstract The authors present necessary and sufficient conditions for the oscillation of a class of second order non-linear neutral dynamic equations with non-positive neutral coefficients by using Krasnosel’skii’s fixed point theorem on time scales. The nonlinear function may be strongly sublinear or strongly superlinear.

In this paper, several oscillation criteria for a class of higher order dynamic equations with superlinear neutral term are established. The proposed results provide a unified platform that adequately covers both discrete and continuous equations and further sufficiently comments on oscillatory behavior of more general class of equations than the ones reported in the literature. We conclude the paper by demonstrating illustrative examples.

Abstract In this paper, new oscillation results for nonlinear third-order difference equations with mixed neutral terms are established. Unlike previously used techniques, which often were based on Riccati transformation and involve limsup or liminf conditions for the oscillation, the main results are obtained by means of a new approach, which is based on a comparison technique. Our new results extend, simplify, and improve existing results in the literature. Two examples with specific values of parameters are offered.

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.

This paper is concerned with the asymptotic behavior of the nonoscillatory solutions of the forced fractional differential equation with positive and negative terms of the form \[^{C}D_{c}^{\alpha}y(t)+f(t,x(t))=e(t)+k(t)x^{\eta}(t)+h(t,x(t)),\] where \(t\geq c \geq 1\), \(\alpha \in (0,1)\), \(\eta \geq 1\) is the ratio of positive odd integers, and \(^{C}D_{c}^{\alpha}y\) denotes the Caputo fractional derivative of \(y\) of order \(\alpha\). The cases \[y(t)=(a(t)(x^{\prime}(t))^{\eta})^{\prime} \quad \text{and} \quad y(t)=a(t)(x^{\prime}(t))^{\eta}\] are considered. The approach taken here can be applied to other related fractional differential equations. Examples are provided to illustrate the relevance of the results obtained.

This work provides new adequate conditions for difference equations with forcing, positive and negative terms to have non-oscillatory solutions. A few mathematical inequalities and the properties of discrete fractional calculus serve as the fundamental foundation to our approach. To help establish the main results, an analogous representation for the main equation, called a Volterra-type summation equation, is constructed. Two numerical examples are provided to demonstrate the validity of the theoretical findings; no earlier publications have been able to comment on their solutions’ non-oscillatory behavior.

Abstract In this paper, we establish new oscillation criteria for nonlinear second order difference equations with mixed neutral terms. The key idea of our approach is to compare with first order equations whose oscillatory behaviors are already known. The obtained results not only improve and extend existing results reported in the literature but also provide a new platform for the investigation of a wide class of nonlinear second order difference equations. The results are supported by examples to demonstrate the validity of the theoretical findings.

The authors present some new oscillation criteria for third order nonlinear differential equations with a nonlinear nonpositive neutral term. The results obtained are new and improve known oscillation criteria appearing in the literature even for equations not having a neutral term. Using comparison methods and integral conditions, the authors obtain five new theorems on the oscillatory behaviour of the solutions. Suggestions for future research are included.

Abstract In the paper, fourth-order delay differential equations of the form (r3(r2(r1y′)′)′)′(t)+q(t)y(τ(t))=0 $$ \bigl(r_{3} \bigl(r_{2} \bigl(r_{1}y' \bigr)' \bigr)' \bigr)'(t) + q(t) y \bigl( \tau (t) \bigr) = 0 $$ under the assumption ∫t0∞dtri(t)

This survey paper is devoted to succinctly reviewing the recent progress in the field of oscillation theory for linear and nonlinear fractional differential equations. The paper provides a fundamental background for all interested researchers who would like to contribute to this topic.

The authors study the oscillatory behaviors of solutions of higher-order nonlinear differential equations with a nonlinear neutral term. The right hand side of their equation contains both an advanced and a delay term, and either (or both) of them can be sublinear or superlinear. The influence of these terms on the oscillatory and asymptotic behaviors of solutions is investigated by using a comparison to first-order advanced and delay differential equations. New oscillation criteria are presented that improve and extend many known oscillation criteria in the literature. An example is provided to illustrate the results.

The present paper is concerned with the asymptotic behavior of solutions to a class of noncanonical third-order Emden–Fowler delay differential equations with a superlinear neutral term. Using a Riccati-type transformation as well as integral criteria, we establish some new sufficient conditions guaranteeing that every solution of the equation considered either oscillates or converges to zero asymptotically. The results are illustrated with two examples.

The oscillation of even-order nonlinear differential equations (NLDiffEqs) with mixed nonlinear neutral terms (MNLNTs) is investigated in this work. New oscillation criteria are obtained which improve, extend, and simplify the existing ones in other previous works. Some examples are also given to illustrate the validity and potentiality of our results.

The authors present some new oscillation criteria for higher order nonlinear difference equations with nonnegative real coefficients of the form Both of the cases n even and n odd are considered. They give examples to illustrate their results.

The authors present a new technique for the linearization of even-order nonlinear differential equations with a sublinear neutral term. They establish some new oscillation criteria via comparison with higher-order linear delay differential inequalities as well as with first-order linear delay differential equations whose oscillatory characters are known. Examples are provided to illustrate the theorems.

In this paper, we present a systematic study concerning the developments of the oscillation results for the fractional difference equations. Essential preliminaries on discrete fractional calculus are stated prior to giving the main results. Oscillation results are presented in a subsequent order and for different types of equations. The investigation was carried out within the delta and nabla operators.

Abstract The paper is devoted to the study of oscillation of solutions to a class of second-order half-linear neutral differential equations with delayed arguments. New oscillation criteria are established, and they essentially improve the well-known results reported in the literature, including those for non-neutral differential equations. The adopted approach refines the classical Riccati transformation technique by taking into account such part of the overall impact of the delay that has been neglected in the earlier results. The effectiveness of the obtained criteria is illustrated via examples.

In this paper, we are concerned with the oscillation of solutions to a class of third-order nonlinear neutral dynamic equations on time scales. New oscillation criteria are presented by employing the Riccati transformation and integral averaging technique. Two examples are shown to illustrate the conclusions.

In this paper, effective oscillation criteria for third-order delay differential equations of the form, r2r1y′′′(t)+q(t)y(τ(t))=0 ensuring that any nonoscillatory solution tends to zero asymptotically, are established. The results become sharp when applied to a Euler-type delay differential equation and, to the best of our knowledge, improve all existing results from the literature. Examples are provided to illustrate the importance of the main results.

In the paper, we provide sufficient conditions for the oscillatory and asymptotic behavior of a new type of third-order nonlinear dynamic equations with mixed nonlinear neutral terms. Our theorems not only improve and extend existing theorems in the literature but also provide a new approach as far as the nonlinear neutral terms are concerned. The main results are illustrated by some particular examples.

This paper discusses oscillatory and asymptotic properties of solutions of a class of third-order nonlinear neutral differential equations. Some new sufficient conditions for a solution of the equation to be either oscillatory or to converges to zero are presented. The results obtained can easily be extended to more general neutral differential equations as well as to neutral dynamic equations on time scales. Two examples are provided to illustrate the results.

This paper is a continuation of the recent study by Bohner et al [9] on oscillation properties of nonlinear third order functional differential equation under the assumption that the second order differential equation is nonoscillatory. We consider both the delayed and advanced case of the studied equation. The presented results correct and extend earlier ones. Several illustrative examples are included.

New sufficient conditions for the oscillation of all solutions to a class of third-order Emden–Fowler differential equations with unbounded neutral coefficients are established. The criteria obtained essentially improve related results in the literature. In particular, as opposed to known results, new criteria can distinguish solutions of third-order differential equations with different behaviors. Examples are also provided to illustrate the results.

The authors establish some new criteria for the oscillation of solutions of second order nonlinear differential equations with a sublinear neutral term by reducing the equation to a linear one. Their results are illustrated with an example.

This note is concerned with the oscillation of third order nonlinear delay differential equations of the form \[\label{*} \left( r_{2}(t)\left( r_{1}(t)y^{\prime}(t)\right)^{\prime}\right)^{\prime}+p(t)y^{\prime}(t)+q(t)f(y(g(t)))=0.\tag{\(\ast\)}\] In the papers [A. Tiryaki, M. F. Aktas, Oscillation criteria of a certain class of third order nonlinear delay differential equations with damping, J. Math. Anal. Appl. 325 (2007), 54-68] and [M. F. Aktas, A. Tiryaki, A. Zafer, Oscillation criteria for third order nonlinear functional differential equations, Applied Math. Letters 23 (2010), 756-762], the authors established some sufficient conditions which insure that any solution of equation (\(\ast\)) oscillates or converges to zero, provided that the second order equation \[\left( r_{2}(t)z^{\prime }(t)\right)^{\prime}+\left(p(t)/r_{1}(t)\right) z(t)=0\tag{\(\ast\ast\)}\] is nonoscillatory. Here, we shall improve and unify the results given in the above mentioned papers and present some new sufficient conditions which insure that any solution of equation (\(\ast\)) oscillates if equation (\(\ast\ast\)) is nonoscillatory. We also establish results for the oscillation of equation (\(\ast\)) when equation (\(\ast\ast\)) is oscillatory.

In this article, we study the asymptotic behavior of non-oscillatory solutions of second-order integro-dynamic equations as well as the oscillatory behavior of forced second order integro-dynamic equations on time scales. The results are new for the continuous and discrete cases. Examples are provided to illustrate the relevance of the results.

We present some new criteria for the oscillation of even order dynamic equation a(t)(xΔn-1(t))αΔ+q(t)(x(t))α=0, on time scale T, where α is the ratio of positive odd integers a and q is a real valued positive rd-continuous functions defined on T.

We will consider the higher order functional dynamic equations with mixed nonlinearities of the form xnt+∑j=0Npjtϕγjxφjt=0, on an above-unbounded time scale T, where n≥2, xi(t)≔ri(t)ϕαixi-1Δ(t), i=1,…,n-1, with x0=x, ϕβ(u)≔uβsgn⁡u, and α[i,j]≔αi⋯αj. The function φi:T→T is a rd-continuous function such that limt→∞φi(t)=∞ for j=0,1,…,N. The results extend and improve some known results in the literature on higher order nonlinear dynamic equations.

This paper discusses oscillatory and asymptotic properties of solutions of a class of second-order nonlinear damped neutral differential equations. Some new sufficient conditions for any solution of the equation to be oscillatory or to converge to zero are given. The results obtained extend and improve some of the related results reported in the literature. The results are illustrated with examples.

We establish some new criteria for the oscillation of third-order difference equations of the form Δ((1/a2(n))(Δ(1/a1(n))(Δx(n))α1)α2)+δq(n)f(x[g(n)])=0, where Δ is the forward difference operator defined by Δx(n)=x(n+1)−x(n).

We study the oscillation behavior for some higher order integrodynamic equations on timescales. We establish some new sufficient conditions guaranteeing that all solutions of theses equations are oscillatory. Some numerical examples in the continuous case are given to validate the theoretical results.

This paper deals with the oscillatory behavior of forced second-order integrodynamic equations on time scales. The results are new for the continuous and discrete cases and can be applied to Volterra integral equation on time scale. We also provide a numerical example in the continuous case to illustrate the results.

A modified q-homotopy analysis method (mq-HAM) was proposed for solving nth-order nonlinear differential equations. This method improves the convergence of the series solution in the nHAM which was proposed in (see Hassan and El-Tawil 2011, 2012). The proposed method provides an approximate solution by rewriting the nth-order nonlinear differential equation in the form of n first-order differential equations. The solution of these n differential equations is obtained as a power series solution. This scheme is tested on two nonlinear exactly solvable differential equations. The results demonstrate the reliability and efficiency of the algorithm developed.