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We provide sufficient conditions that ensure oscillations and nonoscillations for nonautonomous impulsive differential equations with piecewise constant arguments of a generalized type. We cover several cases of differential equations with deviated arguments investigated before as particular cases. [ABSTRACT FROM AUTHOR]

In this paper, we proposed a trigonometrically-fitted fifth order four-step predictor-corrector method based on the four-step Adams-Bashforth method as predictor and five-step Adams- Moulton method as corrector to solve linear ordinary differential equations with oscillatory solutions. This method is constructed which exactly integrate initial value problems whose solutions can be expressed as linear combinations of the set functions fsin(vx); cos(vx)g with v E R, where v represents an approximation of the frequency of the problem. The frequency will be used in the method to raise the accuracy of the solution. Stability of the proposed method is examined and the corresponding region of stability is depicted. The new fifth algebraic order trigonometrically-fitted predictor-corrector method is applied to solve the initial value problems whose solutions involved trigonometric functions. Numerical results presented proved that the prospective method is more efficient than the widely used methods for the numerical solution of linear ordinary differential equations with oscillating solutions. [ABSTRACT FROM AUTHOR]

In this article, we examine the first-order retarded difference equation ∆v(n) + m ∑i=1pi(n)v (δi(n)) = 0, n ≥ 0, where (δi(n))(i = 1, . . ., m) are sequences of positive real numbers such that δi(n) ≤ n for n ≥ 0, and limn→∞ δi(n) = ∞ and (pi(n))(i = 1, . . ., m) are sequences of nonnegative real numbers, When the delay terms δi(n) are not necessarily monotone, we will give an oscillation criterion obtained for the differential equation, but not yet obtained for its discrete analogous. Finally, we give an example to illustrate our result. [ABSTRACT FROM AUTHOR]

This work is aimed at studying some comparison and oscillation properties of boundary value problems (BVP’s) of a new type, which differ from classical problems in that they are defined on two disjoint intervals and include additional transfer conditions that describe the interaction between the left and right intervals. This type of problems we call boundary value-transmission problems (BVTP’s). The main difficulty arises when studying the distribution of zeros of eigenfunctions, since it is unclear how to apply the classical methods of Sturm’s theory to problems of this type. We established new criteria for comparison and oscillation properties and new approaches used to obtain these criteria. The obtained results extend and generalizes the Sturm’s classical theorems on comparison and oscillation.

Abstract In this work, we study the oscillatory behavior of even-order neutral delay differential equations υ n ( l ) + b ( l ) u ( η ( l ) ) = 0 $\upsilon ^{n}(l)+b(l)u(\eta (l))=0$ , where l ≥ l 0 $l\geq l_{0}$ , n ≥ 4 $n\geq 4$ is an even integer and υ = u + a ( u ∘ μ ) $\upsilon =u+a ( u\circ \mu ) $ . By deducing a new iterative relationship between the solution and the corresponding function, new oscillation criteria are established that improve those reported in (T. Li, Yu.V. Rogovchenko in Appl. Math. Lett. 61:35–41, 2016).

Abstract In this paper, we study even-order DEs where we deduce new conditions for nonexistence Kneser solutions for this type of DEs. Based on the nonexistence criteria of Kneser solutions, we establish the criteria for oscillation that take into account the effect of the delay argument, where to our knowledge all the previous results neglected the effect of the delay argument, so our results improve the previous results. The effectiveness of our new criteria is illustrated by examples.

Abstract In this paper, we present a new sufficient condition for the oscillation of all solutions of linear delay differential equations. The obtained result improves known conditions in the literature. We also give an example to illustrate the applicability and strength of the obtained condition over known ones.

This paper deals with the oscillatory results of first order nonlinear delay differential equations with several deviating arguments by employing an iterative process. The results presented here has improved the outcomes of [1, 2, 8]. Various examples are solved in MATLAB software to illustrate the relevance of the main results. [ABSTRACT FROM AUTHOR]

Abstract 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.

In this paper we prove the existence and the multiplicity of radial positive oscillatory solutions for a nonlinear problem governed by the mean curvature operator in the Lorentz-Minkowski space. The problem is set in an N-dimensional ball and is subject to Neumann boundary conditions. The main tool used is the shooting method for ODEs.

The solution of fractional oscillation equations exhibits highly oscillatory behavior. It is a challenging work to find efficient numerical methods for fractional oscillation equations. The objective of this paper is to develop an accurate numerical technique for fractional oscillation equations with oscillatory solutions. The present method combines the variation‐of‐constants formula with the reproducing kernel function approximation. The merit of our approach is that it can preserve the oscillatory structure of the solution to the considered fractional initial value problems. Illustrative examples clearly show the reliability and efficiency of the approach. [ABSTRACT FROM AUTHOR]

First-order compartment models are common tools for modelling many biological processes, including pharmacokinetics. Given the compartments and the transfer rates, solutions for the time-dependent quantity (or concentration) curves can normally be described by a sum of exponentials. This paper investigates cases that go beyond simple sums of exponentials. With specific relations between the transfer rate constants, two exponential rate constants can be equal, in which case the normal solution cannot be used. The conditions for this to occur are discussed, and advice is provided on how to circumvent these cases. An example of an analytic solution is given for the rare case where an exact equality is the expected result. Furthermore, for models with at least three compartments, cases exist where the solution to a real-valued model involves complex-valued exponential rate constants. This leads to solutions with an oscillatory element in the solution for the tracer concentration, i.e., there are cases where the solution is not a simple sum of (real-valued) exponentials but also includes sine and cosine functions. Detailed solutions for three-compartment cases are given. As a tentative conclusion of the analysis, oscillatory solutions appear to be tied to cases with a cyclic element in the model itself. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a class of 4th $4{th}$-order neutral delay differential equations with continuously distributed delay is studied. We establish a new oscillation criterion using the Riccati transformation. An example illustrating the results is also given.

Abstract This paper deals with the oscillation properties of higher-order nonlinear differential equations with distributed delay [b(ℓ)(y(n−1)(ℓ))γ]′+∫cdq(ℓ,ξ)yγ(g(ℓ,ξ))d(ξ)=0,ℓ≥ℓ0, $$ \bigl[ b ( \ell ) \bigl( y^{ ( n-1 ) } ( \ell ) \bigr) ^{\gamma} \bigr] ^{\prime}+ \int_{c}^{d}q ( \ell, \xi ) y^{\gamma} \bigl( g ( \ell, \xi ) \bigr) \,d ( \xi ) =0, \quad \ell\geq\ell_{0}, $$ under the condition ∫ℓ0∞1b1γ(ℓ)dℓ

This work is aimed at studying some comparison and oscillation properties of boundary value problems (BVP's) of a new type, which differ from classical problems in that they are defined on two disjoint intervals and include additional transfer conditions that describe the interaction between the left and right intervals. This type of problems we call boundary value-transmission problems (BVTP's). The main dificulty arises when studying the distribution of zeros of eigenfunctions, since it is unclear how to apply the classical methods of Sturm's theory to problems of this type. We established new criteria for comparison and oscillation properties and new approaches used to obtain these criteria. The obtained results extend and generalizes the Sturm's classical theorems on comparison and oscillation. [ABSTRACT FROM AUTHOR]

In this work, we study the oscillatory behavior of even-order neutral delay differential equations υ n (l) + b (l) u (η (l)) = 0 , where l ≥ l 0 , n ≥ 4 is an even integer and υ = u + a (u ∘ μ) . By deducing a new iterative relationship between the solution and the corresponding function, new oscillation criteria are established that improve those reported in (T. Li, Yu.V. Rogovchenko in Appl. Math. Lett. 61:35–41, 2016). [ABSTRACT FROM AUTHOR]

In this paper, we study even-order DEs where we deduce new conditions for nonexistence Kneser solutions for this type of DEs. Based on the nonexistence criteria of Kneser solutions, we establish the criteria for oscillation that take into account the effect of the delay argument, where to our knowledge all the previous results neglected the effect of the delay argument, so our results improve the previous results. The effectiveness of our new criteria is illustrated by examples. [ABSTRACT FROM AUTHOR]

In this work, some new oscillation criteria are established for a third order nonlinear mixed neutral dynamic equation. Our results improve and extend some known results in the literature. Several examples are given to illustrate the importance of the results. [ABSTRACT FROM AUTHOR]

In this article, we study the existence of monotone bounded solutions and of oscillatory solutions to a second-order differential equation with asymptotic conditions. Such asymptotic conditions arise in the study of the ocean flow in arctic gyres. Our approach relies on functional-analytic techniques.

In this paper, we present a new sufficient condition for the oscillation of all solutions of linear delay differential equations. The obtained result improves known conditions in the literature. We also give an example to illustrate the applicability and strength of the obtained condition over known ones. [ABSTRACT FROM AUTHOR]

The oscillation of the first-order linear difference equations with several non-monotone deviating arguments and nonnegative coefficients is investigated, using an iterative procedure. The conditions obtained by this method achieve a marked improvement on all known conditions in the literature. Examples, numerically solved in MATLAB, are also given to illustrate the applicability and strength of the results. [ABSTRACT FROM AUTHOR]

Fourth‐order advanced differential equations naturally appear in models concerning physical, biological, chemical phenomena applications. The aim of this paper is to study the oscillatory properties of fourth‐order advanced differential equations with p‐Laplacian like operators. We obtain some oscillation criteria for the equation by the theory of comparison. The obtained results improve the well‐known oscillation results in the literature. Some examples to illustrate the results are given. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate oscillatory properties of even-order advanced differential equations. The key idea of our approach is to conduct a comparison with first order equations and use the Riccati transformation technique. Some new oscillation criteria are shown. Two examples are presented in order to clarify the main results. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Criteria for oscillatory behavior of solutions of fourth order half-linear differential equations of the form... where > 0 is a constant and q(t) is positive continuous function on [a,1), are given in terms of an increasing continuously differentiable function !(t) from [a,1) to (0,1) which satisfies... [ABSTRACT FROM AUTHOR]

This paper deals with the oscillation of the fourth-order delay differential equation. New oscillation criteria are obtained by employing a refinement of the generalized Riccati transformations and new comparison principles. An example is included to illustrate the main results. [ABSTRACT FROM AUTHOR]

In this article, studied the properties of the oscillation of fractional difference equations, and we obtain some results. The results we obtained are an expansion and further development of highly known results. Then we showed them with examples. [ABSTRACT FROM AUTHOR]

One of the well-known schemes for the direct numerical integration of second-order initial-value problems is due to Falkner. This paper focuses on the construction of a family of adapted block Falkner methods which are frequency dependent for the direct numerical solution of second-order initial value problems with oscillatory solutions. The techniques of collocation and interpolation are adopted here to derive the new methods. The study of the properties of the proposed adapted block Falkner methods reveals that they are consistent and zero-stable, and thus, convergent. Furthermore, the stability analysis and the algebraic order conditions of the proposed methods are established. As may be seen from the numerical results, the resulting family is efficient and competitive compared to some recent methods in the literature.

Some new oscillatory and asymptotic properties of solutions of neutral differential equations with odd-order are established. Through the new results, we give sufficient conditions for the oscillation of all solutions of the studied equations, and this is an improvement of the relevant results. The efficiency of the obtained criteria is illustrated via example.

In this work, we study the oscillatory behavior of a class of fourth-order differential equations. New oscillation criteria were obtained by employing a refinement of the Riccati transformations. The new theorems complement and improve a number of results reported in the literature. An example is provided to illustrate the main results.

The aim of this work is to study oscillatory properties of a class of fourth-order delay differential equations. New oscillation criteria are obtained by using generalized Riccati transformations. This new theorem complements and improves a number of results reported in the literature. Some examples are provided to illustrate the main results.

In this paper, we deal with the oscillation of fourth-order nonlinear advanced differential equations of the form r t y ‴ t α ′ + p t f y ‴ t + q t g y σ t = 0 . We provide oscillation criteria for this type of equations, and examples to illustrate the criteria.

In this paper, we obtain necessary and sufficient conditions for a Kamenev-type oscillation criterion of a fourth order differential equation of the form r 3 t r 2 t r 1 t y ′ t ′ ′ ′ + q t f y σ t = 0 , where t ≥ t 0 . The results presented here complement some of the known results reported in the literature. Moreover, the importance of the obtained conditions is illustrated via some examples.

This work is concerned with the oscillatory behavior of solutions of even-order neutral differential equations. By using Riccati transformation and the integral averaging technique, we obtain a new oscillation criteria. This new theorem complements and improves some known results from the literature. An example is provided to illustrate the main results.

Sufficient conditions for the monotonicity of the sequences of the absolute values of all local extrema of components of a two-dimensional systems are obtained.

Consider the delay difference equation with continuous time of the form \[x(t)-x(t-1)+\sum_{i=1}^mP_i(t)x(t-k_i(t))=0,\qquad t\ge t_0,\] where $P_i\colon[t_0,\infty)\mapsto\mathbb{R}$, $k_i\colon[t_0,\infty)\mapsto \{2,3,4,\dots\}$ and $\lim_{t\to\infty}(t-k_i(t))=\infty$, for $i=1,2,\dots,m$. We introduce the generalized characteristic equation and its importance in oscillation of all solutions of the considered difference equations. Some results for the existence of positive solutions of considered difference equations are presented as the application of the generalized characteristic equation.

An iterative procedure is used to establish the oscillatory sufficient conditions of first-order linear difference equations with several non-monotone deviating arguments and nonnegative coefficients. The conditions obtained by this method, achieve a marked improvement on all known conditions in the literature. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to derive sufficient conditions for the oscillation of all solutions of a difference equation with several non-monotone deviating arguments and nonnegative coefficients. Corresponding difference equations of both retarded and advanced type are studied. Examples illustrating the significance of the results are also given. [ABSTRACT FROM AUTHOR]

Our aim in the present paper is to employ the Riccatti transformation which differs from those reported in some literature and comparison principles with the second-order differential equations, to establish some new conditions for the oscillation of all solutions of fourth-order differential equations. Moreover, we establish some new criterion for oscillation by using an integral averages condition of Philos-type, also Hille and Nehari-type. Some examples are provided to illustrate the main results.