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The research and results described in this paper belong to the qualitative theory of differential equations (more precisely, the partial differential equations with the one-dimensional p-Laplacian). Using a method whose core is formed by the Prufer technique, we identify a borderline case between oscillatory and non-oscillatory equations. Moreover, we are able to decide whether the studied equations are oscillatory or not even in the so-called critical (i.e., the borderline) case. The advantage of our approach is the fact that we obtain new and strong results for linear and half-linear equations (i.e., the equations with the one-dimensional p-Laplacian) at the same time. In addition, we are able to work with equations whose coefficients are non-constant and non-periodic. The novelty of our results is documented by examples and corollaries.

This paper mainly deals with the oscillation of nonlinear delay differential equation which is used to describe advertising capital model, analytically and numerically. Firstly, the condition of oscillation of the analytic solution is presented by the technique of the theory of characteristic. Secondly, the asymptotic behavior of non-oscillatory analytic solution is verified. Thirdly, the θ-methods are applied to the mentioned equation, some conditions under which the numerical solution oscillates are obtained. Moreover, it is proved that every non-oscillatory numerical solution tends to the steady state of the model. Finally, some numerical simulations for verifying the theoretical findings are provided.

The objective of this paper is to offer sufficient conditions for the oscillation and asymptotic behavior of all solutions of second-order half-linear differential equations with advanced argument of the form ( r ( y ′ ) α ) ′ ( t ) + q ( t ) y α ( σ ( t ) ) = 0 , when ∫ ∞ r − 1 / α ( s ) d s ∞ . Our criteria substantially improve, simplify and complement a number of existing ones. The results obtained are illustrated by an example on the Euler type equations.

As is well known, Mathieu’s equation is a representative of mathematical models describing parametric excitation phenomena. This paper deals with the oscillation problem for Mathieu’s equation with two frequencies. The ratio of these two frequencies is not necessarily a rational number. When the ratio is an irrational number, the coefficient of Mathieu’s equation is quasi-periodic, but not periodic. For this reason, the basic knowledge for linear periodic systems such as Floquet theory is not useful. Whether all solutions of Mathieu’s equation oscillate or not is determined by parameters and frequencies. Our results provide parametric conditions to guarantee that all solutions are nonoscillatory. The advantage of the obtained parametric conditions is that it can be easily checked. Parametric nonoscillation region is drawn to understand these results easily. Finally, several simulations are carried out to clarify the remaining problems.

A new approach is introduced for the oscillatory solutions of the second order non-linear delay dynamic equations on time scale with a super-linear neutral term. The result are obtained through various integral and differential relations. The Kamenev and Philos-type criteria for oscillation are also discussed. The results improve and extend the ones reported in the literature in the sense that the inequalities are easy to verify and results cover various kinds of equations by choosing different time scales. Examples for particular time scales are given for illustration.

New oscillation criteria for third-order delay differential equations with noncanonical operators are presented. Contrary to existing results, oscillation of the studied equation is attained via only two conditions. Our criteria not only improve, extend and significantly simplify existing ones, but the newly proposed approach could hopefully serve as a reference in the less-developed theory of noncanonical equations of higher-order. The importance of the results obtained is illustrated via Euler-type equations.

In this paper, a partly sharp oscillation criterion is established for the first-order delay differential equation x ′ ( t ) + p ( t ) x ( τ ( t ) ) = 0 , t ≥ T 0 , where p ( t ) is continuous and p ( t ) ≥ 0 for all t ≥ T 0 ; τ ( t ) is continuous, non-decreasing, τ ( t ) t for all t ≥ T 0 , and lim t → ∞ τ ( t ) = + ∞ . By improving techniques from previous researches, we show that when α = lim inf t → ∞ ∫ τ ( t ) t p ( s ) d s ∈ ( 0 , 1 e ] , all solutions of the equation are oscillatory if lim sup t → ∞ ∫ τ ( t ) t p ( s ) d s > 2 α + 2 λ 1 − 1 under no additional constraints, where λ 1 is the smaller root of the equation λ = e α λ . This result is proved to be weaker than previous ones, and sharp when α ∈ ( 0 , ln 2 2 ) by constructing a specific example.

In this article, the fault-tolerant control problem against thruster faults is investigated for unmanned marine vehicles. The main approach is, for the first time, based on integral sliding mode control technique where only measurable outputs of unmanned marine vehicles are used. By using matrix full-rank decomposition and adaptive mechanism, an output integral sliding surface is constructed and an integral sliding mode controller with a full-order compensator is designed to attenuate the oscillation amplitudes of the yaw angle and yaw velocity error in the presence of thruster faults and ocean external disturbances. Through a typical floating production ship, the effectiveness of the proposed method is verified in the simulation example.

We study asymptotic behavior of nonoscillatory solutions for the third-order differential equations of the form [ r ( t ) ( y ′ ( t ) ) γ ] ′ ′ + p ( t ) y ( t ) = 0 . For property A of studied equations we extend the information about the asymptotic properties of positive decreasing solutions.

The paper deals with the oscillation and non-oscillation of the Runge-Kutta methods for a differential equation with piecewise continuous arguments of mixed type. The conditions of the oscillation and non-oscillation for the Runge-Kutta method are obtained. It is proved that oscillation of the analytic solution is not preserved by the Runge-Kutta method under any conditions. The conditions under which the non-oscillation of analytic solutions is preserved by the Runge-Kutta method are obtained. Some numerical experiments are given.

Volterra equations governed by oscillatory functions arise frequently in applied fields. To construct efficient algorithms for solving these equations, oscillatory properties of their solutions should be studied. In this paper, oscillatory orders of solutions to a class of highly oscillatory Volterra integral equations are presented. Then some modified solutions are given with the help of asymptotic expansions of highly oscillatory integrals. Finally, theoretical analysis verifies these modified solutions enjoy less oscillation than the original one, and extensions to highly oscillatory Volterra integro-differential equations are considered.

In the paper, we establish new technique for investigation of properties of trinomial differential equations with positive and negative term ( b ( t ) ( a ( t ) x ′ ( t ) ) ′ ) ′ + p ( t ) f ( x ( τ ( t ) ) ) − q ( t ) h ( x ( σ ( t ) ) ) = 0 . We offer new criteria for asymptotic properties of nonoscillatory solutions for the studied equations. We support our results with illustrative examples.

The effect of non-Gaussian noises on stochastic resonance of intracellular Ca 2 + concentration in intracellular calcium oscillation(ICO) system with time delay is investigated by means of second-order stochastic Runge-Kutta type algorithm. By simulating the signal power amplification(SPA), the results indicate: there are respectively continuous values and a value of the parameter p(which is used to control the degree of the departure from the non-Gaussian noise and Gaussian noise.) to enhance reverse resonance in the behavior of SPA vs. p in cytosol and calcium store, namely continuous reverse resonance occurs in cytosol and reverse resonance occurs in calcium store. Moreover, SPA monotonically decreases as non-Gaussian noises strengthen, and SPA fast decays to constant as correlation time of non-Gaussian noises increases.

In the paper, we offer a qualitatively unimprovable oscillation result for half-linear several delay second-order differential equations, which improves and generalizes the one from the very recent study [8]. The sharpness of our newly obtained criterion is illustrated via Euler-type half-linear several delay differential equations.

We present new oscillation tests for the PΔE a u ( m + 1 , n ) + b u ( m , n + 1 ) − c u ( m , n ) + p ( m , n ) u ( m − τ , n − σ ) = 0 , m , n = 0 , 1 , ⋯ , where a, b, c ∈ (0, ∞) and τ, σ ∈ {0, 1, ⋅⋅⋅}, and {p(m, n)} ⊂ [0, ∞). Our main result improves some of the well-known results in the literature. We also present a numerical example, where all the previous results in the literature fail to deliver an answer.

A new oscillation criterion is established for the first-order delay differential equation x ′ ( t ) + p ( t ) x ( τ ( t ) ) = 0 , t ≥ t 0 , where p ( t ) , τ ( t ) ∈ C ( [ t 0 , + ∞ ) , [ 0 , + ∞ ) ) , τ(t) is non-decreasing, τ(t) lim t → ∞ τ ( t ) = ∞ . By discretizing not only x(t) but also ∫ τ ( t ) t p ( s ) d s , we show that all solutions of the equation are oscillatory if lim sup t → ∞ ∫ τ ( t ) t p ( s ) d s > 1 + ln λ 1 λ 1 − α 2 2 1 − α − α 2 2 + α 3 λ 1 3 ! 1 − α − α 2 2 + α 3 λ 1 3 ! + α 4 λ 1 2 4 ! 1 − α − α 2 2 + α 3 λ 1 3 ! + α 4 λ 1 2 4 ! + α 5 λ 1 3 5 ! 1 − α − ⋱ , ( * ) where α = lim inf t → ∞ ∫ τ ( t ) t p ( s ) d s ∈ ( 0 , 1 e ] and λ1 is the smaller root of the equation λ = e α λ . The right-hand side of (*) is proved to be strictly smaller than previous ones with an illustrative example. In particular, when α = 1 e , our result leads to lim sup t → ∞ ∫ τ ( t ) t p ( s ) d s > 0.5621 .

The purpose of this paper is to present a pair of an oscillation theorem and a nonoscillation theorem for Meissner’s equation which is the special case of Hill’s equation. Proof is given by means of the Riccati technique. Furthermore, using Liouville transformation and Sturm’s comparison theorem, we show a relation between Meissner equations and Cauchy-Euler equations. Moreover, by numerical computations, we give an approximate value which is the borderline between oscillation and nonoscillation for Meissner equations.

The aim of this work is to study the oscillatory properties of the solutions of fourth-order delay differential equations with a p-Laplacian like operator of the form ( r ( ν ) | w ″ ′ ( ν ) | p 1 − 2 w ″ ′ ( ν ) ) ′ + q ( ν ) | w ( η ( ν ) ) | p 2 − 2 w ( η ( ν ) ) = 0 . The results obtained are based on the comparison with first and second-order differential equations and the use of a Riccati transformation. Some examples are provided to illustrate the main results.

This paper is concerned with nth-order nonlinear dynamic equations on time scales of the form ( r ( t ) ? γ ( x Δ n - 1 ( t ) ) ) Δ + ? i = 0 k q i ( t ) ? α i ( x ( ? i ( t ) ) ) = 0 with n ? 2. By discussing the signs of ith-order derivatives of eventually positive solutions for i = 1 , ? , n - 1 , and using the generalized Riccati technique and integral averaging technique, we derive new criteria for oscillation and asymptotic behavior of the equation. Our results extend many existing results in the literature.

In the paper, we establish a variant of Kneser oscillation theorem for the second-order half-linear delay differential equation ( r ( t ) | y ′ ( t ) | α − 1 y ′ ( t ) ) ′ + q ( t ) | y ( τ ( t ) ) | α − 1 y ( τ ( t ) ) = 0 , t ≥ t 0 > 0 , under the assumption ∫ t 0 ∞ r − 1 / α ( s ) d s = ∞ , which improves a plenty of results reported in the literature. In case of α ≥ 1, the obtained criterion is sharp in the sense that the oscillation constant is optimal for the corresponding half-linear Euler type equation with delay, while the result for α

In this paper we consider multistage SIS models of infectious diseases, where infected individuals are passing through infectious stages I 1 , I 2 , ⋯ I n and then return to the susceptible compartment. First we calculate the basic reproduction number R 0 , and prove that the disease dies out for R 0 ≤ 1 , while a unique endemic equilibrium exists for R 0 > 1 . Our main result is that the stability of the endemic equilibrium depends on the number of stages: the endemic equilibrium is always stable when n ≤ 3, while for any n > 3 it can be either stable or unstable, depending on the particular choice of the parameters. We generalize previous stability results for SIRS models as well and point out a mistake in the literature for multistage SEIRS models. Our results have important implications on the discretization of infectious periods with varying infectivity.

By using a generalized Riccati substitution, we present a new criterion for oscillation of solutions of fourth-order quasi-linear differential equations (r(t)(x‴(t))α)′ + f ( t , x ( σ ( t ) ) ) = 0 , in noncanonical case ∫ ∞ r − 1 / α ( s ) d s lim sup ( · ) > 1 ), while condition ( lim sup ( · ) = ∞ ) cannot be applied. Our results essentially improve and complement a number of related ones. Examples are provided to illustrate the results.

This paper involves a new improved method for establishing the oscillation of second order advanced difference equation of the form Δ ( a ( l ) Δ υ ( l ) ) + p ( l ) υ ( σ ( l ) ) = 0 using the difference equation Δ ( a ( l ) Δ υ ( l ) ) + q ( l ) υ ( l + 1 ) = 0 . The results that we obtain are new and improve the existing criteria. We illustrate the results by various examples, at the end of the paper.

In this article, we discuss the oscillation of all solutions of a first order linear delay difference equation with several delay arguments and obtain some new oscillation criteria under more relaxed conditions. In particular, we establish a necessary and sufficient condition only based on the coefficient functions and delay arguments when the coefficient functions are non-negative. Examples and tables are given to illustrate our results.

The paper begins with a novel variational formulation of Duffing equation using the extended framework of Hamilton's principle (EHP). Unlike the original Hamilton's principle, this formulation properly accounts for initial conditions, and it recovers all the governing differential equations as its Euler–Lagrange equation. Thus, it serves base for the development of novel computational methods, involving finite element representation over time. For its feasibility, we provide the simplest temporal finite element method by adopting linear temporal shape functions. Numerical examples are included to verify and investigate performance of non-iterative algorithm in the developed method.

In the paper, we give new criteria for the oscillation of third-order delay differential equations with noncanonical operators. Our results essentially improve and simplify the ones in Dzurina et al. (2018). The results are illustrated by a Euler-type delay differential equation.

In this paper we consider the boundary value problem for the canonical one-dimensional Dirac system and investigate the oscillatory properties of the eigenvector-functions of this problem. We find the number of zeros of components of the eigenvector-functions in dependence on the parameters of boundary conditions and give a refinement the asymptotic formula for the eigenvalues of the considered problem.

By employing the generalized Riccati transformation, we establish some new oscillation criteria for second order delay dynamic equations on time scales. The obtained results essentially improve the well-know oscillation results for half-linear dynamic equations such as Kamenev-type and Philos-type oscillation criteria. Some interesting examples are given to illustrate the versatility of our results.

The study of oscillation theory for fractional differential equations has been initiated by Grace et.al. 14. In this paper we establish some new criteria for the oscillation of fractional differential equations with the Caputo derivative of the form c D a α x ( t ) = e ( t ) + f ( t , x ( t ) ) , a 1 , α ? ( 1 , 2 ) .We also present the conditions under which all solutions of this equation are asymptotic to at + b as t ? ∞ for some real numbers a, b. We shall employ a different technique rather than that in 14.

This paper is concerned with oscillation and non-oscillation of numerical solutions for nonlinear delay differential equation of hematopoiesis with unimodal production rate. By investigating the roots of characteristic equation, some conditions under which the numerical solution is oscillatory and non-oscillatory are obtained. The properties of non-oscillatory numerical solutions are investigated. To verify our results, we give numerical experiments.

We investigate the eventual sign changing for the solutions of the linear equation x ( α ) ' + q ( t ) x = 0 , t ? 0 , when the functional coefficient q satisfies the Kamenev-type restriction lim sup t ? + ∞ 1 t e ? t 0 t ( t - s ) e q ( s ) ds = + ∞ for some e 2 , t 0 0 . The operator x ( α ) is the Caputo differential operator and α ? ( 0 , 1 ) .

Consider the first-order retarded difference equation Δ x ( n ) + p ( n ) x ? ( n ) = 0 , n ? N 0 where ( p ( n ) ) n ? 0 is a sequence of nonnegative real numbers, and ( ? ( n ) ) n ? 0 is a sequence of integers such that ? ( n ) ≤ n - 1 , n ? 0 , and lim n ? ∞ ? ( n ) = ∞ . Under the assumption that the retarded argument is non-monotone, a new oscillation criterion, involving lim inf , is established. An example illustrates the case when the result of the paper implies oscillation while previously known results fail.

This paper is concerned with the distribution of zeros of all solutions of the first-order neutral differential equation x ( t ) + p ( t ) x ( t - ? ) ' + Q ( t ) x ( t - ? ) = 0 , t ? t 0 , where p ? C t 0 , ∞ ) , 0 , ∞ ) , Q ? C t 0 , ∞ ) , ( 0 , ∞ ) and ? , ? ? R + . New estimations for the distance between adjacent zeros of this neutral equation are obtained via comparison with a corresponding differential inequality. These results extend some known results from the non-neutral to the neutral case and improve other published results as well.

In the paper, Sturmian comparison theory is developed for the pair of second order differential equations; first of which is the nonlinear differential equations (1) ( m ( t ) y ' ) ' + s ( t ) y ' + ? i = 1 n q i ( t ) | y | α i - 1 y = 0 , with mixed nonlinearities α 1 ? α m 1 α m + 1 ? α n , and the second is the nonselfadjoint differential equations (2) ( k ( t ) x ' ) ' + r ( t ) x ' + p ( t ) x = 0 . Under the assumption that the solution of Eq. (2) has two consecutive zeros, we obtain Sturm-Picone type and Leighton type comparison theorems for Eq. (1) by employing the new nonlinear version of Picone's formula that we derive. Wirtinger type inequalities and several oscillation criteria are also attained for Eq. (1). Examples are given to illustrate the relevance of the results.

By refining the generalized Riccati transformation technique, we obtain new oscillation criteria for a class of second-order half-linear delay dynamic equations with damping on a time scale. Assumptions in our theorems are less restrictive, these results complement and improve related contributions to the subject. In particular, the results obtained amend related results reported in the literature.

Using a matrix linear transformation which preserves oscillatory property, some new oscillation criteria for the following linear matrix Hamiltonian system U ' = ( A ( t ) - λ ( t ) I ) U + B ( t ) V , V ' = C ( t ) U + ( µ ( t ) I - A ? ( t ) ) V , t ? t 0 are obtained, which improve and generalize some recent results in literature.

This paper presents a new sufficient condition for the oscillation of all solutions of difference equations with several deviating arguments and oscillating coefficients. Corresponding difference equations of both retarded and advanced type are studied. Examples illustrating the results are also given.

The nonlinear oscillation in a compressible air-filled two-dimensional cylindrical resonator driven by a loudspeaker is simulated by using the gas-kinetic scheme. The influences of shock wave and higher harmonic on the time and space distribution of acoustic variables are investigated numerically for the practical applications of high-intensity acoustic devices. The validation of the developed model is verified by comparing the numerical results of pressure distribution with the theoretical ones for the finite-amplitude case. And then, the verified gas-kinetic scheme is used to simulate the acoustic field of highly nonlinear standing wave. Some interesting physical phenomena have been revealed for the highly nonlinear case. Sharp velocity spikes accompanied by the saw-tooth pressure waveforms appear at the end of the resonator. Moreover, the pressure at the position of theoretical pressure node is not zero and its frequency is about twice of the resonance frequency. Furthermore, the second harmonic is predominant at the location of pressure node. And nonlinear saturation can be found in tandem as the shock wave appears. Additionally, quasi-one-dimensional distribution accompanied changing flow direction and annular effect is observed for the spatial distribution of x-velocity. In addition, the y-velocity is in an irregular two-dimensional distribution and the y-velocity is not any more negligible relative to the x-velocity. Meanwhile, the important impacts as well as the causes of these nonlinear phenomena are analyzed. The results demonstrate the gas-kinetic scheme is an efficient and appropriate method for simulation of highly nonlinear acoustic oscillation and concerned problems.

Scheme of Phase-fitted and P-stable symmetric extended linear multistep (SELM) methods.Order conditions for modified linear multistep methods.Practical explicit and implicit SELM methods.Eight-step SELM predictor-corrector method of order ten.Numerical experiments showing the effectiveness of the SELM methods. A novel family of phase-fitted and P-stable symmetric extended linear multistep (SELM) methods for solving initial value problems of second-order oscillatory differential equations are developed. SELM methods are naturally P-stable. The order conditions are presented. Based on the harmonic fitting condition and order conditions, new explicit SELM methods of order two, order four, order six and order eight, respectively and implicit SELM methods of order four, order six, order eight and order ten, respectively, are derived and the local error for each method is obtained. A new eight-step SELM predictor-corrector pair of order ten is obtained. Numerical experiments are accompanied to show the high accuracy and competence of the new SELM methods compared with some highly efficient symmetric multistep methods in the literature.

A new Philos-type theorem for a class of third-order nonlinear dynamic equations is presented. Among others, the restrictive condition in terms of the commutativity of the jump and delay operators is removed here.

In this article, a class of fourth-order difference equations with quasi-differences and a deviating argument is considered. We state new oscillation theorems for the super-linear and for the half-linear case and we complete the existing results in the literature.

A class of second order nonlinear damped differential equations is studied and several new oscillation conditions are obtained. Our results extend, improve and unify a number of existing results and handle the cases which are not covered by known criteria.

In this paper we propose a new approach to replace nonlinear ordinary differential equations by approximate cubic-quintic Duffing oscillators in which its coefficients depend on the initial amplitude of oscillation. It is shown that this procedure leads to angular frequency values with relative errors that are lower than those found by previously developed approximate solutions.

In this paper we study the second-order dynamic equation on the time scale T of the form(r(t)y^@D)^@[email protected](t)[email protected](t)y^@s=0,wherer,qare positive rd-continuous periodic functions with inf{r(t),[email protected]?T}>0 and @c is an arbitrary real constant. This equation corresponds to Euler-type differential (resp. Euler-type difference) equation for continuous (resp. discrete) case. Our aim is to prove that this equation is conditionally oscillatory, i.e., there exists a constant @C>0 such that studied equation is oscillatory for @c>@C and non-oscillatory for @c

Sufficient conditions are established for the oscillation of nth order neutral differential equations of the form r ( t ) x ( t ) | x ( t ) | α - 1 + p ( t ) x ( τ ( t ) ) ( n - 1 ) ′ + q ( t ) f x ( σ ( t ) ) = 0 , t ⩾ t 0 , where n ⩾ 2 is even integer, α ⩾ 1 , p , q ∈ C ( [ t 0 , + ∞ ) , R ) , f ∈ C ( R , R ) . The results obtained extend some of the known results.

Consider the first-order linear delay differential equation(1)x^'(t)+p(t)x(@t(t))=0,t>=t"0,where p,@[email protected]?C([t"0,~),R^+), @t(t)=t"0 and lim"t"->"[email protected](t)=~, and the (discrete analogue) difference equation(1')@Dx(n)+p(n)x(@t(n))=0,n=0,1,2,...,where @D denotes the forward difference operator @Dx(n)=x(n+1)-x(n),p(n) is a sequence of nonnegative real numbers and @t(n) is a sequence of integers such that @t(n)==0 and lim"n"->"[email protected](n)=~. The state-of-the-art on the oscillation of all solutions to these equations are established especially in the case of non-monotone arguments. Examples illustrating the results are given.

In the paper, we present several functional inequalities and offer their application for higher order advanced differential equations of the formx^(^n^)(t)+q(t)x(@s(t))=0,to be oscillatory. The conditions obtained essentially improve, complements and extends many other known results and involves unimprovable constants.

This paper presents a new Kamenev-type oscillation criterion for a second-order advanced dynamic equation on a time scale. Two examples are included to show that the result obtained complements and improves those reported in the literature.

In this paper, we present sufficient conditions for the oscillation and asymptotic behaviour of solutions of delay dynamic equations of the form x ( t ) + A ( t ) x α ( t ) Δ n + B ( t ) x β ( t ) = 0 for t ? t 0 , ∞ T , where n ? N , T is a time scale unbounded above, t 0 ? T , A ? C rd ( t 0 , ∞ ) T , R ) satisfies either A ? C rd ( t 0 , ∞ ) T , 0 , 1 ] R ) with limsup t ? ∞ A ( t ) - 1 , B ? C rd ( t 0 , ∞ ) T , R 0 + ) and α , β ? C rd ( t 0 , ∞ ) T , T ) are unbounded nondecreasing functions such that α ( t ) , β ( t ) ≤ t for all t ? t 0 , ∞ ) T under the condition ? t 0 ∞ B ( ? ) Δ ? = ∞ . .

In this paper we prove a Tauberian-like theorem to retrieve the slow oscillation of a real-valued function s(x) from its (C,1) summability of the generator sequence v(x) and one-sidedly boundedness of the general control modulo of integer order m>=1 with respect to a function satisfying an appropriate condition.