Oscillation of second‐order neutral differential equations

We study oscillatory behavior of a class of second‐order neutral differential equations under the assumptions that allow applications to differential equations with both delayed and advanced arguments, and not only. New theorems complement and improve a number of results reported in the literature. Illustrative examples are provided.     

Positive Solutions of One-Dimensional <Emphasis Type="Italic">p</Emphasis>-Laplacian Boundary Value Problems for Fourth-Order Differential Equations with Deviating Arguments

This paper considers the existence of positive solutions of four-point boundary value problems for fourth-order ordinary differential equations with deviating arguments and p-Laplacian. We discuss such problems in the cases when the deviating arguments are delayed or advanced, what may concern optimization issues related to some technical problems. To obtain the existence results, a fixed point theorem for cones due to Avery and Peterson is applied. According to the Author’s knowledge, the results are new. It is a first paper where a fixed point theorem for cones is applied to fourth-order differential equations with deviating arguments and p-Laplacian. An example is included to verify the theoretical results.     

Fite–Hille–Wintner-type oscillation criteria for second-order half-linear dynamic equations with deviating arguments

We study oscillatory behavior of solutions to a class of second-order half-linear dynamic equations with deviating arguments under the assumptions that allow applications to dynamic equations with delayed and advanced arguments. Several improved Fite–Hille–Wintner-type criteria are obtained that do not need some restrictive assumptions required in related results. Illustrative examples and conclusions are presented to show that these criteria are sharp for differential equations and provide sharper estimates for oscillation of corresponding $q$-difference equations.     

Oscillation and Nonoscillation in Delay or Advanced Differential Equations and in Integrodifferential Equations

Some new oscillation and nonoscillation criteria are given for linear delay or advanced differential equations with variable coefficients and not (necessarily) constant delays or advanced arguments. Moreover, some new results on the existence and the nonexistence of positive solutions for linear integrodifferential equations are obtained.     

Multiple Solutions of Boundary-Value Problems for Fourth-Order Differential Equations with Deviating Arguments

This paper considers fourth-order differential equations with four-point boundary conditions and deviating arguments. We establish sufficient conditions under which such boundary-value problems have positive solutions. We discuss such problems in the cases when the deviating arguments are delayed or advanced. In order to obtain the existence of at least three positive solutions, we use a fixed-point theorem due to Avery and Peterson. To the authors’ knowledge, this is a first paper where the existence of positive solutions of boundary-value problems for fourth-order differential equations with deviating arguments is discussed.     

A pair of difference differential equations of Euler-Cauchy type

We study two classes of linear difference differential equations analogous to Euler-Cauchy ordinary differential equations, but in which multiple arguments are shifted forward or backward by fixed amounts. Special cases of these equations have arisen in diverse branches of number theory and combinatorics. They are also of use in linear control theory. Here, we study these equations in a general setting. Building on previous work going back to de Bruijn, we show how adjoint equations arise naturally in the problem of uniqueness of solutions. Exploiting the adjoint relationship in a new way leads to a significant strengthening of previous uniqueness results. Specifically, we prove here that the general Euler-Cauchy difference differential equation with advanced arguments has a unique solution (up to a multiplicative constant) in the class of functions bounded by an exponential function on the positive real line. For the closely related class of equations with retarded arguments, we focus on a corresponding class of solutions, locating and classifying the points of discontinuity. We also provide an explicit asymptotic expansion at infinity.

     

Oscillation analysis of numerical solutions in the θ‐methods for differential equation of advanced type

The paper deals with the oscillation analysis of numerical solution in the θ‐methods for differential equations with piecewise constant arguments of advanced type. The conditions of the oscillation for the θ‐method are obtained. It is proved that the oscillation of the analytic solution is preserved by the θ‐ method. Some numerical experiments are given. Copyright © 2015 John Wiley & Sons, Ltd.     

Oscillatory and asymptotic behavior of strongly superlinear differential equations with deviating arguments

Summary This paper deals with the oscillation and the asymptotic behavior of the solutions of superlinear differential equations with general (retarded, advanced or mixced type) deviating arguments. The equations considered involve a damping term. The results obtained extend known fundamental oscillation criteria for superlinear differential equations without damping terms and especially the recent basic reults of Kitamura and Kusano [5], and Staikos [19, 20].     

Oscillatory properties of first order linear functional differential equations

Oscillatory properties of retarded and advanced functional differential equations are investigated.In the first part, the study concerns equations with piecewise constant arguments, which found applications in certain biomedical problems. Then, results of some authors are generalized for general equations with many argument deviations. Finally, applications are given to equations with linear transformations of the argument.     

A continuous method for nonlocal functional differential equations with delayed or advanced arguments

In the previous works, the authors presented the reproducing kernel method (RKM) for solving various differential equations. However, to the best of our knowledge, there exist no results for functional differential equations. The aim of this paper is to extend the application of reproducing kernel theory to nonlocal functional differential equations with delayed or advanced arguments, and give the error estimation for the present method. Some numerical examples are provided to show the validity of the present method.     

Oscillatory and periodic solutions of differential equations with piecewise constant generalized mixed arguments

We study scalar advanced and delayed differential equations with piecewise constant generalized arguments, in short DEPCAG of mixed type, that is, the arguments are general step functions. It is shown that the argument deviation generates, under certain conditions, oscillations of the solutions, which is an impossible phenomenon for the corresponding equation without the argument deviations. Criteria for existence of periodic solutions of such equations are discussed. New criteria extend and improve related results reported in the literature. The efficiency of our criteria is illustrated via several numerical examples and simulations.     

Comparison theorems on the oscillation of even order nonlinear mixed neutral differential equations

This study purposes to present some new comparison theorems that guarantee the oscillation of all solutions of even order functional differential equations with a neutral term including both advanced and delayed arguments. The obtained results are based on comparisons with associated first-order delay differential inequalities and first-order delay differential equations, and they are applicable to the both cases where the neutral coefficients are unbounded and/or bounded. In the last section, several examples showing the applicability of the results are provided.     

Existence of positive solutions to third order differential equations with advanced arguments and nonlocal boundary conditions

We use a fixed point theorem due to Avery and Peterson to establish the existence of at least three non-negative solutions of some nonlocal boundary value problems to third order differential equations with advanced arguments. An example is given to illustrate the main results.     

Positive solutions for fourth-order differential equations with deviating arguments and integral boundary conditions

In this paper we investigate integral boundary value problems for fourth order differential equations with deviating arguments. We discuss our problem both for advanced or delayed arguments. We establish sufficient conditions under which such problems have positive solutions. To obtain the existence of multiple (at least three) positive solutions, we use a fixed point theorem due to Avery and Peterson. An example is also included to illustrate that corresponding assumptions are satisfied. The results are new.     

Oscillation theory of first order functional differential equations with deviating arguments

Summary New oscillation criteria are established for the first order functional differential equation (*) y'(t)+p(t)y(g(t))=0and its nonlinear analogue. The results are presented so that a remarkable duality existing between the case where (*) is retarded (g(t)t) is apparent. Possible extension of the results for (*) to equations with several deviating arguments is attempted. Finally, it is shown that there exists a class of autonomous equations for which the oscillation situation can be completely characterized.     

Coupled fixed points of multivalued operators and first-order ODEs with state-dependent deviating arguments

We establish a coupled fixed point theorem for a meaningful class of mixed monotone multivalued operators, and then we use it to derive some results on the existence of quasisolutions and unique solutions to first-order functional differential equations with state-dependent deviating arguments. Our results are very general and can be applied to functional equations featuring discontinuities with respect to all of their arguments, but we emphasize that they are new even for differential equations with continuously state-dependent delays.     

Second Order Scalar Functional Differential Equations with Piecewise Constant Arguments

The study of functional differential equations with piecewise constant arguments usually results in a study of certain related difference equations. In this paper we consider certain neutral functional differential equations of this type and the associated difference equations. We give conditions under which such equations with almost periodic time dependence will have unique almost periodic solutions, and for certain autonomous cases, we obtain certain stability results and also conditions for chaotic behavior of solutions. We are particularly concerned with such equations which are partially discretized versions of non-forced Duffing equations.     

Solving 2D time‐fractional diffusion equations by a pseudospectral method and Mittag‐Leffler function evaluation

Two‐dimensional time‐fractional diffusion equations with given initial condition and homogeneous Dirichlet boundary conditions in a bounded domain are considered. A semidiscrete approximation scheme based on the pseudospectral method to the time‐fractional diffusion equation leads to a system of ordinary fractional differential equations. To preserve the high accuracy of the spectral approximation, an approach based on the evaluation of the Mittag‐Leffler function on matrix arguments is used for the integration along the time variable. Some examples along with numerical experiments illustrate the effectiveness of the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd.     

向前型分段连续微分方程的数值解

本文讨论了向前型分段连续微分方程Euler-Maclaurin方法的收敛性和稳定性,给出了Euler-Maclaurin方法的稳定条件,证明了方法的收敛阶是2n+2,并且得到了数值解稳定区域包含解析解稳定区域的条件,最后给出了一些数值例子用以验证本文结论的正确性.     

Runge–Kutta methods for first-order periodic boundary value differential equations with piecewise constant arguments

In this paper we deal with the numerical solutions of Runge–Kutta methods for first-order periodic boundary value differential equations with piecewise constant arguments. The numerical solution is given by the numerical Green’s function. It is shown that Runge–Kutta methods preserve their original order for first-order periodic boundary value differential equations with piecewise constant arguments. We give the conditions under which the numerical solutions preserve some properties of the analytic solutions, e.g., uniqueness and comparison theorems. Finally, some experiments are given to illustrate our results.