On two classes of autonomous third-order nonlinear ordinary differential equations

Two autonomous, nonlinear, third-order ordinary differential equations whose dynamics can be represented by second-order nonlinear ordinary differential equations for the first-order derivative of the solution are studied analytically and numerically. The analytical study includes both the obtention of closed-form solutions and the use of an artificial parameter method that provides approximations to both the solution and the frequency of oscillations. It is shown that both the analytical solution and the accuracy of the artificial parameter method depend greatly on the sign of the nonlinearities and the initial value of the first-order derivative.     

Remarks on Barbashin-Ezeilo Problem on third-order nonlinear differential equations

In this paper, we use the frequency domain criteria to initiate a general result on the Barbashin-Ezeilo Problem on third-order nonlinear differential equation. Earlier ideas of Burkin [I.M. Burkin, Orbital stability of second-kind limiting cycles for dynamical systems with cylindrical phase space, Differential Equations 29 (1993) 1262-1264], Leonov [G.A. Leonov, A frequency criterion for the existence of limit cycles of dynamical systems with cylindrical phase spaces, Differential Equations 23 (1987) 1375-1378] on the problem are being improved upon.     

Compact almost automorphic solutions for some nonlinear dissipative differential equations in Banach spaces

The aim of this work is to prove the existence and uniqueness of compact almost automorphic solutions for some dissipative differential equations in Banach spaces when the input function is only almost automorphic in the sense of Stepanov. Examples and a numerical simulation are provided to illustrate the theoretical findings.     

Pseudo almost periodic solutions of differential equations with piecewise constant arguments

In this paper, we give a definition of pseudo almost periodic sequence and study the existence of pseudo almost periodic sequence to difference equation. Based on these, we investigate the existence of pseudo almost periodic solutions to differential equations with piecewise constant arguments which was considered by K. L. Cooke & J. Wiener and found applications in certain biomedical problems.     

Existence of almost periodic solutions to some stochastic differential equations

The article introduces and studies the concept of p-mean almost periodicity for stochastic processes. Our abstract results are, subsequently, applied to studying the existence of square-mean almost periodic solutions to some semilinear stochastic equations.     

Periodic solutions of some fourth-order nonlinear differential equations

Using inequality techniques and coincidence degree theory, new results are provided concerning the existence of T-periodic solutions for fourth-order nonlinear differential equations. Two illustrative examples are provided to demonstrate that the results in this paper hold under weaker conditions than the existing ones, and are more effective.     

非线性三阶常微分方程的非线性三点边值问题解的存在性

基于上下解方法,在一定条件下,得到了一类带有非线性混合边界条件的三阶常微分方程的非线性三点边值问题解的存在性,作为上述存在性结果的应用,在推论中给出了一类三阶非线性微分方程三点边值问题解的存在性.     

Contractivity and exponential stability of solutions to nonlinear neutral functional differential equations in banach spaces

A series of contractivity and exponential stability results for the solutions to nonlinear neutral functional differential equations (NFDEs) in Banach spaces are obtained,which provide unified theoretical foundation for the contractivity analysis of solutions to nonlinear problems in functional differential equations (FDEs),neutral delay differential equations (NDDEs) and NFDEs of other types which appear in practice.     

Twin monotone positive solutions to a singular nonlinear third-order differential equation

This paper discusses the existence of at least one or two nondecreasing positive solutions for the following singular nonlinear third-order differential equation

     

Almost periodic and pseudo-almost periodic solutions to fractional differential and integro-differential equations

We study the existence of almost periodic (resp., pseudo-almost periodic) mild solutions for fractional differential and integro-differential equations in the case when the forcing term belongs to the class of Stepanov almost (resp., Stepanov-like pseudo-almost) periodic functions.     

Asymptotically almost periodic solutions of stochastic functional differential equations

In this paper, we investigate a class of stochastic functional differential equations of the form

dx(t)=(Ax(t)+F(t,x(t),x_t))dt+G(t,x(t),x_t)°dW(t).     

中立型二阶非线性微分方程振动性的判据

Abstract. In this paper ,the oscillation criteria for the solutions of the nonlinear differential e-quations of neutral type of the forms:     

Existence of almost periodic solutions for strong stable impulsive differential equations

Conditions for strong stability and the existence of almostperiodic solutions of systems of impulsive differential equationswith impulsive effect at fixed moments are obtained. The investigationsare carried out by means of piecewise continuous functions whichare analogues of Lyapunov functions.     

A Volterra integral formulation for determining the periodic solutions of some autonomous, nonlinear, third-order ordinary differential equations

A Volterra integral formulation based on the introduction of a term proportional to the velocity times the square of the (unknown) frequency of oscillation, a new independent variable equal to the original one times the (unknown) frequency of oscillation, the method of variation of parameters and series expansions of both the solution and the frequency of oscillation, is used to determine the periodic solutions to three nonlinear, autonomous, third-order, ordinary differential equations. It is shown that the first term of the series expansion of the frequency of oscillation coincides with that determined from a first-order harmonic balance procedure, whereas the two-term approximation to the frequency of oscillation is shown to be more accurate than that of a parameter perturbation procedure for the second equation considered in this paper. For the third equation, it is shown that the two-term approximation presented in this paper is more accurate than the corresponding one for one of the parameter perturbation methods, and for initial velocities less than one, for the other parameter perturbation approach.     

Non-positivity and oscillations of solutions of nonlinear stochastic difference equations with state-dependent noise

Non-positivity and oscillations of nonlinear stochastic difference equations are discussed to characterize the long-term behaviour of their solutions in almost sure sense. Moreover, we present results on non-positivity with probability less than one and give numerous examples satisfying our main hypotheses.     

A link of stochastic differential equations to nonlinear parabolic equations

Using Girsanov transformation,we derive a new link from stochastic differential equations of Markovian type to nonlinear parabolic equations of Burgers-KPZ type,in such a manner that the obtained BurgersKPZ equation characterizes the path-independence property of the density process of Girsanov transformation for the stochastic differential equation.Our assertion also holds for SDEs on a connected differential manifold.     

On the existence of solutions for strongly nonlinear differential equations

The objectives of this paper are twofold. Firstly, to prove the existence of an approximate solution in the mean for some nonlinear differential equations, we also investigate the behavior of the class of solutions which may be associated with the differential equation. Secondly, we aim to implement the homotopy perturbation method (HPM) to find analytic solutions for strongly nonlinear differential equations.     

Existence, uniqueness and attractiveness of a pseudo almost automorphic solutions for some dissipative differential equations in Banach spaces

We give sufficient conditions ensuring the existence, uniqueness and global attractiveness of a pseudo compact almost automorphic solution of the following differential equation:

x^′(t)=f(t,x(t))