Solvability for a third order discontinuous fully equation with nonlinear functional boundary conditions

We prove an existence and location result for the third order functional nonlinear boundary value problem

     

Regularity of solutions for a third order differential equation in Hilbert spaces

We study regularity of mild and strong solutions of an abstract mathematical model of a flexible space structure under appropriate initial conditions. We apply our results showing qualitative properties of the trajectories in the case of the negative Laplacian operator.     

On the solution of a third order linear homogeneous difference equation with variable coefficents

A closed form solution of a third order linear homogeneous difference equation with variable coefficients is presented. From it, the solution for the special case of an equation with constant coefficients is also obtained.     

Boundedness results for a certain third order nonlinear differential equation

Sufficient conditions for the existence of solutions to boundedness and ultimate boundedness problems associated to a certain third order nonlinear differential equation are given by means of the Lyapunov’s second method. The appropriate Lyapunov function is given explicitly. Our results complement some well known results on the third order differential equations in the literature.     

Extremality and comparison results for discontinuous implicit third order functional initial-boundary value problems

Extremality results are derived for implicit third order functional initial-boundary value problems. Differential equations, initial conditions and boundary conditions of problems may involve discontinuous nonlinearities.     

Nonoscillation and asymptotic behaviour for third order nonlinear differential equations

In this paper we consider the equation y +q(t)y + p(t)h(y)=0, where p, q are real valued continuous functions on [0, ) such that q(t) 0, p(t) 0 and h(y) is continuous in (–, ) such that h(y)y > 0 for y 0. We obtain sufficient conditions for solutions of the considered equation to be nonoscillatory. Furthermore, the asymptotic behaviour of these nonoscillatory solutions is studied.     

Nonoscillation and oscillation of solutions of a class of third order difference equations

In this paper, sufficient conditions are obtained for nonoscillation/oscillation of all solutions of a class of nonlinear third order difference equations of the form

     

EXISTENCE AND UNIQUENESS FOR THIRD ORDER NONLINEAR BOUNDARY VALUE PROBLEMS

Abstract. In this Paper, the existence and uniqueness of solutions for boundary valueproblem     

Monotonic solutions of a quadratic integral equation of fractional order

In the paper we indicate an error made in the proof of the main result of the paper [M.A. Darwish, On quadratic integral equation of fractional orders, J. Math. Anal. Appl. 311 (2005) 112-119]. Moreover, we provide correct proof of a slightly modified version of the mentioned result. The main tool used in our proof is the technique associated with the Hausdorff measure of noncompactness.     

The third order Melnikov function of a quadratic center under quadratic perturbations

We study quadratic perturbations of the integrable system , where H=(x²+y²)/2. We prove that the first three Melnikov functions associated to the perturbed system give rise at most to three limit cycles.     

一类偏积分微分方程二阶差分全离散格式

本给出了数值求解一类偏积分微分方程的二阶全离散差分格式．采用了Crank-Nicolson格式；积分项的离散利用了Lubieh的二阶卷积积分公式；给出了稳定性的证明，误差估计及收敛性的结果．     

One projection method for linear third order equation

In this paper we study the Galyorkin method with a special basis for a linear operator-differential equation of the third order in a separable Hilbert space. The projection method is based on the eigenvectors of the operator similar to the leading operator of equation. We obtain estimates for the convergence rate of approximate solutions in uniform topology.     

一类偏积分微分方程一阶差分全离散格式

本文给出了数值求解一类偏积分微分方程的一阶差分全离散格式。时间方向采用了一阶向后差分格式，空间方向采用二阶差分格式，给出了稳定性的证明，误差估计及收敛性的结果，并给出了数值例子。     

Qualitative behavior of solutions of a third order nonlinear differential equation

In this paper the oscillatory and asymptotic behavior of solutions of a class of nonlinear third order differential equations of the form are studied. Results on the square integrability of solutions and their derivatives are obtained. An example to illustrate the results is included and directions for future research are indicated.     

Conservation laws for a class of third order evolutionary differential systems

Conservation laws of third order quasi-linear scalar evolution equations are studied via exterior differential system and characteristic cohomology. We find a subspace of 2-forms in the infinite prolongation space in which every conservation law has a unique representative. Analysis of the structure of this subspace based upon the symbol of the differential equation leads to a universal integrability condition for an evolution equation to admit any higher order (weight) conservation laws. As an example, we give a complete classification of a class of evolution equations which admit conservation laws of the first three consecutive weights , , . The differential system describing the flow of a curve in the plane by the derivative of its curvature with respect to the arc length is also shown to exhibit the KdV property, i.e., an infinite sequence of conservation laws of distinct weights.

     

高阶非线性差分方程的全局吸引性

研究了一类高阶非线性差分方程所有正解的周期性,不变区间及全局吸引性.证明了方程的正平衡点是在一个依赖于参数的盆里的全局吸引子.     

Second order linear difference equations

We provide the explicit solution of a general second order linear difference equation via the computation of its associated Green function. This Green function is completely characterized and we obtain a closed expression for it using functions of two–variables, that we have called Chebyshev functions due to its intimate relation with the usual one–variable Chebyshev polynomials. In fact, we show that Chebyshev functions become Chebyshev polynomials if constant coefficients are considered.