Periodic Solutions for a Class of Neutral Functional Differential Equations with Distributed and Discrete Delays

Based on Krasnoselskii's fixed point theorem,matrix measure and functional analysis methods,some new sufficient conditions for the existence of periodic solutions of neutral functional differential equations with distributed and discrete delays are obtained. Moreover,we construct an example to illustrate the feasibility of our results.

Oscillation Criteria for Even Order Functional Differential Equations with Damped Term

This paper investigates a class of even order functional differential equations with damped term,and derives two new oscillatory criteria of solution.

Oscillation of Second Order Delay Differential Equations with Nonlinear Neutral Term

By using the averaging technique, we obtain new oscillation criteria for second order delay differential equation with nonlinear neutral term. These results generalize and improve some known results about neutral delay differential equation of second order.

Generalized Higher-Order Algebraic Differential Equations with Admissible Algebroid Solutions

Using the Nevanlinna theory of the value distribution of meromorphic functions, we investigate the existence problem of admissible algebroid solutions of generalized higher order algebraic differential equations.

On the Growth of Solutions of a Class of Higher Order Linear Differential Equations with Meromorphic Coefficients

In this paper, we investigate the growth of solutions of higher order linear differential equations with meromorphic coefficients. Under certain conditions, we obtain precise estimation of growth order and hyper-order of solutions of the equation.

Existence of positive solutions for fourth order singular differential equations with Sturm-Liouville boundary conditions

By using the upper and lower solutions method and fixed point theory,we investigate a class of fourth-order singular differential equations with the Sturm-Liouville Boundary conditions.Some sufficient conditions are obtained for the existence of C2[0,1] positive solutions and C3[0,1] positive solutions.

Growth of Solutions to Higher Order Differential Equation with Meromorphic Coefficients

In this paper, we study the growth of solutions of higher order differential equation with meromorphic coefficients, and find some conditions which guarantee that its every nontrivial solution is of infinite order.

FORCED OSCILLATIONS OF BOUNDARY VALUE PROBLEMS OF HIGHER ORDER FUNCTIONAL PARTIAL DIFFERENTIAL EQUATIONS

In this paper we study the forced oscillations of boundary value problems of a class of higher order functional partial differential equations.The principal tool is an everaging techniqe which enables one to establish oscillation in terms of related functional differential inequallities.

On the Stability of Solutions of Nonlinear Functional Differential Equation of the Fifth-Order

The main purpose of this paper is to investigate global asymptotic stability of the zero solution of the fifth-order nonlinear delay differential equation on the following form By constructing a Lyapunov functional, sufficient conditions for the stability of the zero solution of this equation are established.

Extreme Stability of Functional Differential Equations with Finite Delay

This paper obtained some theorems that can ascertain the zero solution of functional differential equations are extremely uniformly stable, extremely asymptotically stable or extremely uniformly asymptotically stable. In the obtained theorems, the derivative of Liapunov function on t along the solutions of functional differential equations is not required to be always negative, especially, it may be even positive.

Boundedness and Oscillation of Third Order Neutral Differential Equations with Deviating Arguments

we consider the third-order neutral functional differential equations with deviating arguments. A new theorem is presented that improves a number of results reported in the literature. Examples are included to illustrate new results.

Global Solutions and Exponential Stability of Stochastic Functional Differential Equations with Infinite Delay

This paper proves that, under the local Lipschitz condition, the stochastic functional differential equations with infinite delay have global solutions without the linear growth condition. Furthermore, the pth moment exponential stability conditions are given. Finally, one example is presented to illustrate our theory.

GROWTH OF SOLUTIONS OF THREE ORDER DIFFERENTIAL EQUATIONS

In this paper,the precise estimation of the order and hyper-order of solutions of a class of three order homogeneous and non-homogeneous linear differential equations are obtained.The results of M.Ozawa(1980),G.Gundersen(1988) and J.K.Langley(1986) are improved.

AN INTEGRATION METHOD WITH FITTING CUBIC SPLINE FUNCTIONS TO A NUMERICAL MODEL OF 2ND-ORDER SPACE-TIME DIFFERENTIAL REMAINDER——FOR AN IDEAL GLOBAL SIMULATION CASE WITH PRIMITIVE ATMOSPHERIC EQUATIONS

In this paper,the forecasting equations of a 2nd-order space-time differential remainder are deduced from the Navier-Stokes primitive equations and Eulerian operator by Taylor-series expansion.Here we introduce a cubic spline numerical model(Spline Model for short),which is with a quasi-Lagrangian time-split integration scheme of fitting cubic spline/bicubic surface to all physical variable fields in the atmospheric equations on spherical discrete latitude-longitude mesh.A new algorithm of"fitting cubic spline—time step integration—fitting cubic spline—……"is developed to determine their first-and2nd-order derivatives and their upstream points for time discrete integral to the governing equations in Spline Model.And the cubic spline function and its mathematical polarities are also discussed to understand the Spline Model’s mathematical foundation of numerical analysis.It is pointed out that the Spline Model has mathematical laws of"convergence"of the cubic spline functions contracting to the original functions as well as its 1st-order and 2nd-order derivatives.The"optimality"of the 2nd-order derivative of the cubic spline functions is optimal approximation to that of the original functions.In addition,a Hermite bicubic patch is equivalent to operate on a grid for a 2nd-order derivative variable field.Besides,the slopes and curvatures of a central difference are identified respectively,with a smoothing coefficient of 1/3,three-point smoothing of that of a cubic spline.Then the slopes and curvatures of a central difference are calculated from the smoothing coefficient 1/3 and three-point smoothing of that of a cubic spline,respectively.Furthermore,a global simulation case of adiabatic,non-frictional and"incompressible"model atmosphere is shown with the quasi-Lagrangian time integration by using a global Spline Model,whose initial condition comes from the NCEP reanalysis data,along with quasi-uniform latitude-longitude grids and the so-called"shallow atmosphere"Navier-Stokes primitive equations in the spherical coordinates.The Spline Model,which adopted the Navier-Stokes primitive equations and quasi-Lagrangian time-split integration scheme,provides an initial ideal case of global atmospheric circulation.In addition,considering the essentially non-linear atmospheric motions,the Spline Model could judge reasonably well simple points of any smoothed variable field according to its fitting spline curvatures that must conform to its physical interpretation.

The Second-Order Differential Equation System with the Feedback Controls for Solving Convex Programming

In this paper, we establish the second-order differential equation system with the feedback controls for solving the problem of convex programming. Using Lagrange function and projection operator, the equivalent operator equations for the convex programming problems under the certain conditions are obtained. Then a second-order differential equation system with the feedback controls is constructed on the basis of operator equation. We prove that any accumulation point of the trajectory of the second-order differential equation system with the feedback controls is a solution to the convex programming problem. In the end, two examples using this differential equation system are solved. The numerical results are reported to verify the effectiveness of the second-order differential equation system with the feedback controls for solving the convex programming problem.

Subspaces for weak mild solutions of the second order abstract differential equation

The topic on the subspaces for the polynomially or exponentially bounded weak mild solutions of the following abstract Cauchy problem d^2/（dr^2）u（t,x）=Au（t,x）;u（0,x）=x,d/（dt）u（0,x）=0,x∈X is studied, where A is a closed operator on Banach space X. The case that the problem is ill-posed is treated, and two subspaces Y（A, k） and H（A, ω） are introduced. Y（A, k） is the set of all x in X for which the second order abstract differential equation has a weak mild solution v（ t, x） such that ess sup{（1＋t）^-k｜d/（dt）〈v（t,x）,x^＊〉｜：t≥0,x^＊∈X^＊,｜x^＊‖≤1}〈＋∞. H（A, ω） is the set of all x in X for which the second order abstract differential equation has a weak mild solution v（t,x）such that ess sup{e^-ωl｜d/（dt）〈v（t,x）,x^＊）｜：t≥0,x^＊∈X^＊,‖x^＊‖≤1}〈＋∞. The following conclusions are proved that Y（A, k） and H（A, ω） are Banach spaces, and both are continuously embedded in X; the restriction operator A ｜ Y（A,k） generates a once-integrated cosine operator family { C（t） }t≥0 such that limh→0＋^-1/h‖C（t＋h）-C（t）‖Y（A,k）≤M（1＋t）^k,arbitary t≥0; the restriction operator A ｜H（A,ω） generates a once- integrated cosine operator family {C（t）}t≥0 such that limh→0＋^-1/h‖C（t＋h）-C（t）‖H（A,ω）≤≤Me^ωt,arbitary t≥0.

ON STABILITY OF SOLUTIONS OF CERTAIN FOURTH-ORDER DELAY DIFFERENTIAL EQUATIONS

By the use of the Liapunov functional approach, a new result is obtained to ascertain the asymptotic stability of zero solution of a certain fourth-order non-linear differential equation with delay. The established result is less restrictive than those reported in the literature.

ON THE GROWTH OF SOLUTIONS OF A CLASS OF HIGHER ORDER DIFFERENTIAL EQUATIONS

In this paper, authors investigate the order of growth and the hyper order of solutions of a class of the higher order linear differential equation, and improve results of M. Ozawa, G. Gundersen and J.K. Langley, Li Chun-hong.

On the Stability and Boundedness of Solutions of Certain Non-Autonomous Delay Differential Equation of Third Order

In this paper, we study certain non-autonomous third order delay differential equations with continuous deviating argument and established sufficient conditions for the stability and boundedness of solutions of the equations. The conditions stated complement previously known results. Example is also given to illustrate the correctness and significance of the result obtained.

A PRIORI BOUNDS FOR PERIODIC SOLUTIONS OF A KIND OF THIRD-ORDER DELAY DIFFERENTIAL EQUATION

By means of continuation theorem of the coincidence degree theory, sufficient conditions are obtained for the existence of periodic solutions of a kind of third-order neutral delay functional differential equation with deviating arguments.