A deficient spline function approximation to systems of first order differential equations

The stability of the vector-valued spline function approximations S(x) of degree m deficiency 3, i.e., S∈C^m?3, to systems of first order differential equations are investigated. The method will be shown to be A-stable for m=4, unstable and hence divergent for m?6. The method is stable form=5.     

Differentiability of solutions of impulsive differential equations with respect to the impulsive perturbations

The nonlinear impulsive differential equations with fixed moments of impulsive perturbation are the main object of investigation in this paper. Sufficient conditions for these types of equations are obtained, under which their solutions are continuously dependent and differentiable with respect to the initial conditions and the impulsive perturbations. The results are applied to a mathematical model of population dynamics.     

Exponential stability for nonlinear stochastic differential equations with respect to semimartingales

Consider a nonlinear stochastic differential equations with respect to semimartingales (1) dY(t) = F{Y(t),t)dti(t) + G(t)dM(t)+f(Y(t),t)dti(t)+g(Y(t),t)dM(t), which might be regarded as a stochastic perturbed system of (2) dX(t) = F(X(t),t)dfi(t) + G(t)dM(t). Suppose Eq. (2) is exponentially stable almost surely. Under what conditions is Eq. (1) still exponentially stable almost surely? In this paper we will give some sufficient conditions. As an application we also discuss the almost sure exponential stability for semilinear stochastic systems and small stochastic perturbed systems     

Second-order differentiability with respect to parameters for differential equations with adaptive delays

In this paper, we study the second-order differentiability of solutions with respect to parameters in a class of delay differential equations, where the evolution of the delay is governed explicitly by a differential equation involving the state variable and the parameters. We introduce the notion of locally complete triple-normed linear space and obtain an extension of the well-known uniform contraction principle in such spaces. We then apply this extended principle and obtain the second-order differentiability of solutions with respect to parameters in the W ^1,p -norm (1 ⩽ p < ∞).     

Instability in the first approximation for time-dependent linearizations

The concept of Krasovskii stability is introduced. The criterion of Krasovskii instability for time-dependent linearizations is obtained. This criterion is compared with Chetayev's theorem.     

On structural stability of ordinary differential equations with respect to discretization methods

Summary. On compact -dimensional discs, Morse-Smale differential systems having no periodic orbits are considered. The main result is that they are correctly reproduced by one-step discretization methods. For methods of order and stepsize sufficiently small, the time--map of the induced local flow and the -discretized system are joined by a conjugacy -near to the identity. The paper fits well in the rapidly growing list of results stating that hyperbolic/transversal structures are preserved by discretization. The proof relies heavily on techniques elaborated by Robbin (1971) in establishing his structural stability theorem on self-diffeomorphisms of compact manifolds. Received February 24, 1994 / Revised version received February 20, 1995     

Boundary-value problems for differential equations unsolvable with respect to the higher time derivative

For differential equations with constant coefficients unsolvable with respect to the higher time derivative, we establish conditions of the existence and uniqueness of solutions of problems with conditions local in time and periodic in space variables. We prove a metric theorem on lower bounds of small denominators appearing in the construction of solutions of the problems.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 9, pp. 1197–1208, September, 1995.This work was supported by the Foundation for Fundamental Research of the Ukrainian State Committee on Science and Technology.     

On the approximation of stochastic differential equations

The stability properties of stochastic differential equations with respetct to the perturbation of the coefficients and of the driving processes are investigated in the topology of uniform convergence in probability     

On the approximation of eigenvalues associated with functional differential equations

In this paper an approximation method based upon spline functions is developed for the eigenvalue problem associated with functional differential equations. Convergence results are established and the rate of convergence is investigated. Numerical results for cubic and quintic spline based methods are given. The paper concludes with a brief discussion of other possible approximation methods.     

Two-sided approximation to periodic solutions of ordinary differential equations

Summary A two-sided approximation to the periodic orbit of an autonomous ordinary differential equation system is considered. First some results about variational equation systems for periodic solutions are obtained in Sect. 2. Then it is proved that if the periodic orbit is convex and stable, the explicit difference solution approximates the periodic orbit from the outer part and the implicit one from the inner part respectively. Finally a numerical example is given to illustrate our result and to point out that the numerical solution no longer has a one-sided approximation property, if the periodic orbit is not convex.The Work is supported by the National Natural Science Foundation of China     

Existence and uniqueness of solutions of differential equations with respect to non-additive measures

By taking Sugeno-derivative into account, first, we investigate the existence of solutions to the initial value problems (IVP) of first-order differential equations with respect to non-additive measure (more precisely, distorted Lebesgue measure). It particularly occurs in the mathematical modeling of biology. We begin by expressing the differential equation in terms of ordinary derivative and the derivative with respect to the distorted Lebesgue measure. Then, by using the fixed point theorem on cones, we construct an operator and prove the existence of positive non-decreasing solutions on cones in semi-order Banach spaces. In addition, we also use Picard–Lindelöf theorem to prove the existence and uniqueness of the solution of the equation. Second, we investigate the existence of a solution to the boundary value problem (BVP) on cones with integral boundary conditions of a mix-order differential equation with respect to non-additive measures. Moreover, the Krasnoselskii fixed point theorem is also applied to both BVP and IVP and obtains at least one positive non-decreasing solution. Examples with graphs are provided to validate the results.     

Representation of analytic functions by series with respect to solutions of differential equations

Translated from Matematicheskie Zametki, Vol. 47, No. 4, pp. 106–114, April, 1990.