Boundedness of solutions and asymptotic properties of some systems of differential equations

We establish conditions under which solutions of some systems of differential equations are bounded and study their asymptotic properties.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 7, pp. 944–946, July, 1994.     

Boundedness and stability of solutions to difference equations

This paper is concerned with the qualitative behaviour of solutions to difference equations. We focus on boundedness and stability of solutions and we present a unified theory that applies both to autonomous and nonautonomous equations and to nonlinear equations as well as linear equations. Our presentation brings together new, established, and hard-to-find results from the literature and provides a theory that is both memorable and easy to apply. We show how the theoretical results given here relate to some of those in the established literature and by means of simple examples we indicate how the use of Lipschitz constants in this way can provide useful insights into the qualitative behaviour of solutions to some nonlinear problems including those arising in numerical analysis.     

An asymptotic result and a stability criterion for linear nonautonomous delay difference equations

Linear delay difference equations with variable coefficients and constant delays are considered. By the use of an appropriate solution of the so called generalized characteristic equation, an asymptotic result is obtained and a stability criterion is established.     

Absolute asymptotic stability of solutions of linear parabolic differential equations with delay

We establish necessary and sufficient conditions for the absolute asymptotic stability of solutions of linear parabolic differential equations with delay. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 12, pp. 1714–1721, December, 2007.     

Coefficient conditions for the asymptotic stability of solutions of systems of linear difference equations with continuous time and delay

We establish sufficient algebraic coefficient conditions for the asymptotic stability of solutions of systems of linear difference equations with continuous time and delay in the case of a rational correlation between delays. We use (n ² + m)-parameter Lyapunov functions (n is the dimension of the system of equations and m is the number of delays). Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 4, pp. 516–522, April, 1998. This work was partially supported by the Joint Foundation of the Ukrainian Government and the Soros International Science Foundation (grant No. K42199).     

Criteria of the mean-square asymptotic stability of solutions of systems of linear stochastic difference equations with continuous time and delay

We obtain spectral and algebraic coefficient criteria and sufficient conditions for the mean-square asymptotic stability of solutions of systems of linear stochastic difference equations with continuous time and delay. We consider the case of a rational correlation between delays and a “white-noise”-type stochastic perturbation of coefficients. We use the method of Lyapunov functions. Most results are presented in terms of the Sylvester and Lyapunov matrix algebraic equations. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 8, pp. 1073–1081, August, 1998. This work was partially supported by the Joint Foundation of the Ukrainian Government and the Soros International Science Foundation (grant No. K42100).     

脉冲时滞差分方程非振动解的存在性

讨论脉冲时滞差分方程 给出了由时滞差分方程非振动解的存在性刻划出相应的脉冲时滞差分方程的同样性质的一般性脉冲条件。     

The behavior of the solutions of periodic linear neutral delay difference equations

Some new asymptotic, nonoscillation and stability criteria for linear neutral delay difference equations with periodic coefficients and constant delays are given. The results are obtained via a positive root (with suitable properties) of an associated equation which is, in a sense, the corresponding characteristic equation.     

Boundedness of solutions for sublinear reversible systems

We are concerned with the boundedness of all the solutions for second order differential equation

On asymptotic equivalence of linear and quasilinear difference equations

The asymptotic equivalence of systems of difference equations of linear and quasilinear type is investigated. The first result on the asymptotic equivalence of linear systems is a discrete analog of an improved version of the Levinson's well-known theorem on asymptotic equivalence of linear differential equations, while the second one providing conditions for asymptotic equivalence of linear and quasilinear systems is related to that of Yakubovich in differential equations case.     

The asymptotic estimates of solutions of difference equations

The asymptotic estimate of the solution y(t) of linear difference equations with almost constant coefficients and the condition of asymptotic equivalence between y(t) and y₀(t) are given, where y₀(t) is the corresponding solution of linear equations with constant coefficients. A brief and elementary proof of the results is presented. The method of proof carries over to differential equations in which some similar results are stated.     

An asymptotic property of the solutions to second order linear nonautonomous delay differential equations

Second order linear nonautonomous delay differential equations are considered, and a fundamental asymptotic criterion for the solutions is established, by the use of the concept of generalized characteristic equation.     

Boundedness and asymptotic behavior of the solutions of a fuzzy difference equation

In this paper we study the existence, the uniqueness, the boundedness and the asymptotic behavior of the positive solutions of the fuzzy difference equation x_n+1=∑_i=0^kA_i/x_n−i^p_i, where k{1,2,…,}, A_i, i{0,1,…,k}, are positive fuzzy numbers, p_i, i{0,1,…,k}, are positive constants and x_i, i{−k,−k+1,…,0}, are positive fuzzy numbers.     

On the behavior of the solutions for linear autonomous neutral delay difference equations

A class of linear autonomous neutral delay difference equations is considered, and some new results on the asymptotic behavior and the stability are given, via a positive root of the correspondng characteristic equation.     

On the behavior of the solutions to periodic linear delay differential and difference equations

Some new results on the behavior of the solutions to periodic linear delay differential equations as well as to periodic linear delay difference equations are given. These results are obtained by the use of two distinct roots of the corresponding (so called) characteristic equation.     

Boundedness for sublinear Duffing equations with time-dependent potentials

In this paper, we prove a sufficient and necessary condition for the boundedness of all solutions for the sublinear equation , where 0<α<1, p(t) and e(t) are smooth 1-periodic functions.     

A priori bounds on difference quotients of solutions to some linear uniformly elliptic difference equations

Second order difference quotients of solutions to a class of linear uniformly elliptic difference Dirichlet problems are bounded in terms of quantities which depend on the coefficients of the operator, the inhomogenous term, the boundary values and the domain-which we take to be a rectangle. The results we obtain have theoretical and practical applications.     

Priori estimates and asymptotic properties of solutions for some fractional order elliptic equations

In this paper, we obtain estimates for solutions for a class of fractional order elliptic equations in different domains and boundary conditions, and prove some regularity results. Then, we study the qualitative properties of solutions with prescribed Q-curvature.