Asymptotic properties of solutions of multidimensional stochastic differential equations

Summary LetX _t R ^d be the solution of the stochastic equationdX _t =b(X _t )dt+(X _t )dW _t , whereW _t denotes a standard Wiener process. The aim of the paper is to clarify under which conditions the drift term or the diffusion term is of negligible significance for the long term behaviour ofX _t .     

Asymptotic properties of solutions of third-order nonlinear differential equations with deviating argument

The aim of this paper is to establish comparison principles on property A$A$, between a nonlinear differential equation of the third order with deviating argument (with delay, advanced or mixed argument) and the corresponding linear equation without deviating argument. On the basis of these comparison principles the sufficient conditions for delay, advanced and mixed equations to have property A$A$ are presented. The results obtained are compared with existing ones in the framework of the papers.     

Asymptotic properties of solutions to linear non-autonomous neutral differential equations

In this article we study asymptotic properties of solutions to first order linear neutral differential equations with variable coefficients and constant delays. Results are stated in terms of the solution to a characteristic equation. By doing this, we extend some of the results obtained for delay equations in [J.G. Dix, Ch.G. Philos, I.K. Purnaras, An asymptotic property of solutions to linear non-autonomous delay differential equations, Electron. J. Differential Equations 2005 (2005) 1-9] to neutral equations.     

Asymptotic solutions of nonlinear differential equations

An asymptotic expression for solutions of nonlinear differential equations is obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 5, pp. 676–678, May, 1991.     

Almost-periodic solutions to systems of differential equations with fast and slow time in the degenerate case

We consider a system of ordinary first-order differential equations. The right-hand sides of the system are proportional to a small parameter and depend almost periodically on fast time and periodically on slow time. With this system, we associate the system averaged over fast time. We assume that the averaged system has a structurally unstable periodic solution. We prove a theorem on the existence and stability of almost periodic solutions of the original system. Translated fromMatematicheskie Zametki, Vol. 63, No. 3, pp. 451–456, March, 1998.     

Asymptotic properties of differential equations with advanced argument

The paper discusses the asymptotic properties of solutions of the scalar functional differential equation

Asymptotic properties of Markov-modulated random sequences with fast and slow timescales

This work is concerned with asymptotic properties of Markov-modulated random processes having two timescales. The model contains a number of mixing sequences modulated by a switching process that is a discrete-time Markov chain. The motivation of our study stems from applications in manufacturing systems, communication networks and economic systems, in which regime-switching models are used. This paper focuses on asymptotic properties of the Markov-modulated processes under suitable scaling. Our main effort focuses on obtaining a strong approximation result.     

Periodic solutions of monotone differential inclusions with fast and slow variables

We study the solvability of a periodic problem for monotone differential inclusions and the behavior of its solutions as the parameter changes.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 5, pp. 694–703, May, 1993.     

Explicit solutions of Riccati equations appearing in differential games

In this paper an explicit closed form solution of Riccati differential matrix equations appearing in games theory is given.     

Asymptotic representations of solutions of second-order differential equations

We establish asymptotic representations as t → ω (ω ≤ + ∞) of a class of monotone solutions of the second-order differential equation y″ = f(t, y, y′), where f:[a,ω[× Δ _Y0 × Δ _Y1 is a continuous function asymptotically close on the considered class of solutions to a function of the form ±p(t)φ ₀(y)φ ₁(y′) with functions φ ₀ and φ ₁ regularly varying as y → Y ₀ and y′ → Y ₁. Here Δ _Yi , i ∈ {0, 1}, is a one-sided neighborhood of Y _i , and Y _i is either zero or ±∞.     

Asymptotic stability and asymptotic solutions of second-order differential equations

We improve, simplify, and extend on quasi-linear case some results on asymptotical stability of ordinary second-order differential equations with complex-valued coefficients obtained in our previous paper [G.R. Hovhannisyan, Asymptotic stability for second-order differential equations with complex coefficients, Electron. J. Differential Equations 2004 (85) (2004) 1–20]. To prove asymptotic stability of second-order differential equations, we establish stability estimates using integral representations of solutions via asymptotic solutions and error estimates. Several examples are discussed.     

Asymptotic properties of fractional delay differential equations

In this paper we study the asymptotic properties of d-dimensional linear fractional differential equations with time delay. We present necessary and sufficient conditions for asymptotic stability of equations of this type using the inverse Laplace transform method and prove polynomial decay of stable solutions. Two examples illustrate the obtained analytical results.     

Asymptotic properties of solutions of difference equations

Sufficient conditions for somem-th order finite difference equations are presented which have a solution behaving in a precisely specified way like a given polynomial.