Asymptotic forms of solutions of certain linear ordinary differential equations

We analyze the asymptotic behavior as x → ∞ of the product integral Π_x0^xe^A(s)ds, where A(s) is a perturbation of a diagonal matrix function by an integrable function on [x₀,∞). Our results give information concerning the asymptotic behavior of solutions of certain linear ordinary differential equations, e.g., the second order equation y″ = a(x)y.     

Limit behavior of solutions of ordinary linear differential equations

A classification of classes of equivalent linear differential equations with respect to -limit sets of their canonical representations is introduced. Some consequences of this classification with respect to the oscillatory behavior of solution spaces are presented.     

Nonclassical behavior of solutions of linear second-order ordinary differential equations

We present a number of unstable second-order equations of the form

$$y'' + (1 + g(x))y = 0,$$

where the coefficient g(x) satisfies the conditions g(x) ∈ C(0, ∞) and lim_x→+∞ g(x)=0 but the maximum absolute values of solutions grow unboundedly (as power-law functions or even as exponentials).

     

Asymptotic behavior of solutions of nth-order ordinary differential equations with regularly varying nonlinearities

We obtain existence conditions and asymptotic, as t → ω (ω ≤ +∞), representations of a certain class of monotone solutions of an nth-order differential equation whose right-hand side is a sum of terms with regularly varying nonlinearities.     

Asymptotic expansions for solutions of n-th order linear ordinary differential equations with two turning points

Asymptotic expansions for solutions of n-th order linear differential equation with two turning points are constructed in Olver's form. Analytic properties of the coefficients of the series obtained are investigated. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 186, pp. 172–179, 1990.     

Asymptotic behavior of solutions of linear delay equations

The delay differential equations of the formx′(t)=?a(t)x(t?1),t≥0 are considered, wherea(t)≥0 is locally integrable on [0,∞). The main result: Let 0<c(t)≤a(t)≤k(t) for large ∫^∞, andc(t)≤Mc(t′) fort, t′≥T,|t?t′|≤l with some constantsl>0,M>1,T≥0. Then the condition \(k(t) \leqslant \frac{3}{2} + \alpha c(t), t \geqslant T\) with some constant α>0 dependent onl, M, ensures that all solutions of (*) tend to zero ast→∞.     

On periodic solutions of linear degenerate second-order ordinary differential equations

We consider a scalar linear second-order ordinary differential equation whose coefficient of the second derivative may change its sign when vanishing. For this equation, we obtain sufficient conditions for the existence of a periodic solution in the case of arbitrary periodic inhomogeneity.     

Asymptotic behavior of solutions of pseudo-parabolic partial differential equations

Summary Consider a solution to a second-order pseudo-parabolic equation with sufficiently smooth time-independent coefficients in a cylindrical domain. If it vanishes on the cylindrical surface for all times and if its restriction to a fixed instant belongs toC _2+a , then its pointwise values decay exponentially as t→∞ while its Dirichlet norm grows expontially as t→−∞. Similar conclusion still hold for solutions to non-homogeneous equations under non-homogeneous boundary conditions provided the free term and the boundary data posses these asymptotic behaviors. Work of the second named author was partially supported by N.S.F. Grant No. GP-19590. Entrata in Redazione il 29 gennaio 1971.     

Asymptotic behavior of solutions to abstract functional differential equations

In this paper, a new approach is provided to study the asymptotic behavior of functions. A Tauberian theorem is improved and applied to describe the asymptotic behavior of abstract functional differential equations of the form

     

Asymptotic behaviour and zeros of solutions of linear differential equations

Summary. Certain classes of linear differential equations are investigated for which the distribution of zeros of their solutions determines their asymptotic behaviour. These results generalize those already obtained for the second order linear differential equations to equations of arbitrary order.     

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.     

Quasiperiodic solutions of degenerate linear systems of second-order ordinary differential equations

We establish sufficient conditions for the existence of quasiperiodic solutions of a system of ordinary second-order differential equations with degenerate symmetric matrix of the second derivatives for an arbitrary quasiperiodic inhomogeneity.