Asymptotic formulas for solutions of functional differential equations. (English) Zbl 0643.34077

Futák, Ján

Acta Univ. Palacki. Olomuc., Fac. Rerum Nat. 85, Math. 25, 181-192 (1986).

The initial problem \((1.1)\quad y'=A(t)y+f(t,\sigma (y;h(t))),\) (1.2) \(y(0)=y_ 0\), \(y_ 0\) is a constant vector, is considered, where the function \(f: R_+\times R^ n\to R^ n\) satisfies the Carathéodory local conditions, the matrix \(A: R_+\to R^{n\times n}\) is local integrable, \(h: R_+\to R\) is a continuous function, with h(t)\(\leq t\) and \(\sigma\) is an operator defined by \(\sigma (u;t)=u(t)\) for \(t\in R_+\), \(=0\) for \(t<0\). Sufficient conditions are given for that: a) all solutions of (1.1), (1.2) exist on \(R_+\) under small initial conditions \(\| y_ 0\|\) and are of asymptotic representation \((1.4)\quad y(t)=X(t)[c+o(1)]\) as \(t\to \infty\), where c is a constant vector; b) the family of solutions of the form (1.4) of (1.1) is stable (in some sense) with regard to small changes of both initial conditions and right- handside of (1.1); c) any solution of (1.1), (1.2) has the form (1.4) under arbitrary initial conditions \(\| y_ 0\|\).

Reviewer: J.Futák

MSC:

34K25

Asymptotic theory of functional-differential equations

Keywords:

Carathéodory local conditions

References:

[1]	Čanturija T. I.: On the asymptotic properties of solutions of perturbed linear systems of differential equations. Ann. di Mat.Pura ed appl., Serie IV, T. XCIV, 1972, 41-62.
[2]	Futák J.: Ob asimptotičeskom provedenij funkciona\?nodifferencialnych uravnenij vyššich poriadkov. Trudy Instituta prikladnoj matematiky im. I. N. Veka, T. 8, 1980, 68-77.
[3]	Futák J.: On the asymptotic behaviour of the solutions of nonlinear delay differential system. Math. Slovaca, 31, 1981, No. 3, 233-242.
[4]	Futák J.: Ob asimptotičeskom provedenij funkciona\?nych differencialnych uravnenij. Trudy MADI Moskva
[5]	Kiguradze I. T.: Nekotorije singularnyje krajevye zadači dlja obyknovennych differencialnych uravnenij. Tbilisi, Izdatelstvo Tbiliskovo univ., 1975.
[6]	Torošelidze I. A.: Ob asimptotičeskom predstavlenii rešenij nekotorych sistem nelinejnych diff. uravnenij. Soobščenija akademii nauk Gruzinskoj SSR, XLII, 2, 1966, 285-292.