Asymptotic and integral equivalence of multivalued differential systems. (English) Zbl 0854.34011

Futák, Ján; Haščák, Alexander

Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 34, 21-30 (1995).

This paper considers the systems (1) \(y'(t)= A(t) y(t)\) and (2) \(x'(t)\in A(t) x(t)+ F(t, x(t), Sx(t))\) on the time interval \(J= [0, \infty)\). Here \(F\) is a set-valued map with non-empty convex and compact values which is upper semicontinuous for every fixed \(t\), and \(S\) is an operator which maps the set of continuous and \(\psi\)-bounded functions into itself (a function \(z\) is \(\psi\)-bounded if \(\psi^{- 1} z\) is bounded). Systems (1) and (2) are \(\psi\)-asymptotically (integrally) equivalent if for every solution \(x\) of (1) there exists a solution \(y\) of (2) such that \(\psi^{- 1}(t)|x(t)- y(t)|\to 0\) for \(t\to \infty\) (\(\psi^{-1}(t) |x(t)- y(t)|\in L_p([0, \infty)]\)) and conversely. Theorems which establish asymptotic and integral equivalence are given.

Reviewer: V.Křivan (České Budějovice)

MSC:

34A60	Ordinary differential inclusions
34D05	Asymptotic properties of solutions to ordinary differential equations
34E10	Perturbations, asymptotics of solutions to ordinary differential equations

Keywords:

differential inclusions; asymptotic and integral equivalence

References:

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