×

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

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.

MSC:

34A60 Ordinary differential inclusions
34D05 Asymptotic properties of solutions to ordinary differential equations
34E10 Perturbations, asymptotics of solutions to ordinary differential equations
PDFBibTeX XMLCite
Full Text: EuDML

References:

[1] Hallam T. G.: On asymptotic equivalence of the bounded solutions of two systems of differential equations. Michigan Math. J. 16 (1969), 353-363. · Zbl 0191.10401 · doi:10.1307/mmj/1029000319
[2] Haščák A.: Integral equivalence of multivalued differential systems I. Acta Math. Univ. Comeniae (Bratislava) XLVI-XLVII (1985), 205-214. · Zbl 0614.34017
[3] Haščák A.: Integral equivalence of multivalued differential systems II. Colloquia Math. Societatis J. Bolyai Diff. Eq., Szeged (Hungary) 47 (1984), 399-412.
[4] Haščák A.: Asymptotic and integral equivalence of multivalued differential systems. Hiroshima Math. J. 20 (1990), 425-442. · Zbl 0713.34076
[5] Haščák A., Švec M.: Integral equivalence of two systems of differential equations. Czech. Math. J. 32 (1982), 432-436.
[6] Sek Wui Seah: Asymptotic equivalence of multivalued differential systems. Boll. U.M.I. (5) 17-B (1980), 1124-1145. · Zbl 0458.34009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.