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Dynamic equations on time scales and generalized ordinary differential equations. (English) Zbl 1235.34247

The author presents a procedure how to convert an arbitrary dynamic equation

x Δ (t)=f(t,x(t)),t𝕋

into a generalized differential equation. This idea of a generalized differential equation is based on the notion of the Kurzweil–Stieltjes or Perron integral. As a byproduct the author shows that some results concerning stability and continuous dependence on parameters drop out of the general setting. In a final section the special case of variational stability is considered.

MSC:
34N05Dynamic equations on time scales or measure chains
34D05Asymptotic stability of ODE
26A39Special integrals of functions of one real variable
References:
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[8]Kurzweil, J.: Generalized ordinary differential equations and continuous dependence on a parameter, Czechoslovak math. J. 7, No. 82, 418-449 (1957) · Zbl 0090.30002
[9]Schwabik, Š.: Generalized ordinary differential equations, (1992)
[10]Schwabik, Š.: Generalized ordinary differential equations and discrete systems, Arch. math. (Brno) 36, 383-393 (2000) · Zbl 1090.34514 · doi:emis:journals/AM/00-5/schwabik.htm
[11]Schwabik, Š.: The Perron product integral and generalized linear differential equations, Časopis pěst. Mat. 115, 368-404 (1990) · Zbl 0724.26006
[12]Slavík, A.: Product integration on time scales, Dynam. systems appl. 19, 97-112 (2010) · Zbl 1200.26043