Eine direkte methode zur bestimmung der wachstumsordnung der losungen von differentialgleichungssystemen
Titel:
Eine direkte methode zur bestimmung der wachstumsordnung der losungen von differentialgleichungssystemen
Auteur:
Jank, Gerhard Volkmann, Lutz
Verschenen in:
Complex variables and elliptic equations
Paginering:
Jaargang 6 (1986) nr. 1 pagina's 39-49
Jaar:
1986-06
Inhoud:
Uber diese Arbeit wurde auf einem “Tag der Funktionentheorie” am 15. und 16.6.1984 an der RWTH Aachen vom ersten Autor berichtet. This lecture is concerned with the asymptotic behavior of solutions of linear differential equations and iinear systems in the irregular singular case. In contrast to the existing theory of asymptotic integration (compare e.g. Wasow [12]) we use a direct method, i.e. a method which makes no use of a representation of solutions. We treat equations and systems with rational coefficients having entire solutions. This suggests the generalization of the theory of Wiman-Valiron Saxer [8] to entire curves (i.e. vector valued entire functions) and its application to systems of differential equations. Define the order of anentire curve g(z) = col(g1(z),0…,gn(z)) by [image omitted] With M(r,g)=max|z|=r|g(z)||=max|z|=r(maxi-1,…,n|gi(z)|), then we get the following results. If we are given the system zg'(z)=A(z)g(z) with A(z)=ZqAq+zq-1Aq-1 +…+ A0, Aj∈Cn×n, we get in the first part for each entire solution ρ(g)≤q-(1/k), where k is the smallest integer for which Akq=0. In a second part we show that every transcendental solution of the system g' = Ag has a rational order ρ=ρ(g)≥1/n and is of normal type and of perfectly regular growth. i.e. [image omitted] . Further we get that the set of all orders which are taken by any solution of the system is invariant under rational transformations.
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