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Perron integral, Perron product integral and ordinary linear differential equations. (English) Zbl 0619.26006

Differential equations and their applications, Equadiff 6, Proc. 6th Int. Conf., Brno/Czech. 1985, Lect. Notes Math. 1192, 149-154 (1986).
[For the entire collection see Zbl 0595.00009.]
If \(\Delta =\{x_ 0,t_ 1,x_ 1,...,t_ k,x_ k\}\) is a Riemann-type partition of an interval [a,b] (i.e. \(a=x_ 0<x_ 1<...<x_ k=b\) and \(x_{j-1}\leq t_ j\leq x_ j\) \((1\leq j\leq k)),\) M is the ring of real or complex (n\(\times n)\)-matrices, A:[a,b]\(\to M\), let \[ P(A,\Delta)=(I+A(t_ k)(x_ k-x_{k-1}))...(I+A(t_ 1)(x_ 1-x_ 0)), \]
\[ \tilde P(A,\Delta)=\exp (A(t_ k)(x_ k-x_{k-1}))...\exp (A(t_ 1)(x_ 1-x_ 0)). \] It is well known that if A is continuous and U is the matrix solution of the differential equation \(x'=A(t)x,\) with \(U(a)=I,\) then both \(P(A,\Delta)\) and \(\tilde P(A,\Delta)\) converge to U(b) when the mesh of the partition \(\Delta\) tends to zero in the usual Riemann sense.
Kurzweil in 1957 and Henstock in 1960 have extended the Riemann integral, and obtained the Perron integral, by modifying the way in which the mesh of the partition tends to zero. That idea is here applied to the products \(P(A,\Delta)\) to define a concept of Perron product integral. The basic properties of this integral are listed and relations to the ACG\({}_*\)- functions are given. The above differential equation is then studied under the mere assumption that the corresponding Perron product integral exists.
Reviewer: J.Mawhin

MSC:

26A39 Denjoy and Perron integrals, other special integrals
34A30 Linear ordinary differential equations and systems

Citations:

Zbl 0595.00009