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Fundamental theorems for linear measure differential equations. (English) Zbl 0642.34004
Let \({\mathcal P}^ 0\) be the complex Borel measures on \({\mathbb{R}}\) and let \({\mathcal P}^ 1\) be the primitive right continuous distributions of the elements in \({\mathcal P}^ 0\). Let \(\phi\) \(\geq 0\), be continuous such that \(\phi (x)=0\), \(| x| \geq 1\), and \(\int \phi =1\). Let \(\epsilon >0\) and let \(\phi (x,\epsilon)=\phi ((x+\epsilon)/\epsilon)/\epsilon.\) Let A be an \(n\times n\) matrix with elements in \({\mathcal P}^ 0\) and let \(f\in ({\mathcal P}^ 0)^ n\). Let \(A(x,\epsilon)=\int \phi (x-t,\epsilon)dA(t)\) and let \(f(x,\epsilon)=\int \phi (x-t,\epsilon)df(t).\) The following is the main theorem. Let \(\epsilon\), A and f be as above. Let \(a\in {\mathbb{R}}\), \(c\in {\mathbb{C}}^ n\) and let \(g(D)=\sum ^{\infty}_{j=1}(j!)^{-1}D^{j-1},\) where D is an \(n\times n\) matrix with entries in \({\mathbb{C}}\). When \(\epsilon\) \(\to 0\) then the solution u(x,\(\epsilon)\) of \[ u'(x,\epsilon)+A(x,\epsilon)u(x,\epsilon)=f(x,\epsilon),u(a,\epsilon)=c, \] converges pointwise to the unique solution \(u\in ({\mathcal P}^ 1)^ n\) of \[ u(x)=c-\int ^{x}_{a^ +}g(A(\{t\}))dA(t)u(t)+\int ^{x}_{a^ +}g(A(\{t\}))df(t),\quad x\geq a \] and \[ u(x)=c+\int ^{a}_{x^ +}g(A(\{t\}))dA(t)u(t)-\int ^{a}_{x^ +}g(A(\{t\}))df(t),\quad x<a. \] This theorem solves the paradox for measure differential equations which is that in general \(u(x,0)=\lim _{\epsilon \to 0}u(x,\epsilon)\) does not solve the integral equations above with \(g(D)=I\), the identity matrix J. Kurzweil [Czech. Math. J. 8(83), 360-388 (1958; Zbl 0094.058)] has solved the paradox in a case with one point mass admitting non-linearity. See also O. Hájek [Bull. Am. Soc. 12, 272-279 (1985)]. The note also contains a theorem for a combined measure and stochastic differential equation.
Reviewer: J.Perrson

MSC:
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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