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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)
##### Keywords:
complex Borel measures
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