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Linear distributional differential equations in the space of regulated functions. (English) Zbl 0796.34014
The paper deals with the system on the interval $$[0,1]$$, (1) $${\mathbf A}_ 1({\mathbf A}_ 0{\mathbf x})'- {\mathbf A}_ 2'{\mathbf x}= {\mathbf f}'$$ with distributional coefficients and solutions from the space of regulated functions. $${\mathbf A}_ 0$$ and $${\mathbf A}_ 1$$ are $$n\times n$$-matrix valued functions continuous on $$[0,1]$$ with $$\text{det}({\mathbf A}_ 0(t){\mathbf A}_ 1(t))\neq 0$$; $${\mathbf A}_ 1$$, $${\mathbf A}_ 2$$ have bounded variation on $$[0,1]$$ and $${\mathbf f}$$ is regulated on $$[0,1]$$. It is shown that the system (1) is equivalent to an integral equation which is a generalized linear differential equation (in the sense of J. Kurzweil). The authors prove (with the aid of the basic results known for generalized differential equations) theorems on existence and uniqueness of solutions and the variation-of-constants formula for the system (1) and for the system ${\mathbf T}(u)\equiv {\mathbf P}_ 1({\mathbf P}_ 0{\mathbf u}^{(m-1)})'+ {\mathbf P}_ 2'{\mathbf u}^{(m-1)}+\cdots + {\mathbf P}_ m'{\mathbf u}'+ {\mathbf P}_{m+1}'{\mathbf u}= {\mathbf q}',\tag{2}$ where $${\mathbf P}_ 0$$ and $${\mathbf P}_ 1$$ are $$n\times n$$- matrix valued functions on $$[0,1]$$, $$\text{det}({\mathbf P}_ 0(t){\mathbf P}_ 1(t))\neq 0$$ on $$[0,1]$$; $${\mathbf P}_ 1,{\mathbf P}_ 2,\dots,{\mathbf P}_{m+1}$$ are of bounded variation on $$[0,1]$$, $${\mathbf q}$$ is regulated on $$[0,1]$$ and an $$n$$-vector valued function $${\mathbf u}$$ is called a solution to the system (2) if $${\mathbf u},{\mathbf u}',\dots,{\mathbf u}^{(m-1)}$$ are regulated on $$[0,1]$$ and $${\mathbf T}(u)-{\mathbf q}'$$ is the zero $$n$$-vector distribution.
Reviewer: J.Diblík (Brno)

##### MSC:
 34A37 Ordinary differential equations with impulses 46F99 Distributions, generalized functions, distribution spaces 34A30 Linear ordinary differential equations and systems, general
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