Abstract:
We study the possibility of constructing a Sobolev–Schwartz generalized solution to the problem $$
A(t)x'(t)+B(t)x(t)=f(t),\quad t\in T=[0,+\infty],\quad x(0)=a,
$$
whose coefficient $(n\times n)$-matrix of derivatives is degenerate for every $t\in T$ in the situation when there is no classical solution $x(t)\in C^1(T)$ (the initial data do not satisfy the agreement conditions and the right-hand side is not a sufficiently smooth vector-function). We prove that the generalized solution is the limit of a sequence of classical solutions of the Cauchy problem for a system with constant coefficients, obtained by the perturbation method.
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