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Discontinuous solutions of semilinear differential-algebraic equations. I: Distribution solutions. (English) Zbl 0865.34003
Continuing previous work, the authors are studying the existence of classical and generalized solutions of semilinear implicit differential equations of the form: \(A(t) x'=G(t,x)\), where \(A(t)\), \(t\in J\subset\mathbb{R}\) are \(n\times n\) singular \(C^\infty\)-matrices of constant rank \(r<n\).
Assuming that the system is “geometrically nonsingular of index 1” in a rather restrictive sense, the authors prove the existence and uniqueness of classical (smooth) solutions through some initial points as well as the existence of certain one-sided solutions and also the existence of generalized solutions in the sense of distributions through certain impasse points.

MSC:
34A09 Implicit ordinary differential equations, differential-algebraic equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
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