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Time switched differential equations and the Euler polynomials. (English) Zbl 1307.34026

Summary: We discuss differential equations depending non-smoothly on the integration time of the form \[ \begin{aligned} y^{(n)}= {\text{ sgn}} (\sigma (t)) + F(t) \end{aligned} \] where \(n\in \mathbb N, n>0\), and \(F, \sigma \) are piecewise \({\mathcal{C}}^{\infty }\) periodic functions. The main results deal with the existence of periodic solutions of such equations as well as their computation by explicit formulas. No infinite series appear, and it is indeed established that these periodic solutions are explicitly computable by means of finitely many Euler polynomials. We also introduce a wider class of piecewise \({\mathcal{C}}^{\infty }\) equations where the problem of finding periodic solutions is finitely solvable as well.

MSC:

34A36 Discontinuous ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
34A45 Theoretical approximation of solutions to ordinary differential equations

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