Integration of a Dirac comb and the Bernoulli polynomials (Q5962423)

The authors investigate the existence and uniqueness of periodic solutions of maximal regularity for the discontinuous ordinary differential equation NEWLINE\[NEWLINEy^{(n)}=1-\mathrm{III}+F,\leqno(1)NEWLINE\]NEWLINE where \(\mathrm{III}\) is the Dirac comb distribution and \(F\) is a piecewise-\(C^{\infty}\) periodic function with vanishing average. First, they introduce a new class of generalized functions, namely the periodic switched Bernoulli polynomials, and then by using these functions, they give some formulae for the periodic solutions of equation \((1)\). A generalization of this approach to a larger class of differential equations is also presented.