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Abstract
We study
the distributions of periodic solutions of
scalar piecewise equations defined by $x'=f(t,x)$ if $x\geq 0$,
and $x'=g(t,x)$ if $x<0$, where $f,g$ are time periodic $\mathcal C^1$-functions
such that $f(t,0)=g(t,0)$. Thus, these are equations on the cylinder where the vector field is
not necessarily smooth on one of the equatorial circles.
We find that the solutions are $\mathcal C^1$-functions when the
equation restricted to the equatorial line has a finite number of zeroes.
Moreover, if $f$ and $g$ are analytic functions and
the zeroes on the equatorial line are finite and simple, the set of periodic solutions
consists of isolated periodic solutions and a
finite number (determined by the number of zeroes)
of closed "bands" of periodic solutions.
Mathematics Subject Classification: Primary: 34C25, Secondary: 34C05.
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