Abstract

Consider the family $A$ of periodic equations $x˜=f(t,x)$ such that $f\; \in \; C3(R2)$ is $T$-periodic with respect to $t$ (for a fixed period $T>0$), $f$_x³(t,x)<0 and $f(t,x)\; \to \; \mp \; \infty$ as $x\; \to \; \pm \; \infty$ uniformly for $t\; \in \; [0,T]$. By [1], equations in this family have at least one and at most three periodic solutions, taking into account multiplicity in the sense that at most one periodic solution is singular, and if there are three periodic solutions, then none is singular. We shall show that the set of equations with at least one singular periodic solutions, $\Sigma$, is a closed connected submanifold of $A$. Moreover $A\; \setminus \; \Sigma$ is a disjoint union of two open sets; the set of equations with exactly one periodic solutions (non-singular) and the set of equations with exactly three periodic solutions. These results extends the ones obtained by Chen and Li in [2].

Bibliography
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