A bifurcation problem for a class of periodically perturbed autonomous parabolic equations

The paper deals with the problem of the existence of a branch of T−periodic solutions originating from the isolated limit cycle of an autonomous parabolic equation in a Banach space when it is perturbed by a nonlinear T−periodic term of small amplitude. We solve this problem by first introducing a novel integral operator, whose fixed points are T−periodic solutions of the considered equation and vice versa. Then we compute the Malkin bifurcation function associated to this integral operator and we provide conditions under which the well-known assumption of the existence of a simple zero of the Malkin bifurcation function guarantees the existence of the branch.