Periodic bifurcation from families of periodic solutions for semilinear differential equations with Lipschitzian perturbations in Banach spaces

TL;DR

This paper establishes necessary and sufficient conditions for bifurcation of periodic solutions in semilinear differential equations with Lipschitzian perturbations in Banach spaces, extending classical methods to nonsmooth cases.

Contribution

It introduces a modified Mel'nikov approach and extends Lyapunov-Schmidt reduction to handle Lipschitzian perturbations in Banach spaces.

Findings

Derived conditions for bifurcation of T-periodic solutions.

Constructed a bifurcation function using topological index.

Extended classical reduction methods to nonsmooth perturbations.

Abstract

Let A:D(A)\to E be an infinitesimal generator either of an analytic compact semigroup or of a contractive C_0-semigroup of linear operators acting in a Banach space E. In this paper we give both necessary and sufficient conditions for bifurcation of $T$-periodic solutions for the equation x'=Ax+f(t,x)+e g(t,x,e) from a k-parameterized family of T-periodic solutions of the unperturbed equation corresponding to e=0. We show that by means of a suitable modification of the classical Mel'nikov approach we can construct a bifurcation function and to formulate the conditions for the existence of bifurcation in terms of the topological index of the bifurcation function. To do this, since the perturbation term g is only Lipschitzian we need to extend the classical Lyapunov-Schmidt reduction to the present nonsmooth case.