An averaging method for periodic solutions of quasilinear equations in Banach spaces

The aim of the paper is to provide an extension of the averaging method to the periodic problem for quasilinear differential equations $(\phi (t,u^{\mathrm{\prime }}){)}^{\mathrm{\prime }}=\epsilon f(t,u,u^{\mathrm{\prime }},\epsilon )$, $u(0)=u(T)$, $u^{\mathrm{\prime }}(0)=u^{\mathrm{\prime }}(T)$, where $u$ and $u^{\mathrm{\prime }}$ take values in a Banach space $X$ and $\phi$ is a suitable homeomorphism between a neighborhood of $0$ in $X$ and $X$. The approach follows from a given extension of the averaging method to first order systems of the form $u^{\mathrm{\prime }}=g(t,v)$, $v^{\mathrm{\prime }}=h(t,u,v)$, $u(0)=u(T)$, $v(0)=v(T)$, in a product of Banach spaces, obtained from a direct application of the implicit function theorem in Banach spaces to an equivalent problem. We prove that every nondegenerate zero of the corresponding averaged equation generates, for sufficiently small nonzero values of the parameter, a unique nearby periodic solution. An application is given to an infinite system of coupled forced pendulums with relativistic acceleration.