A principle of relatedness for systems with small delays

PABLO AMSTER; MARIEL PAULA KUNA; GONZALO ROBLEDO

Buenos Aires

Congreso; V International Symposium on Nonlinear Equations and Free Boundary Problems; 2017

A general result by Krasnoselskii \cite{KZ} establishes that, if we consider a fixed point operator $K: U\subset C_T(\mathbb R, \mathbb R^N)\to C_T(\mathbb R, \mathbb R^N)$ associated to the problem$$u'(t)=g(t,u(t)), \;\; u(0)=u(T),$$ and $P: G\subset \mathbb R^N \to \mathbb R^N$ is the Poincar\'e map, then, under appropriate hypotheses, the Leray-Schauder degree of $I-K$ in $U$ coincides with the Brouwer degree of $I-P$ in $G$.In this work, we extend this relatedness principle to a system of DDEs\begin{equation} \label{eq} u'(t)=g(u(t),u(t-\tau))+p(t),\end{equation}where $\tau >0$, $g:  \overline\Omega \times \overline\Omega \to \mathbb R^N$ is continuously differentiable and $\Omega \subset \mathbb R^N$.In this case the Poincar\'e map  is defined in the infinite-dimensional space $C([-\tau, 0], \mathbb R^N)$. Based on the result for $\tau=0$, we shall prove that the principle holds for small values of $\tau$.As a consequence, we deduce that, for nearly all, i. e. except a countable set, $T>0$, if $G(u):=g(u,u)$ is an inward pointing field, then the system with $p=0$ has an equilibrium $e\in \Omega$ and, furthermore, the index of the Poincar\'e operator of the linearised system for $\tau = 0$ is equal to  $-1$, then problem (\ref{eq}) has at least two (generically three) $T-$periodic solutions, provided that $p\in C(\mathbb R, \mathbb R^N)$ is $T-$periodic and close to the origin.Moreover, extending another result by Krasnoselskii \cite{K}, we prove that the previous assumptions imply that the equilibrium is unstable.