A principle of relatedness for systems with small delays

MARIEL PAULA KUNA; PABLO AMSTER; 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 $K: Usubset C_T(mathbb R, mathbb R^N) o C_T(mathbb R, mathbb R^N)$ associated to the problem$$u´(t)=g(t,u(t)), ;; u(0)=u(T),$$ and $P: Gsubset mathbb R^N o mathbb R^N$ 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 DDEsegin{equation} label{eq} u´(t)=g(u(t),u(t- au))+p(t),end{equation}where $ au >0$, $g:  overlineOmega imes overlineOmega o mathbb R^N$ is continuously differentiable and $Omega subset mathbb R^N$.In this case the Poincar´e map  is defined in the infinite-dimensional space $C([- au, 0], mathbb R^N)$. Based on the result for $ au=0$, we shall prove that the principle holds for small values of $ au$.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 $ein Omega$ and, furthermore, the index of the Poincar´e operator of the linearised system for $ au = 0$ is equal to  $-1$, then problem ( ef{eq}) has at least two (generically three) $T-$periodic solutions, provided that $pin C(mathbb R, mathbb R^N)$ 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.