Persistence and periodic solutions in systems of delay differential equations

MELANIE BONDOREVSKY; PABLO AMSTER

Buenos Aires

Congreso; Escuela de invierno Luis A. Santaló; 2019

We study semi-dynamical systems associated to delay differential equations.With population models in mind, we consider the delayed differential systemegin{equation}label{eq}x´(t)=f(t,x(t),x(t-au))end{equation}where$f:Rimes [0,+infty)^{2N}o R^N$ is continuous and $auinR^+$ is the delay.An initial condition for (ef{eq}) can be expressedin the following wayegin{equation}label{icsd}x_0=arphi,end{equation}where $arphi:[-au,0]o [0,+infty)^N$ is a continuous function and $x_tin C([-au,0],R^N)$ is defined by $x_t(s)=x(t+s)$.Thus, the flow egin{equation} label{flow} Phi:[0,+infty)imes C([-au,0],R^N)o C([-au,0],R^N),end{equation}given by $Phi(t,arphi)=x_t$, induces a semi-dynamical system.We give sufficient conditions to guarantee uniform persistence, employing guiding functions techniques.In order to find periodic orbits of (ef{flow}) we employ topological degree methods. Since the space of initial conditions is infinite dimensional, the Brouwer degree cannot be applied: we instead use Leray-Schauder degree techniques.More precisely, inspired by Krasnoselskii´s work, we shall consider the positive cone $X$ of $C_T$, the Banach space of continuous $T$-periodic functions, for some $T>0$, and define an appropriate fixed point operator $K:Xo C_T$. medskipThe results are based on the work{em Persistence and periodic solutions in systems of delay differential equations} (to be submitted), of P. Amster and M. Bondorevsky.