On the range of certain semilinear operators for systems at resonance

P. AMSTER; M. P. KUNA

ELECTRONIC JOURNAL OF DIFFERENTIAL EQUATIONS

TEXAS STATE UNIVERSITY-SAN MARCOS

Año: 2012

1072-6691

For a vector function $u:R o R^{N}$ we consider the system egin{equation} label{ecua1} left{ egin{array}{l} u´´(t)+ abla G(u(t))= p(t) u(t)=u(t+T), end{array} ight. end{equation} where $G: R^{N} o R$ is a $C^{1}$ function. We are interested in the problem of finding the set of all possible $T$-periodic forcings $p(t)$ such that ( ef{ecua1}) has at least one solution. In other words, we study the range of the semilinear operator $S:H^{2}_{per} o L^2([0,T],R^N)$ given by $Su= u´´+ abla G(u),$ where $$H^{2}_{per}= { uin H^{2}([0,T], R^N); u(0) - u(T) = u´(0)-u´(T)=0 }.$$ Writing $p(t)= overline p + ilde p(t)$, where $overline p:=rac 1Tint_0^Tp(t), dt$, we present several results concerning the topological structure of the set $$mathcal I( ilde p)=left{ overline p in R^N; overline p + ilde pin Im(S) ight}.$$