Range of semilinear operators for systems at resonance

For a vector function $u:mathbb{R} o mathbb{R}^N $ we consider the system $$displaylines{ u''(t)+ abla G(u(t))= p(t)cr u(t)=u(t+T), }$$ where $G: mathbb{R}^N o mathbb{R}$ is a $C^1$ function. We are interested in finding all possible T-periodic forcing terms p(t) for which there is at least one solution. In other words, we examine the range of the semilinear operator $S:H^2_{ m per} o L^2([0,T],mathbb{R}^N)$ given by $Su= u''+ abla G(u)$, where $$ H^2_{ m per}= { uin H^2([0,T], mathbb{R}^N); u(0) - u(T) = u'(0)-u'(T)=0 }. $$ Writing $p(t)= overline{p} + widetilde{p}(t)$, where $overline{p}:=frac 1Tint_0^Tp(t), dt$, we present several results concerning the topological structure of the set $$ mathcal{I}(widetilde{p})={ overline{p} in mathbb{R}^N; overline{p} + widetilde{p}in operatorname{Im}(S)}. $$