Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 209, pp. 1-13.

Title: Range of semilinear operators for systems at resonance
        
Authors: Pablo Amster (Univ. de Buenos Aires, Argentina)
         Mariel Paula Kuna (Univ. de Buenos Aires, Argentina)

Abstract:
 For a vector function $u:\mathbb{R} \to \mathbb{R}^N $ we consider the system
 $$\displaylines{
 u''(t)+ \nabla G(u(t))= p(t)\cr
 u(t)=u(t+T),
 }$$
 where $G: \mathbb{R}^N \to \mathbb{R}$ 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
 $S:H^2_{\rm per}\to L^2([0,T],\mathbb{R}^N)$ given by $Su= u''+ \nabla G(u)$,
 where
 $$
 H^2_{\rm per}= \{ u\in H^2([0,T], \mathbb{R}^N);  
  u(0) - u(T) = u'(0)-u'(T)=0 \}.
 $$
 Writing
 $p(t)= \overline{p} + \widetilde{p}(t)$, where
 $\overline{p}:=\frac 1T\int_0^Tp(t)\, dt$,
 we present several results concerning the topological structure of the set
 $$
 \mathcal{I}(\widetilde{p})=\{ \overline{p} \in \mathbb{R}^N;
 \overline{p} + \widetilde{p}\in \operatorname{Im}(S)\}.
 $$

Submitted October 7, 2011. Published November 27, 2012.
Math Subject Classifications: 34B15, 34L30.
Key Words: Resonant systems; semilinear operators; critical point theory.