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.