| Abstract: |
| Starting from the pioneering work by Ambrosetti and Prodi [Ann. Mat. Pura Appl., 1972], by means of the expression ``Ambrosetti-Prodi type problems`` one refers to nonlinear parameter-depending boundary value problems (BVPs) of the form $Lu +g(u)=\mu$ where the derivative of the nonlinear term $g$ crosses some eigenvalues of the linear differential operator $L$ and there is a change in the number of the solutions, as the real parameter $\mu$ varies. Motivated by a problem proposed by Ambrosetti [Atti Accad. Naz. Lincei, 2011], which regards the periodic case of these problems, we consider periodic BVPs associated with first order or second order ODEs of the form $x`+g(t,x)=\mu$ or $x``+f(x)x`+g(t,x)=\mu$, respectively. We present results of existence and multiplicity of Ambrosetti-Prodi type assuming local coercivity conditions on $g$, in order to relax assumptions of uniform type previously considered in the literature. The proofs are carried out by both the use of a classical topological argument, that is the Mawhin coincidence degree, and also by means of new tools which permit to deal with the more general case of a local coercive nonlinearity. The results are based on joint works with F. Zanolin (University of Udine, Italy).
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