Let A be a (nonlinear) mapping of monotone type from the real Banach space X into its conjugate space X^*. The solvability of the functional equation Au = 0 in a given open subset of X is investigated under the assumption that A = A₀ is homotopic to an odd mapping A₁. The same problem is then studied for A of the form A = T + B, with T a maximal monotone mapping of X into the set 2^X* of all subsets of X^*, and B of monotone type. The results are used to derive criterions of Fredholm alternative type for the solvability of the equation Au = f, provided A is asymptotically homogeneous and odd. Applications to nonlinear Hammerstein integral equations and to elliptic boundary value problems are given.