the degree defined by p. benevieri and m. furi (see ‘a simple notion of orientability for fredholm maps of index zero between banach manifolds and degree theory’, ann. sci. math. québec 22 (1998) 131–148) is used here to obtain some sharp lower bounds for the number of solutions of the $\lambda$-sections of the compact components of the set of non-trivial solutions of $\mf{f}(\lambda,x)=0$, where $\mf{f}$ is a $\cal{c}^1$ fredholm map of index 1 such that $\mf{f}(\lambda,0)=0$ for all $\lambda\in\mathbb{r}$. these bounds are given in terms of the parity of the linearized fredholm family $d_2\mf{f}(\cdot,0)$. the parity is a local invariant measuring the change of the orientation of $d_2\mf{f}(\lambda,0)$ as $\lambda$ crosses an interval. the set of eigenvalues of $d_2\mf{f}(\cdot,0)$ is not assumed to be discrete. therefore, even in the classical situation when $\mf{f}$ is a compact perturbation of the identity, the results presented here are completely new.