The article deals with some elementary results on the coincidence degree for mappings of type \(L - N\) between Banach spaces \(X\) and \(Y\), where \(L\) is a Fredholm operator of index \(0\) and \(N\) is a continuous and compact operator. The coincidence degree \(d(L - N,\Omega)\) of such a mapping on the boundary of an open bounded set of \(X\) is defined as Leray-Schauder degree of a completely continuous vector field \(I - (L + JP)^{-1}(N + JP)\), where \(J\) is an isomorphism between \(\text{Ker} \, L\) and some complementary to \(\text{Im} \, L\) subspace \(Y_1\), \(P\) is a projection of \(X\) onto \(\text{Ker} \, L\). This degree has standard properties of coincidence degree. The paper's main results are concerned with the case when \(N\) is a homogeneous operator of order \(\alpha > 0\). As an application, the \(\omega\)-periodic problem for the second order functional-differential equation \[ x''(t) = Fx(t)+F_px(t) \] is considered. Here, \(F,F_p:C_\omega^1 \to L_\omega^1\) are continuous operators, \(F\) is homogeneous of order \(\alpha\) and \(F_p\) asymptotically zero either at \(0\) or at the infinity.