The author studies surjectivity results for \(A\)-proper mappings \(T:X\to Y\) between Banach spaces such that \(\{x_n\mid n\in\mathbb N\}\) is bounded whenever \((Tx_n)_{n\in\mathbb N}\) is convergent. In particular, he considers the case \(T=A+N\), where \(A\) is a linear Fredholm operator of index 0 and \(N\) is continuously differentiable such that \(A+tN'(x)\) is \(A\)-proper for each \(t\in X\) and \(t\in[0,1]\).