Our study focuses on the location and approximation of a fixed point for an operator \(\mathbb {H}\) . To do this, we consider the Krasnoselskij operator associated with the operator \(\mathbb {H}\) and locate a domain \(\mathcal {W}\) for Krasnoselskij operator where we establish the existence and uniqueness of a fixed point using restricted fixed-point theorem. For this, first, we prove that the Krasnoselskij operator is contractive under certain conditions. In addition, it is shown that after modifying the contractivity condition for the operator \(\mathbb {H}\) , we can obtain new results of fixed point for the operator \(\mathbb {H}\) . Furthermore, by assuming that \(\mathbb {H}\) is m-times Fréchet differentiable operator, we examined some new fixed-point results. To finish, we apply our theoretical results to Bratu’s boundary value problem.
