On bounded solutions of nonlinear first- and second-order equations with a Carathéodory function

In the paper we study the existence and uniqueness of bounded solutions for differential equations of the form: x′−Ax=f(t,x), x″−Ax=f(t,x), where A∈L(Rm), is a Carathéodory function and the homogeneous equations x′−Ax=0, x″−Ax=0 have nontrivial solutions bounded on R. Using a perturbation of the equations, the Leray–Schauder Topological Degree and Fixed Point Theory, we overcome the difficulty that the linear problems are non-Fredholm in any reasonable Banach space.