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), f :R × Rm→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. © 2007 Elsevier Inc. All rights reserved.