On bounded solutions of nonlinear first- and second-order equations with a Caratheodory 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 epsilon L(R-m), f : R x R-m -> R-m is a Caratheodory 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. (c) 2007 Elsevier Inc. All rights reserved.