Boundary-value problems for nonautonomous nonlinear systems on the half-line

A method is presented for proving the existence of solutions for boundary-value problems on the half line. The problems under study are nonlinear, nonautonomous systems of ODEs with the possibility of some prescribed value at $t=0$ and with the condition that solutions decay to zero as $t$ grows large. The method relies upon a topological degree for proper Fredholm maps. Specific conditions are given to ensure that the boundary-value problem corresponds to a functional equation that involves an operator with the required smoothness, properness, and Fredholm properties (including a calculable Fredholm index). When the Fredholm index is zero and the solutions are bounded a priori, then a solution exists. The method is applied to obtain new existence results for systems of the form $dot{v}+g(t,w)=f_1(t)$ and $dot{w}+h(t,v)=f_2(t)$