Existence and Asymptotic Properties of the Solution of a Nonlinear Boundary-Value Problem on the Real Axis

We consider a nonlinear system of ordinary differential equations defined on the entire real axis with Dirichlet-type boundary conditions at˙±∞. It is assumed that the linear part of the system has the property of nonuniform strong exponential dichotomy. To prove the existence theorem, we apply a Schauder–Tikhonov-type fixed-point principle. In addition, we also establish conditions under which the obtained solution has the same asymptotic properties as the solution of the inhomogeneous linearized system. © 2022, Springer Science+Business Media, LLC, part of Springer Nature.