Conditions for the existence of bounded solutions of nonlinear differential and functional differential equations

Abstract

Let E be a finite-dimensional Banach space, let C0(R; E) be a Banach space of functions continuous and bounded on R and taking values in E; let K:C 0(R ,E) → C 0(R, E) be a c-continuous bounded mapping, let A: E → E be a linear continuous mapping, and let h ∈ C 0(R, E). We establish conditions for the existence of bounded solutions of the nonlinear equation $$ \frac{{dx(t)}}{{dt}} + \left( {Kx} \right)(t)Ax(t) = h(t),\quad t \in \mathbb{R} $$