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

Abstract

Let $E$ be a finite-dimensional Banach space, let $C^0(R; E)$ 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 $$\frac{dx(t)}{dt} + (Kx)(t)Ax(t) = h(t),\;t ∈ R.$$