  We prove results of existence of a solution (resp. existence and uniqness of
a solution) for nonlinear differential equations of type $x'(t) +G(x,t) x(t) =
F(x,t),$ in an abstract Banach subspace $X$ of the space of bounded real-valued
continuous functions, satisfying some general and natural property. In our
work, the functions $F$ and $G$ jointly depend on the variables $(x,t)\in
X\times \R$. Several examples will be given, in various function spaces, to
illustrate our results. The vector-valued framework is also considered.
