Conditions are given under
which a quasi-linear differential equation has at least one solution in a given compact
interval that satisfies a given system of homogeneous or nonhomogeneous linear
constraints. These conditions are not formulated in the space in which the solutions
take their values, as is usually done; instead they involve the set of continuous
mappings subject to the constraints and the set of forcing terms for which
the associated nonhomogeneous linear differential equation has solutions
satisfying the constraints. The latter set is, under mild conditions, a topological
direct summand of the space of continuous mappings. This occurs in the
problem of the existence of periodic solutions which is discussed in detail as
illustration.
