  Using Leray-Schauder degree or degree for $\alpha$-condensing maps we obtain
the existence of at least one solution for the boundary value problem of the
type \[ \left\{\begin{array}{lll} (\varphi(u' ))' = f(t,u,u') & & \\
u(T)=0=u'(0), & & \quad \quad \end{array}\right. \] where $\varphi:
X\rightarrow X $ is a homeomorphism with reverse Lipschitz such that
$\varphi(0)=0$, $f:\left[0, T\right]\times X \times X \rightarrow X $ is a
continuous function, $T$ a positive real number and $X$ is a real Banach space.
