Problems with mixed boundary conditions in Banach spaces

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 $$\{\begin{array}{lll}(\phi (u^{\mathrm{\prime }}){)}^{\mathrm{\prime }}=f(t,u,u^{\mathrm{\prime }})& & \\ u(T)=0=u^{\mathrm{\prime }}(0),& & \end{array}$$ where $\varphi: X\rightarrow X $isahomeomorphismwithreverseLipschitzsuchthat$\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.