Existence and controllability for neutral partial differential inclusions nondenselly defined on a half-line

In this article, we study the existence of the integral solution to the neutral functional differential inclusion $${\frac{d}{dt}\mathcal{D}y_t-A\mathcal{D}y_t-Ly_t \in F(t,y_t), \quad\text{for a.e. }t \in J:=[0,\infty),\\  y_0=\phi \in C_E=C([-r,0];E),\quad r>0,}$$ and the controllability of the corresponding neutral inclusion $${\frac{d}{dt}\mathcal{D}y_t-A\mathcal{D}y_t-Ly_t \in F(t,y_t)+Bu(t),\quad  \text{for a.e. } t \in J,\\ y_0=\phi \in C_E,}$$ on a half-line via the nonlinear alternative of Leray-Schauder type for contractive multivalued mappings given by Frigon. We illustrate our results with  applications to a neutral partial differential inclusion with diffusion, and to a  neutral functional partial differential equation with obstacle constrains.