We propose a necessary and sufficient condition about the existence of variations, i.e., of non trivial solutions $\eta \in W_0^{1,\infty }\left(\Omega \right)$ to the differential inclusion $\nabla \eta \left(x\right)\in -\nabla u\left(x\right)+𝐃$.

We consider the mixed problem for the quasilinear partial functional differential equation with unbounded delay $D_{t}z(t,x)={\sum }_{i=1}^n{f}_i(t,x,z_{(t,x)})D_{x_i}z(t,x)+h(t,x,z_{(t,x)})$, where $z_{(t,x)}\in X{̶}_0$ is defined by $z_{(t,x)}(\tau ,s)=z(t+\tau ,x+s)$, $(\tau ,s)\in (-\infty ,0]\times [0,r]$, and the phase space $X{̶}_0$ satisfies suitable axioms. Using the method of bicharacteristics and the fixed-point method we prove a theorem on the local existence and uniqueness of Carathéodory solutions of the mixed problem.

We consider viscosity solutions for first order differential-functional equations. Uniqueness theorems for initial, mixed, and boundary value problems are presented. Our theorems include some results for generalized ("almost everywhere") solutions.