Let E be a Banach space. We consider a Cauchy problem of the type ⎧ $D_t^{k}u+{\sum }_{j=0}^{k-1}{\sum }_{\left|\alpha \right|\le m}{A}_{j,\alpha }\left(D_t^j{D}_x^{\alpha }u\right)=f$ in ${ℝ}^{n+1}$, ⎨ ⎩ $D_t^{j}u(0,x)={\phi }_j\left(x\right)$ in ${ℝ}^n$, j=0,...,k-1, where each $A_{j,\alpha }$ is a given continuous linear operator from E into itself. We prove that if the operators $A_{j,\alpha }$ are nilpotent and pairwise commuting, then the problem is well-posed in the space of all functions $u\in C^{\infty }({ℝ}^{n+1},E)$ whose derivatives are equi-bounded on each bounded subset of ${ℝ}^{n+1}$.

We prove that Perron's method and the method of half-relaxed limits of Barles-Perthame works for the so called B-continuous viscosity solutions of a large class of fully nonlinear unbounded partial differential equations in Hilbert spaces. Perron's method extends the existence of B-continuous viscosity solutions to many new equations that are not of Bellman type. The method of half-relaxed limits allows limiting operations with viscosity solutions without any a priori estimates. Possible applications...