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We establish the existence of solutions of fully nonlinear parabolicsecond-order equations like $\partial_{t}u+H(v,Dv,D^{2}v,t,x)=0$ in smoothcylinders without requiring $H$ to be convex or concave with respect to thesecond-order derivatives. Apart from ellipticity nothing is required of $H$ atpoints at which $|D^{2}v|\leq K$, where $K$ is any fixed constant. For large$|D^{2}v|$ some kind of relaxed convexity assumption with respect to $D^{2}v$mixed with a VMO condition with respect to $t,x$ are still imposed. Thesolutions are sought in Sobolev classes. We also establish the solvabilitywithout almost any conditions on $H$, apart from ellipticity, but of a"cut-off" version of the equation $\partial_{t}u+H(v,Dv,D^{2}v,t,x)=0$.

On the existence of $W^{1,2}_{p}$ solutions for fully nonlinear parabolic equations under either relaxed or no convexity assumptions