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We prove an existence theorem of Cauchy-Kovalevskaya type for the equation $D_{t}u(t,z)=f(t,z,u\left({\alpha }^{\left(0\right)}(t,z)\right),D_{z}u\left({\alpha }^{\left(1\right)}(t,z)\right),...,D_z^{k}u\left({\alpha }^{\left(k\right)}(t,z)\right))$ where f is a polynomial with respect to the last k variables.

We answer some questions concerning Perron and Kamke comparison functions satisfying the Carathéodory condition. In particular, we show that a Perron function multiplied by a constant may not be a Perron function, and that not every comparison function is bounded by a comparison function with separated variables. Moreover, we investigate when a sum of Perron functions is a Perron function. We also present a criterion for comparison functions with separated variables.

It is proved that nonincreasing and satisfying the Volterra condition right-hand side of a functional differential equation does not guarantee the uniqueness of solutions.

We obtain existence of absolutely continuous extremal solutions of the problem $u^{\text{'}}\left(x\right)=F(x,u\left(x\right),u\left(h\left(x\right)\right))$, $u\left(0\right)=u_0$, and the Darboux problem for $u_{xy}(x,y)=G(x,y,u(x,y),u\left(H(x,y)\right))$, where $h$ and $H$ are arbitrary continuous deviated arguments.