We prove an existence theorem of Cauchy-Kovalevskaya type for the equation $D_t u(t,z) = f(t,z,u(α^{(0)}(t,z)), D_z u(α^{(1)}(t,z)),...,D_z^k u(α^{(k)}(t,z)))$ where f is a polynomial with respect to the last k variables.
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The aim of our paper is to study oscillatory and asymptotic properties of solutions of nonlinear differential equations of the third order with deviating argument. In particular, we prove a comparison theorem for properties A and B as well as a comparison result on property A between nonlinear equations with and without deviating arguments. Our assumptions on nonlinearity f are related to its behavior only in a neighbourhood of zero and/or of infinity.
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