Under the assumptions of continuity and the existence of first-order partial derivatives of the solutions it is proved that the Cauchy problem

$$
z_x^{(i)} (x, y) = f^{(i)} (x, y, z (x, y), z, z_y^{(i)} (x, y))
$$

$$
z^{(i)} (x, y) = \varphi_i (x, y) for (x, y) \in [–\tau_0, 0] \times \mathbb R^n \;\; (i= 1, \dots, m)
$$

admits at most one solution if the function $f = (f^{(i)}, \dots, f^{(m)}$ of the variables $(x, y, p, z, q)$ satisfies a Lipschitz condition with respect to $(p, z, q)$, or a Lipschitz condition with respect to $(p, z)$ and a Hölder condition with respect to $q$.

On Solutions of First-Order Partial Differential-Functional Equations in an Unbounded Domain

Zdzisław Kamont

Katarzyna Prządka

Abstract

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