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Zeitschrift für Analysis und ihre Anwendungen


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Volume 6, Issue 2, 1987, pp. 121–132
DOI: 10.4171/ZAA/235

Published online: 1987-04-30

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

Zdzisław Kamont[1] and Katarzyna Prządka

(1) University of Gdansk, Poland

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 $q$.

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Kamont Zdzisław, Prządka Katarzyna: On Solutions of First-Order Partial Differential-Functional Equations in an Unbounded Domain. Z. Anal. Anwend. 6 (1987), 121-132. doi: 10.4171/ZAA/235