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

Zdzisław Kamont; Katarzyna Prządka

doi:10.4171/zaa/235

JournalszaaVol. 6, No. 2pp. 121–132

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

Zdzisław Kamont
University of Gdansk, Poland
Katarzyna Prządka
University of Gdansk, Poland

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Abstract

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) = φ_{i} (x, y) f or (x, y) \in [- τ_{0}, 0] \times 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 $q$ .

Cite this article

Zdzisław Kamont, Katarzyna Prządka, On Solutions of First-Order Partial Differential-Functional Equations in an Unbounded Domain. Z. Anal. Anwend. 6 (1987), no. 2, pp. 121–132

DOI 10.4171/ZAA/235