# 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

## Abstract

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

$z_{x}(x,y)=f_{(i)}(x,y,z(x,y),z,z_{y}(x,y))$

$z_{(i)}(x,y)=φ_{i}(x,y)for(x,y)∈[–τ_{0},0]×R_{n}(i=1,…,m)$

admits at most one solution if the function $f=(f_{(i)},…,f_{(m)}$ 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