# On the Mixed Problem for Quasilinear Partial Differential-Functional Equations of the First Order

### Tomasz Człapiński

University of Gdansk, Poland

## Abstract

We consider the mixed problem for the quasilinear partial differential-functional equation of the first order

$D_{z}(x,y)=i=1∑n f_{i}(x,y,z_{(x,y)}D_{y,i}z(x,y)+G(x,y,z_{(x,y)})$

$z(x,y)=ϕ(x,y)((x,y)∈[−r,a]x[−b,b+h]\[0,a]×[−b,b])$

where $z_{(x,y)}:[−r,0]×[0,h]→R$ is a function defined by $z_{(x,y)}(t,s)=z(x+t,y+s)$ for $(t,s)∈[−r,0]×[0,h]$. Using the method of characteristics and the fixed-point method we prove, under suitable assumptions, a theorem on the local existence and uniqueness of solutions of the problem.

## Cite this article

Tomasz Człapiński, On the Mixed Problem for Quasilinear Partial Differential-Functional Equations of the First Order. Z. Anal. Anwend. 16 (1997), no. 2, pp. 463–478

DOI 10.4171/ZAA/773