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Zeitschrift für Analysis und ihre Anwendungen


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Volume 16, Issue 2, 1997, pp. 463–478
DOI: 10.4171/ZAA/773

Published online: 1997-06-30

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

Tomasz Człapiński[1]

(1) University of Gdansk, Poland

We consider the mixed problem for the quasilinear partial differential-functional equation of the first order $$D_z(x,y) = \sum^n_{i=1}f_i(x, y, z_{(x,y)} D_{y,i}z(x, y) + G(x, y, z_{(x,y)}) $$ $$z(x,y) = \phi(x,y) \ \ \ ((x,y) \in [-r,a] x [-b,b + h] \backslash [0,a] \times [-b,b])$$ where $z_{(x,y)} : [-r,0] \times [0,h] \to \mathbb R$ is a function defined by $z_{(x,y)}(t,s) = z(x + t,y + s)$ for $(t,s) \in [-r,0] \times [0,h]$. 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.

Keywords: Partial differential-functional equations, classical solutions, local existence, bicharacteristics, fixed-point theorem

Człapiński Tomasz: On the Mixed Problem for Quasilinear Partial Differential-Functional Equations of the First Order. Z. Anal. Anwend. 16 (1997), 463-478. doi: 10.4171/ZAA/773