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

Tomasz Człapiński

doi:10.4171/zaa/773

JournalszaaVol. 16, No. 2pp. 463–478

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

Tomasz Człapiński
University of Gdansk, Poland

Download PDF

Abstract

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

D_{z} (x, y) = i = 1 \sum 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) \in [- r, a] x [- b, b + h] \ [0, a] \times [- b, b])

where $z_{(x, y)} : [- r, 0] \times [0, h] \to R$ 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.

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