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

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

Dz(x,y)=i=1nfi(x,y,z(x,y)Dy,iz(x,y)+G(x,y,z(x,y))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)=ϕ(x,y)   ((x,y)[r,a]x[b,b+h]\[0,a]×[b,b])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]×[0,h]Rz_{(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)z_{(x,y)}(t,s) = z(x + t,y + s) for (t,s)[r,0]×[0,h](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