**
Czechoslovak Mathematical Journal, Vol. 49, No. 4, pp. 791-809, 1999
**

#
On the mixed problem for hyperbolic

partial differential-functional equations

of the first order

##
Tomasz Czlapinski

Institute of Mathematics, University of Gdansk, Wit Stwosz Str. 57, 80-952 Gdansk, Poland

**Abstract:** We consider the mixed problem for the hyperbolic partial differential-functional equation of the first order

D_xz(x,y) = f(x,y,z_{(x,y)}, D_yz(x,y)),

where $z_{(x,y)} [-\tau,0] \times[0,h] \rightarrow\Bbb R$ is a function defined by $z_{(x,y)}(t,s) = z(x+t, y+s)$, $(t,s) \in[-\tau,0] \times[0,h]$. Using the method of bicharacteristics and the method of successive approximations for a certain integral-functional system we prove, under suitable assumptions, a theorem of the local existence of generalized solutions of this problem.

**Keywords:** partial differential-functional equations, mixed problem, generalized solutions, local existence, bicharacteristics, successive approximations

**Classification (MSC 1991):** 35D05, 35L60, 35R10

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