\documentstyle[12pt]{article}
\title{One-step difference methods for mixed type
       diferrential-functional equations}
\author{Danuta Jaruszewska-Walczak}
\date{}
\begin{document}
\maketitle

Suppose that $E=(0,a) \times (-b,b) $ where $a>0,\;b=(b_{1},...,b_{n})
\in R^{n},\;b_{i}>0,\;i=1,...,n$ and $B=[-\tau_{0},0] \times [-\tau,\tau],\;
\tau_{0} \in R_{+},\;\tau=(\tau_{1},...,\tau_{n}) \in R^{n}_{+}.$
For $c=b+\tau$ let $E_{0}=[-\tau_{0},0] \times [-c,c]$ and
$\partial_{0}E=(0,a) \times ([-c,c] \setminus (-b,b))$.

For a function $z:[-\tau_{0},a) \times [-c,c] \rightarrow R$ and for a point
$(x,y) \in E$ we define the function $z_{(x,y)}: B \rightarrow R$ by the
formula
\[z_{(x,y)}(t,s)=z(x+t,y+s),\;(t,s) \in B.\]

Suppose that $f:E \times C(B,R) \rightarrow R$ and $\varphi:E_{0} \cup
\partial_{0}E \rightarrow R$ are given functions. Consider the initial -
boundary value problem
\begin{eqnarray}
D_{x}z(x,y)&=&f(x,y,z_{(x,y)})\nonumber\\
z(x,y)&=&\varphi(x,y),\;(x,y) \in E_{0} \cup \partial_{0}E,\label{rr}
\end{eqnarray}
where $D_{x}z$ denotes the derivative of $z$ with respect to $x$.

We consider the Euler difference - functional method for the problem
(\ref{rr}) and the one - step methods more effective than this one. We
give a constructive way to obtain such methods.
\end{document}