The method of lines for first order partial differential-functional equations

The Cauchy problem for nonlinear first order partial differential-functiona l equations in unbounded domains is treated with a general class of the met hod of lines. Existence and convergence properties of the method are invest igated under the assumption that the right-hand side of the equation satisf ies the Lipschitz condition with respect to the functional argument. The th eorems are proved by means of the differential-difference inequalities tech nique. Examples of differential-functional problems and corresponding metho ds of lines are given.