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

The Cauchy problem for nonlinear first order partial differential-functional equations in unbounded domains is treated with a general class of the method of lines. Existence and convergence properties of the method are investigated under the assumption that the right-hand side of the equation satisfies the Lipschitz condition with respect to the functional argument. The theorems are proved by means of the differential-difference inequalities technique. Examples of differential-functional problems and corresponding methods of lines are given.