This paper gives sufficient conditions for existence and uniqueness of solutions and for the convergence of Picard iterations and more general waveform relaxation methods for differential-algebraic systems of neutral type. The results are obtained by the contraction mapping principle on Banach spaces with weighted norms and by the use of the Perron-Frobenius theory of nonnegative and nonreducible matrices. It is demonstrated that waveform relaxation methods are convergent faster than the classical Picard iterations.

We investigate continuous-time and discretized waveform relaxation iterations for functional differential systems of neutral type. It is demonstrated that continuous-time iterations converge linearly for neutral equations and superlinearly when the right hand side is independent of the history of the derivative of the solution. The error bounds for discretized iterations are also obtained and some implementation aspects are discussed. Numerical results are presented which indicate a potential speedup of this technique as compared with the classical approach based on discrete variable methods.

I. We consider the following two-points boundary value problem for functionaldifferential equation of neutral typeThe problems of thid type were considered in the literature, when the function f does not depend on the third argument (see for instance [a], [5]). In the present paper we consider existence and uniqueness of solution of problem (1) -(2), by the method which is some modification of the idea described in [5], (chap. 2, $ 2 , 3 ) . First let us change the unknown function x in equation (1) by putting t (3) 4) = Zo + J 4 s ) ds Y 0where xo E R R is fixed, then we get the equivalent problem: in the following form This means: if the pair (z0, z) is a solution of the system (1')-(2') then z defined by (3) is a solution of (1) -(2) and conversely.