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We denote by R" the n-dimensional real vector space with the norm Its II = max1515lsl (s = s) € R') and by M(m,k) the space of real m x k matrices. Furthermore by C(X,Y) we denote the usual space of continuous functions from Xto Yand by L1(I,R+) the usual space of Lebesgue integrable functions, I C R being an interval and R+ = [0, +co). Let Jbe the set of all functions (P = (p .... . pm) E C([-h, 0] x R T Rm ), h "?: 0, such that, for some A, r € R, and W € L 1 ([-h, 01, R+),

On the Cauchy Problem for Quasilinear Hyperbolic Systems of Partial Differential-Functional Equations of the First Order