Perron's method for quasilinear hyperbolic systems, Part II

In part I (P. Smith, Perron's method for quasilinear hyperbolic systems, part I, J. Math. Anal., in press) of this paper we defined a notion of viscosity solution (sub- (super-)solution) for these systems, proved a comparison principle for viscosity sub- and supersolutions. Here, in part II, we prove existence of viscosity solutions to the Cauchy problem, using a Perron-like method, for long time, and for all time.