On Hyperbolic Partial Differential Equations

Abstract

where p = zx, q = zy, it is assumed that f is continuous in (x, y, z, p, q) and satisfies a uniform Lipschitz conditioni with respect to (z, p, q). It will be shown (Section 2) that the assumption of a Lipschitz. condition with respect to z can be omitted in these existence theorems, though not in the uniqueness theorems. On the other hand, it will be shown in Section 3 by an example that the Lipschitz condition with respect to (p, q) cannot be omitted and, what is more, that there exist continiuous f such that (1) has no solution whatsoever (in a vicinity of a given point of the (x, y) -plane). The existence theorem to be proved for (1) leads to improvements of some of the results of H. Lewy ([14]; cf. the presentation in [6], pp. 487508). For example, it will be shown (Section 5) that if F(x, y, z, p, q, r, s, t) is a function of class C2, and if the partial differential equation