Hyperbolic partial differential equations in a complete linear normed vector space

Abstract

Existence and uniqueness results for the abstract partial differential equation of the form uxy=F( x,y,u,ux,uy), where values of the unknown function lies in an arbitrary Banach space. The solution is approximated by abstract"hyperbolic paraboloids", in a manner analogous to the classical method used in the existence proof for the ordinary differential equation dy/dx=F(x,y) using Cauchy-Euler polygons. Convergence is proven using the abstract versions of the theorems of Ascoli and Arzela; the latter in the case where derivatives of the solution are involved. Also used is a little known theorem of Mazur in proving pointwise compactness (as required by the Ascoli and Arzela theorems) of the aforementioned approximating sequences.