On higher order hyperbolic partial differential equations (Q1173905)

The problem \[ \partial^{n+m}u(x,y)/\partial x^ n\partial y^ m=F(x,y,u(x,y),\partial^ nu(x,y)/\partial x^ n,\partial^ mu(x,y)/\partial y^ m) \] with given initial data for \(0\leq j\leq m-1\), \(0\leq i\leq n-1\) \[ \partial^ ju(x,0)/\partial y^ j=\alpha_ j(x), \quad 0<x<a; \quad \partial^ iu(0,y)/\partial x^ i=\beta_ i(y), \quad 0<y<b, \] is studied. The problem is converted into an integral equation, which is solved by direct iterations, given some rather technical conditions upon \(F\). Uniqueness and continuous dependence upon data is also discussed.