Generalized solutions of the Cauchy problem for a nonlinear functional partial differential equation (Q1116014)

The aim of the paper is to seek generalized solutions in the sense ``almost everywhere'' for the following problem \[ (1)\quad D_ xu(x,y)=F(x,y,u(x,y),(Vu)(x,y),D_ yu(x,y)),\quad (2)\quad u(0,y)=\phi (y). \] The method applied is of fixed point type. It is based on defining an operator, whose range consists of solutions of suitable equations without functional argument. By applying differential inequalities it is proved that this operator is a contraction, and its fixed point is a solution of problem (1), (2).