A monotone method for the equation uxyz=f

摘要

Comparison theorems for the nonlinear boundary value problemuxyz =f(x, y, z, u, ux, uy, uz, uxy, uyz, uzx)with the boundary conditionsu(o, y, z) = α(y,z), u(x,0,z) = β(z, x), u(x, y, 0) = γ(x, y),α(y, 0) = γ(0, y), α(0, z), = β(z, 0), β(0, x) = γ(x, 0)are established under certain conditions. Then it is shown that there exist sequences of lower and upper solutions which converge uniformly and monotonically to minimal and maximal solutions respectively.