Nonlinear equations in non-reflexive Banach spaces and strongly nonlinear differential equations

In this paper, we study strongly nonlinear degenerate elliptic and parabolic equations of the form F(x,u,Du,...,D(2m-1)u,Lu) = 0 and ut = F(x,t,u,Du,...,D(2m-1)u,Lu), respectively, where L is a linear operator of the derivatives of highest (i.e., of 2m-th) order. Under very weak restrictions on the growth of F with respect to the derivatives of u, existence results for weak solutions are proved. These existence results are based on general solvability results for nonlinear operator equations in Banach spaces which will be proved in this paper