Higher-order estimates for fully nonlinear difference equations. I

The purpose of this work is to establish a priori C-2,C-alpha estimates for mesh function solutions of nonlinear positive difference equations in full y nonlinear form on a uniform mesh, where the fully nonlinear finite-differ ence operator F-h is concave in the second-order variables. The estimate is an analogue of the corresponding estimate for solutions of concave fully n onlinear elliptic partial differential equations. We deal here with the spe cial case that the operator does not depend explicitly upon the independent variables. We do this by discretizing the approach of Evans for fully nonl inear elliptic partial differential equations using the discrete linear the ory of Kuo and Trudinger. The result in this special case forms the basis f or a more general result in part II. We also derive the discrete interpolat ion inequalities needed to obtain estimates for the interior C-2,C-alpha se mi-norm terms of the C-0 norm.