chapter  13
46 Pages

Chapter 13

Difference Inequalities in Several Independent Variables

Several Independent Variables Inequalities developed in Chapter 4 have natural extensions for functions

of 111 independent variables. These inequalities are used as a fundamental tool in the study of related partial difference equations. We begin this chapter with the recently established discrete analog of Riemann's function. This function is repeatedly used to study linear Gronwall type inequalities. Next we shall provide an upper estimate ou the Riemann's function which is quite adequate in practical applications and provides Wendroff's type estimates rather easily. This is followed by several nonlinear inequalities. Inequalities involving higher order differences in two independent variables arc also directly considered. For this the relevant Taylor's formula in two independent variables is induded. Next we move to multidimensional linear as well as nonlinear discrete inequalities, and wherever pos..o;iblc provide upper hounds in terms of known functions. This is followed by convolution type inequalities. Here the upper estimate appears in terms of discrete resolvent function. Finally, we shall develop Opial's and Wirtinger's type inequalities in two independent variables.