ABSTRACT
It is well recognized that the inequalities furnish a very general comparison principle in studying many qualitative as well as quantitative properties of solutions of related equations. The celebrated Gronwall's inequality is but one of the examples for a monotone operator /C in which the exact solution of w = p + /Cw provides an upper bound on all solutions of the inequality u ~ p + /Cu. On the basis of various motivations this inequality has been extended and applied in various contexts. We begin this ehapter with Gronwall type inequalities, and include, in particular, the pradkally important case of weakly singular disc::rete kernels. This is followed by several nonlinear versions of Gronwall inequality which have been established recently and arc of immense value. To deal with inequalities involving higher order differences a usual procedure is to convert them to their equivalent systems and then, either obtain the estimates in terms of maximal solutions of the related difference systems; or use a suitable norm and treat the resulting inequalities as in the sc::alar case, which provides uniform bounds for all the components of the systems. In Section 4.3 we shall deal with these type of inequalities directly and obtain the estimates in terms of known functions. Then, we shall move to finite linear as well as nonlinear difference inequalities and wherever possible provide upper bounds in terms of known quantities. In Sections 4.5 and 4.6, respectively, we shall consider discrete Opial and Wirtinger type inequalities.
