Gronwall-OuIang-Type Integral Inequalities on Time Scales

OuIang inequalities and their generalizations have proved to be useful tools in oscillation theory, boundedness theory, stability theory, and other applications of differential and difference equations. A nice introduction to continuous and discrete OuIang inequalities can be found in 1, 2 , and studies in 3–5 give some of their generalizations to multiple integrals and higher-dimensional spaces. Like Gronwall’s inequality, OuIang’s inequality is also used to obtain a priori bounds on unknown functions. Therefore, integral inequalities of this type are usually known as Gronwall-OuIang-type inequalities 6 . The calculus on time scales has been introduced by Hilger 7 in order to unify discrete and continuous analysis. For the general basic ideas and background, we refer to 8, 9 . In this paper, we are concerned with Gronwall-OuIang-type integral inequalities on time scales, which unify and extend the corresponding continuous inequalities and their discrete analogues. We also provide a more useful and explicit bound than that in 10–12 .