Bounds for Certain Delay Integral Inequalities on Time Scales

The unification and extension of differential equations, difference equations, q-difference equations, and so on to the encompassing theory of dynamic equations on time scales was initiated by Hilger 1 in his Ph.D. thesis in 1988. During the last few years, some integral inequalities on time scales related to certain inequalities arising in the theory of dynamic equations had been established by many scholars. For example, we refer the reader to literatures 2–8 and the references therein. However, nobody studied the delay integral inequalities on time scales, as far as we know. In this paper, we investigate some delay integral inequalities on time scales, which provide explicit bounds on unknown functions. Our results extend some known results in 9 . Throughout this paper, a knowledge and understanding of time scales and time scale notation is assumed. For an excellent introduction to the calculus on time scales, we refer the reader to monographes 10, 11 .