Abstract
Abstract We establish some new dynamic Opial-type diamond alpha inequalities in time scales. Our results in special cases yield some of the recent results on Opial's inequality and also provide new estimates on inequalities of this type. Also, we introduce an example to illustrate our result. Mathematics Subject Classification 2000: 39A12; 26D15; 49K05
Highlights
Introduction and PreliminariesIn 1960, the Polish Mathematician Zdzidlaw Opial [1] published an inequality involving integrals of functions and their derivatives; h h| x(t)x (t)|dt ≤ h | x (t)|2dt 4 (1:1)where x Î C1[0, h], x(0) = x(h) = 0 and x(t) >0 in (0, h), and the constant h/4 is the best possible.Inequalities which involve integrals of functions and their derivatives are of great importance in mathematics with applications in the theory of differential equations, approximations and probability
The monograph [2] is the first book dedicated to the theory of Opial type inequalities
First we introduce a set of Opial type Diamond-alpha Inequalities obtained by Bohner-Duman [11]
Summary
The positivity requirement of x(t) in the original proof of Opial was shown to be unnecessary later by Olech [3] where he proved that the inequality (1.1) holds even for functions x(t) which are only absolutely continuous in [0, h]. (Wong’s inequality) Let {xi}τi=0 be a non-decreasing sequence of nonnegative numbers, and x0 = 0. Time-scale set-up of basic Opial type inequality For convenience we recall the following easiest versions of Opial’s inequality.
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