SOME WEIGHTED OPIAL-TYPE INEQUALITIES ON TIME SCALES

Some Opial-Type Inequalities on Time Scales

and Applied Analysis 3 where r t is positive and continuous function with ∫X a dt/r t < ∞, and if x b 0, then ∫b X |x t |∣∣x′ t ∣∣dt ≤ 1 2 ∫b

Opial inequalities on time scales

We present a version of Opial’s inequality for time scales and point out some of its applications to so-called dynamic equations. Such dynamic equations contain both differential equations and difference equations as special cases. Various extensions of our inequality are offered as well.

On some Opial-type inequalities

* Correspondence: chjzhao@163. com Department of Mathematics, China Jiliang University, Hangzhou 310018, PR China Full list of author information is available at the end of the article Abstract In the present paper we establish some new Opial-type inequalities involving higherorder partial derivatives. Our results in special cases yield some of the recent results on Opial’s inequality and also ...

Some New Opial Dynamic Inequalities with Weighted Functions on Time Scales

In this paper we prove some new dynamic inequalities with two weight functions and some new dynamic inequalities with two unknown functions of Opial type on time scales. The main results will be proved by employing Hölder’s inequality, the chain rule and some basic algebraic inequalities. Mathematics subject classification (2010): 26A15, 26D10, 26D15, 39A13, 34A40.

Some Pachpatte type inequalities on time scales