(1971). Integral Inequalities Involving a Function and its Derivative. The American Mathematical Monthly: Vol. 78, No. 7, pp. 705-741.

Integral Inequalities Involving a Function and its Derivative

Opial-type integral inequalities involving several higher order derivatives

On Opial-type integral inequalities

In this paper some new Opial-type integrodifferential inequalities in one variable are established. These generalize the existing ones which have a wide range of applications in the study of differential and integral equations.

Some new Opial-type inequalities

Motivated essentially by several recent investigations which claimed to have generalized, improved and extended such classical inequalities as the well-known Opial's inequality, here we establish some general weighted Opial-type inequalities on time scales. We also provide counterexamples, corrections and modifications of the aforementioned recent claims by Wong {\it et al.} \noindent [{\em Taiwanese J. Math.}, \textbf{12} (2008), 463-471].

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Some weighted opial-type inequalities on time scales

rb 1 b (1) fJ |yy(n) I dx (b ? a)n1 I y(n) 12dx. 2 (He employs (1) to prove uniqueness of the initial value problem for linear differential equations of order n.) Also there are generalizations of Opial's original inequality in other directions. (See, for instance, Calvert [2] and Yang [6].) The purpose of this note is to obtain a sharper version of (1) and other generalizations. THEOREM 1. Let yE C(n-1) [a, b ] be such that y() (a) = O for i=O, 1, n -1 where n ? 1. Let y(n-1) be absolutely continuous and y (n) 1 2 dx < oo . Then,

/pdf/an-inequality-similar-to-opial-s-inequality-1xwqbveaxp.pdf

K. M. Das

Papers

An inequality similar to Opial’s inequality