Some new Opial-type inequalities
TL;DR: In this paper, some new Opial-type integrodifferential inequalities in one variable are established, which generalize the existing ones which have a wide range of applications in the study of differential and integral equations.
Abstract: 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.
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TL;DR: In this article, some new Opial-type integral inequalities of many functions in many variables are established, which generalize the existing ones which have a wide range of applications in the study of differential and integral equations.
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TL;DR: In this paper, the Opial's inequality was generalized to the case of functions of n variables, m, n ≥ 1, where m is the number of variables in the function.
Abstract: In this paper, the Opial's inequality, which has a wide range of applications in the study of differential and integral equations, is generalized to the case involving m functions of n variables, m, n ≥ 1.
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TL;DR: In this article, a very general Opial-type inequalities involving higher-order derivatives of two functions are presented, and extended and improved versions of several recent results are derived from these inequalities.
Abstract: In this paper we shall offer very general Opial-type inequalities involving higher order derivatives of two functions. From these inequalities we then deduce extended and improved versions of several recent results.
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43 citations
01 Jan 1969
TL;DR: In this article, a sharper version of Opial's original inequality was obtained for linear differential equations of order n, where y(n-1) = O for i=O, 1, n -1 where n? 1.
Abstract: 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,
41 citations