SOME RESULTS OF OPIAL-TYPE INTEGRAL INEQUALITIES

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Date

2018-03

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UNIVERSITY OF ILORIN

Abstract

Inequalities are essential branches of Mathematics and are useful tools in some theory of analyses. Some inequalities such as Wirtinger’s, Holder’s, Cauchy’s, Minkowski’s, Hardy’s and Opial’s inequalities are more currently used by researchers and these are of particular interest in this study. Opial inequality and its generalisations have various applications in the theories of differential and difference equations. Opial inequality on time-scale (an arbitrary non-empty closed subset of real numbers) was introduced by Hilger in order to unify discrete and continuous analyses. Little work has been done on the determination of best possible constant for Opial-type inequalities on time-scales, multi-function and higher order delta derivatives. This study aimed at generalising Opial-type inequalities with best possible constant. The objectives of the study were to: (i) obtain some deductions from Opial-type inequalities of Shum, Pachpatte, Olech, Calvert, Fabelurin, Oguntuase, Beesack, Yang, and Maroni; (ii) investigate the nature of the deduced Opial-type inequalities; (iii) obtain sharp bound of the new Opial-type inequalities; and (iv) generalise Opial-type inequalities on time-scales. The methodology adopted was based on the definite and indefinite integrals with modified Jensen’s inequality. The theorem relating to this inequality states that: Let be convex. Then, for each , there exist such that Then implies The findings of the study were that: (i) deductions from Shum(1974), Pachpatte(1986), Olech (1962), Calvert(1967), Fabelurin(2010), Oguntuase(2009), Beesack(1962), Yang(1983), and Maroni(1967) results were generalised; (ii) the properties of Opial-type inequalities involving integral of functions and their derivatives were verified; (iii) sharp bounds for the new Opial-type inequalities were obtained; and (iv) Opial-type inequalities were generalised on time-scales. The study concluded that modified Jensen’s inequality was useful in refining and extending Opial-type inequalities in order to generalise some results on time-scales. The results recommended that refined Jensen’s inequality is sufficient in obtaining Opial-type inequalities.

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Keywords

OPIAL-TYPE, INTEGRAL INEQUALITIES, Inequalities, Opial inequality

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