- Open Access
On Opial's type inequalities for an integral operator with homogeneous kernel
© Zhao and Cheung; licensee Springer. 2012
- Received: 8 February 2012
- Accepted: 30 May 2012
- Published: 30 May 2012
In this article, we establish some new Opial's type integral inequalities for an integral operator with homogeneous kernel. The results in special cases yield some of the interrelated results and provide new estimates on inequalities of this type.
MS (2000) Subject Classification: 26D15.
- Opial's inequality
- Hölder's inequality
- Jensen's inequality
- integral operator
- homogeneous kernel
In 1960, Opial  established the following inequality:
The Opial's type inequality was first established by Willett :
A non-trivial generalization of Theorem 1.2 was established by Hua :
A sharper inequality was established by Godunova :
Opial's inequality and its generalizations, extensions, and discretizations play an important role in establishing the existence and uniqueness of initial and boundary value problems for ordinary and partial differential equations as well as difference equations [5–13]. For Opial-type integral inequalities involving high-order partial derivatives, we refer the readers to see [14, 15]. For an extensive survey on these inequalities, see . Mitrinović and Pečarić  proved some new extensions of Opial's type inequalities. The aim of this article is to establish some Opial-type inequalities, which are some extensions of Godunova and Levin's, Mitrnović and Pečarić inequalities.
where f(x) is a continuous function on [0, + ∞), and K(y, x) is a non-negative homogeneous kernel function defined on [0, + ∞) × [0, + ∞), such that T(f)y > 0 if f(x) > 0, where x ∈ [0, + ∞).
then y(t) is the derivative of order λ of x(t) in the sense of Riemann-Liouville. Thus, if x(t) is differentiable, then for λ = 1, it follows that y(t) = x'(t).
Further, taking for f(x, y) = f(x) and K(t, s) = K λ (t, s) in (2.3), (2.3) reduces to a result of Godunova and Levin .
This is just a new result established by Mitrinović and Pečarić .
This completes the proof.
Proof of Theorem 2.4 From the hypotheses of Theorem 2.4 and in view of Hölder's inequality,
This completes the proof.
The research of Chang-Jian Zhao was supported by the National Natural Science Foundation of China (10971205), and the research of Wing-Sum Cheung was partially supported by a HKU URC grant.
- Opial Z: Sur une inégalité. Ann Polon Math 1960, 8: 29–32.MathSciNetGoogle Scholar
- Willett D: The existence-uniqueness theorem for an n th order linear ordinary differential equation. Am Math Monthly 1968, 75: 174–178. 10.2307/2315901MathSciNetView ArticleGoogle Scholar
- Hua LK: On an inequality of Opial. Sci China 1965, 14: 789–790.Google Scholar
- Godunova EK: Integral'nye neravenstva s proizvodnysi i proizvol'nymi vypuklymi funkcijami. Uc Zap Mosk Gos Ped In-ta im Lenina 1972, 460: 58–65.Google Scholar
- Rozanova GI: On an inequality of Maroni (Russian). Math Zametki 1967, 2: 221–224.Google Scholar
- Das KM: An inequality similar to Opial's inequality. Proc Am Math Soc 1969, 22: 258–261.Google Scholar
- Agarwal RP, Thandapani E: On some new integrodifferential inequalities. Anal sti Univ ''Al I Cuza'' din Iasi 1982, 28: 123–126.MathSciNetGoogle Scholar
- Yang GS: A note on inequality similar to Opial inequality. Tamkang J Math 1987, 18: 101–104.MathSciNetGoogle Scholar
- Agarwal RP, Lakshmikantham V: Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations. World Scientific, Singapore; 1993.View ArticleGoogle Scholar
- Bainov D, Simeonov P: Integral Inequalities and Applications. Kluwer Academic Publishers, Dordrecht; 1992.View ArticleGoogle Scholar
- Li JD: Opial-type integral inequalities involving several higher order derivatives. J Math Anal Appl 1992, 167: 98–100. 10.1016/0022-247X(92)90238-9MathSciNetView ArticleGoogle Scholar
- Cheung WS: On Opial-type inequalities in two variables. Aequationes Math 1989, 38: 236–244. 10.1007/BF01840008MathSciNetView ArticleGoogle Scholar
- Cheung WS: Some generalized Opial-type inequalities. J Math Anal Appl 1991, 162: 317–321. 10.1016/0022-247X(91)90152-PMathSciNetView ArticleGoogle Scholar
- Zhao CJ, Cheung WS: Sharp integral inequalities involving high-order partial derivatives. J Inequal Appl 2008, 2008: 1–10. Article ID 571417Google Scholar
- Agarwal RP, Pang PYH: Sharp opial-type inequalities in two variables. Appl Anal 1996, 56(3):227–242.MathSciNetGoogle Scholar
- Agarwal RP, Pang PYH: Opial Inequalities with Applications in Differential and Difference Equations. Kluwer Academic Publishers, Dordrecht; 1995.View ArticleGoogle Scholar
- Mitrinović DS, Pečarić JE: Generalizations of two inequalities of Godunova and Levin. Polish Acad Sci Math 1988, 36: 645–648.Google Scholar
- Godunova EK, Levin VI: O neikotoryh integralnyh nerabenstvah, soderzascih proizvodnye (Russian). Izv vuzov Matem 1969, 91: 20–24.Google Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.