Abstract
Here we study L p; p > 0; fractional Opial-type inequalities subject to high-order boundary conditions. They involve the right and left Caputo, Riemann–Liouville fractional derivatives. These derivatives are blended together into the balanced Caputo, Riemann–Liouville, respectively, fractional derivative. We give an application to a balanced fractional boundary value problem by proving uniqueness of the solution. This chapter relies on [7].
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
R. P. Agarwal and P. Y. H. Pang, Opial Inequalities with Applications in Differential and Difference Equations, Klower, Dordrecht, London, 1995.
C.D. Aliprantis and O. Burkinshaw “Principles of Real Analysis”, 3rd Edition, Academic Press, Boston, New York, 1998.
G. A. Anastassiou, Fractional Differentiation Inequalities, Research Monograph, Springer, New York, 2009.
G. A. Anastassiou, On Right Fractional Calculus, “Chaos, Solitons and Fractals”, 42(2009), 365-376.
G. A. Anastassiou, “Balanced fractional Opial inequalities”, Chaos Solitons Fractals, 42(2009), no.3, 1523-1528.
G. A. Anastassiou, Fractional Korovkin Theory, 42(2009), 2080-2094.
G. A. Anastassiou, Opial-type inequalities for functions and their ordinary and balanced fractional derivatives, submitted, 2011.
P. R. Beesack, On an integral inequality of Z. Opial, Trans. Amer. Math. Soc. 104 (1962), 470-475.
Kai Diethelm, Fractional Differential Equations, online: http://www.tubs.de/  ˜ diethelm/lehre/ f-dg102/fde-skript.ps.gz.
A. M. A. El-Sayed and M. Gaber, On the finite Caputo and finite Riesz derivatives, Electronic Journal of Theoretical Physics, Vol. 3, No. 12(2006), 81-95.
G. S. Frederico and D. F. M. Torres, Fractional Optimal Control in the sense of Caputo and the fractional Noether’s theorem, International Mathematical Forum, Vol.3, No. 10 (2008), 479-493.
R. Gorenflo and F. Mainardi, Essentials of Fractional Calculus, 2000, Maphysto Center, http://www.maphysto.dk/oldpages/events/LevyCAC2000/Mainardi Notes/fm2k0a.ps
C. Olech, A simple proof of a certain result of Z. Opial, Ann. Polon. Math. 8 (1960), 61-63.
Z. Opial, Sur une inegalite, Ann. Polon. Math. 8 (1960), 29-32.
S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, (Gordon and Breach, Amsterdam, 1993) [English translation from the Russian, Integrals and Derivatives of Fractional Order and Some of Their Applications (Nauka i Tekhnika, Minsk, 1987)].
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2011 George A. Anastassiou
About this chapter
Cite this chapter
Anastassiou, G.A. (2011). Opial-Type Inequalities for Balanced Fractional Derivatives. In: Advances on Fractional Inequalities. SpringerBriefs in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0703-4_1
Download citation
DOI: https://doi.org/10.1007/978-1-4614-0703-4_1
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-0702-7
Online ISBN: 978-1-4614-0703-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)