Wirtinger type inequalities via fractional integral operators
DOI:
https://doi.org/10.24193/subbmath.2019.1.04Keywords:
Fractional derivative, fractional integral, Wirtinger inequalityAbstract
In this study, we shall present Wirtinger type inequality in the fractional
case with comformable fractional operators.
References
bibitem{podlubny} Podlubny, Igor Fractional differential equations. An
introduction to fractional derivatives, fractional differential equations,
to methods of their solution and some of their applications. Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999. xxiv+340 pp.
bibitem{baleanu1} Baleanu, Dumitru; Diethelm, Kai; Scalas, Enrico;
Trujillo, Juan J. Fractional calculus. Models and numerical methods. Series
on Complexity, Nonlinearity and Chaos, 3. World Scientific Publishing Co.
Pte. Ltd., Hackensack, NJ, 2012. xxiv+400 pp.
bibitem{diethelm} Diethelm, Kai The analysis of fractional differential
equations. An application-oriented exposition using differential operators
of Caputo type. Lecture Notes in Mathematics, 2004. Springer-Verlag, Berlin, 2010. viii+247 pp.
bibitem{khalil} Khalil, R.; Al Horani, M.; Yousef, A.; Sababheh, M. A new
definition of fractional derivative. J. Comput. Appl. Math. 264 (2014),
--70.
bibitem{hardy} Hardy, G. H.; Littlewood, J. E.; P'{o}lya, G. Inequalities. Reprint of the 1952 edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1988. xii+324 pp.
bibitem{mitrinovic} Mitrinovi'{c}, D. S.; Pev{c}ari'{c}, J. E.; Fink, A.
M. Classical and new inequalities in analysis. Mathematics and its
Applications (East European Series), 61. Kluwer Academic Publishers Group, Dordrecht, 1993. xviii+740 pp.
bibitem{pachpatte} Pachpatte, B. G. Mathematical inequalities.
North-Holland Mathematical Library, 67. Elsevier B. V., Amsterdam, 2005.
xii+591 pp.
bibitem{agarwal} Agarwal, Ravi P. Difference equations and inequalities.
Theory, methods, and applications. Monographs and Textbooks in Pure and Applied Mathematics, 155. Marcel Dekker, Inc., New York, 1992. xiv+777 pp.
bibitem{saker} Agarwal, Ravi; O'Regan, Donal; Saker, Samir Dynamic
inequalities on time scales. Springer, Cham, 2014. x+256 pp.
bibitem{ferreira} Ferreira, Rui A. C. A discrete fractional Gronwall
inequality. Proc. Amer. Math. Soc. 140 (2012), no. 5, 1605--1612.
bibitem{anastassiou} Anastassiou, George A. Multivariate fractional
representation formula and Ostrowski type inequality. Sarajevo J. Math.
(22) (2014), no. 1, 27--35.
bibitem{peterson} G"{u}venilir, A. Feza; Kaymakc{c}alan, Billur;
Peterson, Allan C.; Tac{s}, Kenan Nabla discrete fractional Gr"{u}ss type inequality. J. Inequal. Appl. 2014, 2014:86, 9 pp.
bibitem{sarikaya} Budak, H"{u}seyin; Sarikaya, Mehmet Z. An inequality of Ostrowski-Grüss type for double integrals. Stud. Univ. Babeş-Bolyai Math. 62 (2017), no. 2, 163-173.
bibitem{sarikaya2} Sarikaya, Mehmet Zeki. Gronwall type inequalities for conformable fractional integrals. Konuralp J. Math. 4 (2016), no. 2,
--222.
bibitem{akin} Akin, Elvan; Asli y"{u}ce, Serkan; G"{u}venilir, A. Feza; Kaymakcalan, Billur Discrete Gr"{u}ss type inequality on fractional calculus. J. Inequal. Appl. 2015, 2015:174, 7 pp.
bibitem{seuret} Seuret, A.; Gouaisbaut, F. Wirtinger-based integral
inequality: application to time-delay systems. Automatica J. IFAC 49 (2013), no. 9, 2860--2866.
bibitem{roman} Hilscher, Roman A time scales version of a Wirtinger-type inequality and applications. Dynamic equations on time scales. J. Comput. Appl. Math. 141 (2002), no. 1-2, 219--226.
bibitem{kun} Liu, Kun; Fridman, Emilia Wirtinger's inequality and
Lyapunov-based sampled-data stabilization. Automatica J. IFAC 48 (2012), no. 1, 102--108.
bibitem{hinton} Hinton, Don B. and Lewis, Roger T. Discrete spectra
criteria for singular differential operators with middle terms. Math. Proc.
Cambridge Philos. Soc. 77 (1975), 337--347.
bibitem{pena} Pena, Simon. Discrete spectra criteria for singular
difference operators. Math. Bohem. 124 (1999), no. 1, 35--44.
bibitem{thabet} Abdeljawad, Thabet. On conformable fractional calculus. J. Comput. Appl. Math. 279 (2015), 57--66.
bibitem{baleanu2} Atangana, Abdon, and Baleanu, Dumitru; Alsaedi, Ahmed New properties of conformable derivative. Open Math. 13 (2015), 889--898.
Downloads
Additional Files
Published
Issue
Section
License
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Transfer of copyright agreement: When the article is accepted for publication, the authors and the representative of the coauthors, hereby agree to transfer to Studia Universitatis Babeș-Bolyai Mathematica all rights, including those pertaining to electronic forms and transmissions, under existing copyright laws, except for the following, which the authors specifically retain: the authors can use the material however they want as long as it fits the NC ND terms of the license. The authors have all rights for reuse according to the license.