×

A discrete fractional gronwall inequality. (English) Zbl 1243.26012

Gronwall type inequalities within the discrete fractional settings are derived and discussed.

MSC:

26D15 Inequalities for sums, series and integrals
26A33 Fractional derivatives and integrals
39A12 Discrete version of topics in analysis
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ravi P. Agarwal, Difference equations and inequalities, 2nd ed., Monographs and Textbooks in Pure and Applied Mathematics, vol. 228, Marcel Dekker, Inc., New York, 2000. Theory, methods, and applications. · Zbl 0952.39001
[2] Elvan Akin, Cauchy functions for dynamic equations on a measure chain, J. Math. Anal. Appl. 267 (2002), no. 1, 97 – 115. · Zbl 1006.39015 · doi:10.1006/jmaa.2001.7753
[3] George A. Anastassiou, Nabla discrete fractional calculus and nabla inequalities, Math. Comput. Modelling 51 (2010), no. 5-6, 562 – 571. · Zbl 1190.26001 · doi:10.1016/j.mcm.2009.11.006
[4] Ferhan M. Atici and Paul W. Eloe, A transform method in discrete fractional calculus, Int. J. Difference Equ. 2 (2007), no. 2, 165 – 176. · Zbl 1157.81315
[5] Ferhan M. Atici and Paul W. Eloe, Initial value problems in discrete fractional calculus, Proc. Amer. Math. Soc. 137 (2009), no. 3, 981 – 989. · Zbl 1166.39005
[6] Ferhan M. Atıcı and Sevgi Şengül, Modeling with fractional difference equations, J. Math. Anal. Appl. 369 (2010), no. 1, 1 – 9. · Zbl 1204.39004 · doi:10.1016/j.jmaa.2010.02.009
[7] Ferhan M. Atıcı and Paul W. Eloe, Two-point boundary value problems for finite fractional difference equations, J. Difference Equ. Appl. 17 (2011), no. 4, 445 – 456. · Zbl 1215.39002 · doi:10.1080/10236190903029241
[8] Nuno R. O. Bastos, Rui A. C. Ferreira, and Delfim F. M. Torres, Necessary optimality conditions for fractional difference problems of the calculus of variations, Discrete Contin. Dyn. Syst. 29 (2011), no. 2, 417 – 437. · Zbl 1209.49020 · doi:10.3934/dcds.2011.29.417
[9] Z. Denton and A. S. Vatsala, Fractional integral inequalities and applications, Comput. Math. Appl. 59 (2010), no. 3, 1087 – 1094. · Zbl 1189.26044 · doi:10.1016/j.camwa.2009.05.012
[10] J. B. Díaz and T. J. Osler, Differences of fractional order, Math. Comp. 28 (1974), 185 – 202. · Zbl 0282.26007
[11] Christopher S. Goodrich, Continuity of solutions to discrete fractional initial value problems, Comput. Math. Appl. 59 (2010), no. 11, 3489 – 3499. · Zbl 1197.39002 · doi:10.1016/j.camwa.2010.03.040
[12] Christopher S. Goodrich, Solutions to a discrete right-focal fractional boundary value problem, Int. J. Difference Equ. 5 (2010), no. 2, 195 – 216.
[13] Henry L. Gray and Nien Fan Zhang, On a new definition of the fractional difference, Math. Comp. 50 (1988), no. 182, 513 – 529. · Zbl 0648.39002
[14] T. H. Gronwall, Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Ann. of Math. (2) 20 (1919), no. 4, 292 – 296. · doi:10.2307/1967124
[15] R. L. Magin, Fractional Calculus in Bioengineering, Begell House, 2006.
[16] Mikeladze, Sh.E., De la résolution numérique des équations intégrales, Bull. Acad. Sci. URSS VII (1935), 255-257 (in Russian).
[17] Kenneth S. Miller and Bertram Ross, Fractional difference calculus, Univalent functions, fractional calculus, and their applications (Kōriyama, 1988) Ellis Horwood Ser. Math. Appl., Horwood, Chichester, 1989, pp. 139 – 152. · Zbl 0693.39002
[18] Kenneth S. Miller and Bertram Ross, An introduction to the fractional calculus and fractional differential equations, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1993. · Zbl 0789.26002
[19] B. G. Pachpatte, Integral and finite difference inequalities and applications, North-Holland Mathematics Studies, vol. 205, Elsevier Science B.V., Amsterdam, 2006. · Zbl 1104.26015
[20] Haiping Ye, Jianming Gao, and Yongsheng Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl. 328 (2007), no. 2, 1075 – 1081. · Zbl 1120.26003 · doi:10.1016/j.jmaa.2006.05.061
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.