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A discrete fractional Gronwall inequality
Author:
Rui A. C. Ferreira
Journal:
Proc. Amer. Math. Soc. 140 (2012), 1605-1612
MSC (2010):
Primary 26D15, 26A33; Secondary 39A12
Posted:
January 9, 2012
MathSciNet review:
2869144
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
Abstract: In this work we derive a Gronwall type inequality within the discrete fractional setting. Our results constitute generalizations on the inequalities of discrete calculus.
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- R. P. Agarwal, Difference equations and inequalities, second edition, Monographs and Textbooks in Pure and Applied Mathematics, 228, Dekker, New York, 2000. MR 1740241 (2001f:39001)
- 2.
- E. Akin, Cauchy functions for dynamic equations on a measure chain, J. Math. Anal. Appl. 267 (2002), no. 1, 97-115. MR 1886819 (2002k:34005)
- 3.
- G. A. Anastassiou, Nabla discrete fractional calculus and nabla inequalities, Math. Comput. Modelling 51 (2010), no. 5-6, 562-571. MR 2594707 (2011a:26007)
- 4.
- F. M. Atici and P. W. Eloe, A transform method in discrete fractional calculus, Int. J. Difference Equ. 2 (2007), no. 2, 165-176. MR 2493595 (2010a:26008)
- 5.
- F. M. Atici and P. W. Eloe, Initial value problems in discrete fractional calculus, Proc. Amer. Math. Soc. 137 (2009), no. 3, 981-989. MR 2457438 (2009g:39037)
- 6.
- F. M. Atici and S. Şengül, Modeling with fractional difference equations, J. Math. Anal. Appl., 369 (2010), no. 1, 1-9. MR 2643839 (2011c:39002)
- 7.
- F. M. Atici, P. W. Eloe, Two-point boundary value problems for finite fractional difference equations, J. Differ. Equ. Appl. 17 (2011), 445-456. MR 2783359
- 8.
- N. R. O. Bastos, R. A. C. Ferreira and D. 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. MR 2728463 (2011j:49044)
- 9.
- Z. Denton and A. S. Vatsala, Fractional integral inequalities and applications, Comput. Math. Appl. 59 (2010), no. 3, 1087-1094. MR 2579473 (2011a:26009)
- 10.
- J. B. Díaz and T. J. Osler, Differences of fractional order, Math. Comp. 28 (1974), 185-202. MR 0346352 (49:11077)
- 11.
- C. S. Goodrich, Continuity of solutions to discrete fractional initial value problems, Comput. Math. Appl. 59 (2010), no. 11, 3489-3499. MR 2646320 (2011e:39002)
- 12.
- C. S. Goodrich, Solutions to a discrete right-focal fractional boundary value problem, Int. J. Difference Equ. 5 (2010), 195-216. MR 2771325
- 13.
- H. L. Gray and N. F. Zhang, On a new definition of the fractional difference, Math. Comp. 50 (1988), no. 182, 513-529. MR 929549 (89i:39008)
- 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. MR 1502565
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- R. L. Magin, Fractional Calculus in Bioengineering, Begell House, 2006.
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- Mikeladze, Sh.E., De la résolution numérique des équations intégrales, Bull. Acad. Sci. URSS VII (1935), 255-257 (in Russian).
- 17.
- K. S. Miller and B. Ross, Fractional difference calculus, in Univalent functions, fractional calculus, and their applications (Kōriyama, 1988), 139-152, Horwood, Chichester, 1989. MR 1199147 (93h:26010)
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- K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley, New York, 1993. MR 1219954 (94e:26013)
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- B. G. Pachpatte, Integral and finite difference inequalities and applications, Mathematics Studies 205, Elsevier, 2006. MR 2286291 (2008a:26003)
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- H. Ye, J. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl. 328 (2007), no. 2, 1075-1081. MR 2290034
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Additional Information
Rui A. C. Ferreira
Affiliation:
Department of Mathematics, Lusophone University of Humanities and Technologies, 1749-024 Lisbon, Portugal
Email:
ruiacferreira@ulusofona.pt
DOI:
http://dx.doi.org/10.1090/S0002-9939-2012-11533-3
PII:
S 0002-9939(2012)11533-3
Keywords:
Discrete fractional calculus,
Gronwall inequality,
comparison theorem.
Received by editor(s):
December 27, 2010
Posted:
January 9, 2012
Additional Notes:
The author was supported by the Portuguese Foundation for Science and Technology (FCT) through the R&D unit Center of Research and Development in Mathematics and Applications (CIDMA)
Communicated by:
Varghese Mathai
Article copyright:
© Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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