The discrete fractional Karamata theorem and its applications
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Pavel Řehák
Abstract
We establish a fractional extension of the discrete Karamata integration theorem for two types of fractional sum operators. This result, along with other properties of regularly varying sequences and the tools such as comparison theorems and a fixed point theorem in FK-space, is then used to study asymptotic properties of solutions to a nonlinear fractional difference equation. We also establish and utilize a fractional extension of the Stolz–Cesáro theorem, i.e., of the discrete l’Hospital rule. Both, the discrete fractional Karamata theorem and the discrete fractional l’Hospital rule, are believed to find applications in a broader context within the discrete fractional calculus.
The research has been supported by the Brno University of Technology grant number FSI-S-23-8161.
Acknowledgement
The author thanks the anonymous reviewer for her/his insightful comments and suggestions.
Communicated by Michal Fečkan
References
[1] Atici, F. M.—Eloe, P. V.: A transform method in discrete fractional calculus, Int. J. Difference Equ. 2 (2007), 165–176.Search in Google Scholar
[2] Bingham, N. H.—Goldie, C. M.—Teugels, J. L.: Regular Variation. Encyclopedia Math. Appl., Vol. 27, Cambridge Univ. Press, 1987.10.1017/CBO9780511721434Search in Google Scholar
[3] Bojanić, R.—Seneta, E.: A unified theory of regularly varying sequences, Math. Z. 134 (1973), 91–106.10.1007/BF01214468Search in Google Scholar
[4] Čermák, J.—Kisela, T.—Nechvátal, L.: Stability and asymptotic properties of a linear fractional difference equation, Adv. Difference Equ. 2012 (2012), Art. No. 122.10.1186/1687-1847-2012-122Search in Google Scholar
[5] Chen, F.—Liu, Z.: Asymptotic stability results for nonlinear fractional difference equations, J. Appl. Math. 2012 (2012), Art. ID 879657.10.1155/2012/879657Search in Google Scholar
[6] Czornik, A.—Anh, P. T.—Babiarz, A.—Siegmund, S.: Asymptotic behavior of discrete fractional systems. In: Fractional dynamical systems: Methods, Algorithms and Applications; Stud. Syst. Decis. Control, Vol. 402, Springer, Cham, 2022, pp. 135–173.10.1007/978-3-030-89972-1_5Search in Google Scholar
[7] Erbe, L.—Goodrich, C. S.—Jia, B.—Peterson, A. C.: Survey of the qualitative properties of fractional difference operators: monotonicity, convexity, and asymptotic behavior of solutions, Adv. Difference Equ. 2016 (2016), Art. No. 43.10.1186/s13662-016-0760-3Search in Google Scholar
[8] Ferreira, R. A. C.: Discrete Fractional Calculus and Fractional Difference Equations, Springer, 2022.10.1007/978-3-030-92724-0Search in Google Scholar
[9] Goodrich, C. S.—Peterson, A. C.: Discrete Fractional Calculus, Springer, Cham, 2015.10.1007/978-3-319-25562-0Search in Google Scholar
[10] Granas, A.—Dugundji, J.: Fixed Point Theory, Springer-Verlag, New York, 2003.10.1007/978-0-387-21593-8Search in Google Scholar
[11] Gray, H. L.—Zhang, N. F.: On a new definition of the fractional difference, Math. Comp. 50 (1988), 513–529.10.1090/S0025-5718-1988-0929549-2Search in Google Scholar
[12] Marić, V.: Regular Variation and Differential Equations. Lecture Notes in Math., Vol. 1726, Springer-Verlag, Berlin-Heidelberg-New York, 2000.10.1007/BFb0103952Search in Google Scholar
[13] Miller, K. S.—Ross, B.: Fractional difference calculus. In: Univalent Functions, Fractional Calculus, and their Applications (Kōriyama, 1988), Ellis Horwood Ser. Math. Appl., Horwood, Chichester, 1989, pp 139–152.Search in Google Scholar
[14] Řehák, P.: Asymptotic formulae for solutions of linear second-order difference equations, J. Difference Equ. Appl. 22 (2016), 107–139.10.1080/10236198.2015.1077815Search in Google Scholar
[15] Řehák, P.: Superlinear solutions of sublinear fractional differential equations and regular variation, Fract. Calc. Appl. Anal. 26 (2023), 989–1015.10.1007/s13540-023-00156-1Search in Google Scholar
[16] Wu, H.: Asymptotic behavior of solutions of fractional nabla difference equations, Appl. Math. Lett. 105 (2020), Art. ID 106302.10.1016/j.aml.2020.106302Search in Google Scholar
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- On the rate of convergence for the q-Durrmeyer polynomials in complex domains
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