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Caputo delta weakly fractional difference equations

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Abstract

In this paper, solutions of fractional difference equations with Caputo-type delta-based fractional difference operator of order \(\mu \sim 1\) are compared with solutions of corresponding limit difference equations with usual first-order forward difference. It is shown that the limit initial value problems differ substantially when \(\mu \rightarrow 1^-\) and \(\mu \rightarrow 1^+\). To derive convergence results, Gronwall type inequalities are proved for suitable fractional sum inequalities of general noninteger order. An illustrative example is also given.

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Acknowledgements

MF was supported in part by the Grants VEGA-MS 1/0358/20, VEGA-SAV 2/0127/20 and APVV-18-0308. MP was supported in part by the Grants VEGA-MS 1/0358/20, VEGA-SAV 2/0127/20 and APVV-18-0308. JRW was supported in part by the National Natural Science Foundation of China (12161015).

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Authors and Affiliations

  1. Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University, Mlynská dolina, 842 48, Bratislava, Slovakia

    Michal Fečkan & Michal Pospíšil

  2. Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73, Bratislava, Slovakia

    Michal Fečkan & Michal Pospíšil

  3. STAR-UBB Institute, Babes-Bolyai University, 400084, Cluj-Napoca, Romania

    Marius-F. Danca

  4. Romanian Institute of Science and Technology, 400504, Cluj-Napoca, Romania

    Marius-F. Danca

  5. Department of Mathematics, Guizhou University, Guiyang, 550025, Guizhou, China

    **Rong Wang

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  1. Michal Fečkan
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  2. Michal Pospíšil
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  3. Marius-F. Danca
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  4. **Rong Wang
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Correspondence to Michal Pospíšil.

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Fečkan, M., Pospíšil, M., Danca, MF. et al. Caputo delta weakly fractional difference equations. Fract Calc Appl Anal 25, 2222–2240 (2022). https://doi.org/10.1007/s13540-022-00093-5

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  • DOI: https://doi.org/10.1007/s13540-022-00093-5

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