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On Cauchy Problems of Caputo Fractional Differential Inclusion with an Application to Fractional Non-Smooth Systems

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  • Jimin Yu

    (School of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
    College of Automation, Chongqing University of Posts and Telecommunications, Chongqing 400065, China)

  • Zeming Zhao

    (School of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China)

  • Yabin Shao

    (School of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China)

Abstract

In this innovative study, we investigate the properties of existence and uniqueness of solutions to initial value problem of Caputo fractional differential inclusion. In the study of existence problems, we considered the case of convex and non-convex multivalued maps. We obtained the existence results for both cases by means of the appropriate fixed point theorem. Furthermore, the uniqueness corresponding to both cases was also determined. Finally, we took a non-smooth system, the modified Murali–Lakshmanan–Chua (MLC) fractional-order circuit system, as an example to verify its existence and uniqueness conditions, and through several sets of simulation results, we discuss the implications.

Suggested Citation

  • Jimin Yu & Zeming Zhao & Yabin Shao, 2023. "On Cauchy Problems of Caputo Fractional Differential Inclusion with an Application to Fractional Non-Smooth Systems," Mathematics, MDPI, vol. 11(3), pages 1-18, January.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:3:p:653-:d:1048776
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    References listed on IDEAS

    as
    1. Srinivasan, K. & Chandrasekar, V.K. & Venkatesan, A. & Raja Mohamed, I., 2016. "Duffing–van der Pol oscillator type dynamics in Murali–Lakshmanan–Chua (MLC) circuit," Chaos, Solitons & Fractals, Elsevier, vol. 82(C), pages 60-71.
    2. Li, Chunguang & Chen, Guanrong, 2004. "Chaos and hyperchaos in the fractional-order Rössler equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 341(C), pages 55-61.
    3. Juan J. Nieto & Abdelghani Ouahab & P. Prakash, 2013. "Extremal Solutions and Relaxation Problems for Fractional Differential Inclusions," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-9, September.
    4. Fu, Shihui & Liu, Yuan, 2020. "Complex dynamical behavior of modified MLC circuit," Chaos, Solitons & Fractals, Elsevier, vol. 141(C).
    5. Mikhail Kamenskii & Valeri Obukhovskii & Garik Petrosyan & Jen-Chih Yao, 2021. "On the Existence of a Unique Solution for a Class of Fractional Differential Inclusions in a Hilbert Space," Mathematics, MDPI, vol. 9(2), pages 1-19, January.
    6. Vasily E. Tarasov, 2020. "Mathematical Economics: Application of Fractional Calculus," Mathematics, MDPI, vol. 8(5), pages 1-3, April.
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