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Existence of Positive Solutions for a Higher-Order Fractional Differential Equation with Multi-Term Lower-Order Derivatives

Author

Listed:
  • Weiwei Liu

    (School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China
    These authors contributed equally to this work.)

  • Lishan Liu

    (School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China
    These authors contributed equally to this work.)

Abstract

This paper deals with the study of the existence of positive solutions for a class of nonlinear higher-order fractional differential equations in which the nonlinear term contains multi-term lower-order derivatives. By reducing the order of the highest derivative, the higher-order fractional differential equation is transformed into a lower-order fractional differential equation. Then, combining with the properties of left-sided Riemann–Liouville integral operators, we obtain the existence of the positive solutions of fractional differential equations utilizing some weaker conditions. Furthermore, some examples are given to demonstrate the validity of our main results.

Suggested Citation

  • Weiwei Liu & Lishan Liu, 2021. "Existence of Positive Solutions for a Higher-Order Fractional Differential Equation with Multi-Term Lower-Order Derivatives," Mathematics, MDPI, vol. 9(23), pages 1-23, November.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:23:p:3031-:d:688500
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    References listed on IDEAS

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    1. Tofighi, Ali, 2003. "The intrinsic damping of the fractional oscillator," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 329(1), pages 29-34.
    2. Jahanshahi, Hadi & Sajjadi, Samaneh Sadat & Bekiros, Stelios & Aly, Ayman A., 2021. "On the development of variable-order fractional hyperchaotic economic system with a nonlinear model predictive controller," Chaos, Solitons & Fractals, Elsevier, vol. 144(C).
    3. Agarwal, Ravi P. & Ahmad, Bashir & Garout, Doa’a & Alsaedi, Ahmed, 2017. "Existence results for coupled nonlinear fractional differential equations equipped with nonlocal coupled flux and multi-point boundary conditions," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 149-161.
    4. Ahmed, E. & Elgazzar, A.S., 2007. "On fractional order differential equations model for nonlocal epidemics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 379(2), pages 607-614.
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