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Analysis of some Katugampola fractional differential equations with fractional boundary conditions

  • Received: 23 July 2021 Accepted: 10 August 2021 Published: 27 August 2021
  • In this work, some class of the fractional differential equations under fractional boundary conditions with the Katugampola derivative is considered. By proving the Lyapunov-type inequality, there are deduced the conditions for existence, and non-existence of the solutions to the considered boundary problem. Moreover, we present some examples to demonstrate the effectiveness and applications of the new results.

    Citation: Barbara Łupińska, Ewa Schmeidel. Analysis of some Katugampola fractional differential equations with fractional boundary conditions[J]. Mathematical Biosciences and Engineering, 2021, 18(6): 7269-7279. doi: 10.3934/mbe.2021359

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  • In this work, some class of the fractional differential equations under fractional boundary conditions with the Katugampola derivative is considered. By proving the Lyapunov-type inequality, there are deduced the conditions for existence, and non-existence of the solutions to the considered boundary problem. Moreover, we present some examples to demonstrate the effectiveness and applications of the new results.



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    [1] A. M. Lyapunov, Probleme General de la Stabilite du Mouvement, Princeton University Press, 1948.
    [2] R. C. Brown, D. B. Hinton, Lyapunov inequalities and their applications, in Survey on Classical Inequalities, Dordrecht, (2000), 1–25.
    [3] G. Borg, On a Liapunoff criterion of stability, Am. J. Math., 71 (1949), 67–70. doi: 10.2307/2372093
    [4] R. S. Dahiya, B. Singh, A Lyapunov inequality and nonoscillation theorem for a second order nonlinear differential-difference equations, J. Math. Phys. Sci., 7 (1973), 163–170.
    [5] S. Clark, D. B. Hinton, A Liapunov inequality for linear Hamiltonian systems, Math. Inequalities Appl., 1 (1998), 201–209.
    [6] Q. M. Zhang, X. H. Tang, Lyapunov inequalities and stability for discrete linear Hamiltonian systems, J. Differ. Equation Appl., 18 (2012), 1467–1484. doi: 10.1080/10236198.2011.572071
    [7] F. M. Atici, G. S. Guseinov and B. Kaymakcalan, On Lyapunov inequality in stability theory for Hill's equation on time scales, J. Inequal. Appl., 5 (2000), 603–620.
    [8] R. A. C. Ferreira, On a Lyapunov-type inequality and the zeros of a certain Mittag-Leffler function, J. Math. Anal. Appl., 412 (2014), 1058–1063. doi: 10.1016/j.jmaa.2013.11.025
    [9] B. Łupińska, T. Odzijewicz, A Lyapunov-type inequality with the Katugampola fractional derivative, Math. Methods Appl. Sci., 41 (2018), 8985–8996. doi: 10.1002/mma.4782
    [10] R. A. C. Ferreira, A Lyapunov-type inequality for a fractional boundary value problem, Fract. Calc. Appl. Anal., 16 (2013), 978–984. doi: 10.2478/s13540-013-0060-5
    [11] Q. Ma, C. Ma, J. Wang, A Lyapunov type inequality for a fractional differential equation with Hadamard derivative, J. Math. Inequalities, 11 (2017), 135–141.
    [12] M. Jleli, B. Samet, Lyapunov-type inequalities for a fractional differential equation with mixed boundary conditions, Math. Inequal. Appl., 18 (2015), 443–451.
    [13] A. Guezane-Lakoud, R. Khaldi, D. F. M. Torres, Lyapunov-type inequality for a fractional boundary value problem with natural conditions, SeMA J., 75 (2018), 157–162. doi: 10.1007/s40324-017-0124-2
    [14] J. Rong, C. Bai, Lyapunov-type inequality for a fractional differential equation with fractional boundary conditions, Adv. Differ. Equation, 82 (2015), 1–10.
    [15] U. N. Katugampola, New approach to a genaralized fractional integral, Appl. Math. Comput., 218 (2011), 860–865.
    [16] U. N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. App., 6 (2014), 1–15.
    [17] B. Łupińska, Properties of the Katugampola fractional operators, Tatra Mt. Math. Publ., 81 (2021), 1–14.
    [18] R. W. Ibrahim, On generalized Srivastava-Owa fractional operators in the unit disk, Adv. Differ. Equation, 1 (2011), 1–10.
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