Existence and uniqueness results for a nonlinear singular fractional differential equation of order $ \sigma\in(1, 2) $

Sinan Serkan Bilgici; Müfit ŞAN; Sinan Serkan Bilgici; Müfit ŞAN

doi:10.3934/math.2021754

AIMS Mathematics

2021, Volume 6, Issue 12: 13041-13056. doi: 10.3934/math.2021754

Previous Article Next Article

Research article

Existence and uniqueness results for a nonlinear singular fractional differential equation of order $ \sigma\in(1, 2) $

Sinan Serkan Bilgici ,
Müfit ŞAN ^,

Department of Mathematics, Çankırı Karatekin University, Faculty of Sicence, Uluyazı Campus, 18100, Çankırı, Turkey

Received: 16 June 2021 Accepted: 10 September 2021 Published: 14 September 2021
MSC : 34A08, 37C25, 34A12, 74G20, 26A33

The first objective of this article is to discuss the local existence of the solution to an initial value problem involving a non-linear differential equation in the sense of Riemann-Liouville fractional derivative of order $ \sigma\in(1, 2), $ when the nonlinear term has a singularity at zero of its independent argument. Hereafter, by using some tools of Lebesgue spaces such as Hölder inequality, we obtain Nagumo-type, Krasnoselskii-Krein-type and Osgood-type uniqueness theorems for the problem.
- fractional differential equations,
- Riemann-Liouville derivative,
- existence and uniqueness,
- Hölder inequality,
- Lebesgue spaces
Citation: Sinan Serkan Bilgici, Müfit ŞAN. Existence and uniqueness results for a nonlinear singular fractional differential equation of order $ \sigma\in(1, 2) $[J]. AIMS Mathematics, 2021, 6(12): 13041-13056. doi: 10.3934/math.2021754

Related Papers:

Abstract

The first objective of this article is to discuss the local existence of the solution to an initial value problem involving a non-linear differential equation in the sense of Riemann-Liouville fractional derivative of order $ \sigma\in(1, 2), $ when the nonlinear term has a singularity at zero of its independent argument. Hereafter, by using some tools of Lebesgue spaces such as Hölder inequality, we obtain Nagumo-type, Krasnoselskii-Krein-type and Osgood-type uniqueness theorems for the problem.

References

[1]	A. A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Vol. 204, Elsevier Science Limited, 2006.
[2]	K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, A Wiley-Interscience Publication, John Wiley Sons Inc., New York, 1993.
[3]	I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
[4]	S. G. Samko, A. A. A Kilbas, O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, Switzerland, 1993.
[5]	S. Hristova, A. Dobreva, Existence, continuous dependence and finite time stability for Riemann-Liouville fractional differential equations with a constant delay, AIMS Math. 5 (2020), 3809–3824.
[6]	J. M. Shen, S. Rashid, M. A. Noor, R. Ashraf, Y. M. Chu, Certain novel estimates within fractional calculus theory on time scales, AIMS Math., 5 (2020), 6073–6086. doi: 10.3934/math.2020390
[7]	M. Şan, U. Sert, Some analysis on a fractional differential equation with a right-hand side which has a discontinuity at zero, Hacet. J. Math. Stat., 49 (2020), 1718–1725.
[8]	F. Yoruk, T. G. Bhaskar, R. P. Agarwal, New uniqueness results for fractional differential equations, Appl. Anal., 2 (2013), 259–269.
[9]	C. Li, S. Sarwar, Existence and continuation of solutions for Caputo type fractional differential equations, Electron. J. Differ. Equ., 2016 (2016), 1–14. doi: 10.1186/s13662-015-0739-5
[10]	M. Şan, Complex variable approach to the analysis of a fractional differential equation in the real line, C. R. Math., 356 (2018), 293–300. doi: 10.1016/j.crma.2018.01.008
[11]	J. R. L. Webb, Weakly singular Gronwall inequalities and applications to fractional differential equations, J. Math. Anal. Appl., 471 (2019), 692–711. doi: 10.1016/j.jmaa.2018.11.004
[12]	J. R. L. Webb, Initial value problems for Caputo fractional equations with singular nonlinearities, Electron. J. Differ. Equ., 2019 (2019), 1–32. doi: 10.1186/s13662-018-1939-6
[13]	R. P. Agarwal, V. Lakshmikantham, Uniqueness and nonuniqueness criteria for ordinary differential equations, (Vol. 6), World Scientific, 1993.
[14]	K. Diethelm, The mean value theorems and a Nagumo-type uniqueness theorem for Caputo's fractional calculus, Fract. Calc. Appl. Anal., 2 (2012), 304–313.
[15]	V. Lakshmikantham, S. Leela, Nagumo-type uniqueness result for fractional differential equations, Nonlinear Anal. Theory Methods Appl., 7-8 (2009), 2886–2889.
[16]	V. Lakshmikantham, S. Leela, Krasnoselskii–Krein-type uniqueness result for fractional differential equations, Nonlinear Anal. Theory Methods Appl., 7-8 (2009), 3421–3424.
[17]	D. Delbosco, Fractional calculus and function spaces, J. Fract. Calc., 6 (1994), 45–53.
[18]	E. Zeidler, Nonlinear Functional Analysis and its Applications, I: Fixed-point Theorems, New York: Springer-Verlag, 1985.

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)