The new general solution for a class of fractional-order impulsive differential equations involving the Riemann-Liouville type Hadamard fractional derivative

Pinghua Yang; Caixia Yang; Pinghua Yang; Caixia Yang

doi:10.3934/math.2023599

AIMS Mathematics

2023, Volume 8, Issue 5: 11837-11850. doi: 10.3934/math.2023599

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The new general solution for a class of fractional-order impulsive differential equations involving the Riemann-Liouville type Hadamard fractional derivative

Pinghua Yang ,
Caixia Yang ^,

School of Computer Engineering, Guangzhou City University of Technology, Guangzhou 510800, China

Received: 28 October 2022 Revised: 05 March 2023 Accepted: 08 March 2023 Published: 20 March 2023
MSC : 34A08, 34A37

In this paper, the new general solution for a class of higher-order impulsive fractional differential equations (IFDEs) involving the Riemann-Liouville (R-L) type Hadamard fractional derivative (FD) is presented. Specifically, the necessary and sufficient conditions of the solution are obtained by converting boundary value problems (BVPs) into integral equations and applying analytical techniques. The results in the paper provide a new method for converting BVPs or initial value problems (IVPs) for IFDEs to integral equations. Finally, some examples are devoted to explaining the application of the theorem.
Citation: Pinghua Yang, Caixia Yang. The new general solution for a class of fractional-order impulsive differential equations involving the Riemann-Liouville type Hadamard fractional derivative[J]. AIMS Mathematics, 2023, 8(5): 11837-11850. doi: 10.3934/math.2023599

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Abstract

In this paper, the new general solution for a class of higher-order impulsive fractional differential equations (IFDEs) involving the Riemann-Liouville (R-L) type Hadamard fractional derivative (FD) is presented. Specifically, the necessary and sufficient conditions of the solution are obtained by converting boundary value problems (BVPs) into integral equations and applying analytical techniques. The results in the paper provide a new method for converting BVPs or initial value problems (IVPs) for IFDEs to integral equations. Finally, some examples are devoted to explaining the application of the theorem.

References

[1]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, 2006.
[2]	I. Podlubny, Fractional differential equations, Academic Press, 1999.
[3]	J. Sabatier, O. P. Agrawal, J. A. Tenreiro Machado, Advances in fractional calculus: Theoretical developments and applications in physics and engineering, Dordrecht: Springer, 2007. http://dx.doi.org/10.1007/978-1-4020-6042-7
[4]	A. A. Kilbas, O. I. Marichev, S. G. Samko, Fractional integrals and derivatives: Theory and applications, New York: Gordon and Breach, 1993.
[5]	R. P. Agarwal, B. Andrade, C. Cuevas, Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations, Nonlinear Anal. Real, 11 (2010), 3532–3554. http://dx.doi.org/10.1016/j.nonrwa.2010.01.002 doi: 10.1016/j.nonrwa.2010.01.002
[6]	A. Aghajani, Y. Jalilian, J. J. Trujillo, On the existence of solutions of fractional integro-differential equations, Fract. Calc. Appl. Anal., 15 (2012), 44–69. http://dx.doi.org/10.2478/s13540-012-0005-4 doi: 10.2478/s13540-012-0005-4
[7]	R. P. Agarwal, Z. Yong, Y. He, Existence of fractional neutral functional differential equations, Comput. Math. Appl., 59 (2010), 1095–1100. http://dx.doi.org/10.1016/j.camwa.2009.05.010 doi: 10.1016/j.camwa.2009.05.010
[8]	K. Balachandran, J. J. Trujillo, The nonlocal Cauchy problem for nonlinear fractional integro-differential equations in Banach spaces, Nonlinear Anal. Theor., 72 (2010), 4587–4593. http://dx.doi.org/10.1016/j.na.2010.02.035 doi: 10.1016/j.na.2010.02.035
[9]	Y. Liu, Y. Zhu, Z. Lu, On Caputo-Hadamard uncertain fractional differential equations, Chaos Soliton. fract., 146 (2021), 110894. http://dx.doi.org/10.1016/j.chaos.2021.110894 doi: 10.1016/j.chaos.2021.110894
[10]	S. Zhang, X. Su, Unique existence of solution to initial value problem for fractional differential equations involving with fractional derivative of variable order, Chaos Soliton. fract., 148 (2021), 111040. http://dx.doi.org/10.1016/j.chaos.2021.111040 doi: 10.1016/j.chaos.2021.111040
[11]	R. Agarwal, S. Hristova, D. ORegan, Stability of solutions to impulsive Caputo fractional differential equations, Electron. J. Differ. Equ., 58 (2016), 1–22.
[12]	R. P. Agarwal, M. Benchohra, S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math., 109 (2010), 973–1033. http://dx.doi.org/10.1007/s10440-008-9356-6 doi: 10.1007/s10440-008-9356-6
[13]	R. P. Agarwal, M. Benchhra, B. A. Slimani, Existence results for differential equations with fractional order and impulses, Mem. Differ. Equ. Math., 44 (2008), 1–21.
[14]	B. Ahmad, S. Sivasundaram, Existence of solutions for impulsive integral boundary value problems of fractional order, Nonlinear Anal. Hybri., 4 (2010), 134–141. http://dx.doi.org/10.1016/j.nahs.2009.09.002 doi: 10.1016/j.nahs.2009.09.002
[15]	Y. Liu, On piecewise continuous solutions of higher order impulsive fractional differential equations and applications, Appl. Math. Comput., 287-288 (2016), 38–49. http://dx.doi.org/10.1016/j.amc.2016.03.041 doi: 10.1016/j.amc.2016.03.041
[16]	Y. Liu, S. Li, Periodic boundary value problems for singular fractional differential equations with impulse effects, Malaya J. Mat., 3 (2015), 423–490.
[17]	J. Wang, Y. Yang, W. Wei, Nonlocal impulsive problems for fractional differential equations with time varying generating operators in Banach spaces, Opusc. math., 30 (2010), 361–381. http://dx.doi.org/10.7494/OpMath.2010.30.3.361 doi: 10.7494/OpMath.2010.30.3.361
[18]	M. Feckan, Y. Zhou, J. Wang, On the concept and existence of solution for impulsive fractional differential equations, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 3050–3060. http://dx.doi.org/10.1016/j.cnsns.2011.11.017 doi: 10.1016/j.cnsns.2011.11.017
[19]	M. Feckan, Y. Zhou, J. Wang, Response to "Comments on the concept of existence of solution for impulsive fractional differential equations [Commun Nonlinear Sci Numer Simul 2014;19:401-C3.]", Commun. Nonlinear Sci., 19 (2014), 4213–4215. http://dx.doi.org/10.1016/j.cnsns.2014.04.014 doi: 10.1016/j.cnsns.2014.04.014
[20]	G. Wang, B. Ahmad, L. Zhang, J. J. Nieto, Comments on the concept of existence of solution for impulsive fractional differential equations, Commun. Nonlinear Sci., 19 (2014), 401–403. http://dx.doi.org/10.1016/j.cnsns.2013.04.003 doi: 10.1016/j.cnsns.2013.04.003
[21]	J. Wang, Y. Zhou, M. Feckan, On recent developments in the theory of boundary value problems for impulsive fractional differential equations, Comput. Math. Appl., 64 (2012), 3008–3020. http://dx.doi.org/10.1016/j.camwa.2011.12.064 doi: 10.1016/j.camwa.2011.12.064
[22]	G. Wang, L. Zhang, G. Song, Systems of first order impulsive functional differential equations with deviating arguments and nonlinear boundary conditions, Nonlinear Anal. Theor., 74 (2011), 974–982. http://dx.doi.org/10.1016/j.na.2010.09.054 doi: 10.1016/j.na.2010.09.054
[23]	X. Zhang, X. Zhang, Z. Liu, W. Ding, H. Cao, T. Shu, On the general solution of impulsive systems with Hadamard fractional derivatives, Math. Probl. Eng., 2016 (2016), 2814310. http://dx.doi.org/10.1155/2016/2814310 doi: 10.1155/2016/2814310
[24]	F. Jarad, T. Abdeljawad, D. Baleanu, Caputo type modification of the Hadamard fractional derivatives, Adv. Differ. Equ., 2012 (2012), 142. http://dx.doi.org/10.1186/1687-1847-2012-142 doi: 10.1186/1687-1847-2012-142
[25]	Y. Gambo, F. Jarad, D. Baleanu, T. Abdeljawad, On Caputo modification of the Hadamard fractional derivatives, Adv. Differ. Equ., 2014 (2014), 10. http://dx.doi.org/10.1186/1687-1847-2014-10 doi: 10.1186/1687-1847-2014-10
[26]	W. Yukunthorn, S. Suantai, S. K. Ntouyas, J. Tariboon, Boundary value problems for impulsive multi-order Hadamard fractional differential equations, Bound. Value Probl., 2015 (2015), 148. http://dx.doi.org/10.1186/s13661-015-0414-5 doi: 10.1186/s13661-015-0414-5
[27]	G. Wang, S. Liu, D. Baleanu, L. Zhang, A new impulsive multi-orders fractional differential equation involving multi point fractional integral boundary conditions, Abstr. Appl. Anal., 2014 (2014), 932747. http://dx.doi.org/10.1155/2014/932747 doi: 10.1155/2014/932747
[28]	W. Yukunthorn, B. Ahmad, S. K. Ntouyas, J. Tariboon, On Caputo-Hadamard type fractional impulsive hybrid systems with nonlinear fractional integral conditions, Nonlinear Anal. Hybri., 19 (2016), 77–92. http://dx.doi.org/10.1016/j.nahs.2015.08.001 doi: 10.1016/j.nahs.2015.08.001
[29]	J. Wang, Y. Zhang, On the concept and existence of solutions for fractional impulsive systems with Hadamard derivatives, Appl. Math. Lett., 39 (2015), 890. http://dx.doi.org/10.1016/j.aml.2014.08.015 doi: 10.1016/j.aml.2014.08.015
[30]	X. Zhang, The general solution of differential equations with Caputo-Hadamard fractional derivatives and impulsive effect, Adv. Differ. Equ., 2015 (2015), 215. http://dx.doi.org/10.1186/s13662-015-0552-1 doi: 10.1186/s13662-015-0552-1
[31]	X. Zhang, T. Shu, H. Cao, Z. Liu, W. Ding, The general solution for impulsive differential equations with Hadamard fractional derivative of order $q\in$ (1, 2), Adv. Differ. Equ., 2016 (2016), 14. http://dx.doi.org/10.1186/s13662-016-0744-3 doi: 10.1186/s13662-016-0744-3

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