Existence and uniqueness results of solution for the initial value problem of Hadamard fractional sequential differential systems

Volume 9, Issue 2, April 2024     |     PP. 31-45      |     PDF (1202 K)    |     Pub. Date: April 18, 2024
DOI: 10.54647/mathematics110481    56 Downloads     4458 Views  

Author(s)

Ala Eddine TAIER, School of Mathematical Sciences, Anhui University, Hefei 230039, China
Ranchao Wu, School of Mathematical Sciences, Anhui University, Hefei 230039, China

Abstract
In this paper, we study the existence and uniqueness of solutions for Hadamard fractional sequential differential systems involving the Hadamard fractional derivative with orders α ∈ (1,2] and β ∈ (2,3]. The main tools in our study are Banach fixed point theorem and schauder fixed point theorem. An example is provided to illustrate our main results.

Keywords
Hadamard fractional sequential differential system, Banach fixed point theorem, schauder fixed point theorem, existence and uniquness of solution.

Cite this paper
Ala Eddine TAIER, Ranchao Wu, Existence and uniqueness results of solution for the initial value problem of Hadamard fractional sequential differential systems , SCIREA Journal of Mathematics. Volume 9, Issue 2, April 2024 | PP. 31-45. 10.54647/mathematics110481

References

[ 1 ] A. A. Kilbas, H. M. Srivastava, J. J Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Sudies Elsevier Science B.V. Amsterdam the Netherlands, 2006.
[ 2 ] V. Lakshmikantham, J.V. Devi, Theory of fractional differential equations in a Banach space, Eur. J. Pure Appl. Math. 1(2008), 38-45.
[ 3 ] K. S. Miller, B. Ross, An Introduction to Fractional Calculus and Fractional Differential Equations, wiley, New YorK, 1993.
[ 4 ] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1993.
[ 5 ] V.E. Tarasov, Fractional Dynamics: Application of Frcational Calculus to Dynamics of Particals, Fields and Media, Springer, Beijing, 2011.
[ 6 ] B. Ahmad, S.K. Ntouyas, J. Tariboon, Existence results for mixed Hadamard and Riemann-Liouville frac- tional integro-differential equations, Adv. Difference Equ, 2015 (2015), 293.
[ 7 ] M. Benchohra, S. Hamani, S.K. Ntouyas, Boundary value problems for differential equations with fractional order, Surv. Math. Appl. 3 (2008), 1-12.
[ 8 ] B. Ahmad, S.K. Ntouyas, An existence theorem for fractional hybrid differential inclusions of Hadamard type with Dirichlet boundary conditions, Abst. Appl. Anal. 2014 (2014), Article ID 705809.
[ 9 ] B. Ahmad, S.K. Ntouyas, J. Tariboon, A nonlocal hybrid boundary value problem of Caputo fractional integro-differential equations, Acta Math. Sci. 36B (2016), 1631-1640.
[ 10 ] A.E.M. Herzallah, D. Baleanu, On fractional order hybrid differential equations, Abst. Appl. Anal. 2014 (2014), Article ID 389386.
[ 11 ] K. Hilal, A. Kajouni, Boundary value problems for hybrid differential equations with fractional order, Adv. Difference Equ. 2015 (2015), 183.
[ 12 ] N. Mahmudov, M. Matar, Existence of mild solutions for hybrid differential equations with arbitrary frac- tional order, TWMS J. Pure Appl. Math. 8 (2017), 160-169.
[ 13 ] S. Sitho, S. K Ntouyas, J. Tariboon, Existence results for hybrid fractional integro-differential equations, Bound. Value Probl. 2015 (2015), 113.
[ 14 ] S. Sun, Y. Zhao, Z. Han, Y. Li, The existence of solutions for boundary value problem of fractional hybrid differential equations, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), 4961-4967.
[ 15 ] Y. Zhao, Y. Wang, Existence of solutions to boundary value problem of a class of nonlinear fractional differ- ential equations, Adv. Difference Equ. 2014 (2014), 174.
[ 16 ] Wang, J., Zhang, Y.: On the concept and existence of solutions for fractional impulsive systems with Hadamard derivatives. Appl. Math. Lett. 39, 8590 (2015)
[ 17 ] Huang, H., Liu, W.: Positive solutions for a class of nonlinear Hadamard fractional differential equations with a parameter. Adv. Differ. Equ. 2018, 96 (2018)
[ 18 ] Zhai, C., Wang, W., Li, H.: A uniqueness method to a new Hadamard fractional differential system with four-point boundary conditions. J. Inequal. Appl. 2018, 207 (2018)
[ 19 ] Yang, W.: Positive solutions for singular coupled integral boundary value problems of nonlinear Hadamard fractional differential equations. J. Nonlinear Sci. Appl. 8, 110129 (2015)
[ 20 ] Yang, W.: Positive solutions for singular Hadamard fractional differential system with four-point coupled boundary conditions. J. Appl. Math. Comput. 49, 357381 (2015)
[ 21 ] Li, Y.L., Lin, S.Y.: Positive solution for the nonlinear Hadamard type fractional differential equation with p-Laplacian. J. Funct. Spaces Appl. 2013, Article ID 951643 (2013)
[ 22 ] Wang, G., Wang, T.: On a nonlinear Hadamard type fractional differential equation with pLaplacian operator and strip condition. J. Nonlinear Sci. Appl. 9, 50735081 (2016)
[ 23 ] Zhang, K., Wang, J., Ma, W.: Solutions for integral boundary value problems of nonlinear Hadamard fractional differential equations. J. Funct. Spaces 2018, Article ID 2193234 (2018)
[ 24 ] Li, S., Zhai, C.: Positive solutions for a new class of Hadamard fractional differential equations on infinite intervals. J. Inequal. Appl. 2019, 50 (2019)
[ 25 ] Zhang, W., Liu, W.: Existence of solutions for several higher-order Hadamard-type fractional differential equations with integral boundary conditions on infinite interval. Bound. Value Probl. 2018, 134 (2018)
[ 26 ] Ahmad, B., Ntouyas, S.K.: A fully Hadamard type integral boundary value problem of a coupled system of fractional differential equations. Fract. Calc. Appl. Anal. 17, 348360 (2014)
[ 27 ] Aljoudi, S., Ahmad, B., Nieto, J.J., Alsaedi, A.: On coupled Hadamard type sequential fractional differential equations with variable coefficients and nonlocal integral boundary conditions. Filomat 31, 60416049 (2017)
[ 28 ] Zhang, W., Liu, W.: Existence, uniqueness, and multiplicity results on positive solutions for a class of Hadamard-type fractional boundary value problem on an infinite interval. Math. Methods Appl. Sci. (2019).
[ 29 ] Jiang, J., ORegan, J., Xu, J., Fu, Z.: Positive solutions for a system of nonlinear Hadamard fractional differential equations involving coupled integral boundary conditions. J. Inequal. Appl. 2019, 204 (2019)
[ 30 ] Zhai, C., Wang, W.: Solutions for a system of Hadamard fractional differential equations with integral conditions. Numer. Funct. Anal. Optim. 41(7), 121 (2019)
[ 31 ] Asawasamrit, S., Ntouyas, S.K., Tariboon, J., Nithiarayaphaks, W.: Coupled systems of sequential Caputo and Hadamard fractional differential equations with coupled separated boundary conditions. Symmetry 10, 701 (2018). https://doi.org/10.3390/sym10120701
[ 32 ] Alesemi, M.: Solvability for a class of nonlinear Hadamard fractional differential equations with parameters. Bound. Value Probl. 2019, 101 (2019)
[ 33 ] Alesemi, M.: Solvability for a class of nonlinear Hadamard fractional differential equations with parameters. Bound. Value Probl. 2019, 101 (2019)