# Existence results for a multipoint boundary value problem of nonlinear sequential Hadamard fractional differential equations

### Abstract

In this paper, existence and uniqueness results are established for a nonlinear sequential Hadamard fractional differential equation with multi-point boundary conditions, via Banach and Krasnosel'skiǐ's fixed point theorems and Leray-Schauder nonlinear alternative. An example illustrating the existence of a unique solution is also constructed.

### References

B. Ahmad and S. K. Ntouyas, “Some fractional-order one-dimensional semi-linear problems under nonlocal integral boundary conditions”, Rev. R. Acad. Cienc. Exactas, Fís. Nat. Ser. A Mat. RACSAM, vol. 110, no. 1 , pp. 159-172, 2016.

B. Ahmad and S. K. Ntouyas, “A fully Hadamard type integral boundary value problem of a coupled system of fractional differential equations”, Fract. Calc. Appl. Anal., vol. 17, no. 2, pp. 348-360, 2014.

B. Ahmad, A. Alsaedi, S. K. Ntouyas and J. Tariboon, Hadamard-type fractional differential equations, inclusions and inequalities, Cham, Switzerland: Springer, 2017.

S. Aljoudi, B. Ahmad, J. J. Nieto and A. Alsaedi, “A coupled system of Hadamard type sequential fractional differential equations with coupled strip conditions”, Chaos Solitons Fractals, vol. 91, pp. 39-46, 2016.

S. Aljoudi, B. Ahmad, J. J. Nieto and A. Alsaedi, “On coupled Hadamard type sequential fractional differential equations with variable coefficients and nonlocal integral boundary conditions”, Filomat, vol. 31, no. 19, pp. 6041-6049, 2017.

S. Aljoudi, B. Ahmad and A. Alsaedi, “Existence and uniqueness results for a coupled system of Caputo-Hadamard fractional differential equations with nonlocal Hadamard type integral boundary conditions”, Fractal Fract. vol. 4, no. 13, 15 pages, 2020.

I. Area, J. Losada and J. J. Nieto, “A note on the fractional logistic equation”, Phys. A, vol. 444, pp. 182-187, 2016.

D. Babusci, G. Dattoli and D. Sacchetti, “The Lamb-Bateman integral equation and the fractional derivatives”, Fract. Calc. Appl. Anal., vol. 14, pp. 317-320, 2011.

Y. Ding, Z. Wei, J. Xu and D. O’Regan, “Extremal solutions for nonlinear fractional boundary value problems with p-Laplacian”, J. Comput. Appl. Math., vol. 288, pp. 151-158, 2015.

X. Du, Y. Meng and H. Pang, “Iterative positive solutions to a coupled Hadamard-type fractional differential system on infinite domain with the multistrip and multipoint mixed boundary conditions”, J. of Funct. Spaces, Art. ID 6508075, 16 pages, 2020.

R. Garra and F. Polito, “On some operators involving Hadamard derivatives”, Integral Transforms Spec. Funct., vol. 24, no. 10, pp. 773-782, 2013.

R. Garra, E. Orsingher and F. Polito, “A note on Hadamard fractional differential equations with varying coefficients and their applications in probability”, Mathematics, vol. 6, no. 4, 10 pages, 2018.

A. Granas and J. Dugundji, Fixed Point Theory, Springer Monogr. in Math., New York: Springer-Verlag, 2003.

J. Hadamard, “Essai sur l’´étude des fonctions données par leur developpment de Taylor”, J. de Math. Pures Appl., vol. 8, pp. 101-186, 1892.

J. Henderson and N. Kosmatov, “Eigenvalue comparison for fractional boundary value problems with the Caputo derivative”, Fract. Calc. Appl. Anal., vol. 17, pp. 872-880, 2014.

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204., Amsterdam: Elsevier Science B.V., 2006.

J. Klafter, S. C Lim and R. Metzler (Editors), Fractional Dynamics in Physics, Singapore: World Scientific, 2011.

L. Ma, “On the kinetics of Hadamard-type fractional differential systems”, Fract. Calc. Appl. Anal., vol. 23, pp. 553-570, 2020.

Q. Ma, R. Wang, J. Wang and Y. Ma, “Qualitative analysis for solutions of a certain more generalized two-dimensional fractional differential system with Hadamard derivative”, Appl. Math. Comput., vol. 257, pp. 436-445, 2015.

R. K. Saxena, R. Garra and E. Orsingher, “Analytical solution of space-time fractional telegraph-type equations involving Hilfer and Hadamard derivatives”, Integral Transforms Spec. Funct., vol. 27, no. 1, pp. 30-42, 2016.

J. Tariboon, S. K. Ntouyas, S. Asawasamrit and C. Promsakon, “Positive solutions for Hadamard differential systems with fractional integral conditions on an unbounded domain”, Open Math., vol. 15, no. 1, pp. 645-666, 2017.

J. R. Wang, Y. Zhou and M. Medved, “Existence and stability of fractional differential equations with Hadamard derivative”, Topol. Methods Nonlinear Anal., vol. 41, no 1, pp. 113-133, 2013.

J. R. Wang and Y. Zhang, “On the concept and existence of solutions for fractional impulsive systems with Hadamard derivatives”, Appl. Math. Lett., vol. 39, pp. 85-90, 2015.

G. Wang, X. Ren, L. Zhang and B. Ahmad, “Explicit iteration and unique positive solution for a Caputo-Hadamard fractional turbulent flow model”, IEEE Access, vol. 7, pp. 109833-109839, 2019.

W. Yukunthorn, B. Ahmad, S. K. Ntouyas and J. Tariboon, “On Caputo-Hadamard type fractional impulsive hybrid systems with nonlinear fractional integral conditions”, Nonlinear Anal. Hybrid Syst., vol. 19, pp. 77-92, 2016.

C. Zhai, W. Wang and H. Li, “A uniqueness method to a new Hadamard fractional differential system with four-point boundary conditions”, J. Inequal. Appl., Paper No. 207, 16 pages, 2018.

C. Zhai and L. Xu, “Properties of positive solutions to a class of four-point boundary value problem of Caputo fractional differential equations with a parameter”, Commun. Nonlinear Sci. Numer. Simul., vol. 19, pp. 2820-2827, 2014.

W. Zhang and J. Ni, “New multiple positive solutions for Hadamard-type fractional differential equations with nonlocal conditions on an infinite interval”, Appl. Math. Lett., vol. 118, ID 107165, 10 pages, 2021.

*CUBO*, vol. 23, no. 2, pp. 225–237, Aug. 2021.

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