### BOUNDARY VALUE PROBLEM FOR NONLINEAR CAPUTO-HADAMARD FRACTIONAL DIFFERENTIAL EQUATION WITH HADAMARD FRACTIONAL INTEGRAL AND ANTI-PERIODIC CONDITIONS

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#### Abstract

The aim of this work is to study a class of boundary value problem including a fractional order differential equation involving the Caputo-Hadamard fractional derivative. Suffcient conditions will be presented to guarantee the existence and uniqueness of solution of this fractional boundary value problem. The boundary conditions introduced in this work are of quite general nature and reduce to many special cases by fixing the parameters involved in the conditions.

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M. A. Abdo, H. A. Wahash and S. K. Panchat: Positive solutions of a fractional differential equation with integral boundary conditions. Journal of Applied Mathematics and Computational Mechanics 17(3) (2018), 5-15.

R. P. Agarwal, M. Meehan and D. O'Regan: Fixed Point Theory and Applications, Cambridge Tracts in Mathematics, 141, Cambridge University Press, Cambridge, 2001.

B. Ahmad, M. Alghanmi, H. M. Srivastava and S. K. Ntouyas: The

Langevin equation in terms of generalized Liouville-Caputo derivatives with non-local boundary conditions involving a generalized fractional integral, Mathematics 7 (2019), Article ID 533,1-10.

B. Ahmad and S. K. Ntouyas: On Hadamard fractional integro-differential boundary value problems, J. Appl. Math. Comput 2015 (2015).

B. Ahmad and S. K. Ntouyas: Initial value problems of fractional

order Hadamard-type functional differential equations, Electron. J. Differ. Equ.2015(2015), 77.

A. Ahmadkhanlu: Existence and Uniqueness Results for a Class of Fractional Differential Equations with an Integral Fractional Boundary Condition, Filomat 31:5 (2017), 1241-1249.

A. Alsaedi, M. Alsulami, H. M. Srivastava, B. Ahmad and S. K. Ntouyas: Existence theory for nonlinear third-order ordinary differential equations with non-local multi-point and multi-strip boundary conditions, Symmetry 11(2019), Article ID 281,1-18.

A. Ardjouni and A. Djoudi: Positive solutions for nonlinear Caputo-Hadamard fractional differential equations with integral boundary conditions, Open J. Math. Anal. 2019, 3(1), 62-69.

Z. Baitiche, M. Benbachir and K. Guerbati: Solvability for multi-point bvp of nonlinear fractional differential equations at resonance with three dimensional kernels, Kragujevac Journal of Mathematics Volume 45(5) (2021), pp: 761-780.

M. Benchohra, S. Hamani and S. K. Ntouyas: Boundary value problems for differential equations with fractional order, Surveys in Mathematics and its Applications vol, 3 (2008), 1-12.

W. Benhamida, J. R. Graef and S. Hamani: Boundary Value Problems for Fractional Differential Equations with Integral and Anti-Periodic Conditions in a Banach Space, Progr. Fract. Differ. Appl. 4, No. 2, 65-70 (2018)

W. Benhamida and S. Hamani: Measure of Noncompactness and Caputo-Hadamard Fractional Differential Equations in Banach Spaces, Eurasian Bulletin Of Mathematics EBM (2018), Vol. 1, No. 3, 98-106

W. Benhamida, S. Hamani and J. Henderson: Boundary Value Problems For Caputo-Hadamard Fractional Differential Equations, Advances in the Theory of Nonlinear Analysis and its Applications 2 (2018) No. 3, 138-145.

D. W. Boyd and J. S. W. Wong: On nonlinear contractions. Proc. Am. Math. Soc.20, 458-464 (1969).

D. Boutiara, K. Guerbati and M. Benbachir: Caputo-Hadamard fractional differential equation with three-point boundary conditions in Banach spaces; AIMS Mathematics, 5(1): 259272., (2019).

B. Hazarika, H. M. Srivastava, R. Arab and M. Rabbani: Existence of solution for an infinite system of nonlinear integral equations via measure of non-compactness and homotopy perturbation method to solve it, J. Comput. Appl. Math. 343 (2018), 341-352.

D. Delbosco and L. Rodino: Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl. 204 (1996), 609-625.

K. Diethelm and A. D. Freed: On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity, Scientic Computing in Chemical Engineering II. Computational Fluid Dynamics, Reaction Engineering and Molecular Properties (F. Keil, W. Mackens, H. Voss and J. Werther, eds.), Springer-Verlag, Heidelberg, 1999, pp. 217-224.

A. M. A. EL-Sayed and E. O. Bin-Taher: Positive solutions for a nonlocal multi-point boundary-value problem of fractional and second order, Electron. J. Differential Equations, Number 64, (2013), 1-8.

Y. Gambo et al: On Caputo modification of the Hadamard fractional derivatives, Adv. Difference Equ. 2014 (2014), Paper No. 10, 12 p.

A. Granas and J. dugundji: Fixed Point Theory, Springer-Verlag, New York, 2003.

F. Jarad, D. Baleanu and A. Abdeljawad: Caputo-type modification of the Hadamard fractional derivatives, Adv. Differ. Equ. 2012 (2012).

A. A. Kilbas and S. A. Marzan: Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions, Differential Equations 41 (2005), 84-89.

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo: Theory and Applications of Fractional Differential Equations, Elsevier Science B.V. Amsterdam, 2006.

F. Mainardi: Fractional calculus: some basic problems in continuum and statistical mechanics, Fractals and Fractional Calculus in Continuum Mechanics (A. Carpinteri and F. Mainardi, eds.), Springer-Verlag, Wien, 1997, pp. 291-348.

K. S. Miller and B. Ross: An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993.

I. Podlubny, I. Petras and B. M. Vinagre and P. O'Leary and L. Dorcak: Analogue realizations of fractional-order controllers. Fractional order calculus and its applications, Nonlinear Dynam. 29 (2002), 281-296.

S. G. Samko, A. A. Kilbas and O. I. Marichev: Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon 1993.

H.M. Srivastava, Fractional-order derivatives and integrals: Introductory overview and recent developments, Kyungpook Math. J. 60 (2020), 73-116

H.M. Srivastava, Diabetes and its resulting complications: Mathematical modeling via fractional calculus, Public Health Open Access 4 (3) (2020), Article ID 2, 1-5.

P. Thiramanus and S. K. Ntouyas and J. Tariboon: Existence and uniqueness results for Hadamard-type fractional differential equations with nonlocal fractional integral boundary conditions, Abstr. Appl. Anal. (2014).

A. Yacine and B. Nouredine: boundary value problem for Caputo-Hadamard fractional differential equations, Surveys in Mathematics and its Applications, Volume 12 (2017), 103-115.

H. Zhang: Nonlocal boundary value problems of fractional order at resonance with integral conditions. Adv. Differ. Equ.2017, 326 (2017)

DOI: https://doi.org/10.22190/FUMI191022054B

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