Abstract
In this paper, we investigate some Cauchy problems involving a left-sided Hadamard-type fractional derivative. A theorem on the existence of a unique solution to a nonlinear problem is proved. The main result is obtained using a fixed point theorem due to Banach, as well as the Bielecki norm. A Cauchy formula for the solution of the linear problem is derived.
(Communicated by Michal Fečkan)
References
[1] Ahmad, B.—Ntouyas, S. K.: Initial value problems for hybrid Hadamard fractional differential equations, Electron. J. Differential Equations 161 (2014), 1–8.Search in Google Scholar
[2] Ahmad, B.—Ntouyas, S. K.: An existence theorem for fractional hybrid differential inclusions of Hadamard type with Dirichlet boundary conditions, Abstr. Appl. Anal. 2014 (2014), Article ID 705809, 7 pp.10.1155/2014/705809Search in Google Scholar
[3] Ahmad, B.—Ntouyas, S. K.: On three-point Hadamard-type fractional boundary value problems, Int. Electron. J. Pure Appl. Math. 8 (2014), 31–42.10.12732/iejpam.v8i4.4Search in Google Scholar
[4] Ahmad, B.—Ntouyas, S. K.: An existence theorem for fractional hybrid differential inclusions of Hadamard type, Discuss. Math. Differ. Incl. Control Optim. 34 (2014), 207–218.10.7151/dmdico.1161Search in Google Scholar
[5] Ahmad, B.—Ntouyas, S. K.: Initial value problems of fractional order Hadamard-type functional differential equations, Electron. J. Differential Equations 77 (2015), 1–9.10.1007/978-3-319-52141-1_2Search in Google Scholar
[6] Ahmad, B.—Ntouyas, S. K.: Nonlocal boundary value problems for hybrid fractional differential equations and inclusions of Hadamard type, Fractional Differ. Calc. 5 (2015), 107–123.10.7153/fdc-05-10Search in Google Scholar
[7] Ahmad, B.—Ntouyas, S. K.: Hadamard-type fractional functional differential equations and inclusions with retarded and advanced arguments, Adv. Difference Equ. 2016(80) (2016).10.1186/s13662-016-0807-5Search in Google Scholar
[8] Ahmad, B.—Ntouyas, S. K.: Initial value problems for functional and neutral functional Hadamard type fractional differential inclusions, Miskolc Math. Notes 17 (2016), 15–27.10.18514/MMN.2016.1632Search in Google Scholar
[9] 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 (2014), 348–360.10.2478/s13540-014-0173-5Search in Google Scholar
[10] Ahmad, B.—Ntouyas, S. K.: Boundary value problems of Hadamard-type fractional differential equations and inclusions with nonlocal conditions, Vietnam J. Math. 45(3) (2017), 409–423.10.1007/s10013-016-0213-zSearch in Google Scholar
[11] Ahmad, B.—Ntouyas, S. K.—Alsaedi, A.: New results for boundary value problems of Hadamard-type fractional differential inclusions and integral boundary conditions, Bound. Value Probl. 2013:275 (2013).10.1186/1687-2770-2013-275Search in Google Scholar
[12] Ahmad, B.— Ntouyas, S. K.—Tariboon, J.: Existence results for mixed Hadamard and Riemann-Liouville fractional integro-differential equations, Adv. Difference Equ. 2015:293 (2015).10.1186/s13662-015-0625-1Search in Google Scholar
[13] Ahmad, B.— Ntouyas, S. K.—Tariboon, J.: Existence results for mixed Hadamard and Riemann-Liouville fractional integro-differential inclusions, J. Nonlinear Sci. Appl. 9 (2016), 6333–6347.10.22436/jnsa.009.12.34Search in Google Scholar
[14] Ahmad, B.—Alsaedi, A.—Ntouyas, S. K.—Tariboon, J.: Hadamard-Type Fractional Differential Equations, Inclusions and Inequalities, Springer, 2017.10.1007/978-3-319-52141-1Search in Google Scholar
[15] Alsaedi, A.—Ntouyas, S. K.—Ahmad, B.—Hobiny, A.: Nonlinear Hadamard fractional differential equations with Hadamard type nonlocal non-conserved conditions, Adv. Difference Equ. 2015:285 (2015).10.1186/s13662-015-0589-1Search in Google Scholar
[16] Benchohra, M.—Lazreg, J. E.: Existence and Ulam stability for nonlinear implicit fractional differential equations with Hadamard derivative, Stud. Univ. Babes-Bolyai Math. 62(1) (2017), 27–38.10.24193/subbmath.2017.0003Search in Google Scholar
[17] Benchohra, M.—Bouriah, S.—Lazreg, J. E.—Nieto, J. J.: Nonlinear implicit Hadamard’s fractional differential equations with delay in Banach space, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 55(1) (2016), 15–26.Search in Google Scholar
[18] Carpinteri, A.—Mainardi, F.: Fractals and Fractional Calculus in Continuum Mechanics, Springer, Berlin, 1997.10.1007/978-3-7091-2664-6Search in Google Scholar
[19] Guerraiche, N.—Hamani, S.—Henderson, J.: Initial value problems for fractional functional differential inclusions with Hadamard type derivative, Arch. Math. (Brno) 52 (2016), 263–273.10.5817/AM2016-4-263Search in Google Scholar
[20] Hadamard, J.: Essai sur l’etude des fonctions donnees par leur developpment de Taylor, J. Pure Appl. Math. 4(8) (1892), 101–186.Search in Google Scholar
[21] Hilfer, R.: Applications of Fractional Calculus in Physics, World Sci. Publishing, River Edge, NJ, 2000.10.1142/3779Search in Google Scholar
[22] Idczak, D.—Kamocki, R.: On the existence and uniqueness and formula for the solution of R-L fractional Cauchy problem in ℝn, Fract. Calc. Appl. Anal. 14(4) (2011), 538–553.10.2478/s13540-011-0033-5Search in Google Scholar
[23] Kassim, M. D.—Furati, K. M.—Tatar, N.-E.: On a differential equation involving Hilfer-Hadamard fractional derivative, Abstr. Appl. Anal. 2012 (2012), 17 pp.10.1155/2012/391062Search in Google Scholar
[24] Kassim, M. D.—Tatar, N.-E.: Well-posedness and stability for a differential problem with Hilfer-Hadamard fractional derivative, Abstr. Appl. Anal. 2014 (2014), 1–7.10.1155/2013/605029Search in Google Scholar
[25] Kilbas, A. A.: Hadamard-type fractional calculus, J. Korean Math. Soc. 38(6) (2001), 1191–1204.Search in Google Scholar
[26] Kilbas, A. A.: Hadamard-type integral equations and fractional calculus operators, Oper. Theory Adv. Appl. 142 (2003), 175–188.10.1007/978-3-0348-8007-7_10Search in Google Scholar
[27] Kilbas, A. A.—Srivastava, H. M.—Trujillo, J. J.: Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.Search in Google Scholar
[28] Malinowska, A. B.—Torres, D. F. M.: Introduction to the Fractional Calculus of Variations, Imperial College Press, London, 2012.10.1142/p871Search in Google Scholar
[29] Samko, S. G.—Kilbas, A. A.—Marichev, O. I.: Fractional Integrals and Derivatives – Theory and Applications, Gordon and Breach, Amsterdam, 1993.Search in Google Scholar
[30] West, BJ.—Grigolini, P.: Applications of Fractional Calculus in Physics, World Scientific, Singapore, 1998.Search in Google Scholar
[31] Zhang, X.—Zhang, X.—Liu, Z.—Ding, W.—Cao, H.—Shu, T.: On the general solution of impulsive systems with Hadamard fractional derivatives, Math. Probl. Eng. 2016 (2016), Article ID 2814310, 12 pp.10.1155/2016/2814310Search in Google Scholar
[32] Zhang, X.—Shu, T.—Cao, H.—Liu, Z.—Ding, W.: The general solution for impulsive differential equations with Hadamard fractional derivative of order q ∈ (1,2), Adv. Difference Equ. 2016:14 (2016), 36 pp.10.1186/s13662-016-1008-ySearch in Google Scholar
© 2018 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Idempotents, group membership and their applications
- Residuation in non-associative MV-algebras
- An extension of F. Šik’s theorem on modular lattices
- Weak pseudo-BCK algebras
- Witt functor of a quadratic order
- A modification of a problem of Diophantus
- Cauchy problems involving a Hadamard-type fractional derivative
- Caratheodory’s solution of the Cauchy problem and a question of Z. Grande
- Sturm-Picone comparison theorems for nonlinear impulsive differential equations
- On oscillatory fourth order nonlinear neutral differential equations – III
- Local-periodic solutions for functional dynamic equations with infinite delay on changing-periodic time scales
- On the representation of involutive Jamesian functions
- Refinements of the heinz inequalities for operators and matrices
- Generalizations of Reid inequality
- Probabilistic convergence transformation groups
- Cluster sets and topology
- Some characterizations for Markov processes as mixed renewal processes
- Equivalent conditions of complete convergence for weighted sums of ANA random variables
Articles in the same Issue
- Idempotents, group membership and their applications
- Residuation in non-associative MV-algebras
- An extension of F. Šik’s theorem on modular lattices
- Weak pseudo-BCK algebras
- Witt functor of a quadratic order
- A modification of a problem of Diophantus
- Cauchy problems involving a Hadamard-type fractional derivative
- Caratheodory’s solution of the Cauchy problem and a question of Z. Grande
- Sturm-Picone comparison theorems for nonlinear impulsive differential equations
- On oscillatory fourth order nonlinear neutral differential equations – III
- Local-periodic solutions for functional dynamic equations with infinite delay on changing-periodic time scales
- On the representation of involutive Jamesian functions
- Refinements of the heinz inequalities for operators and matrices
- Generalizations of Reid inequality
- Probabilistic convergence transformation groups
- Cluster sets and topology
- Some characterizations for Markov processes as mixed renewal processes
- Equivalent conditions of complete convergence for weighted sums of ANA random variables