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An existence theorem for fractional hybrid differential inclusions of Hadamard type

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This paper studies the existence of solutions for fractional hybrid differential inclusions of Hadamard type by using a fixed point theorem due to Dhage. The main result is illustrated with the aid of an example.
  • Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
  • Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece
  • [1] I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999).
  • [2] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204 (Elsevier Science B.V., Amsterdam, 2006).
  • [3] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models (Imperial College Press, 2010).
  • [4] K. Diethelm, The Analysis of Fractional Differential Equations. An Application-oriented Exposition Using Differential Operators of Caputo Type, Lecture Notes in Mathematics, 2004 (Springer-Verlag, Berlin, 2010).
  • [5] D. Baleanu, K. Diethelm, E. Scalas and J.J. Trujillo, Fractional Calculus Models and Numerical Methods. Series on Complexity, Nonlinearity and Chaos (World Scientific, Boston, 2012).
  • [6] B. Ahmad and S.K. Ntouyas, Some existence results for boundary value problems for fractional differential inclusions with non-separated boundary conditions, Electron. J. Qual. Theory Differ. Equ. (2010), No. 71, 1-17.
  • [7] J.R. Wang and Y. Zhou, Existence and controllability results for fractional semilinear differential inclusions, Nonlinear Anal. Real World Appl. 12 (2011) 3642-3653. doi: 10.1016/j.nonrwa.2011.06.021
  • [8] B. Ahmad, S.K. Ntouyas and A. Alsaedi, New existence results for nonlinear fractional differential equations with three-point integral boundary conditions, Adv. Differ. Equ. (2011) Art. ID 107384, 11 pp.
  • [9] J.R. Graef, L. Kong and Q. Kong, Application of the mixed monotone operator method to fractional boundary value problems, Fract. Calc. Differ. Calc. 2 (2011), 554-567.
  • [10] B. Ahmad and J.J. Nieto, Sequential fractional differential equations with three-point boundary conditions, Comput. Math. Appl. 64 (2012) 3046-3052. doi: 10.1016/j.camwa.2012.02.036
  • [11] R. Sakthivel, N.I. Mahmudov and J.J. Nieto, Controllability for a class of fractional-order neutral evolution control systems, Appl. Math. Comput. 218 (2012) 10334-10340. doi: 10.1016/j.amc.2012.03.093
  • [12] R.P. Agarwal, D. O'Regan and S. Stanek, Positive solutions for mixed problems of singular fractional differential equations, Math. Nachr. 285 (2012) 27-41. doi: 10.1002/mana.201000043
  • [13] G. Wang, B. Ahmad, L. Zhang and R.P. Agarwal, Nonlinear fractional integro-differential equations on unbounded domains in a Banach space, J. Comput. App. Math. 249 (2013) 51-56. doi: 10.1016/
  • [14] B. Ahmad, S.K. Ntouyas and A. Alsaedi, A study of nonlinear fractional differential equations of arbitrary order with Riemann-Liouville type multistrip boundary conditions, Math. Probl. Eng. 2013, Art. ID 320415, 9 pp.
  • [15] J.J. Nieto, A. Ouahab and P. Prakash, Extremal solutions and relaxation problems for fractional differential inclusions, Abstr. Appl. Anal. 2013, Art. ID 292643, 9 pp.
  • [16] J.R. Wang, Y. Zhou and W. Wei, Fractional sewage treatment models with impulses at variable times, Appl. Anal. 92 (2013) 1959-1979. doi: 10.1080/00036811.2012.715150
  • [17] C. Zhai and M. Hao, Mixed monotone operator methods for the existence and uniqueness of positive solutions to Riemann-Liouville fractional differential equation boundary value problems, Bound. Value Probl. 85 (2013) 13 pp.
  • [18] F. Punzo and G. Terrone, On the Cauchy problem for a general fractional porous medium equation with variable density, Nonlinear Anal. 98 (2014) 27-47. doi: 10.1016/
  • [19] G. Wang, B. Ahmad and L. Zhang, Existence of extremal solution to a nonlinear system with the right-handed Riemann-Liouville fractional derivative, Appl. Math. Lett. 31 (2014) 1-6. doi: 10.1016/j.aml.2014.01.004
  • [20] J. Hadamard, Essai sur l'etude des fonctions donnees par leur developpment de Taylor, J. Mat. Pure Appl. Ser. 8 (1892) 101-186.
  • [21] P.L. Butzer, A.A. Kilbas and J.J. Trujillo, Compositions of Hadamard-type fractional integration operators and the semigroup property, J. Math. Anal. Appl. 269 (2002) 387-400. doi: 10.1016/S0022-247X(02)00049-5
  • [22] P.L. Butzer, A.A. Kilbas and J.J. Trujillo, Fractional calculus in the Mellin setting and Hadamard-type fractional integrals, J. Math. Anal. Appl. 269 (2002) 1-27. doi: 10.1016/S0022-247X(02)00001-X
  • [23] P.L. Butzer, A.A. Kilbas and J.J. Trujillo, Mellin transform analysis and integration by parts for Hadamard-type fractional integrals, J. Math. Anal. Appl. 270 (2002) 1-15. doi: 10.1016/S0022-247X(02)00066-5
  • [24] A.A. Kilbas, Hadamard-type fractional calculus, J. Korean Math. Soc. 38 (2001) 1191-1204.
  • [25] A.A. Kilbas and J.J. Trujillo, Hadamard-type integrals as G-transforms, Integral Transform. Spec. Funct. 14 (2003) 413-427. doi: 10.1080/1065246031000074443
  • [26] M. El Borai and M. Abbas, On some integro-differential equations of fractional orders involving Carathéodory nonlinearities, Int. J. Mod. Math. 2 (2007) 41-52.
  • [27] Y. Zhao, S. Sun, Z. Han and Q. Li, Theory of fractional hybrid differential equations, Comput. Math. Appl. 62 (2011) 1312-1324. doi: 10.1016/j.camwa.2011.03.041
  • [28] S. Sun, Y. Zhao, Z. Han and Y. Li, The existence of solutions for boundary value problem of fractional hybrid differential equations, Commun. Nonlinear Sci. Numer. Simul. 17 (2012) 4961-4967. doi: 10.1016/j.cnsns.2012.06.001
  • [29] M. Ammi, E. El Kinani and D. Torres, Existence and uniqueness of solutions to functional integro-differential fractional equations, Electron. J. Differ. Eq. 2012 (103) (2012) 1-9.
  • [30] B.C. Dhage and S.K. Ntouyas, Existence results for boundary value problems for fractional hybrid differential inclucions, Topol. Methods Nonlinar Anal. 44 (2014) 229-238.
  • [31] B. Ahmad and S.K. Ntouyas, Initial value problems for hybrid Hadamard fractional differential equations, Electron. J. Differ. Eq. 2014 (161) (2014) 1-8.
  • [32] G.A. Anastassiou, Fractional Differentiation Inequalities (Springer Publishing Company, New York, 2009). doi: 10.1007/978-0-387-98128-4
  • [33] K. Deimling, Multivalued Differential Equations, Walter De Gruyter (Berlin-New York, 1992). doi: 10.1515/9783110874228
  • [34] Sh. Hu and N. Papageorgiou, Handbook of Multivalued Analysis, Theory I (Kluwer, Dordrecht, 1997). doi: 10.1007/978-1-4615-6359-4
  • [35] A. Lasota and Z. Opial, An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 13 (1965) 781-786.
  • [36] B. Dhage, Existence results for neutral functional differential inclusions in Banach algebras, Nonlinear Anal. 64 (2006) 1290-1306. doi: 10.1016/
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