Boundary value problems for fractional differential inclusions with Hadamard type derivatives in Banach spaces

Authors

  • John R. GRAEF Department of Mathematics University of Tennessee at Chattanooga Chattanooga, TN 37403-2504, USA, e-mail: John-Graef@utc.edu https://orcid.org/0000-0002-8149-4633
  • Nassim GUERRAICHE Laboratoire des Math´ematiques Appliqu´es et Pures Universit´e de Mostaganem B.P. 227, 27000, Mostaganem, Algerie, e-mail: hamani_samira@yahoo.fr https://orcid.org/0000-0003-2018-3388
  • Samira HAMANI Laboratoire des Math´ematiques Appliqu´es et Pures Universit´e de Mostaganem B.P. 227, 27000, Mostaganem, Algerie, e-mail: nassim.guerraiche@univ-mosta.dz

DOI:

https://doi.org/10.24193/subbmath.2017.4.02

Keywords:

Fractional differential inclusion, Hadamard-type fractional derivative, fractional integral, M¨onch’s fixed point theorem, Kuratowski measure of noncompacteness.

Abstract

The authors establish sufficient conditions for the existence of solutions to boundary value problems for fractional differential inclusions involving the Hadamard type fractional derivative of order α ∈ (1, 2] in Banach spaces. Their approach uses M¨onch’s fixed point theorem and the Kuratowski measure of noncompacteness.

Mathematics Subject Classification (2010): 26A33, 34A08, 34A60, 34B15.

References

Agarwal, R.P., Benchohra, M., Seba, D., An the application of measure of noncompactness to the existence of solutions for fractional differential equations, Results Math. 55(2009), 221-230.

Ahmed, B., Ntouyas, S.K., Initial value problems for hybrid Hadamard fractional equations, Electron. J. Differential Equations, 2014(2014), no. 161, 1-8.

Akhmerov, R.R., Kamenski, M.I., Patapov, A.S., Rodkina, A.E., Sadovski, B.N., Measures of Noncompactness and Condensing Operators (Translated from the 1986 Russian original by A. Iacop), Operator theory: Advances and Applications, 55, Birkha¨user Ver- lag, Bassel, 1992.

Aubin, J.P., Cellina, A., Differential Inclusions, Springer-Verlag, Berlin-Heidelberg, New York, 1984.

Aubin, J.P., Frankowska, H., Set-Valued Analysis, Birkha¨user, Boston, 1990.

Banas, J., Goebel, K., Measure of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, Vol. 60, Dekker, New York.

Benchohra, M., Henderson, J., Seba, D., Measure of noncompactness and fractional differential equations in Banach spaces, Commun. Appl. Anal., 12(2008), 419-428.

Benchohra, M., Henderson, J., Seba, D., Boundary value problems for fractional differential inclusions in Banach space, Fract. Differ. Calc., 2(2012), 99-108.

Benchohra, M., Nieto, J.J., Seba, D., Measure of noncompactness and fractional and hyperbolic partial fractional differential equations in Banach space, Panamer. Math. J., 20(2010), 27-37.

Benhamida, W., Graef, J.R., Hamani, S., Boundary value problems for fractional differential equations with integral and anti-periodic conditions in a Banach space, (to appear).

Butzer, P.L., Kilbas, A.A., Trujillo, J.J., Composition of Hadamard-type fractional integration operators and the semigroup property, J. Math. Anal. Appl., 269(2002), 387-400.

Butzer, P.L., Kilbas, A.A., Trujillo, J.J., Fractional calculus in the Mellin setting and Hadamard-type fractional integrals, J. Math. Anal. Appl., 269(2002), 1-27.

Butzer, P.L., Kilbas, A.A., Trujillo, J.J., Mellin transform analysis and integration by parts for Hadamard-type fractional integrals, J. Math. Anal. Appl., 270(2002), 1-15.

Castaing, C., Valadier, M., Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics 580, Springer-Verlag, Berlin-Heidelberg-New York, 1977.

Deimling, K., Multivalued Differential Equations, De Gruyter, Berlin-New York, 1992. [16] Hadamard, J., Essai sur l’etude des fonctions donnees par leur development de Taylor, J. Mat. Pure Appl., 8(1892), 101-186.

Heinz, H.P., On the behavior of measure of noncompactness with respect of differentiation and integration of vector-valued function, Nonlinear. Anal, 7(1983), 1351-1371.

Hilfer, R., Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.

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

Lasota, A., Opial, Z., An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equation, Bull. Accd. Pol. Sci. Ser. Sci. Math. Astronom. Phys., 13(1965), 781-786.

Momani, S.M., Hadid, S.B., Alawenh, Z.M., Some analytical properties of solutions of diifferential equations of noninteger order, Int. J. Math. Math. Sci., 2004(2004), 697-701.

O’Regan, D., Precup, R., Fixed point theorems for set-valued maps and existence principles for integral inclusions, J. Math. Anal. Appl., 245(2000), 594-612.

Podlubny, I., Fractional Differential Equation, Academic Press, San Diego, 1999.

Thiramanus, P., Ntouyas, S.K., Tariboon, J., Existence and uniqueness results for Hadamard-type fractional differential equations with nonlocal fractional integral boundary conditions, Abstr. Appl. Anal., (2014), Art. ID 902054, 9 pp.

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Published

2017-12-30

How to Cite

GRAEF, J. R., GUERRAICHE, N., & HAMANI, S. (2017). Boundary value problems for fractional differential inclusions with Hadamard type derivatives in Banach spaces. Studia Universitatis Babeș-Bolyai Mathematica, 62(4), 427–438. https://doi.org/10.24193/subbmath.2017.4.02

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