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2015 | 35 | 2 | 105-122
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Upper and lower solutions method for partial Hadamard fractional integral equations and inclusions

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In this paper we use the upper and lower solutions method combined with Schauder's fixed point theorem and a fixed point theorem for condensing multivalued maps due to Martelli to investigate the existence of solutions for some classes of partial Hadamard fractional integral equations and inclusions.
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autor
  • Laboratory of Mathematics, University of Saïda, P.O. Box 138, 20000 Saïda, Algeria
  • Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
  • Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
  • Laboratory of Mathematics, University of Sidi Bel-Abbès, P.O. Box 89, Sidi Bel-Abbès 22000, Algeria
  • Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
Bibliografia
  • [1] S. Abbas and M. Benchohra, Fractional order integral equations of two independent variables, Appl. Math. Comput. 227 (2014), 755-761. doi: 10.1016/j.amc.2013.10.086
  • [2] S. Abbas and M. Benchohra, Upper and lower solutions method for the Darboux problem for fractional order partial differential inclusions, Int. J. Modern Math. 5 (3) (2010), 327-338.
  • [3] S. Abbas and M. Benchohra, Upper and lower solutions method for impulsive partial hyperbolic differential equations with fractional order, Nonlinear Anal. Hybrid Syst. 4 (2010), 406-413. doi: 10.1016/j.nahs.2009.10.004
  • [4] S. Abbas and M. Benchohra, Upper and lower solutions method for Darboux problem for fractional order implicit impulsive partial hyperbolic differential equations, Acta Univ. Palacki. Olomuc. 51 (2) (2012), 5-18.
  • [5] S. Abbas and M. Benchohra, The method of upper and lower solutions for partial hyperbolic fractional order differential inclusions with impulses, Discuss. Math. Differ. Incl. Control Optim. 30 (1) (2010), 141-161. doi: 10.7151/dmdico.1116
  • [6] S. Abbas, M. Benchohra and A. Hammoudi, Upper, lower solutions method and extremal solutions for impulsive discontinuous partial fractional differential inclusions, Panamerican Math. J. 24 (1) (2014), 31-52.
  • [7] S. Abbas, M. Benchohra and G.M. N'Guérékata, Topics in Fractional Differential Equations (Springer, New York, 2012). doi: 10.1007/978-1-4614-4036-9
  • [8] S. Abbas, M. Benchohra and G.M. N'Guérékata, Advanced Fractional Differential and Integral Equations (Nova Science Publishers, New York, 2015).
  • [9] S. Abbas, M. Benchohra and J.J. Trujillo, Upper and lower solutions method for partial fractional differential inclusions with not instantaneous impulses, Prog. Frac. Diff. Appl. 1 (1) (2015), 11-22.
  • [10] J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces (Marcel-Dekker, New York, 1980).
  • [11] M. Benchohra, J. Henderson, S.K. Ntouyas and A. Ouahab, Existence results for functional differential equations of fractional order, J. Math. Anal. Appl. 338 (2008), 1340-1350. doi: 10.1016/j.jmaa.2007.06.021
  • [12] M. Benchohra, J. Henderson and S.K. Ntouyas, Impulsive Differential Equations and Inclusions (Hindawi Publishing Corporation, Vol. 2, New York, 2006). doi: 10.1155/9789775945501
  • [13] 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
  • [14] 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
  • [15] K. Deimling, Multivalued Differential Equations (Walter de Gruyter, Berlin-New York, 1992). doi: 10.1515/9783110874228
  • [16] S. Djebali, L. Górniewicz and A. Ouahab, Solution Sets for Differential Equations and Inclusions, de Gruyter Series in Nonlinear Analysis and Applications (Walter de Gruyter & Co., Berlin, 2013). doi: 10.1515/9783110293562
  • [17] L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings, Mathematics and its Applications, 495 (Kluwer Academic Publishers, Dordrecht, 1999). doi: 10.1007/978-94-015-9195-9
  • [18] A. Granas and J. Dugundji, Fixed Point Theory (Springer-Verlag, New York, 2003). doi: 10.1007/978-0-387-21593-8
  • [19] J. Hadamard, Essai sur l'étude des fonctions données par leur développment de Taylor, J. Pure Appl. Math. 4 (8) (1892), 101-186.
  • [20] Sh. Hu and N. Papageorgiou, Handbook of Multivalued Analysis, Volume I: Theory (Kluwer, Dordrecht, Boston, London, 1997). doi: 10.1007/978-1-4615-6359-4
  • [21] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier Science B.V., Amsterdam, 2006).
  • [22] M. Kisielewicz, Differential Inclusions and Optimal Control (Kluwer Academic Publishers, Dordrecht, Netherlands, 1991).
  • [23] G.S. Ladde, V. Lakshmikanthan and A.S. Vatsala, Monotone Iterative Techniques for Nonliner Differential Equations (Pitman Advanced Publishing Program, London, 1985).
  • [24] A. Lasota and Z. Opial, An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations, Bull. Acad. Pol. Sci. Ser. Sci. Math. Astronom. Phys. 13 (1965), 781-786.
  • [25] M. Martelli, A Rothe's type theorem for noncompact acyclic-valued map, Boll. Un. Math. Ital. 11 (1975), 70-76.
  • [26] K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations (John Wiley, New York, 1993).
  • [27] B.G. Pachpatte, Monotone methods for systems of nonlinear hyperbolic problems in two independent variables, Nonlinear Anal. 30 (1997), 235-272.
  • [28] S. Pooseh, R. Almeida and D. Torres, Expansion formulas in terms of integer-order derivatives for the hadamard fractional integral and derivative, Numer. Funct. Anal. Optim. 33 (3) (2012), 301-319. doi: 10.1080/01630563.2011.647197
  • [29] S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives. Theory and Applications (Gordon and Breach, Yverdon, 1993).
  • [30] A.N. Vityuk, On solutions of hyperbolic differential inclusions with a nonconvex right-hand side, (Russian) Ukran. Mat. Zh. 47 (4) (1995), 531-534; translation in Ukrainian Math. J. 47 (4) (1995), 617-621 (1996).
  • [31] A.N. Vityuk and A.V. Golushkov, Existence of solutions of systems of partial differential equations of fractional order, Nonlinear Oscil. 7 (2004), 318-325. doi: 10.1007/s11072-005-0015-9
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1172
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