Upper and lower solutions method for partial Hadamard fractional integral equations and inclusions

• Autorzy: Saïd Abbas , Eman Alaidarous , Wafaa Albarakati , Mouffak Benchohra
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

functional integral equation; integral inclusion; Hadamard partial fractional integral; condensing multivalued map; existence; upper solution; lower solution; fixed point

Kategorie tematyczne

• 34A08: Fractional differential equations
• 34K05: General theory

• Wydawca: Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra
• Czasopismo: Discussiones Mathematicae, Differential Inclusions, Control and Optimization
• Rocznik: 2015
• Tom: 35
• Numer: 2
• Strony: 105-122

Daty

wydano

2015

otrzymano

2015-07-27

Twórcy

autor

Saïd Abbas

• Laboratory of Mathematics, University of Saïda, P.O. Box 138, 20000 Saïda, Algeria

autor

Eman Alaidarous

• Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

autor

Wafaa Albarakati

• Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

autor

Mouffak Benchohra

• 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

Identyfikatory

DOI

10.7151/dmdico.1172