Upper and lower solutions method for partial discontinuous fractional differential inclusions with not instantaneous impulses

Saïd Abbas ; Mouffak Benchohra ; Mohamed Abdalla Darwish

Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 36 (2016), p. 155-179 / Harvested from The Polish Digital Mathematics Library

Résumé

In this paper, we use the upper and lower solutions method combined with a fixed point theorem for multivalued maps in Banach algebras due to Dhage for investigations of the existence of solutions of a class of discontinuous partial differential inclusions with not instantaneous impulses. Also, we study the existence of extremal solutions under Lipschitz, Carath´eodory and certain monotonicity conditions

Détail

Publié le : 2016-01-01
EUDML-ID : urn:eudml:doc:289596

Bibliographie

[000] [1] S. Abbas, R.P. Agarwal and M. Benchohra, Impulsive discontinuous partial hyperbolic differential equations of fractional order on Banach Algebras, Electron. J. Differential Equations 2010 (2010), 1-17.

[001] [2] 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.

[002] [3] 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.

[003] [4] 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

[004] [5] S. Abbas and M. Benchohra, Impulsive partial hyperbolic functional differential equations of fractional order with state-dependent delay, Frac. Calc. Appl. Anal. 13 (3) (2010), 225-244.

[005] [6] S. Abbas and M. Benchohra, Uniqueness and Ulam stabilities results for partial fractional differential equations with not instantaneous impulses, Appl. Math. Comput. 257 (2015), 190-198. doi: 10.1016/j.amc.2014.06.073

[006] [7] 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.

[007] [8] 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

[008] [9] S. Abbas, M. Benchohra and G.M. N'Guérékata, Advanced Fractional Differential and Integral Equations (Nova Science Publishers, New York, 2015).

[009] [10] S. Abbas, M. Benchohra, G.M. N'Guérékata and B.A. Slimani, Darboux problem for fractional order discontinuous hyperbolic partial differential equations in Banach algebras, Complex Var. Elliptic Equ. 57 (2012), 337-350. doi: 10.1080/17476933.2011.555542

[010] [11] S. Abbas, M. Benchohra and A.N. Vityuk, On fractional order derivatives and Darboux problem for implicit differential equations, Frac. Calc. Appl. Anal. 15 (2) (2012), 168-182. doi: 10.2478/s13540-012-0012-5

[011] [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

[012] [13] M. Benchohra and S.K. Ntouyas, The method of lower and upper solutions to the Darboux problem for partial differential inclusions, Miskolc Math. Notes 4 (2) (2003), 81-88.

[013] [14] M.A. Darwish, J. Henderson and D. O'Regan, Existence and asymptotic stability of solutions of a perturbed fractional functional-integral equation with linear modification of the argument, Bull. Korean Math. Soc. 48 (3) (2011), 539-553. doi: 10.4134/BKMS.2011.48.3.539

[014] [15] M.A. Darwish and J. Henderson, Nondecreasing solutions of a quadratic integral equation of Urysohn-Stieltjes type, Rocky Mountain J. Math. 42 (2) (2012), 545-566. doi: 10.1216/RMJ-2012-42-2-545

[015] [16] M.A. Darwish and J. Banaś, Existence and characterization of solutions of nonlinear Volterra-Stieltjes integral equations in two vriables, Abstr. Appl. Anal. 2014, Art. ID 618434, 11 pp.

[016] [17] B.C. Dhage, Existence results for neutral functional differential inclusions in Banach algebras, Nonlinear Anal. 64 (2006), 1290-1306. doi: 10.1016/j.na.2005.06.036

[017] [18] B.C. Dhage, A fixed point theorem for multi-valued mappings on ordered Banach spaces with applications} II, Panamer. Math. J. 15 (2005), 15-34.

[018] [19] K. Diethelm and N.J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl. 265 (2002), 229-248. doi: 10.1006/jmaa.2000.7194

[019] [20] S. Heikkila and V. Lakshmikantham, Monotone Iterative Technique for Nonlinear Discontinuous Differential Equations (Marcel Dekker Inc., New York, 1994).

[020] [21] E. Hernández and D. O'Regan, On a new class of abstract impulsive differential equations, Proc. Amer. Math. Soc. 141 (2013), 1641-1649. doi: 10.1090/S0002-9939-2012-11613-2

[021] [22] Sh. Hu and N. Papageorgiou, Handbook of Multivalued Analysis, Volume I: Theory (Kluwer, Dordrecht, Boston, London, 1997), .

[022] [23] A.A. Kilbas and S.A. Marzan, Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions, Diff. Equ. 41 (2005), 84-89. doi: 10.1007/s10625-005-0137-y

[023] [24] 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).

[024] [25] G.S. Ladde, V. Lakshmikanthan and A.S. Vatsala, Monotone Iterative Techniques for Nonliner Differential Equations (Pitman Advanced Publishing Program, London, 1985).

[025] [26] 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.

[026] [27] M. Pierri, D. O'Regan and V. Rolnik, Existence of solutions for semi-linear abstract differential equations with not instantaneous, Appl. Math. Comput. 219 (2013), 6743-6749. doi: 10.1016/j.amc.2012.12.084

[027] [28] A.N. Vityuk and A.V. Golushkov, Existence of solutions of systems of partial differential equations of fractional order, Nonlinear Oscil. 7 (3) (2004), 318-325. doi: 10.1007/s11072-005-0015-9

[028] [29] A.N. Vityuk and A.V. Mykhailenko, The Darboux problem for an implicit fractional-order differential equation, J. Math. Sci. 175 (4) (2011), 391-401. doi: 10.1007/s10958-011-0353-3