An existence theorem for fractional hybrid differential inclusions of Hadamard type with Dirichlet boundary conditions. (English) Zbl 1474.34094

Ahmad, Bashir; Ntouyas, Sotiris K.

Abstr. Appl. Anal. 2014, Article ID 705809, 7 p. (2014).

Summary: This paper studies the existence of solutions for a boundary value problem of nonlinear fractional hybrid differential inclusions by using a fixed point theorem due to B. C. Dhage [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 64, No. 6, 1290–1306 (2006; Zbl 1105.34051)]. The main result is illustrated with the aid of an example.

MSC:

34A60	Ordinary differential inclusions
34A08	Fractional ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations

Citations:

Zbl 1105.34051

References:

[1]	Podlubny, I., Fractional Differential Equations. Fractional Differential Equations, Mathematics in Science and Engineering, 198 (1999), San Diego, Calif, USA: Academic Press, San Diego, Calif, USA · Zbl 0924.34008
[2]	Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations. Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204 (2006), Amsterdam, The Netherlands: Elsevier Science B.V., Amsterdam, The Netherlands · Zbl 1092.45003
[3]	Sabatier, J.; Agrawal, O. P.; Machado, J. A. T., Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering (2007), Dordrecht, The Netherlands: Springer, Dordrecht, The Netherlands · Zbl 1116.00014
[4]	Ahmad, B.; Nieto, J. J., Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions, Boundary Value Problems, 2011 (2011) · Zbl 1275.45004
[5]	Alsaedi, A.; Ntouyas, S. K.; Ahmad, B., Existence results for Langevin fractional differential inclusions involving two fractional orders with four-point multiterm fractional integral boundary conditions, Abstract and Applied Analysis, 2013 (2013) · Zbl 1276.26008 · doi:10.1155/2013/869837
[6]	Băleanu, D.; Mustafa, O. G.; Agarwal, R. P., On \(L^p\)-solutions for a class of sequential fractional differential equations, Applied Mathematics and Computation, 218, 5, 2074-2081 (2011) · Zbl 1235.34008 · doi:10.1016/j.amc.2011.07.024
[7]	Agarwal, R. P.; Ahmad, B., Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions, Computers & Mathematics with Applications, 62, 3, 1200-1214 (2011) · Zbl 1228.34009 · doi:10.1016/j.camwa.2011.03.001
[8]	Graef, J. R.; Kong, L.; Kong, Q., Application of the mixed monotone operator method to fractional boundary value problems, Fractional Differential Calculus, 2, 1, 87-98 (2012) · Zbl 1412.34086
[9]	Akyildiz, F. T.; Bellout, H.; Vajravelu, K.; Van Gorder, R. A., Existence results for third order nonlinear boundary value problems arising in nano boundary layer fluid flows over stretching surfaces, Nonlinear Analysis: Real World Applications, 12, 6, 2919-2930 (2011) · Zbl 1231.35155 · doi:10.1016/j.nonrwa.2011.02.017
[10]	Ahmad, B.; Nieto, J. J., Sequential fractional differential equations with three-point boundary conditions, Computers & Mathematics with Applications, 64, 10, 3046-3052 (2012) · Zbl 1268.34006 · doi:10.1016/j.camwa.2012.02.036
[11]	Bai, Z.; Sun, W., Existence and multiplicity of positive solutions for singular fractional boundary value problems, Computers & Mathematics with Applications, 63, 9, 1369-1381 (2012) · Zbl 1247.34006 · doi:10.1016/j.camwa.2011.12.078
[12]	Sakthivel, R.; Mahmudov, N. I.; Nieto, J. J., Controllability for a class of fractional-order neutral evolution control systems, Applied Mathematics and Computation, 218, 20, 10334-10340 (2012) · Zbl 1245.93022 · doi:10.1016/j.amc.2012.03.093
[13]	Ahmad, B.; Nieto, J. J., Anti-periodic fractional boundary value problems with nonlinear term depending on lower order derivative, Fractional Calculus and Applied Analysis, 15, 3, 451-462 (2012) · Zbl 1281.34005 · doi:10.2478/s13540-012-0032-1
[14]	Agarwal, R. P.; O’Regan, D.; Staněk, S., Positive solutions for mixed problems of singular fractional differential equations, Mathematische Nachrichten, 285, 1, 27-41 (2012) · Zbl 1232.26005 · doi:10.1002/mana.201000043
[15]	Wang, J.; Zhou, Y.; Medveud, M., Qualitative analysis for nonlinear fractional differential equations via topological degree method, Topological Methods in Nonlinear Analysis, 40, 2, 245-271 (2012) · Zbl 1282.34013
[16]	Cabada, A.; Wang, G., Positive solutions of nonlinear fractional differential equations with integral boundary value conditions, Journal of Mathematical Analysis and Applications, 389, 1, 403-411 (2012) · Zbl 1232.34010 · doi:10.1016/j.jmaa.2011.11.065
[17]	Zhang, L.; Ahmad, B.; Wang, G.; Agarwal, R. P., Nonlinear fractional integro-differential equations on unbounded domains in a Banach space, Journal of Computational and Applied Mathematics, 249, 51-56 (2013) · Zbl 1302.45019 · doi:10.1016/j.cam.2013.02.010
[18]	Zhou, W.-X.; Chu, Y.-D.; Băleanu, D., Uniqueness and existence of positive solutions for a multi-point boundary value problem of singular fractional differential equations, Advances in Difference Equations, 2013 (2013) · Zbl 1380.34024 · doi:10.1186/1687-1847-2013-114
[19]	Ahmad, B.; Ntouyas, S. K.; Alsaedi, A., A study of nonlinear fractional differential equations of arbitrary order with Riemann-Liouville type multistrip boundary conditions, Mathematical Problems in Engineering, 2013 (2013) · Zbl 1296.34011 · doi:10.1155/2013/320415
[20]	Ahmad, B.; Ntouyas, S. K., Existence results for higher order fractional differential inclusions with multi-strip fractional integral boundary conditions, Electronic Journal of Qualitative Theory of Differential Equations, 20 (2013) · Zbl 1340.34055
[21]	Faieghi, M.; Kuntanapreeda, S.; Delavari, H.; Baleanu, D., LMI-based stabilization of a class of fractional-order chaotic systems, Nonlinear Dynamics, 72, 1-2, 301-309 (2013) · Zbl 1268.93121 · doi:10.1007/s11071-012-0714-6
[22]	Nieto, J. J.; Ouahab, A.; Prakash, P., Extremal solutions and relaxation problems for fractional differential inclusions, Abstract and Applied Analysis, 2013 (2013) · Zbl 1293.34012 · doi:10.1155/2013/292643
[23]	O’Regan, D.; Staněk, S., Fractional boundary value problems with singularities in space variables, Nonlinear Dynamics, 71, 4, 641-652 (2013) · Zbl 1268.34023 · doi:10.1007/s11071-012-0443-x
[24]	Wang, J.; Zhou, Y.; Wei, W., Fractional sewage treatment models with impulses at variable times, Applicable Analysis: An International Journal, 92, 9, 1959-1979 (2013) · Zbl 1279.26021 · doi:10.1080/00036811.2012.715150
[25]	Zhai, C.; Hao, M., Mixed monotone operator methods for the existence and uniqueness of positive solutions to Riemann-Liouville fractional differential equation boundary value problems, Boundary Value Problems, 2013 (2013) · Zbl 1300.34024 · doi:10.1186/1687-2770-2013-85
[26]	Ahmad, B.; Ntouyas, S. K., Existence results for fractional differential inclusions arising from real estate asset securitization and HIV models, Advances in Difference Equations, 2013 (2013) · Zbl 1379.34024 · doi:10.1186/1687-1847-2013-216
[27]	Ahmad, B.; Ntouyas, S. K., New existence results for differential inclusions involving Langevin equation with two indices, Journal of Nonlinear and Convex Analysis, 14, 3, 437-450 (2013) · Zbl 1278.26008
[28]	Wang, J.; Zhou, Y.; Fečkan, M., On the nonlocal Cauchy problem for semilinear fractional order evolution equations, Central European Journal of Mathematics, 12, 6, 911-922 (2014) · Zbl 1296.26035 · doi:10.2478/s11533-013-0381-y
[29]	Graef, J. R.; Kong, L.; Wang, M., Existence and uniqueness of solutions for a fractional boundary value problem on a graph, Fractional Calculus and Applied Analysis, 17, 2, 499-510 (2014) · Zbl 1308.34012 · doi:10.2478/s13540-014-0182-4
[30]	Punzo, F.; Terrone, G., On the Cauchy problem for a general fractional porous medium equation with variable density, Nonlinear Analysis: Theory, Methods & Applications, 98, 27-47 (2014) · Zbl 1284.35005 · doi:10.1016/j.na.2013.12.007
[31]	Liu, X.; Liu, Z.; Fu, X., Relaxation in nonconvex optimal control problems described by fractional differential equations, Journal of Mathematical Analysis and Applications, 409, 1, 446-458 (2014) · Zbl 1311.49033 · doi:10.1016/j.jmaa.2013.07.032
[32]	Razminia, K.; Razminia, A.; Tenreiro Machado, J. A., Analysis of diffusion process in fractured reservoirs using fractional derivative approach, Communications in Nonlinear Science and Numerical Simulation, 19, 9, 3161-3170 (2014) · Zbl 1458.76096 · doi:10.1016/j.cnsns.2014.01.025
[33]	Choudhary, S.; Daftardar-Gejji, V., Nonlinear multi-order fractional differential equations with periodic/anti-periodic boundary conditions, Fractional Calculus and Applied Analysis, 17, 2, 333-347 (2014) · Zbl 1305.34009 · doi:10.2478/s13540-014-0172-6
[34]	Hadamard, J., Essai sur l’etude des fonctions donnees par leur developpment de Taylor, Journal de Mathématiques Pures et Appliquées Série, 8, 101-186 (1892) · JFM 24.0359.01
[35]	Dhage, B. C., Existence results for neutral functional differential inclusions in Banach algebras, Nonlinear Analysis: Theory, Methods & Applications, 64, 6, 1290-1306 (2006) · Zbl 1105.34051 · doi:10.1016/j.na.2005.06.036
[36]	El Borai, M. M.; Abbas, M. I., On some integro-differential equations of fractional orders involving Carathéodory nonlinearities, International Journal of Modern Mathematics, 2, 1, 41-52 (2007) · Zbl 1152.47060
[37]	Zhao, Y.; Sun, S.; Han, Z.; Li, Q., Theory of fractional hybrid differential equations, Computers & Mathematics with Applications, 62, 3, 1312-1324 (2011) · Zbl 1228.45017 · doi:10.1016/j.camwa.2011.03.041
[38]	Sun, S.; Zhao, Y.; Han, Z.; Li, Y., The existence of solutions for boundary value problem of fractional hybrid differential equations, Communications in Nonlinear Science and Numerical Simulation, 17, 12, 4961-4967 (2012) · Zbl 1352.34011 · doi:10.1016/j.cnsns.2012.06.001
[39]	Sidi Ammi, M. R.; El Kinani, E. H.; Torres, D. F. M., Existence and uniqueness of solutions to functional integro-differential fractional equations, Electronic Journal of Differential Equations, 103 (2012) · Zbl 1253.26005
[40]	Dhage, B. C.; Ntouyas, S. K., Existence results for boundary value problems for fractional hybrid differential inclucions
[41]	Kassim, M. D.; Tatar, N.-E., Well-posedness and stability for a differential problem with Hilfer-Hadamard fractional derivative, Abstract and Applied Analysis, 2013 (2013) · Zbl 1470.34017 · doi:10.1155/2013/605029
[42]	Ahmad, B.; Ntouyas, S. K.; Alsaedi, A., On fractional differential inclusions with anti-periodic type integral boundary conditions, Boundary Value Problems, 2013 (2013) · Zbl 1296.34010 · doi:10.1186/1687-2770-2013-82
[43]	Ahmad, B.; Ntouyas, S. K., A fully Hadamard type integral boundary value problem of a coupled system of fractional differential equations, Fractional Calculus and Applied Analysis, 17, 2, 348-360 (2014) · Zbl 1312.34005 · doi:10.2478/s13540-014-0173-5
[44]	Ahmad, B.; Ntouyas, S. K., On Hadamard fractional integro-differential boundary value problems, Journal of Applied Mathematics and Computing (2014) · Zbl 1328.34006 · doi:10.1007/s12190-014-0765-6
[45]	Deimling, K., Multivalued Differential Equations. Multivalued Differential Equations, de Gruyter Series in Nonlinear Analysis and Applications, 1 (1992), Berlin, Germany: Walter de Gruyter, Berlin, Germany · Zbl 0760.34002 · doi:10.1515/9783110874228
[46]	Hu, S.; Papageorgiou, N. S., Handbook of Multivalued Analysis. Vol. I, Theory. Handbook of Multivalued Analysis. Vol. I, Theory, Mathematics and its Applications, 419 (1997), Dordrecht, The Netherlands: Kluwer Academic, Dordrecht, The Netherlands · Zbl 0887.47001 · doi:10.1007/978-1-4615-6359-4
[47]	Lasota, A.; Opial, Z., An application of the Kakutani—Ky Fan theorem in the theory of ordinary differential equations, Bulletin de l’Académie Polonaise des Sciences: Série des Sciences Mathématiques, Astronomiques et Physiques, 13, 781-786 (1965) · Zbl 0151.10703