Mathematica Bohemica, Vol. 142, No. 3, pp. 309-321, 2017
Nonlinear fractional differential inclusion with nonlocal fractional integro-differential boundary conditions in Banach spaces
Received March 14, 2016. First published January 5, 2017.
Abstract: We consider a nonlinear fractional differential inclusion with nonlocal fractional integro-differential boundary conditions in a Banach space. The existence of at least one solution is proved by using the set-valued analog of Mönch fixed point theorem associated with the technique of measures of noncompactness.
References:  R. P. Agarwal, B. Ahmad: Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions. Comput. Math. Appl. 62 (2011), 1200-1214. DOI 10.1016/j.camwa.2011.03.001 | MR 2824708 | Zbl 1228.34009  R. P. Agarwal, M. Benchohra, D. Seba: On the application of measure of noncompactness to the existence of solutions for fractional differential equations. Result. Math. 55 (2009), 221-230. DOI 10.1007/s00025-009-0434-5 | MR 2571191 | Zbl 1196.2600  B. Ahmad, A. Alsaedi: Nonlinear fractional differential equations with nonlocal fractional integro-differential boundary conditions. Bound. Value Probl. (electronic only) (2012), Article ID 124, 10 pages. DOI 10.1186/1687-2770-2012-124 | MR 3017351 | Zbl 1281.34004  B. Ahmad, J. J. Nieto: Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti-periodic boundary conditions. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 69 (2008), 3291-3298. DOI 10.1016/j.na.2007.09.018 | MR 2450538 | Zbl 1158.34049  R. R. Akhmerov, M. I. Kamenskiĭ, A. S. Potapov, A. E. Rodkina, B. N. Sadovskiĭ: Measures of Noncompactness and Condensing Operators. Operator Theory: Advances and Applications 55. Birkhäuser, Basel (1992). DOI 10.1007/978-3-0348-5727-7 | MR 1153247 | Zbl 0748.47045  A. Alsaedi, S. K. Ntouyas, B. Ahmad: Existence results for Langevin fractional differential inclusions involving two fractional orders with four-point multiterm fractional integral boundary conditions. Abstr. Appl. Anal. 2013 (2013), Article ID 869837, 17 pages. DOI 10.1155/2013/869837 | MR 3049420 | Zbl 1276.26008  H. H. Alsulami: Application of fixed point theorems for multivalued maps to anti-periodic problems of fractional differential inclusions. Filomat 28 (2014), 91-98. DOI 10.2298/FIL1401091A | MR 3359985  K. Balachandran, J. Y. Park, J. J. Trujillo: Controllability of nonlinear fractional dynamical systems. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 75 (2012), 1919-1926. DOI 10.1016/j.na.2011.09.042 | MR 2870885 | Zbl 1277.34006  J. Banaś, K. Goebel: Measures of Noncompactness in Banach Spaces. Lecture Notes in Pure and Applied Mathematics 60. Marcel Dekker, New York (1980). MR 0591679 | Zbl 0441.47056  M. Benchohra, J. Henderson, D. Seba: Measure of noncompactness and fractional differential equations in Banach spaces. Commun. Appl. Anal. 12 (2008), 419-427. MR 2494987 | Zbl 1182.26007  M. Benchohra, J. Henderson, D. Seba: Boundary value problems for fractional differential inclusions in Banach spaces. Fract. Differ. Calc. 2 (2012), 99-108. DOI 10.7153/fdc-02-07 | MR 3003005  M. Benchohra, G. M. N'Guérékata, D. Seba: Measure of noncompactness and nondensely defined semilinear functional differential equations with fractional order. Cubo 12 (2010), 35-48. DOI 10.4067/S0719-06462010000300003 | MR 2779372 | Zbl 1219.34100  Y. Cui: Uniqueness of solution for boundary value problems for fractional differential equations. Appl. Math. Lett. 51 (2016), 48-54. DOI 10.1016/j.aml.2015.07.002 | MR 3396346 | Zbl 1329.34005  D. Guo, V. Lakshmikantham, X. Liu: Nonlinear Integral Equations in Abstract Spaces. Mathematics and Its Applications 373. Kluwer Academic Publishers, Dordrecht (1996). DOI 10.1007/978-1-4613-1281-9 | MR 1418859 | Zbl 0866.45004  J. Han, Y. Liu, J. Zhao: Integral boundary value problems for first order nonlinear impulsive functional integro-differential differential equations. Appl. Math. Comput. 218 (2012), 5002-5009. DOI 10.1016/j.amc.2011.10.067 | MR 2870024 | Zbl 1246.45006  H.-P. Heinz: On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions. Nonlinear Anal., Theory Methods Appl. 7 (1983), 1351-1371. DOI 10.1016/0362-546X(83)90006-8 | MR 0726478 | Zbl 0528.47046  A. A. Kilbas, H. M. Srivastava, J. J. Trujillo: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies 204. Elsevier, Amsterdam (2006). DOI 10.1016/s0304-0208(06)x8001-5 | MR 2218073 | Zbl 1092.45003  V. Lakshmikantham, S. Leela, J. Vasundhara Devi: Theory of Fractional Dynamic Systems. Cambridge Scientific Publishers, Cambridge (2009). Zbl 1188.37002  A. Lasota, Z. Opial: An application of the Kakutani - Ky Fan theorem in the theory of ordinary differential equations. Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 13 (1965), 781-786. MR 0196178 | Zbl 0151.10703  H. Mönch: Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces. Nonlinear Anal., Theory Methods Appl. 4 (1980), 985-999. DOI 10.1016/0362-546X(80)90010-3 | MR 0586861 | Zbl 0462.34041  S. K. Ntouyas: Existence results for nonlocal boundary value problems for fractional differential equations and inclusions with fractional integral boundary conditions. Discuss. Math., Differ. Incl. Control Optim. 33 (2013), 17-39. DOI 10.7151/dmdico.1146 | MR 3136580 | Zbl 1307.34016  S. K. Ntouyas, J. Tariboon: Nonlocal boundary value problems for Langevin fractional differential inclusions with Riemann-Liouville fractional integral boundary conditions. Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 22 (2015), 123-141. MR 3360149 | Zbl 1326.34022  D. O'Regan, R. Precup: Fixed point theorems for set-valued maps and existence principles for integral inclusions. J. Math. Anal. Appl. 245 (2000), 594-612. DOI 10.1006/jmaa.2000.6789 | MR 1758558 | Zbl 0956.47026  I. Podlubny: Geometric and physical interpretation of fractional integration and fractional differentiation. Fract. Calc. Appl. Anal. 5 (2002), 367-386; Correction: "Geometric and physical interpretation of fractional integration and fractional differentiation", ibid. 6 (2003), 109-110. MR 1967839 | Zbl 1042.26003  J. Sabatier, O. P. Agrawal, J. A. Tenreiro Machado (eds.): Advances in Fractional Calculus - Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht (2007). DOI 10.1007/978-1-4020-6042-7 | MR 2432163 | Zbl 1116.00014  S. G. Samko, A. A. Kilbas, O. I. Marichev: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, New York (1993). MR 1347689 | Zbl 0818.26003  S. Szufla: On the application of measure of noncompactness to existence theorems. Rend. Sem. Mat. Univ. Padova 75 (1986), 1-14. MR 0847653 | Zbl 0589.45007  J. Xu, Z. Wei, W. Dong: Uniqueness of positive solutions for a class of fractional boundary value problems. Appl. Math. Lett. 25 (2012), 590-593. DOI 10.1016/j.aml.2011.09.065 | MR 2856039 | Zbl 1247.34011
Access to the full text of journal articles on this site is restricted to the subscribers of Myris Trade. To activate your access, please contact Myris Trade at email@example.com. Subscribers of Springer need to access the articles on their site, which is http://mb.math.cas.cz/.
Affiliations: Djamila Seba, Department of Mathematics, Faculty of Sciences, University M'Hamed Bougara, Route de la Gare Ferroviaire, 35000 Boumerdès, Algérie, e-mail: firstname.lastname@example.org