# zbMATH — the first resource for mathematics

Existence of solutions for fractional differential inclusions with separated boundary conditions in Banach space. (English) Zbl 1272.34007
Summary: We discuss the existence of solutions for a class of some separated boundary differential inclusions of fractional orders $$2 < \alpha < 3$$ involving the Caputo derivative. In order to obtain necessary conditions for the existence result, we apply the fixed point technique, fractional calculus, and multivalued analysis.

##### MSC:
 34A08 Fractional ordinary differential equations 34G25 Evolution inclusions 34B15 Nonlinear boundary value problems for ordinary differential equations 47N20 Applications of operator theory to differential and integral equations
Full Text:
##### References:
 [1] R. P. Agarwal, D. O’Regan, and S. Stan\vek, “Positive solutions for mixed problems of singular fractional differential equations,” Mathematische Nachrichten, vol. 285, no. 1, pp. 27-41, 2012. · Zbl 1232.26005 [2] B. Ahmad, J. J. Nieto, and J. Pimentel, “Some boundary value problems of fractional differential equations and inclusions,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1238-1250, 2011. · Zbl 1228.34011 [3] B. Ahmad and S. K. Ntouyas, “A note on fractional differential equations with fractional separated boundary conditions,” Abstract and Applied Analysis, vol. 2012, Article ID 818703, 11 pages, 2012. · Zbl 1244.34004 [4] Z. Bai and W. Sun, “Existence and multiplicity of positive solutions for singular fractional boundary value problems,” Computers & Mathematics with Applications, vol. 63, no. 9, pp. 1369-1381, 2012. · Zbl 1247.34006 [5] J. Caballero, J. Harjani, and K. Sadarangani, “Positive solutions for a class of singular fractional boundary value problems,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1325-1332, 2011. · Zbl 1235.34010 [6] I. J. Cabrera, J. Harjani, and K. B. Sadarangani, “Existence and uniqueness of positive solutions for a singular fractional three-point boundary value problem,” Abstract and Applied Analysis, vol. 2012, Article ID 803417, 18 pages, 2012. · Zbl 1246.34006 [7] J. Jin, X. Liu, and M. Jia, “Existence of positive solutions for singular fractional differential equations with integral boundary conditions,” Electronic Journal of Differential Equations, vol. 2012, no. 63, pp. 1-14, 2012. · Zbl 1261.34005 [8] D. O’Regan and S. Stanek, “Fractional boundary value problems with singularities in space variables,” Nonlinear Dynamics, vol. 71, no. 4, pp. 641-652, 2013. · Zbl 1268.34023 [9] B. Ahmad and S. K. Ntouyas, “Boundary value problems for n-th order differential inclusions with four-point integral boundary conditions,” Opuscula Mathematica, vol. 32, no. 2, pp. 205-226, 2012. · Zbl 1252.34024 [10] A. Boucherif and N. Al-Malki, “Solvability of Neumann boundary-value problems with Carathéodory nonlinearities,” Electronic Journal of Differential Equations, vol. 2004, no. 51, pp. 1-7, 2004. · Zbl 1095.34516 [11] G. A. Chechkin, D. Cioranescu, A. Damlamian, and A. L. Piatnitski, “On boundary value problem with singular inhomogeneity concentrated on the boundary,” Journal de Mathématiques Pures et Appliquées, vol. 98, no. 2, pp. 115-138, 2012. · Zbl 1277.35142 [12] A. Nouy, M. Chevreuil, and E. Safatly, “Fictitious domain method and separated representations for the solution of boundary value problems on uncertain parameterized domains,” Computer Methods in Applied Mechanics and Engineering, vol. 200, no. 45-46, pp. 3066-3082, 2011. · Zbl 1230.65135 [13] M. Bragdi and M. Hazi, “Existence and uniqueness of solutions of fractional quasilinear mixed integrodifferential equations with nonlocal condition in Banach spaces,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 2012, no. 51, pp. 1-16, 2012. · Zbl 1243.93017 [14] A. Debbouche and D. Baleanu, “Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1442-1450, 2011. · Zbl 1228.45013 [15] A. Debbouche and D. Baleanu, “Exact null controllability for fractional nonlocal integrodifferential equations via implicit evolution system,” Journal of Applied Mathematics, vol. 2012, Article ID 931975, 17 pages, 2012. · Zbl 1251.93029 [16] A. Debbouche, D. Baleanu, and R. P. Agarwal, “Nonlocal nonlinear integrodifferential equations of fractional orders,” Boundary Value Problems, vol. 2012, article 78, pp. 1-10, 2012. · Zbl 1277.35337 [17] C. Kou, H. Zhou, and C. Li, “Existence and continuation theorems of Riemann-Liouville type fractional differential equations,” International Journal of Bifurcation and Chaos, vol. 22, no. 4, Article ID 1250077, 12 pages, 2012. · Zbl 1258.34016 [18] C. Li and Y. Ma, “Fractional dynamical system and its linearization theorem,” Nonlinear Dynamics, vol. 71, no. 4, pp. 621-633, 2013. · Zbl 1268.34019 [19] C. P. Li and F. R. Zhang, “A survey on the stability of fractional differential equations,” The European Physical Journal: Special Topics, vol. 193, no. 1, pp. 27-47, 2011. [20] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier, Amsterdam, The Netherlands, 2006. · Zbl 1092.45003 [21] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1993. · Zbl 0918.34010 [22] T. M. Atanackovic and B. Stankovic, “Generalized wave equation in nonlocal elasticity,” Acta Mechanica, vol. 208, no. 1-2, pp. 1-10, 2009. · Zbl 1397.74100 [23] M. Caputo, “Linear models of dissipation whose q is almost frequency independent-part II,” Geophysical Journal of the Royal Astronomical Society, vol. 13, pp. 529-539, 1967. [24] D. Delbosco and L. Rodino, “Existence and uniqueness for a nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 204, no. 2, pp. 609-625, 1996. · Zbl 0881.34005 [25] T. Qiu and Z. Bai, “Existence of positive solutions for singular fractional differential equations,” Electronic Journal of Differential Equations, vol. 2008, no. 146, pp. 1-9, 2008. · Zbl 1172.34313 [26] M. Aitaliobrahim, “Neumann boundary-value problems for differential inclusions in banach spaces,” Electronic Journal of Differential Equations, vol. 2010, no. 104, pp. 1-5, 2010. · Zbl 1200.34066
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.