Ergören, Hilmi; Kiliçman, Adem Some existence results for impulsive nonlinear fractional differential equations with closed boundary conditions. (English) Zbl 1261.34004 Abstr. Appl. Anal. 2012, Article ID 387629, 15 p. (2012). Summary: We investigate some existence results for the solutions to impulsive fractional differential equations having closed boundary conditions. Our results are based on the contracting mapping principle and the Burton-Kirk fixed point theorem. Cited in 8 Documents MSC: 34A08 Fractional ordinary differential equations 34A37 Ordinary differential equations with impulses 47N20 Applications of operator theory to differential and integral equations × Cite Format Result Cite Review PDF Full Text: DOI References: [1] K. Diethelm, The Analysis of Fractional Differential Equations, Springer, 2010. · Zbl 1215.34001 · doi:10.1007/978-3-642-14574-2 [2] N. Heymans and I. Podlubny, “Physical interpretation of initial conditions for fractional differential equationswith Riemann Liouville fractional derivatives,” Rheologica Acta, vol. 45, no. 5, pp. 765-772, 2006. [3] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. · Zbl 0998.26002 [4] 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 Science B.V., Amsterdam, The Netherlands, 2006. · Zbl 1092.45003 [5] V. Lakshmikantham, S. Leela, and J. Vasundhara Devi, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, UK, 2009. · Zbl 1188.37002 [6] J. Sabatier, O. P. Agrawal, and J. A. T. Machado, Eds., Advances in Fractional Calculus: Theoretical Developmentsand Applications in Physics and Engineering, Springer, Dordrecht, The Netherlands, 2007. · Zbl 1116.00014 [7] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach Science, Yverdon, Switzerland, 1993. · Zbl 0818.26003 [8] R. P. Agarwal, M. Benchohra, and S. Hamani, “Boundary value problems for fractional differential equations,” Georgian Mathematical Journal, vol. 16, no. 3, pp. 401-411, 2009. · Zbl 1179.26011 · doi:10.1007/s10440-008-9356-6 [9] R. P. Agarwal, M. Benchohra, and S. Hamani, “A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions,” Acta Applicandae Mathematicae, vol. 109, no. 3, pp. 973-1033, 2010. · Zbl 1198.26004 · doi:10.1007/s10440-008-9356-6 [10] Z. Bai and H. Lü, “Positive solutions for boundary value problem of nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 495-505, 2005. · Zbl 1079.34048 · doi:10.1016/j.jmaa.2005.02.052 [11] M. Belmekki, J. J. Nieto, and R. Rodríguez-López, “Existence of periodic solution for a nonlinear fractional differential equation,” Boundary Value Problems, vol. 2009, Article ID 324561, 18 pages, 2009. · Zbl 1181.34006 · doi:10.1155/2009/324561 [12] M. Benchohra, S. Hamani, and S. K. Ntouyas, “Boundary value problems for differential equations with fractional order and nonlocal conditions,” Nonlinear Analysis, vol. 71, no. 7-8, pp. 2391-2396, 2009. · Zbl 1198.26007 · doi:10.1016/j.na.2009.01.073 [13] H. A. H. Salem, “On the fractional order m-point boundary value problem in reflexive Banach spaces and weak topologies,” Journal of Computational and Applied Mathematics, vol. 224, no. 2, pp. 565-572, 2009. · Zbl 1176.34070 · doi:10.1016/j.cam.2008.05.033 [14] W. Zhong and W. Lin, “Nonlocal and multiple-point boundary value problem for fractional differential equations,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1345-1351, 2010. · Zbl 1189.34036 · doi:10.1016/j.camwa.2009.06.032 [15] V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. · Zbl 0719.34002 [16] Y. V. Rogovchenko, “Impulsive evolution systems: main results and new trends,” Dynamics of Continuous, Discrete and Impulsive Systems, vol. 3, no. 1, pp. 57-88, 1997. · Zbl 0879.34014 [17] A. M. Samoĭlenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995. · Zbl 0837.34003 · doi:10.1142/9789812798664 [18] B. Ahmad and S. Sivasundaram, “Existence of solutions for impulsive integral boundary value problems of fractional order,” Nonlinear Analysis, vol. 4, no. 1, pp. 134-141, 2010. · Zbl 1187.34038 · doi:10.1016/j.nahs.2009.09.002 [19] M. Benchohra and B. A. Slimani, “Existence and uniqueness of solutions to impulsive fractional differential equations,” Electronic Journal of Differential Equations, vol. 10, pp. 1-11, 2009. · Zbl 1178.34004 [20] G. Wang, B. Ahmad, and L. Zhang, “Some existence results for impulsive nonlinear fractional differential equations with mixed boundary conditions,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1389-1397, 2011. · Zbl 1228.34021 · doi:10.1016/j.camwa.2011.04.004 [21] T. A. Burton and C. Kirk, “A fixed point theorem of Krasnoselskii-Schaefer type,” Mathematische Nachrichten, vol. 189, pp. 23-31, 1998. · Zbl 0896.47042 · doi:10.1002/mana.19981890103 [22] A. Granas and J. Dugundji, Fixed Point Theory, Springer Monographs in Mathematics, Springer, New York, NY, USA, 2003. · Zbl 1025.47002 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.