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Existence of solutions for integrodifferential equations of fractional order with antiperiodic boundary conditions. (English) Zbl 1203.45005

Summary: We discuss the existence of solutions for a nonlinear antiperiodic boundary value problem of integrodifferential equations of fractional order \(q\in (1,2]\). The contraction mapping principle and Krasnoselskii’s fixed point theorem are applied to establish the results.

MSC:

45J05 Integro-ordinary differential equations
45G10 Other nonlinear integral equations
26A33 Fractional derivatives and integrals
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[1] V. Daftardar-Gejji and S. Bhalekar, “Boundary value problems for multi-term fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 345, no. 2, pp. 754-765, 2008. · Zbl 1151.26004 · doi:10.1016/j.jmaa.2008.04.065
[2] A. Yang and W. Ge, “Positive solutions for boundary value problems of N-dimension nonlinear fractional differential system,” Boundary Value Problems, vol. 2008, Article ID 437453, 15 pages, 2008. · Zbl 1167.34314 · doi:10.1155/2008/437453
[3] S. Z. Rida, H. M. El-Sherbiny, and A. A. M. Arafa, “On the solution of the fractional nonlinear Schrödinger equation,” Physics Letters A, vol. 372, no. 5, pp. 553-558, 2008. · Zbl 1217.81068 · doi:10.1016/j.physleta.2007.06.071
[4] B. Ahmad, “Some existence results for boundary value problems of fractional semilinear evolution equations,” Electronic Journal of Qualitative Theory of Differential Equations, no. 28, pp. 1-7, 2009. · Zbl 1183.34079
[5] B. Ahmad and J. J. Nieto, “Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations,” Abstract and Applied Analysis, vol. 2009, Article ID 494720, 9 pages, 2009. · Zbl 1186.34009 · doi:10.1155/2009/494720
[6] B. Ahmad and S. Sivasundaram, “Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations,” Nonlinear Analysis: Hybrid Systems, vol. 3, no. 3, pp. 251-258, 2009. · Zbl 1193.34056 · doi:10.1016/j.nahs.2009.01.008
[7] Y.-K. Chang and J. J. Nieto, “Some new existence results for fractional differential inclusions with boundary conditions,” Mathematical and Computer Modelling, vol. 49, no. 3-4, pp. 605-609, 2009. · Zbl 1165.34313 · doi:10.1016/j.mcm.2008.03.014
[8] V. Lakshmikantham, S. Leela, and J. V. Devi, Theory of Fractional Dynamic Systems, Cambridge Academic, Cambridge, UK, 2009. · Zbl 1188.37002
[9] X. Su and S. Zhang, “Solutions to boundary-value problems for nonlinear differential equations of fractional order,” Electronic Journal of Differential Equations, vol. 2009, no. 26, p. 115, 2009. · Zbl 1173.34011
[10] V. Gafiychuk, B. Datsko, V. Meleshko, and D. Blackmore, “Analysis of the solutions of coupled nonlinear fractional reaction-diffusion equations,” Chaos, Solitons and Fractals, vol. 41, no. 3, pp. 1095-1104, 2009. · Zbl 1198.35123 · doi:10.1016/j.chaos.2008.04.039
[11] M. Benchohra, A. Cabada, and D. Seba, “An existence result for nonlinear fractional differential equations on Banach spaces,” Boundary Value Problems, vol. 2009, Article ID 628916, 11 pages, 2009. · Zbl 1181.34007 · doi:10.1155/2009/628916
[12] A. M. A. El-Sayed and H. H. G. Hashem, “Monotonic solutions of functional integral and differential equations of fractional order,” Electronic Journal of Qualitative Theory of Differential Equations, no. 7, pp. 1-8, 2009. · Zbl 1181.45016
[13] M. Benchohra and S. Hamani, “The method of upper and lower solutions and impulsive fractional differential inclusions,” Nonlinear Analysis: Hybrid Systems, vol. 3, no. 4, pp. 433-440, 2009. · Zbl 1221.49060 · doi:10.1016/j.nahs.2009.02.009
[14] A. Arara, M. Benchohra, N. Hamidi, and J. J. Nieto, “Fractional order differential equations on an unbounded domain,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 2, pp. 580-586, 2010. · Zbl 1179.26015 · doi:10.1016/j.na.2009.06.106
[15] Y. Tian and A. Chen, “The existence of positive solution to three-point singular boundary value problem of fractional differential equation,” Abstract and Applied Analysis, vol. 2009, Article ID 314656, 18 pages, 2009. · Zbl 1190.34004 · doi:10.1155/2009/314656
[16] B. Ahmad and B. S. Alghamdi, “Approximation of solutions of the nonlinear Duffing equation involving both integral and non-integral forcing terms with separated boundary conditions,” Computer Physics Communications, vol. 179, no. 6, pp. 409-416, 2008. · Zbl 1197.34023 · doi:10.1016/j.cpc.2008.04.008
[17] B. Ahmad, “On the existence of T-periodic solutions for Duffing type integro-differential equations with p-Laplacian,” Lobachevskii Journal of Mathematics, vol. 29, no. 1, pp. 1-4, 2008. · Zbl 1166.45300 · doi:10.1007/s12202-008-1001-2
[18] S. Mesloub, “On a mixed nonlinear one point boundary value problem for an integrodifferential equation,” Boundary Value Problems, vol. 2008, Article ID 814947, 8 pages, 2008. · Zbl 1262.34076 · doi:10.1155/2008/814947
[19] B. Ahmad and J. J. Nieto, “Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions,” Boundary Value Problems, vol. 2009, Article ID 708576, 11 pages, 2009. · Zbl 1167.45003 · doi:10.1155/2009/708576
[20] Y. K. Chang and J. J. Nieto, “Existence of solutions for impulsive neutral integrodifferential inclusions with nonlocal initial conditions via fractional operators,” Numerical Functional Analysis and Optimization, vol. 30, no. 3, pp. 227-244, 2009. · Zbl 1176.34096 · doi:10.1080/01630560902841146
[21] Y. Chen, J. J. Nieto, and D. O/Regan, “Anti-periodic solutions for fully nonlinear first-order differential equations,” Mathematical and Computer Modelling, vol. 46, no. 9-10, pp. 1183-1190, 2007. · Zbl 1142.34313 · doi:10.1016/j.mcm.2006.12.006
[22] B. Liu, “An anti-periodic LaSalle oscillation theorem for a class of functional differential equations,” Journal of Computational and Applied Mathematics, vol. 223, no. 2, pp. 1081-1086, 2009. · Zbl 1166.34323 · doi:10.1016/j.cam.2008.03.040
[23] B. Ahmad and V. Otero-Espinar, “Existence of solutions for fractional differential inclusions with antiperiodic boundary conditions,” Boundary Value Problems, vol. 2009, Article ID 625347, 11 pages, 2009. · Zbl 1172.34004 · doi:10.1155/2009/625347
[24] B. Ahmad and J. J. Nieto, “Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray-Schauder degree theory,” to appear in Topological Methods in Nonlinear Analysis. · Zbl 1245.34008
[25] Y. Q. Chen, D. O/Regan, F. L. Wang, and S. L. Zhou, “Antiperiodic boundary value problems for finite dimensional differential systems,” Boundary Value Problems, vol. 2009, Article ID 541435, 11 pages, 2009. · Zbl 1182.34022 · doi:10.1155/2009/541435
[26] I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. · Zbl 0924.34008
[27] R. Hilfer, Ed., Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. · Zbl 0998.26002
[28] 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
[29] D. R. Smart, Fixed Point Theorems, Cambridge University Press, Cambridge, UK, 1980. · Zbl 0427.47036
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