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Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions. (English) Zbl 1167.45003

Summary: This paper deals with some existence results for a boundary value problem involving a nonlinear integrodifferential equation of fractional order \(q\in (1,2]\) with integral boundary conditions. Our results are based on contraction mapping principle and Krasnosel’skiĭ’s fixed point theorem.

MSC:

45J05 Integro-ordinary differential equations
45G10 Other nonlinear integral equations
26A33 Fractional derivatives and integrals
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[1] Ahmad B, Nieto JJ: Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. preprint · Zbl 1205.34003
[2] Ahmad B, Sivasundaram S: Existence and uniqueness results for nonlinear boundary value problems of fractional differential equations with separated boundary conditions. to appear in Dynamic Systems and Applications · Zbl 1180.34003
[3] Araya D, Lizama C: Almost automorphic mild solutions to fractional differential equations. Nonlinear Analysis: Theory, Methods & Applications 2008, 69(11):3692-3705. 10.1016/j.na.2007.10.004 · Zbl 1166.34033
[4] Bai Z, Lü H: Positive solutions for boundary value problem of nonlinear fractional differential equation. Journal of Mathematical Analysis and Applications 2005, 311(2):495-505. 10.1016/j.jmaa.2005.02.052 · Zbl 1079.34048
[5] Belmekki M, Nieto JJ, Rodríguez-López R: Existence of periodic solution for a nonlinear fractional differential equation. preprint · Zbl 1113.45007
[6] Benchohra M, Hamani S, Nieto JJ, Slimani BA: Existence results for differential inclusions with fractional order and impulses. preprint · Zbl 1194.26008
[7] Bonilla B, Rivero M, Rodríguez-Germá L, Trujillo JJ: Fractional differential equations as alternative models to nonlinear differential equations. Applied Mathematics and Computation 2007, 187(1):79-88. 10.1016/j.amc.2006.08.105 · Zbl 1120.34323
[8] Chang Y-K, Nieto JJ: Some new existence results for fractional differential inclusions with boundary conditions. Mathematical and Computer Modelling 2009, 49(3-4):605-609. 10.1016/j.mcm.2008.03.014 · Zbl 1165.34313
[9] Gafiychuk V, Datsko B, Meleshko V: Mathematical modeling of time fractional reaction-diffusion systems. Journal of Computational and Applied Mathematics 2008, 220(1-2):215-225. 10.1016/j.cam.2007.08.011 · Zbl 1152.45008
[10] Daftardar-Gejji V: Positive solutions of a system of non-autonomous fractional differential equations. Journal of Mathematical Analysis and Applications 2005, 302(1):56-64. 10.1016/j.jmaa.2004.08.007 · Zbl 1064.34004
[11] Daftardar-Gejji V, Bhalekar S: Boundary value problems for multi-term fractional differential equations. Journal of Mathematical Analysis and Applications 2008, 345(2):754-765. 10.1016/j.jmaa.2008.04.065 · Zbl 1151.26004
[12] El-Shahed, M., Positive solutions for boundary value problem of nonlinear fractional differential equation, No. 2007, 8 (2007) · Zbl 1149.26012
[13] Ibrahim RW, Darus M: Subordination and superordination for univalent solutions for fractional differential equations. Journal of Mathematical Analysis and Applications 2008, 345(2):871-879. 10.1016/j.jmaa.2008.05.017 · Zbl 1147.30009
[14] Jafari H, Seifi S: Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation. Communications in Nonlinear Science and Numerical Simulation 2009, 14(5):2006-2012. 10.1016/j.cnsns.2008.05.008 · Zbl 1221.65278
[15] Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies. Volume 204. Elsevier Science B.V., Amsterdam, The Netherlands; 2006:xvi+523.
[16] Ladaci S, Loiseau JJ, Charef A: Fractional order adaptive high-gain controllers for a class of linear systems. Communications in Nonlinear Science and Numerical Simulation 2008, 13(4):707-714. 10.1016/j.cnsns.2006.06.009 · Zbl 1221.93128
[17] Lazarević MP: Finite time stability analysis of fractional control of robotic time-delay systems.[InlineEquation not available: see fulltext.]Mechanics Research Communications 2006, 33(2):269-279. 10.1016/j.mechrescom.2005.08.010 · Zbl 1192.70008
[18] Podlubny I: Fractional Differential Equations, Mathematics in Science and Engineering. Volume 198. Academic Press, San Diego, Calif, USA; 1999:xxiv+340. · Zbl 0924.34008
[19] Rida SZ, El-Sherbiny HM, Arafa AAM: On the solution of the fractional nonlinear Schrödinger equation. Physics Letters A 2008, 372(5):553-558. 10.1016/j.physleta.2007.06.071 · Zbl 1217.81068
[20] Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science, Yverdon, Switzerland; 1993:xxxvi+976. · Zbl 0818.26003
[21] Varlamov V: Differential and integral relations involving fractional derivatives of Airy functions and applications. Journal of Mathematical Analysis and Applications 2008, 348(1):101-115. 10.1016/j.jmaa.2008.06.052 · Zbl 1155.33005
[22] Zhang S: Positive solutions for boundary-value problems of nonlinear fractional differential equations. Electronic Journal of Differential Equations 2006, 2006(36):1-12.
[23] Ahmad B, Sivasundaram S: Some existence results for fractional integrodifferential equations with nonlinear conditions. Communications in Applied Analysis 2008, 12: 107-112. · Zbl 1179.45009
[24] Ahmad B, Alghamdi BS: Approximation of solutions of the nonlinear Duffing equation involving both integral and non-integral forcing terms with separated boundary conditions. Computer Physics Communications 2008, 179(6):409-416. 10.1016/j.cpc.2008.04.008 · Zbl 1197.34023
[25] Ahmad B: On the existence of [InlineEquation not available: see fulltext.]-periodic solutions for Duffing type integro-differential equations with -Laplacian. Lobachevskii Journal of Mathematics 2008, 29(1):1-4. · Zbl 1166.45300
[26] Chang YK, Nieto JJ: Existence of solutions for impulsive neutral integrodifferential inclusions with nonlocal initial conditions via fractional operators. to appear in Numerical Functional Analysis and Optimization · Zbl 1176.34096
[27] Luo Z, Nieto JJ: New results for the periodic boundary value problem for impulsive integro-differential equations. Nonlinear Analysis: Theory, Methods & Applications 2009, 70(6):2248-2260. 10.1016/j.na.2008.03.004 · Zbl 1166.45002
[28] Mesloub, S., On a mixed nonlinear one point boundary value problem for an integrodifferential equation, No. 2008, 8 (2008) · Zbl 1262.34076
[29] Nieto JJ, Rodríguez-López R: New comparison results for impulsive integro-differential equations and applications. Journal of Mathematical Analysis and Applications 2007, 328(2):1343-1368. 10.1016/j.jmaa.2006.06.029 · Zbl 1113.45007
[30] Ahmad B, Alsaedi A, Alghamdi BS: Analytic approximation of solutions of the forced Duffing equation with integral boundary conditions. Nonlinear Analysis: Real World Applications 2008, 9(4):1727-1740. 10.1016/j.nonrwa.2007.05.005 · Zbl 1154.34311
[31] Ahmad B, Alsaedi A: Existence of approximate solutions of the forced Duffing equation with discontinuous type integral boundary conditions. Nonlinear Analysis: Real World Applications 2009, 10(1):358-367. 10.1016/j.nonrwa.2007.09.004 · Zbl 1154.34314
[32] Benchohra M, Hamani S, Nieto JJ: The method of upper and lower solutions for second order differential inclusions with integral boundary conditions. Rocky Mountain Journal of Mathematics. In press · Zbl 1205.34013
[33] Boucherif A: Second-order boundary value problems with integral boundary conditions. Nonlinear Analysis: Theory, Methods & Applications 2009, 70(1):364-371. 10.1016/j.na.2007.12.007 · Zbl 1169.34310
[34] Chang Y-K, Nieto JJ, Li W-S: On impulsive hyperbolic differential inclusions with nonlocal initial conditions. Journal of Optimization Theory and Applications 2009, 140(3):431-442. 10.1007/s10957-008-9468-1 · Zbl 1159.49042
[35] Chang YK, Nieto JJ, Li WS: Controllability of semilinear differential systems with nonlocal initial conditions in Banach spaces. to appear in Journal of Optimization Theory and Applications · Zbl 1178.93029
[36] Feng M, Du B, Ge W: Impulsive boundary value problems with integral boundary conditions and one-dimensional -Laplacian. Nonlinear Analysis: Theory, Methods & Applications. In press · Zbl 1169.34022
[37] Yang Z: Existence of nontrivial solutions for a nonlinear Sturm-Liouville problem with integral boundary conditions. Nonlinear Analysis: Theory, Methods & Applications 2008, 68(1):216-225. 10.1016/j.na.2006.10.044 · Zbl 1132.34022
[38] Krasnosel’skiĭ MA: Two remarks on the method of successive approximations. Uspekhi Matematicheskikh Nauk 1955, 10(1(63)):123-127.
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