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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 |

### Keywords:

boundary value problem; nonlinear integrodifferential equation of fractional order; integral boundary conditions; contraction mapping principle; Krasnosel’skiĭ’s fixed point theorem
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\textit{B. Ahmad} and \textit{J. J. Nieto}, Bound. Value Probl. 2009, Article ID 708576, 11 p. (2009; Zbl 1167.45003)

### References:

[1] | Ahmad B, Nieto JJ: Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. preprint |

[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, 69, 3692-3705, (2008) · 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, 311, 495-505, (2005) · 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 1262.34076 |

[7] | Bonilla, B; Rivero, M; Rodríguez-Germá, L; Trujillo, JJ, Fractional differential equations as alternative models to nonlinear differential equations, Applied Mathematics and Computation, 187, 79-88, (2007) · Zbl 1120.34323 |

[8] | Chang, Y-K; Nieto, JJ, Some new existence results for fractional differential inclusions with boundary conditions, Mathematical and Computer Modelling, 49, 605-609, (2009) · Zbl 1165.34313 |

[9] | Gafiychuk, V; Datsko, B; Meleshko, V, Mathematical modeling of time fractional reaction-diffusion systems, Journal of Computational and Applied Mathematics, 220, 215-225, (2008) · Zbl 1152.45008 |

[10] | Daftardar-Gejji, V, Positive solutions of a system of non-autonomous fractional differential equations, Journal of Mathematical Analysis and Applications, 302, 56-64, (2005) · Zbl 1064.34004 |

[11] | Daftardar-Gejji, V; Bhalekar, S, Boundary value problems for multi-term fractional differential equations, Journal of Mathematical Analysis and Applications, 345, 754-765, (2008) · 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, 345, 871-879, (2008) · 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, 14, 2006-2012, (2009) · 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, 13, 707-714, (2008) · Zbl 1221.93128 |

[17] | Lazarević, MP, Finite time stability analysis of [inlineequation not available: see fulltext.] fractional control of robotic time-delay systems, Mechanics Research Communications, 33, 269-279, (2006) · 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. |

[19] | Rida, SZ; El-Sherbiny, HM; Arafa, AAM, On the solution of the fractional nonlinear Schrödinger equation, Physics Letters A, 372, 553-558, (2008) · 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, 348, 101-115, (2008) · Zbl 1155.33005 |

[22] | Zhang, S, Positive solutions for boundary-value problems of nonlinear fractional differential equations, Electronic Journal of Differential Equations, 2006, 1-12, (2006) |

[23] | Ahmad, B; Sivasundaram, S, Some existence results for fractional integrodifferential equations with nonlinear conditions, Communications in Applied Analysis, 12, 107-112, (2008) · 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, 179, 409-416, (2008) · Zbl 1197.34023 |

[25] | Ahmad, B, On the existence of [inlineequation not available: see fulltext.]-periodic solutions for Duffing type integro-differential equations with [inlineequation not available: see fulltext.]-Laplacian, Lobachevskii Journal of Mathematics, 29, 1-4, (2008) · 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 1120.34323 |

[27] | Luo, Z; Nieto, JJ, New results for the periodic boundary value problem for impulsive integro-differential equations, Nonlinear Analysis: Theory, Methods & Applications, 70, 2248-2260, (2009) · 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, 328, 1343-1368, (2007) · 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, 9, 1727-1740, (2008) · 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, 10, 358-367, (2009) · 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, 70, 364-371, (2009) · 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, 140, 431-442, (2009) · 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 |

[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, 68, 216-225, (2008) · Zbl 1132.34022 |

[38] | Krasnosel’skiĭ, MA, Two remarks on the method of successive approximations, Uspekhi Matematicheskikh Nauk, 10, 123-127, (1955) |

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