×

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
PDF BibTeX XML Cite
Full Text: DOI EuDML

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)
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.