Existence of periodic solution for a nonlinear fractional differential equation. (English) Zbl 1181.34006

Summary: We consider the following nonlinear fractional differential equation of the form
\[ D^\delta u(t)-\lambda u(t)=f(t,u(t)),\quad t\in J:=(0,1], \;0<\delta<1,\tag{1.1} \]
where \(D^\delta\) is the standard Riemann-Liouville fractional derivative, \(f\) is continuous, and \(\lambda\in\mathbb R\).
Due to the singularity of the possible solutions, we introduce a new and proper concept of periodic boundary value conditions. We present Green’s function and give some existence results for the linear case and then we study the nonlinear problem.


34A08 Fractional ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
Full Text: DOI


[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.; Nieto, JJ, Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions, No. 2009, 11 (2009) · Zbl 1167.45003
[3] Benchohra, M.; Cabada, A.; Seba, D., An existence result for nonlinear fractional differential equations on Banach spaces, No. 2009, 11 (2009) · Zbl 1181.34007
[4] 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 · doi:10.1016/j.amc.2006.08.105
[5] 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 · doi:10.1016/j.jmaa.2008.04.065
[6] 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 · doi:10.1016/j.jmaa.2008.06.052
[7] Lazarević MP, Spasić AM: Finite-time stability analysis of fractional order time-delay systems: Gronwall’s approach. Mathematical and Computer Modelling 2009, 49(3-4):475-481. 10.1016/j.mcm.2008.09.011 · Zbl 1165.34408 · doi:10.1016/j.mcm.2008.09.011
[8] 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. · Zbl 1092.45003
[9] Kiryakova V: Generalized Fractional Calculus and Applications, Pitman Research Notes in Mathematics Series. Volume 301. Longman Scientific & Technical, Harlow, UK; 1994:x+388. · Zbl 0882.26003
[10] Miller KS, Ross B: An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication. John Wiley & Sons, New York, NY, USA; 1993:xvi+366. · Zbl 0789.26002
[11] Podlubny I: Fractional Differential Equations, Mathematics in Science and Engineering. Volume 198. Academic Press, San Diego, Calif, USA; 1999:xxiv+340. · Zbl 0924.34008
[12] Oldham KB, Spanier J: The Fractional Calculus. Academic Press, New York, NY, USA; 1974:xiii+234. · Zbl 0292.26011
[13] 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
[14] Diethelm, K.; Freed, AD; Keil, F. (ed.); Mackens, W. (ed.); Voss, H. (ed.); Werther, J. (ed.), On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity, 217-224 (1999), Heidelberg, Germany
[15] Diethelm K, Ford NJ: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 2002, 265(2):229-248. 10.1006/jmaa.2000.7194 · Zbl 1014.34003 · doi:10.1006/jmaa.2000.7194
[16] Diethelm K, Walz G: Numerical solution of fractional order differential equations by extrapolation. Numerical Algorithms 1997, 16(3-4):231-253. · Zbl 0926.65070 · doi:10.1023/A:1019147432240
[17] Mainardi, F.; Carpinteri, A. (ed.); Mainardi, F. (ed.), Fractional calculus: some basic problems in continuum and statistical mechanis, 291-348 (1997), Wien, Austria · Zbl 0917.73004 · doi:10.1007/978-3-7091-2664-6_7
[18] Metzler R, Schick W, Kilian H-G, Nonnenmacher TF: Relaxation in filled polymers: a fractional calculus approach. The Journal of Chemical Physics 1995, 103(16):7180-7186. 10.1063/1.470346 · doi:10.1063/1.470346
[19] Podlubny I, Petráš I, Vinagre BM, O’Leary P, Dorčák L: Analogue realizations of fractional-order controllers. Nonlinear Dynamics 2002, 29(1-4):281-296. · Zbl 1041.93022 · doi:10.1023/A:1016556604320
[20] 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 · doi:10.1016/j.na.2007.10.004
[21] Benchohra M, Hamani S, Nieto JJ, Slimani BA: Existence results for fractional differential inclusions with fractional order and impulses. to appear in Computers & Mathematics with Applications · Zbl 1194.26008
[22] 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 · doi:10.1016/j.mcm.2008.03.014
[23] Chang Y-K, Nieto JJ: Existence of solutions for impulsive neutral integro-differential inclusions with nonlocal initial conditions via fractional operators. Numerical Functional Analysis and Optimization 2009, 30: 227-244. 10.1080/01630560902841146 · Zbl 1176.34096 · doi:10.1080/01630560902841146
[24] 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 · doi:10.1016/j.cnsns.2008.05.008
[25] Li J, Nieto JJ, Shen J: Impulsive periodic boundary value problems of first-order differential equations. Journal of Mathematical Analysis and Applications 2007, 325(1):226-236. 10.1016/j.jmaa.2005.04.005 · Zbl 1110.34019 · doi:10.1016/j.jmaa.2005.04.005
[26] Nieto JJ: Differential inequalities for functional perturbations of first-order ordinary differential equations. Applied Mathematics Letters 2002, 15(2):173-179. 10.1016/S0893-9659(01)00114-8 · Zbl 1014.34060 · doi:10.1016/S0893-9659(01)00114-8
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.