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Unbounded solutions to a boundary value problem of fractional order on the half-line. (English) Zbl 1217.34045
Summary: This paper deals with a boundary value problem of a fractional differential equation with the nonlinear term dependent on a fractional derivative of lower order on the semi-infinite interval. An appropriate compactness criterion is established, such that we can use Schauder’s fixed point theorem on an unbounded domain to obtain the existence result for solutions. Moreover, a suitable choice of a Banach space allows the solutions to be unbounded. An example illustrating our main result is also given.
MSC:
34B40Boundary value problems for ODE on infinite intervals
26A33Fractional derivatives and integrals (real functions)
34A08Fractional differential equations
34C11Qualitative theory of solutions of ODE: growth, boundedness
45J05Integro-ordinary differential equations
47N20Applications of operator theory to differential and integral equations
References:
[1]Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations, (2006)
[2]Podlubny, I.: Fractional differential equations, Mathematics in science and engineering 198 (1999) · Zbl 0924.34008
[3]Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives: theory and applications, (1993) · Zbl 0818.26003
[4]Agarwal, R. P.; Lakshmikantham, V.; Nieto, J. J.: On the concept of solution for fractional differential equations with uncertainty, Nonlinear anal. 72, 2859-2862 (2010) · Zbl 1188.34005 · doi:10.1016/j.na.2009.11.029
[5]Araya, D.; Lizama, C.: Almost automorphic mild solutions to fractional differential equations, Nonlinear anal. 69, 3692-3705 (2008) · Zbl 1166.34033 · doi:10.1016/j.na.2007.10.004
[6]Benchohra, M.; Henderson, J.; Ntouyas, S. K.; Ouahab, A.: Existence results for fractional order functional differential equations with infinite delay, J. math. Anal. appl. 338, 1340-1350 (2008) · Zbl 1209.34096 · doi:10.1016/j.jmaa.2007.06.021
[7]Eidelman, S. D.; Kochubei, A. N.: Cauchy problem for fractional diffusion equations, J. differential equations 199, 211-255 (2004) · Zbl 1068.35037 · doi:10.1016/j.jde.2003.12.002
[8]El-Borai, M. M.: Semigroups and some nonlinear fractional differential equations, Appl. math. Comput. 149, 823-831 (2004) · Zbl 1046.34079 · doi:10.1016/S0096-3003(03)00188-7
[9]Jaradat, O. K.; Al-Omari, A.; Momani, S.: Existence of the mild solution for fractional semilinear initial value problems, Nonlinear anal. 69, 3153-3159 (2008) · Zbl 1160.34300 · doi:10.1016/j.na.2007.09.008
[10]Lakshmikantham, V.: Theory of fractional functional differential equations, Nonlinear anal. 69, 3337-3343 (2008) · Zbl 1162.34344 · doi:10.1016/j.na.2007.09.025
[11]Lakshmikantham, V.; Vatsala, A. S.: General uniqueness and monotone iterative technique for fractional differential equations, Appl. math. Lett. 21, 828-834 (2008) · Zbl 1161.34031 · doi:10.1016/j.aml.2007.09.006
[12]Zhang, S.: Monotone iterative method for initial value problem involving Riemann–Liouville fractional derivatives, Nonlinear anal. 71, 2087-2093 (2009) · Zbl 1172.26307 · doi:10.1016/j.na.2009.01.043
[13]Zhou, Y.: Existence and uniqueness of fractional functional differential equations with unbounded delay, Int. J. Dyn. syst. Differ. equ. 1, 239-244 (2008) · Zbl 1175.34081 · doi:10.1504/IJDSDE.2008.022988
[14]Ahmad, B.: Existence of solutions for irregular boundary value problems of nonlinear fractional differential equations, Appl. math. Lett. 23, 390-394 (2010) · Zbl 1198.34007 · doi:10.1016/j.aml.2009.11.004
[15]Bai, Z. B.; Lü, H. S.: Positive solutions for boundary value problem of nonlinear fractional differential equation, J. math. Anal. appl. 311, 495-505 (2005) · Zbl 1079.34048 · doi:10.1016/j.jmaa.2005.02.052
[16]Benchohra, M.; Hamani, S.; Ntouyas, S. K.: Boundary value problems for differential equations with fractional order and nonlocal conditions, Nonlinear anal. 71, 2391-2396 (2009) · Zbl 1198.26007 · doi:10.1016/j.na.2009.01.073
[17]Gejji, V. D.; Bhalekar, S.: Boundary value problems for multi-term fractional differential equations, J. math. Anal. appl. 345, 754-765 (2008) · Zbl 1151.26004 · doi:10.1016/j.jmaa.2008.04.065
[18]Jafari, H.; Gejji, V. D.: Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method, Appl. math. Comput. 180, 700-706 (2006) · Zbl 1102.65136 · doi:10.1016/j.amc.2006.01.007
[19]Salem, H. A. H.: On the fractional m-point boundary value problem in reflexive Banach space and the weak topologies, J. comput. Appl. math. 224, 565-572 (2009) · Zbl 1176.34070 · doi:10.1016/j.cam.2008.05.033
[20]Su, X.: Boundary value problem for a coupled system of nonlinear fractional differential equations, Appl. math. Lett. 22, 64-69 (2009) · Zbl 1163.34321 · doi:10.1016/j.aml.2008.03.001
[21]Ahmad, B.; Nieto, J. J.: Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions, Comput. math. Appl. 58, 1838-1843 (2009) · Zbl 1205.34003 · doi:10.1016/j.camwa.2009.07.091
[22]Su, X.; Zhang, S.: Solutions to boundary-value problems for nonlinear differential equations of fractional order, Electron. J. Differential equations 2009, No. 26, 1-15 (2009) · Zbl 1173.34011 · doi:emis:journals/EJDE/Volumes/2009/26/abstr.html
[23]Arara, A.; Benchohra, M.; Hamidi, N.; Nieto, J. J.: Fractional order differential equations on an unbounded domain, Nonlinear anal. 72, 580-586 (2010) · Zbl 1179.26015 · doi:10.1016/j.na.2009.06.106
[24]Zhao, X. K.; Ge, W. G.: Unbounded solutions for a fractional boundary value problem on the infinite interval, Acta appl. Math. 109, 495-505 (2010) · Zbl 1193.34008 · doi:10.1007/s10440-008-9329-9