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Solutions to boundary value problem of fractional order on unbounded domains in a Banach space. (English) Zbl 1250.34007

Summary: By means of Darbo’s fixed point theorem, we establish the existence of solutions to a boundary value problem of a fractional differential equation on the half-line in a Banach space. An example illustrating our main result is given.

MSC:

34A08 Fractional ordinary differential equations
34B40 Boundary value problems on infinite intervals for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
34G20 Nonlinear differential equations in abstract spaces
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[1] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations (2006), Elsevier B.V.: Elsevier B.V. Amsterdam · Zbl 1092.45003
[2] Podlubny, I., (Fractional Differential Equations. Fractional Differential Equations, Mathematics in Science and Engineering, vol. 198 (1999), Academic Press: Academic Press New York, London, Toronto) · Zbl 0918.34010
[3] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives: Theory and Applications (1993), Gordon and Breach: Gordon and Breach Yverdon · Zbl 0818.26003
[4] 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
[5] Balachandran, K.; Park, J. Y., Nonlocal Cauchy problem for abstract fractional semilinear evolution equations, Nonlinear Anal. TMA, 71, 4471-4475 (2009) · Zbl 1213.34008
[6] Balachandran, K.; Trujillo, J. J., The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces, Nonlinear Anal. TMA, 72, 4587-4593 (2010) · Zbl 1196.34007
[7] Balachandran, K.; Kiruthika, S.; Trujillo, J. J., Existence results for fractional impulsive integrodifferential equations in Banach spaces, Commun. Nonlinear Sci. Numer. Simul., 16, 1970-1977 (2011) · Zbl 1221.34215
[8] N’Guérékata, G. M., A Cauchy problem for some fractional abstract differential equation with non local conditions, Nonlinear Anal. TMA, 70, 1873-1876 (2009) · Zbl 1166.34320
[9] Salem, H. A.H., On the fractional calculus in abstract spaces and their applications to the Dirichlet-type problem of fractional order, Comput. Math. Appl., 59, 1278-1293 (2010) · Zbl 1241.34011
[10] Salem, H. A.H., Multi-term fractional differential equation in reflexive Banach space, Math. Comput. Modelling, 49, 829-834 (2009) · Zbl 1165.34388
[11] Arara, A.; Benchohra, M.; Hamidi, N.; Nieto, J. J., Fractional order differential equations on an unbounded domain, Nonlinear Anal. TMA, 72, 580-586 (2010) · Zbl 1179.26015
[12] Lakshmikantham, V.; Leela, S., Nonlinear Differential Equations in Abstract Space (1981), Pergamon Press: Pergamon Press Oxford · Zbl 0456.34002
[13] Guo, D. J.; Lakshmikantham, V.; Liu, X., Nonlinear Integral Equations in Abstract Spaces (1996), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht
[14] 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
[15] Guo, F.; Liu, L. S.; Wu, Y. H.; Siew, P., Global solutions of initial value problems for nonlinear second-order impulsive integro-differential equations of mixed type in Banach spaces, Nonlinear Anal. TMA, 61, 1363-1382 (2005) · Zbl 1081.34077
[16] Zhang, X. G.; Liu, L. S.; Wu, Y. H., Global solutions of nonlinear second-order impulsive integro-differential equations of mixed type in Banach spaces, Nonlinear Anal. TMA, 67, 2335-2349 (2007) · Zbl 1121.45004
[17] Agarwal, R. P.; O’Regan, D., Infinite Interval Problems for Differential, Difference and Integral Equations (2001), Kluwer Academic Publisher: Kluwer Academic Publisher Netherlands · Zbl 1003.39017
[18] Chen, S. Z.; Zhang, Y., Singular boundary value problems on a half-line, J. Math. Anal. Appl., 195, 449-468 (1995) · Zbl 0852.34019
[19] Eloe, P. W.; Kaufmann, E. R.; Tisdell, C. C., Multiple solutions of a boundary value problem on an unbounded domain, Dyn. Syst. Appl., 15, 53-63 (2006) · Zbl 1108.34024
[20] Gomes, J. M.; Sanchez, L., A variational approach to some boundary value problems in the half-line, Z. Angew. Math. Phys., 56, 192-209 (2005) · Zbl 1073.34026
[21] Guseinov, G. Sh.; Yaslan, I., Boundary value problems for second order nonlinear differential equations on infinite intervals, J. Math. Anal. Appl., 290, 620-638 (2004) · Zbl 1054.34045
[22] Lian, H. R.; Wang, P. G.; Ge, W. G., Unbounded upper and lower solutions method for Sturm-Liouville boundary value problem on infinite intervals, Nonlinear Anal. TMA, 70, 2627-2633 (2009) · Zbl 1167.34320
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