×

Nonlinear fractional integro-differential equations on unbounded domains in a Banach space. (English) Zbl 1302.45019

Summary: By employing the fixed point theory and the monotone iterative technique, we investigate the existence of a unique solution for a class of nonlinear fractional integro-differential equations on semi-infinite domains in a Banach space. An explicit iterative sequence for approximating the solution of the boundary value problem is derived. An error estimate is also given.

MSC:

45J05 Integro-ordinary differential equations
45N05 Abstract integral equations, integral equations in abstract spaces
34A08 Fractional ordinary differential equations
26A33 Fractional derivatives and integrals
34K37 Functional-differential equations with fractional derivatives
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press San Diego · Zbl 0918.34010
[2] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., (Theory and Applications of Fractional Differential Equations. Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204 (2006), Elsevier Science B.V: Elsevier Science B.V Amsterdam) · Zbl 1092.45003
[3] Lakshmikantham, V.; Leela, S.; Devi, J. V., Theory of Fractional Dynamic Systems (2009), Cambridge Scientific Publishers: Cambridge Scientific Publishers Cambridge · Zbl 1188.37002
[4] (Sabatier, J.; Agrawal, O. P.; Machado, J. A.T., Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering (2007), Springer: Springer Dordrecht) · Zbl 1116.00014
[5] Baleanu, D.; Diethelm, K.; Scalas, E.; Trujillo, J. J., Fractional calculus models and numerical methods, (Series on Complexity, Nonlinearity and Chaos (2012), World Scientific: World Scientific Boston) · Zbl 1248.26011
[6] Hernandez, E.; O’Regan, D.; Balachandran, K., On recent developments in the theory of abstract differential equations with fractional derivatives, Nonlinear Anal., 73, 10, 3462-3471 (2010) · Zbl 1229.34004
[7] Ford, N. J.; Morgado, M. L., Fractional boundary value problems: analysis and numerical methods, Fract. Calc. Appl. Anal., 14, 554-567 (2011) · Zbl 1273.65098
[8] Ahmad, B.; Nieto, J. J., Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions, Bound. Value Probl., 2011, 36 (2011), 9 pages · Zbl 1275.45004
[9] Wang, G.; Agarwal, R. P.; Cabada, A., Existence results and the monotone iterative technique for systems of nonlinear fractional differential equations, Appl. Math. Lett., 25, 1019-1024 (2012) · Zbl 1244.34008
[10] Bai, Z. B.; Sun, W., Existence and multiplicity of positive solutions for singular fractional boundary value problems, Comput. Math. Appl., 63, 1369-1381 (2012) · Zbl 1247.34006
[11] Sakthivel, R.; Mahmudov, N. I.; Nieto, J. J., Controllability for a class of fractional-order neutral evolution control systems, Appl. Math. Comput., 218, 10334-10340 (2012) · Zbl 1245.93022
[12] Wang, G.; Ahmad, B.; Zhang, L., Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order, Nonlinear Anal., 74, 792-804 (2011) · Zbl 1214.34009
[13] Ahmad, B.; Ntouyas, S. K., Existence results for nonlocal boundary value problems for fractional differential equations and inclusions with strip conditions, Bound. Value Probl., 2012, 55 (2012), 21 pages · Zbl 1279.26010
[14] Agarwal, R. P.; O’Regan, D.; Stanek, S., Positive solutions for mixed problems of singular fractional differential equations, Math. Nachr., 285, 1, 27-41 (2012) · Zbl 1232.26005
[15] Arara, A.; Benchohra, M.; Hamidia, N.; Nieto, J. J., Fractional order differential equations on an unbounded domain, Nonlinear Anal., 72, 580-586 (2010) · Zbl 1179.26015
[16] 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
[17] Liang, S.; Zhang, J., Existence of three positive solutions for \(m\)-point boundary value problems for some nonlinear fractional differential equations on an infinite interval, Comput. Math. Appl., 61, 3343-3354 (2011) · Zbl 1235.34079
[18] Su, X., Solutions to boundary value problem of fractional order on unbounded domains in a Banach space, Nonlinear Anal., 74, 2844-2852 (2011) · Zbl 1250.34007
[19] Agarwal, R. P.; Benchohra, M.; Hamani, S.; Pinelas, S., Boundary value problems for differential equations involving Riemann-Liouville fractional derivative on the half-line, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 18, 2, 235-244 (2011) · Zbl 1208.26012
[20] Liang, S.; Zhang, J., Existence of multiple positive solutions for \(m\)-point fractional boundary value problems on an infinite interval, Math. Comput. Modelling, 54, 1334-1346 (2011) · Zbl 1235.34023
[21] Chen, F.; Zhou, Y., Attractivity of fractional functional differential equations, Comput. Math. Appl., 62, 1359-1369 (2011) · Zbl 1228.34017
[22] Su, X.; Zhang, S., Unbounded solutions to a boundary value problem of fractional order on the half-line, Comput. Math. Appl., 61, 1079-1087 (2011) · Zbl 1217.34045
[23] Wang, G.; Ahmad, B.; Zhang, L., A coupled system of nonlinear fractional differential equations with multi-point fractional boundary conditions on an unbounded domain, Abstr. Appl. Anal., 2012 (2012), 11 pages, Article ID 248709
[24] Wang, G., Monotone iterative technique for boundary value problems of a nonlinear fractional differential equations with deviating arguments, J. Comput. Appl. Math., 236, 2425-2430 (2012) · Zbl 1238.65077
[25] Wang, G.; Baleanu, D.; Zhang, L., Monotone iterative method for a class of nonlinear fractional differential equations, Fract. Calc. Appl. Anal., 15, 244-252 (2012) · Zbl 1273.34021
[26] McRae, F. A., Monotone iterative technique and existence results for fractional differential equations, Nonlinear Anal., 71, 6093-6096 (2009) · Zbl 1260.34014
[27] Ramirez, J. D.; Vatsala, A. S., Monotone iterative technique for fractional differential equations with periodic boundary boundary conditions, Opuscula Math., 29, 289-304 (2009) · Zbl 1197.26007
[28] Wei, Z.; Li, G.; Che, J., Initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative, J. Math. Anal. Appl., 367, 260-272 (2010) · Zbl 1191.34008
[29] Ladde, G. S.; Lakshmikantham, V.; Vatsala, A. S., Monotone Iterative Techniques for Nonlinear Differential Equations (1985), Pitman: Pitman London · Zbl 0658.35003
[30] Guo, Dajun; Lakshmikantham, V.; Liu, Xinzhi, Nonlinear Integral Equations in Abstract Spaces (1996), Kluwer Academic Publishers · Zbl 0866.45004
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.