Approximating solutions of nonlinear hybrid Caputo fractional integro-differential equations via Dhage iteration principle. (English) Zbl 1463.45028

Summary: In this article, we prove the existence and approximation of solutions of the initial value problems of nonlinear hybrid Caputo fractional integro-differential equations. The main tool employed here is the Dhage iteration principle in a partially ordered normed linear space. An example is also given to illustrate the main results.


45J05 Integro-ordinary differential equations
34A08 Fractional ordinary differential equations
45L05 Theoretical approximation of solutions to integral equations
Full Text: DOI MNR


[1] Ahmad B., Ntouyas S. K., “Nonlocal boundary value problems for hybrid fractional differential equations and inclusions of Hadamard type”, Fractional Differ. Calc., 5:2 (2015), 107-123 · Zbl 1412.34006
[2] Ahmad B., Ntouyas S. K., “Initial-value problems for hybrid Hadamard fractional differential equations”, Electron. J. Differential Equations, 2014:161 (2014), 1-8 · Zbl 1300.34012
[3] Boulares H., Ardjouni A., Laskri Y., “Positive solutions for nonlinear fractional differential equations”, Positivity, 21:3 (2017), 1201-1212 · Zbl 1377.26006
[4] Boulares H., Ardjouni A., Laskri Y., “Stability in delay nonlinear fractional differential equations”, Rend. Circ. Mat. Palermo (2), 65:2 (2016), 243-253 · Zbl 1373.34114
[5] Dhage B. C., “Hybrid fixed point theory in partially ordered normed linear spaces and applications to fractional integral equations”, Differ. Equ. Appl., 5:2 (2013), 155-184 · Zbl 1279.45005
[6] Dhage B. C., “Partially condensing mappings in ordered normed linear spaces and applications to functional integral equations”, Tamkang J. Math., 45:4 (2014), 397-426 · Zbl 1343.45004
[7] Dhage B. C., Dhage S. B., Ntouyas S. K., “Approximating solutions of nonlinear hybrid differential equations”, Appl. Math. Lett., 34 (2014), 76-80 · Zbl 1314.34036
[8] Dhage B. C., Dhage S. B., Ntouyas S. K., “Approximating solutions of nonlinear second order ordinary differential equations via Dhage iteration principle”, Malaya J. Mat., 4:1 (2016), 8-18
[9] Dhage B. C., Khurpe G. T., Shete A. Y., Salunke J. N., “Existence and approximate solutions for nonlinear hybrid fractional integro-differential equations”, Int. J. Anal. Appl., 11:2 (2016), 157-167 · Zbl 1379.65099
[10] Dhage B. C., Lakshmikantham V., “Basic results on hybrid differential equations”, Nonlinear Anal. Hybrid Syst., 4:3 (2010), 414-424 · Zbl 1206.34020
[11] Ge F., Kou C., “Stability analysis by Krasnoselskii”s fixed point theorem for nonlinear fractional differential equations”, Appl. Math. Comput., 257 (2015), 308-316 · Zbl 1338.34103
[12] Ge F., Kou C., “Asymptotic stability of solutions of nonlinear fractional differential equations of order \(1<\alpha<2\)”, J. Shanghai Normal Univ. Nat. Sci., 44:3 (2015), 284-290
[13] Gomoyunov M. I., “Fractional derivatives of convex Lyapunov functions and control problems in fractional order systems”, Fract. Calc. Appl. Anal., 21:5 (2018), 1238-1261 · Zbl 1426.34012
[14] Heikkilä S., Lakshmikantham V., Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations, Marcel Dekker inc., New York, 1994, 536 pp. · Zbl 0804.34001
[15] Kilbas A. A., Srivastava H. H., Trujillo J. J., Theory and Applications of Fractional Differential Equations, Elsevier Science, Amsterdam, 2006, 540 pp. · Zbl 1092.45003
[16] Kou C., Zhou H., Yan Y., “Existence of solutions of initial value problems for nonlinear fractional differential equations on the half-axis”, Nonlinear Anal., 74:17 (2011), 5975-5986 · Zbl 1235.34022
[17] Lakshmikantham V., Vatsala A. S., “Basic theory of fractional differential equations”, Nonlinear Anal., 69:8 (2008), 2677-2682 · Zbl 1161.34001
[18] Li N., Wang C., “New existence results of positive solution for a class of nonlinear fractional differential equations”, Acta Math. Sci., 33:3 (2013), 847-854 · Zbl 1299.34015
[19] Podlubny I., Fractional Differential Equations, Academic Press, San Diego, 1999, 340 pp. · Zbl 0924.34008
[20] Zhao Y., Sun S., Han Z., Li Q., “Theory of fractional hybrid differential equations”, Comput. Math. Appl., 62:3 (2011), 1312-1324 · Zbl 1228.45017
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.