Abstract
In this paper, we prove the existence of solutions of a nonlocal boundary value problem for nonlinear integro-differential equations of fractional order given by
where \({q \in (m-1, m], m \in \mathbb{N}, m \ge 2}$, $0< \eta <1\) , and \({\phi x}\) and \({\psi x}\) are integral operators. The existence results are established by means of the contraction mapping principle and Krasnoselskii’s fixed point theorem. An illustrative example is also presented.
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References
Coppel, W.: Disconjugacy. Lecture Notes in Mathematics, vol. 220. Springer, Berlin (1971)
Zhang Z., Wang J.: Positive solutions to a second order three-point boundary value problem. J. Math. Anal. Appl. 285, 237–249 (2003)
Eloe P.W., Ahmad B.: Positive solutions of a nonlinear nth order boundary value problem with nonlocal conditions. Appl. Math. Lett. 18, 521–527 (2005)
Webb J.R.L., Infante G.: Positive solutions of nonlocal boundary value problems: a unified approach. J. Lond. Math. Soc. 74, 673–693 (2006)
Infante G., Webb J.R.L.: Nonlinear nonlocal boundary value problems and perturbed Hammerstein integral equations. Proc. Edinb. Math. Soc. 49, 637–656 (2006)
Du Z., Lin X., Ge W.: Nonlocal boundary value problem of higher order ordinary differential equations at resonance. Rocky Mt. J. Math. 36, 1471–1486 (2006)
Graef J.R., Yang B.: Positive solutions of a third order nonlocal boundary value problem. Discret. Contin. Dyn. Syst. 1, 89–97 (2008)
Webb J.R.L.: Nonlocal conjugate type boundary value problems of higher order. Nonlinear Anal. 71, 1933–1940 (2009)
Webb J.R.L., Infante G.: Nonlocal boundary value problems of arbitrary order. J. Lond. Math. Soc. 79, 238–258 (2009)
Ahmad B., Nieto J.J.: Existence of solutions for nonlocal boundary value problems of higher order nonlinear fractional differential equations. Abstr. Appl. Anal. Art. ID 494720, 9 (2009)
Samko S.G., Kilbas A.A., Marichev O.I.: Fractional Integrals and Derivatives, Theory and Applications. Gordon and Breach, New Jersey (1993)
Podlubny I.: Fractional Differential Equations. Academic Press, San Diego (1999)
Hilfer, R. (eds): Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)
Kilbas A.A., Srivastava H.M., Trujillo J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies 204, Elsevier (2006)
Daftardar-Gejji V., Bhalekar S.: Boundary value problems for multi-term fractional differential equations. J. Math. Anal. Appl. 345, 754–765 (2008)
Gafiychuk V., Datsko B., Meleshko V.: Mathematical modeling of time fractional reaction-diffusion systems. J. Comput. Appl. Math. 220, 215–225 (2008)
Rida S.Z., El-Sherbiny H.M., Arafa A.A.M.: On the solution of the fractional nonlinear Schrödinger equation. Phys. Lett. A 372, 553–558 (2008)
Agarwal R.P., Benchohra M., Slimani B.A.: Existence results for differential equations with fractional order and impulses. Mem. Differ. Equ. Math. Phys. 44, 1–21 (2008)
N’Guerekata G.M.: A Cauchy problem for some fractional abstract differential equation with non local conditions. Nonlinear Anal. 70, 1873–1876 (2009)
Ahmad B., Sivasundaram S.: Existence and uniqueness results for nonlinear boundary value problems of fractional differential equations with separated boundary conditions. Commun. Appl. Anal. 13, 121–228 (2009)
Ahmad B., Nieto J.J.: Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions. Bound. Value Probl. Art. ID 708576, 11 (2009)
Lakshmikantham V., Leela S., Vasundhara Devi J.: Theory of Fractional Dynamic Systems. Cambridge Academic Publishers, Cambridge (2009)
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)
Ahmad B., Graef J.R.: Coupled systems of nonlinear fractional differential equations with nonlocal boundary conditions. Panamer. Math. J. 19, 29–39 (2009)
Ahmad B., Sivasundaram S.: Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations. Nonlinear Anal. Hybrid Syst. 3, 251–258 (2009)
Ahmad B., Otero-Espinar V.: Existence of solutions for fractional differential inclusions with anti-periodic boundary conditions. Bound. Value Probl. Art. ID 625347, 11 (2009)
Mophou G.M.: Existence and uniqueness of mild solutions to impulsive fractional differential equations. Nonlinear Anal. 72, 1604–1615 (2010)
Ahmad B., Nieto J.J.: Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray-Schauder degree theory. Topol. Methods Nonlinear Anal. 35, 295–304 (2010)
Ahmad B.: Existence of solutions for fractional differential equations of order \({{q \in (2,3]}}\) with anti-periodic boundary conditions. J. Appl. Math. Comput 34, 385– (2010)
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)
Smart, D.R.: Fixed Point Theorems. Cambridge University Press (1980)
Ji Y., Guo Y.: The existence of countably many positive solutions for nonlinear nth-order three-point boundary value problems. Bound. Value Probl. Art. ID 572512, 18 (2009)
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Ahmad, B. On Nonlocal Boundary Value Problems for Nonlinear Integro-differential Equations of Arbitrary Fractional Order. Results. Math. 63, 183–194 (2013). https://doi.org/10.1007/s00025-011-0187-9
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DOI: https://doi.org/10.1007/s00025-011-0187-9