Results in Mathematics

, Volume 63, Issue 1–2, pp 183–194

On Nonlocal Boundary Value Problems for Nonlinear Integro-differential Equations of Arbitrary Fractional Order

Article

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
$$ \begin{array}{ll} ^cD^qx(t) = f(t,x(t),(\phi x)(t),(\psi x)(t)), \quad 0 < t < 1,\\x(0) = \beta x(\eta), x'(0) =0, x''(0) =0, \ldots, x^{(m-2)}(0) =0, x(1)= \alpha x(\eta), \end{array}$$
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.

Mathematics Subject Classification (2000)

Primary 26A33 Secondary 34B15 

Keywords

Nonlinear fractional differential equations nonlocal boundary conditions existence fixed point theorem 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Coppel, W.: Disconjugacy. Lecture Notes in Mathematics, vol. 220. Springer, Berlin (1971)Google Scholar
  2. 2.
    Zhang Z., Wang J.: Positive solutions to a second order three-point boundary value problem. J. Math. Anal. Appl. 285, 237–249 (2003)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    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)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Webb J.R.L., Infante G.: Positive solutions of nonlocal boundary value problems: a unified approach. J. Lond. Math. Soc. 74, 673–693 (2006)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Infante G., Webb J.R.L.: Nonlinear nonlocal boundary value problems and perturbed Hammerstein integral equations. Proc. Edinb. Math. Soc. 49, 637–656 (2006)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    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)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Graef J.R., Yang B.: Positive solutions of a third order nonlocal boundary value problem. Discret. Contin. Dyn. Syst. 1, 89–97 (2008)MathSciNetMATHGoogle Scholar
  8. 8.
    Webb J.R.L.: Nonlocal conjugate type boundary value problems of higher order. Nonlinear Anal. 71, 1933–1940 (2009)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Webb J.R.L., Infante G.: Nonlocal boundary value problems of arbitrary order. J. Lond. Math. Soc. 79, 238–258 (2009)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    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)Google Scholar
  11. 11.
    Samko S.G., Kilbas A.A., Marichev O.I.: Fractional Integrals and Derivatives, Theory and Applications. Gordon and Breach, New Jersey (1993)MATHGoogle Scholar
  12. 12.
    Podlubny I.: Fractional Differential Equations. Academic Press, San Diego (1999)MATHGoogle Scholar
  13. 13.
    Hilfer, R. (eds): Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)MATHGoogle Scholar
  14. 14.
    Kilbas A.A., Srivastava H.M., Trujillo J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies 204, Elsevier (2006)MATHGoogle Scholar
  15. 15.
    Daftardar-Gejji V., Bhalekar S.: Boundary value problems for multi-term fractional differential equations. J. Math. Anal. Appl. 345, 754–765 (2008)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Gafiychuk V., Datsko B., Meleshko V.: Mathematical modeling of time fractional reaction-diffusion systems. J. Comput. Appl. Math. 220, 215–225 (2008)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    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)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    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)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    N’Guerekata G.M.: A Cauchy problem for some fractional abstract differential equation with non local conditions. Nonlinear Anal. 70, 1873–1876 (2009)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    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)MathSciNetMATHGoogle Scholar
  21. 21.
    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)Google Scholar
  22. 22.
    Lakshmikantham V., Leela S., Vasundhara Devi J.: Theory of Fractional Dynamic Systems. Cambridge Academic Publishers, Cambridge (2009)MATHGoogle Scholar
  23. 23.
    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)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Ahmad B., Graef J.R.: Coupled systems of nonlinear fractional differential equations with nonlocal boundary conditions. Panamer. Math. J. 19, 29–39 (2009)MathSciNetMATHGoogle Scholar
  25. 25.
    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)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    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)Google Scholar
  27. 27.
    Mophou G.M.: Existence and uniqueness of mild solutions to impulsive fractional differential equations. Nonlinear Anal. 72, 1604–1615 (2010)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    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)MathSciNetMATHGoogle Scholar
  29. 29.
    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)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    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)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Smart, D.R.: Fixed Point Theorems. Cambridge University Press (1980)Google Scholar
  32. 32.
    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)Google Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceKing Abdulaziz UniversityJeddahSaudi Arabia

Personalised recommendations