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Existence and uniqueness of positive solutions to nonlinear fractional differential equation with integral boundary conditions

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Abstract

In this paper, we consider the following nonlinear fractional three-point boundary-value problem:

$$ \begin{array}{*{20}{c}} {D_{0 + }^\alpha u(t) + f\left( {t,u(t)} \right) = 0,\,\,\,\,0 < t < 1,} \\ {u(0) = u'(0) = 0,\,\,\,\,u'(1) = \int\limits_0^\eta {u(s){\text{d}}s,} } \\ \end{array} $$

whereD α 0+ is the standard Riemann-Liouville fractional derivative. By the properties of the Green function, the lower and upper solution method, and fixed-point theorem in partially ordered sets, we establish some new results on the existence and uniqueness of positive solutions to the above boundary-value problem. As applications, examples are presented to illustrate the main results.

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References

  1. C. Bai, Positive solutions for nonlinear fractional differential equations with coefficient that changes sign, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods, 64:677–685, 2006.

    Article  MATH  Google Scholar 

  2. C. Bai, Triple positive solutions for a boundary-value problem of nonlinear fractional differential equation, Electron. J. Qual. Theory Differ. Equ., 2008(24):1–10, 2008.

    Google Scholar 

  3. Z. Bai and H. Lü, Positive solutions for boundary-value problem of nonlinear fractional differential equation, J. Math. Anal. Appl., 311:495–505, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  4. P.R. Beesack, On the Green function of an n-point boundary-value problem, Pac. J. Math., 12:801–812, 1962.

    MathSciNet  MATH  Google Scholar 

  5. R. Bellman and R. Kalaba, Quasilinearization and Nonlinear Boundary-Value Problems, Elsevier, New York, 1965.

    MATH  Google Scholar 

  6. M. Benchohra, J. Henderson, S.K. Ntouyas, and A. Ouahab, Existence results for fractional-order functional differential equations with infinite delay, J. Math. Anal. Appl., 338:1340–1350, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  7. J. Caballero Mena, J. Harjani, and K. Sadarangani, Existence and uniqueness of positive and nondecreasing solutions for a class of singular fractional boundary-value problems, Bound. Value Probl., 2009, Article ID 421310, 10 pp., 2009.

  8. E.S. Chichkin, A theorem on a differential inequality for multipoint boundary-value problems, Izv. Vyssh. Uchebn. Zaved., Mat., 2:170–179, 1962.

    Google Scholar 

  9. A.M.A. El-Sayed, A.E.M. El-Mesiry, and H.A.A. El-Saka, On the fractional-order logistic equation, Appl. Math. Lett., 20:817–823, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  10. M. El-Shahed, Positive solutions for boundary-value problem of nonlinear fractional differential equation, Abstr. Appl. Anal., 2007, Article ID 10368, 8 pp., 2007.

  11. M.Q. Feng, X.M. Zhang, and W.G. Ge, New existence results for higher-order nonlinear fractional differential equation with integral boundary conditions, Bound. Value Probl., 2011, Article ID 720702, 20 pp., 2011.

  12. Y. Guo and W. Ge, Positive solutions for three-point boundary-value problems with dependence on the first-order derivative, J. Math. Anal. Appl., 290:291–301, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  13. C.P. Gupta, Solvability of a three-point nonlinear boundary-value problem for a second-order ordinary differential equations, J. Math. Anal. Appl., 168:540–551, 1998.

    Article  Google Scholar 

  14. X. Han, Positive solutions for a three-point boundary-value problem, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods, 66:679–688, 2007.

    Article  MATH  Google Scholar 

  15. J. Harjani and K. Sadarangani, Fixed-point theorems for weakly contractive mappings in partially ordered sets, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods, 71:3403–3410, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  16. V.A. Il’in and E.I. Moiseev, Nonlocal boundary-value problem of the first kind for a Sturm–Liouville operator in its differential and finite difference aspects, Differ. Equ., 23:803–810, 1987.

    MATH  Google Scholar 

  17. K.A. Khasseinov, The Multipoint and Adjoint Boundary-Value Problems and Their Applications, Nauka, Moscow, 2006 (in Russian).

    Google Scholar 

  18. A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud., Vol. 204, Elsevier, Amsterdam, 2006.

  19. V. Lakshmikantham, S. Leela, and J. Vasundhara Devi, Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers Ltd, Cambridge, UK, 2009.

    MATH  Google Scholar 

  20. V. Lakshmikantham and A.S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods, 69:2677–2682, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  21. V. Lakshmikantham and A.S. Vatsala, General uniqueness and monotone iterative technique for fractional differential equations, Appl. Math. Lett., 21:828–834, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  22. A.Yu. Levin, On the differential properties of the Green’s function for multipoint boundary problem, Dokl. Akad. Nauk SSSR, 136:1022–1025, 1961 (in Russian).

    Google Scholar 

  23. C.F. Li, X.N. Luo, and Y. Zhou, Existence of positive solutions of the boundary-value problem for nonlinear fractional differential equations, Comput. Math. Appl., 59:1363–1375, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  24. S. Liang and L. Mu, Multiplicity of positive solutions for singular three-point boundary-value problems at resonance, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods, 71:2497–2505, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  25. S. Liang and J.H. Zhang, Positive solutions for boundary-value problems of nonlinear fractional differential equation, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods, 71:5545–5550, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  26. R. Ma, Positive solutions of a nonlinear three-point boundary-value problem, Electron. J. Differ. Equ., 1999(34):1–8, 1999.

    Google Scholar 

  27. R. Ma, Multiplicity of positive solutions for second-order three-point boundary-value problems, Comput. Math. Appl., 40:193–204, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  28. J.J. Nieto and R. Rodríguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 22:223–239, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  29. J.J. Nieto and R. Rodríguez-López, Fixed-point theorems in ordered abstract spaces, Proc. Am. Math. Soc., 135(8):2505–2517, 2007.

    Article  MATH  Google Scholar 

  30. I. Podlubny, Fractional Differential Equations, Math. Sci. Eng., Vol. 198, Academic Press, San Diego, 1999.

  31. Yu.V. Pokornyi, On estimates for the Green’s function for a multipoint boundary problem, Mat. Zametki, 4(5):533–540, 1968 (in Russian).

    MathSciNet  MATH  Google Scholar 

  32. D. O’Regan and A. Petrusel, Fixed-point theorems for generalized contractions in ordered metric spaces, J. Math. Anal. Appl., 341:1241–1252, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  33. B.O. Salyga, Multipoint Problems for Differential Equations with Partial Derivatives, Lviv, 1985.

  34. S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon, 1993.

  35. S. Zhang, Positive solutions to singular boundary-value problem for nonlinear fractional differential equation, Comput. Math. Appl., 59:1300–1309, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  36. Y. Zhou, Existence and uniqueness of fractional functional differential equations with unbounded delay, Int. J. Dyn. Syst. Differ. Equ., 1:239–244, 2008.

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. College of Mathematics, Changchun Normal University, Changchun, 130032, Jilin, PR China

    Sihua Liang

  2. Research Department, Changchun Normal University, Changchun, 130032, Jilin, PR China

    Yueqiang Song

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  1. Sihua Liang
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  2. Yueqiang Song
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Correspondence to Sihua Liang.

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The author is supported by the Research Fundation during the 12st Five-Year Plan Period of Department of Education of Jilin Province, China and the Natural Science Foundation of Changchun Normal University.

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Liang, S., Song, Y. Existence and uniqueness of positive solutions to nonlinear fractional differential equation with integral boundary conditions. Lith Math J 52, 62–76 (2012). https://doi.org/10.1007/s10986-012-9156-6

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  • DOI: https://doi.org/10.1007/s10986-012-9156-6

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