Existence of Solutions of Nonlinear and Non-local Fractional Boundary Value Problems

  • Alberto Cabada
  • Suzana Aleksić
  • Tatjana V. Tomović
  • Sladjana DimitrijevićEmail author


In this paper, we establish new results for non-local boundary value problems. In particular, we study a fractional differential equation where the associated integral equation has a kernel that is not bounded above and changes its sign, so that, the positive sign of the possible solutions is generally not ensured. We provide some examples which support the theory and illustrate the applicability of the obtained results.


Fractional differential equation non-local boundary value problem positive solution Caputo’s fractional derivative Green’s function 

Mathematics Subject Classification

34A08 34B10 34B15 



First author partially is supported by Xunta de Galicia (Spain), project EM2014/032 and AIE Spain and FEDER, Grant MTM2016-75140-P.

Second, third and fourth authors are supported by Serbian Ministry of Science and Technology (Grants 174024 and 174015).

The authors are very grateful to the reviewer of the work. His/her suggestions have been fundamental for the improvement of this article.


  1. 1.
    Ahmad, B., Agarwal, R.: On nonlocal fractional boundary value problems. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 18, 535–544 (2011)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Ahmad, B., Nieto, J.J.: Anti-periodic fractional boundary value problems with nonlinear term depending on lower order derivative. Fract. Calc. Appl. Anal. 15, 451–462 (2012)MathSciNetzbMATHGoogle Scholar
  3. 3.
    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)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ahmad, B., Nieto, J.J.: Riemann–Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions. Bound. Value Probl. 36, 9 (2011)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Ahmad, B., Nieto, J.J., Alsaedi, A., El-Shahed, M.: A study of nonlinear Langevin equation involving two fractional orders in different intervals. Nonlinear Anal. Real World Appl. 13(2), 599–606 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Baleanu, D., Agarwal, R.P., Khan, H., Khan, R.A., Jafari, H.: On the existence of solution for fractional differential equations of order \(3 <\delta _1\leqslant 4\). Adv. Differ. Equ. 362, 9 (2015)zbMATHGoogle Scholar
  7. 7.
    Baleanu, D., Nazemi, S.Z., Rezapour, S.: A \(k\)-dimensional system of fractional neutral functional differential equations with bounded delay. Abstr. Appl. Anal. (2014). zbMATHGoogle Scholar
  8. 8.
    Benchohra, M., Cabada, A., Seba, D.: An existence result for nonlinear fractional differential equations on Banach spaces. Bound. Value Probl. (2009). MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Butzer, P.L., Westphal, U.: An introduction to fractional calculus. In: Hilfer, R. (ed.) Applications of Fractional Calculus in Physics. World Scientific, New Jersey (2000)zbMATHGoogle Scholar
  10. 10.
    Cabada, A., Cid, J.A., Infante, G.: New criteria for the existence of non-trivial fixed points in cones. Fixed Point Theory Appl. 125, 12 (2013)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Cabada, A., Hamdi, A.: Nonlinear fractional differential equations with integral boundary value conditions. Appl. Math. Comput. 228, 251–257 (2014)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Cabada, A., Infante, G., Tojo, F.A.F.: Nonzero solutions of perturbed Hammerstein integral equations with deviated arguments and applications. Topol. Methods Nonlinear Anal. 47, 265–287 (2016)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Cabada, A., Infante, G.: Positive solutions of a nonlocal Caputo fractional BVP. Dyn. Syst. Appl. 23(4), 715–722 (2014)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Cabada, A., Saavedra, L.: Existence of solutions for \(n^{th}\)-order nonlinear differential boundary value problems by means of fixed point theorems. Nonlinear Anal. Real World Appl. 12, 180–206 (2018)CrossRefGoogle Scholar
  15. 15.
    Cabada, A., Wang, G.: Positive solutions of nonlinear fractional differential equations with integral boundary value conditions. J. Math. Anal. Appl. 389, 403–411 (2012)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Fan, H., Ma, R.: Loss of positivity in a nonlinear second order ordinary differential equations. Nonlinear Anal. 71(1), 437–444 (2009)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Gorenflo, R., Mainardi, F.: Fractional calculus: integral and differential equations of fractional order. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics, pp. 223–276. Springer-Verlag, New York (1997)CrossRefGoogle Scholar
  18. 18.
    Guidotti, P., Merino, S.: Gradual loss of positivity and hidden invariant cones in a scalar heat equation. Differ. Integral Equ. 13, 1551–1568 (2000)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Infante, G.: Nonlocal boundary value problems with two nonlinear boundary conditions. Commun. Appl. Anal. 12(3), 279–288 (2008)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Infante, G.: Positive solutions of some nonlinear BVPs involving singularities and integral BCs. Discrete Contin. Dyn. Syst. 1, 99–106 (2008)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Infante, G., Webb, J. R. L.: Loss of positivity in a nonlinear scalar heat equation. NoDEA Nonlinear Differ. Equ. Appl. MathSciNetCrossRefGoogle Scholar
  22. 22.
    Infante, G., Webb, J.R.L.: Nonlinear non-local boundary-value problems and perturbed Hammerstein integral equations. Proc. Edinb. Math. Soc. 49(3), 637–656 (2006)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Infante, G., Webb, J.R.L.: Three point boundary value problems with solutions that change sign. J. Integral Equ. Appl. 15, 37–57 (2003)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Infante, G., Pietramala, P.: Perturbed Hammerstein integral inclusions with solutions that change sign. Comment. Math. Univ. Carolin. 50(4), 591–605 (2009)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Karatsompanis, I., Palamides, P.K.: Polynomial approximation to a non-local boundary value problem. Comput. Math. Appl. 60, 3058–3071 (2010)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Kilbas, A., Srivastava, H.M., Trujillo, J.: Theory and Applications of Fractional Differential Equations, North Holland Mathematics Studies 204. Elsevier, Amsterdam (2006)Google Scholar
  27. 27.
    Lloyd, N.G.: Degree Theory, Cambridge Tracts in Mathematics 73. Cambridge University Press, Cambridge (1978)Google Scholar
  28. 28.
    Mainardi, F.: Fractional calculus: some basic problems in continuum and statistical mechanics. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics, pp. 291–348. Springer, New York (1997)CrossRefGoogle Scholar
  29. 29.
    Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)zbMATHGoogle Scholar
  30. 30.
    Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)zbMATHGoogle Scholar
  31. 31.
    Palamides, P.K., Infante, G., Pietramala, P.: Nontrivial solutions of a nonlinear heat flow problem via Sperner lemma. Appl. Math. Lett. 22, 1444–1450 (2009)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press, San Diego (1999)zbMATHGoogle Scholar
  33. 33.
    Samko, S., Kilbas, A., Maricev, O.: Fractional Integrals and Derivatives. Gordon & Breach, New York (1993)Google Scholar
  34. 34.
    Webb, J.R.L.: Multiple positive solutions of some nonlinear heat flow problems. Discrete Contin. Dyn. Syst. (2005). MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Webb, J.R.L.: Existence of positive solutions for a thermostat model. Nonlinear Anal. Real World Appl. 13, 923–938 (2012)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Webb, J.R.L.: Optimal constants in a nonlocal boundary value problem. Nonlinear Anal. 63, 672–685 (2005)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Zhang, S.: Positive solutions for boundary-value problems of nonlinear fractional differential equations. Electron. J. Differ. Equ. 2006, 1–12 (2006)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Departamento de Estatística, Análise Matemática e Optimización Instituto de Matemáticas, Facultade de MatemáticasUniversidade de Santiago de CompostelaSantiago de CompostelaSpain
  2. 2.Department of Mathematics and InformaticsFaculty of Science, University of KragujevacKragujevacSerbia

Personalised recommendations