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Nonlinear Functional Boundary Value Problems for Conformable Fractional Dynamic Equations on Time Scales

  • Bouharket BendoumaEmail author
  • Ahmed Hammoudi
Article
  • 21 Downloads

Abstract

In this paper, we establish the existence of solutions for the conformable fractional dynamic equations on time scales, with nonlinear functional boundary value conditions. We first obtain the exact expression of the fractional Green’s function related to the linear conformable fractional dynamic problem and then we study the nonlinear functional boundary problems, by means of the upper and lower solutions method together with Schauder’s fixed-point theorem.

Keywords

Conformable fractional calculus on time scales conformable fractional dynamic equation nonlinear functional boundary conditions Green’s function upper and lower solutions 

Mathematics Subject Classification

34N05 34A08 34K37 34B15 26A33 

Notes

References

  1. 1.
    Abdeljawad, T.: On conformable fractional calculus. J. Comput. Appl. Math. 279, 57–66 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Agarwal, R.P., Otero-Espinar, V., Perera, K., Vivero, D.R.: Basic properties of Sobolev’s spaces on time scales. Adv. Differ. Equ. 2006, 14 (2006)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Ahmadkhanlu, A., Jahanshahi, M.: On the existence and uniqueness of solution of initial value problem for fractional order differential equations on time scales. Bull. Iran. Math. Soc. 38(1), 241–252 (2012)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Anderson, D.R., Avery, R.I.: Fractional-order boundary value problem with Sturm–Liouville boundary conditions. Electron. J. Differ. Equ. 2015(29), 10 (2015)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Atici, M.F., Guseinov, G.S.: On Green’s functions and positive solutions for boundary value problems on time scales. J. Comput. Appl. Math. 141(1–2), 75–99 (2002)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Aulbach, B., Hilger, S.: A unified approach to continuous and discrete dynamics. In: Qualitative theory of differential equations (Szeged, 1988), Colloq. Math. Soc. János Bolyai, 53, pp. 37–56, North-Holland, Amsterdam (1990)Google Scholar
  7. 7.
    Batarfi, H., Losada, J., Nieto, J.J., Shammakh, W.: Three-point boundary value problems for conformable fractional differential equations. J. Funct. Spaces 2015, 6 (2015)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bayour, B., Torres, D.F.M.: Existence of solution to a local fractional nonlinear differential equation. J. Comput. Appl. Math. 312, 127–133 (2016)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Benchohra, M., Cabada, A., Seba, D.: An existence result for nonlinear fractional differential equations on Banach spaces. Bound. Value Probl. 2009, Article ID 628916, 11 (2009)Google Scholar
  10. 10.
    Benchohra, M., Hamani, S., Ntouyas, S.K.: Boundary value problems for differential equations with fractional order. Surv. Math. Appl. 3, 1–12 (2008)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Benchohra, M., Henderson, J., Ntouyas, S.K., Ouahab, A.: Existence results for functional differential equations of fractional order. J. Math. Anal. Appl. 338, 1340–1350 (2008)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Bendouma, B., Cabada, A., Hammoudi, A.: Existence of solutions for conformable fractional problems with nonlinear functional boundary conditions (submitted)Google Scholar
  13. 13.
    Benkhettou, N., Hammoudi, A., Torres, D.F.M.: Existence and uniqueness of solution for a fractional Riemann–Liouville initial value problem on time scales. J. King Saud Univ. Sci. 28(1), 87–92 (2016)CrossRefGoogle Scholar
  14. 14.
    Benkhettou, N., Hassani, S., Torres, D.F.M.: A conformable fractional calculus on arbitrary time scales. J. King Saud Univ. Sci. 28(1), 93–98 (2016)CrossRefGoogle Scholar
  15. 15.
    Bohner, M., Peterson, A.: Dynamic equations on time scales. Birkhauser, Boston (2001)CrossRefGoogle Scholar
  16. 16.
    Bohner, M., Peterson, A.: Advances in dynamic equations on time scales. Birkhauser, Boston (2003)CrossRefGoogle Scholar
  17. 17.
    Cabada, A.: The monotone method for first-order problems with linear and nonlinear boundary conditions. Appl. Math. Comput. 63, 163–188 (1994)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Cabada, A., Vivero, D.R.: Existence of solutions of first-order dynamic equations with nonlinear functional boundary value conditions. Nonlinear Anal. Theory Methods Appl. 63(5–7), 697–706 (2005)CrossRefGoogle Scholar
  19. 19.
    Cabada, A., Vivero, D.R.: Criterions for absolute continuity on time scales. J. Differ. Equ. Appl. 11, 1013–1028 (2005)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Gilbert, H.: Existence theorems for first-order equations on time scales with \( \Delta \)-Carathéodory functions. Adv. Differ. Equ. 2010, 20 (2010). Article ID 650827zbMATHGoogle Scholar
  21. 21.
    Gözütok, N.Y., Gözütok, U.: Multivariable conformable fractional calculus, math.CA (2017)Google Scholar
  22. 22.
    Gulsen, T., Yilmaz, E., Goktas, S.: Conformable fractional Dirac system on time scales. J. Inequal. Appl. 2017(1), 161 (2017).  https://doi.org/10.1186/s13660-017-1434-8 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Guseinov, Sh: Integration on time scales. J. Math. Anal. Appl. 285(1), 107–127 (2003)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Heikkilä, S., Lakshmikantham, V.: Monotone iterative techniques for discontinuous nonlinear differential equations. Marcel Dekker, New York (1994)zbMATHGoogle Scholar
  25. 25.
    Jankowski, T.: Boundary problems for fractional differential equations. Appl. Math. Lett. 28, 14–19 (2014)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Katugampola, U.N.: A new fractional derivative with classical properties. arXiv:1410.6535v2 (2014)
  27. 27.
    Khalil, R., Al Horani, M., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Kilbas, A., Srivastava, M.H., Trujillo, J.J.: Theory and application of fractional differential equations. North Holland Mathematics Studies, vol. 204. Elsevier Sci. B.V., Amsterdam (2006)Google Scholar
  29. 29.
    Nwaeze, E.R.: A mean value theorem for the conformable fractional calculus on arbitrary time scales. J. Progr. Fract. Differ. Appl. 2(4), 287–291 (2016)CrossRefGoogle Scholar
  30. 30.
    Podlubny, I.: Fractional differential equations. Academic Press, San Diego (1999)zbMATHGoogle Scholar
  31. 31.
    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Yverdon (1993)zbMATHGoogle Scholar
  32. 32.
    Shugui, K., Huiqing, C., Yaqing, Y., Ying, G.: Existence and uniqueness of the solutions for the fractional initial value problem. Electron. J. Shanghai Normal Univ. (Natural Sciences) 45(3), 313–319 (2016)zbMATHGoogle Scholar
  33. 33.
    Shi, A., Zhang, S.: Upper and lower solutions method and a fractional differential equation boundary value problem. Electron. J. Qual. Theory Differ. Equ. 30, 13 (2009)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Tarasov, V.E.: Fractional dynamics: Application of fractional calculus to dynamics of particles, fields and media, Springer, Heidelberg; Higher Education Press, Beijing (2010)Google Scholar
  35. 35.
    Wang, Y., Zhou, J., Li, Y.: Fractional Sobolev’s spaces on time scales via conformable fractional calculus and their application to a fractional differential equation on time scales. Adv. Math. Phys. 2016, 1–21 (2016)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Wei, Z., Li, Q., Che, J.: Initial value problems for fractional differential equations involving Riemann–Liouville sequential fractional derivative. J. Math. Anal. Appl. 367(1), 260–272 (2010)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Yaslan, I., Liceli, O.: Three-point boundary value problems with delta Riemann–Liouville fractional derivative on time scales. Fract. Differ. Calc. 6(1), 1–16 (2016)MathSciNetGoogle Scholar
  38. 38.
    Yan, R.A., Sun, S.R., Han, Z.L.: Existence of solutions of boundary value problems for Caputo fractional differential equations on time scales. Bull. Iran. Math. Soc. 42(2), 247–262 (2016)MathSciNetzbMATHGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Ibn Khaldoun, Tiaret UniversityTiaretAlgeria
  2. 2.University of Sidi Bel AbbèsSidi Bel AbbèsAlgeria
  3. 3.Laboratory of MathematicsAin Témouchent UniversityAin TémouchentAlgeria

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