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Existence and Asymptotic Behavior of Positive Solutions for a Coupled Fractional Differential System

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Abstract

In this paper, we take up the existence and the asymptotic behavior of positive and continuous solutions to the following coupled fractional differential system

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle D^{\alpha } u=a(x)\displaystyle u^{p }\displaystyle v^{r}\quad \text { in }(0,1) , \\ \displaystyle D^{\beta } v=b(x)\displaystyle u^{s }\displaystyle v^{q}\quad \text { in }(0,1) , \\ u(0)= u(1)= D^{\alpha -3}u(0)= u^{\prime }(1)=0,\\ v(0)= v(1)= D^{\beta -3}v(0)= v^{\prime }(1)=0, \end{array} \right. \end{aligned}$$

where \( \alpha , \beta \in (3,4]\), \(p, q\in (-1,1)\), \(r, s\in \mathbb {R}\) such that \((1-|p|)(1-|q|)-|rs|> 0\), D is the standard Riemann–Liouville differentiation and a, b are nonnegative and continuous functions in (0, 1) allowed to be singular at \(x=0\) and \(x=1\) and they are required to satisfy some appropriate conditions related to Karamata regular variation theory.

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References

  1. Agarwal, R.P., O’Regan, D., Stanek, S.: Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations. J. Math. Anal. Appl. 371, 57–68 (2010)

    Article  MathSciNet  Google Scholar 

  2. 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)

    Article  MathSciNet  Google Scholar 

  3. Alfifi, H., Ben Saad, I., Turki, S., Zine El Abidine, Z.: Existence and asymptotic behavior of positive solutions for a coupled system of semilinear fractional differential equations. Results Math. (2016). doi:10.1007/s00025-016-0528-9

  4. Alsaedi, R.S.: Existence and global behavior of positive solutions for some fourth order boundary value problems. In: Abstract and Applied Analysis, vol. 2014, p. 5 (2014)

  5. Bachar, I., Mâagli, H., Toumi, F., Zine El Abidine, Z.: Existence and global asymptotic behavior of positive solutions for sublinear and superlinear fractional boundary value problems. Chin. Ann. Math. B 36(6), 1–28 (2015)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Bai, Z., Fang, J.: The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations. Appl. Math. Comput. 150, 611–621 (2004)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. Chemmam, R., Mâagli, H., Masmoudi, S., Zribi, M.: Combined effects in nonlinear singular elliptic problems in a bounded domain. Adv. Nonlinear Anal. 1, 301–318 (2012)

    MathSciNet  MATH  Google Scholar 

  10. Chen, Y., An, H.: Numerical solutions of coupled Burgers equations with time and space fractional derivatives. Appl. Math. Comput. 200, 87–95 (2008)

    MathSciNet  MATH  Google Scholar 

  11. Gafiychuk, V., Datsko, B., Meleshko, V.: Mathematical modeling of time fractional reaction–diffusion systems. J. Comput. Appl. Math. 220, 215–225 (2008)

    Article  MathSciNet  Google Scholar 

  12. Graef, J.R., Kong, L., Kong, Q., Wang, M.: Existence and uniqueness of solutions for a fractional boundary value problem with Dirichlet boundary condition. Electron. J. Qual. Theory Differ. Equ. 55, 1–11 (2013)

    Article  MathSciNet  Google Scholar 

  13. Kaufmann, E.R., Mboumi, E.: Positive solutions of a boundary value problem for a nonlinear fractional differential equation. Electron. J. Qual. Theory Differ. Equ. 3, 1–11 (2008)

    Article  MathSciNet  Google Scholar 

  14. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations, vol. 204 of North-Holland Mathematics Studies. Elsevier Science, Amsterdam, The Netherlands (2006)

  15. Lakshmikantham, V., Leela, S., Vasundhara Devi, J.: Theory of Fractional Dynamic Systems. Cambridge Academic Publishers, Cambridge (2006)

    MATH  Google Scholar 

  16. Lazarevi, M.P.: Finite time stability analysis of \(PD^{\alpha }\) fractional control of robotic time-delay systems. Mech. Res. Commun. 33, 269–279 (2006)

    Article  MathSciNet  Google Scholar 

  17. Liang, S., Zhang, J.: Positive solutions for boundary value problems of nonlinear fractional differential equation. Nonlinear Anal. 71, 5545–5550 (2009)

    Article  MathSciNet  Google Scholar 

  18. Marić, V.: Regular Variation and Differential Equations, vol. 1726 of Lecture Notes in Mathematics. Springer, Berlin (2000)

  19. Nyamoradi, N.: Multiple positive solutions for fractional differential systems. Ann. Univ. Ferrara 58, 359–369 (2012)

    Article  MathSciNet  Google Scholar 

  20. Podlubny, I.: Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering. Academic Press, San Diego (1999)

  21. Sabatier, J., Agrawal, O.P., Machado, J.A.T. (eds.): Advances in Factional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht (2007)

    Google Scholar 

  22. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, New York (1993)

    MATH  Google Scholar 

  23. Seneta, R.: Regular Varying Functions. Lectures Notes in Math., vol. 508. Springer, Berlin (1976)

    Book  Google Scholar 

  24. Su, X.: Boundary value problem for a coupled system of nonlinear fractional differential equations. Appl. Math. Lett. 22, 64–69 (2009)

    Article  MathSciNet  Google Scholar 

  25. Wang, G., Agarwal, R.P., Cabada, A.: Existence results and monotone iterative technique for systems of nonlinear fractional differential equations. Appl. Math. Lett. 25, 1019–1024 (2012)

    Article  MathSciNet  Google Scholar 

  26. Wang, G., Ahmad, B., Zhang, L.: A Coupled system of nonlinear fractional differential equations with multipoint fractional boundary conditions on an unbounded domain. In: Abstract and Applied Analysis, vol. 2012, p. 11 (2012)

  27. Wang, J., Xiang, H., Liu, Z.: Positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations. Int. J. Differ. Equ. 2010, 1–12 (2010)

  28. Xu, X., Jiang, D., Yuan, C.: Multiple positive solutions for the boundary value problem of a nonlinear fractional differenteial equation. Nonlinear Anal. 71, 4676–4688 (2009)

    Article  MathSciNet  Google Scholar 

  29. Yang, A., Ge, W.: Positive solutions for boundary value problems of N-dimension nonlinear fractional differential system. Bound. Value Probl. 2008, 1–15 (2008)

  30. Zhang, X., Liu, L., Wu, Y.: Multiple positive solutions of a singular fractional differential equation with negatively perturbed term. Math. Comput. Model. 55, 1263–1274 (2012)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We thank Professors Habib Mâagli and Malek Zribi for their interest in this work. We also thank the referee for his / her careful reading of the manuscript.

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  1. Université de Tunis El Manar, Faculté des Sciences de Tunis, UR11ES22 Potentiels et Probabilités, 2092, Tunis, Tunisie

    Sonia Ben Makhlouf, Majda Chaieb & Zagharide Zine El Abidine

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  1. Sonia Ben Makhlouf
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  2. Majda Chaieb
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  3. Zagharide Zine El Abidine
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Correspondence to Zagharide Zine El Abidine.

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Ben Makhlouf, S., Chaieb, M. & Zine El Abidine, Z. Existence and Asymptotic Behavior of Positive Solutions for a Coupled Fractional Differential System. Differ Equ Dyn Syst 28, 953–998 (2020). https://doi.org/10.1007/s12591-017-0358-6

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