Existence Result for a Superlinear Fractional Navier Boundary Value Problems

  • Habib Mâagli
  • Abdelwaheb Dhifli
  • Abdulah Khamis Alzahrani
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Abstract

In this paper, we study the following fractional Navier boundary value problem
$$\begin{aligned} \left\{ \begin{array}{lllc} D^{\beta }(D^{\alpha }u)(x)=u(x)g(u(x)),\quad x\in (0,1), \\ \displaystyle \lim _{x\longrightarrow 0}x^{1-\beta }D^{\alpha }u(x)=-a,\quad \,\,u(1)=b, \end{array} \right. \end{aligned}$$
where \(\alpha ,\beta \in (0,1]\) such that \(\alpha +\beta >1\), \(D^{\beta }\) and \(D^{\alpha }\) stand for the standard Riemann–Liouville fractional derivatives and ab are nonnegative constants such that \(a+b>0\). The function g is a nonnegative continuous function in \([0,\infty )\) that is required to satisfy some suitable integrability condition. Using estimates on the Green’s function and a perturbation argument, we prove the existence of a unique positive continuous solution, which behaves like the unique solution of the homogeneous problem.

Keywords

Fractional Navier differential equations positive solutions Green’s function perturbation arguments 

Mathematics Subject Classification

34A08 34B15 34B18 34B27 

Notes

Acknowledgements

We thank the referee for his/her careful reading of the paper and helpful comments and remarks.

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© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Habib Mâagli
    • 1
  • Abdelwaheb Dhifli
    • 2
  • Abdulah Khamis Alzahrani
    • 3
  1. 1.Department of Mathematics, College of Sciences and ArtsKing Abdulaziz UniversityRabighSaudi Arabia
  2. 2.Département de MathématiquesFaculté des Sciences de TunisTunisTunisia
  3. 3.Department of Mathematics, Faculty of SciencesKing Abdulaziz UniversityJeddahSaudi Arabia

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