Positivity

, Volume 21, Issue 3, pp 1201–1212 | Cite as

Positive solutions for nonlinear fractional differential equations

  • Hamid Boulares
  • Abdelouaheb Ardjouni
  • Yamina Laskri
Article

Abstract

We study the existence and uniqueness of positive solutions of the nonlinear fractional differential equation
$$\begin{aligned} \left\{ \begin{array}{l} ^{C}D^{\alpha }x\left( t\right) =f(t,x(t))+^{C}D^{\alpha -1}g\left( t,x\left( t\right) \right) ,\ 0<t\le T,\\ x\left( 0\right) =\theta _{1}>0,\ x^{\prime }\left( 0\right) =\theta _{2}>0, \end{array} \right. \end{aligned}$$
where \(1<\alpha \le 2\). In the process we convert the given fractional differential equation into an equivalent integral equation. Then we construct appropriate mapping and employ Schauder fixed point theorem and the method of upper and lower solutions to show the existence of a positive solution of this equation. We also use the Banach fixed point theorem to show the existence of a unique positive solution. The results obtained here extend the work of Matar (AMUC 84(1):51–57, 2015 [7]). Finally, an example is given to illustrate our results.

Keywords

Fractional differential equations Positive solutions Upper and lower solutions Existence Uniqueness Fixed point theorems 

Mathematics Subject Classification

Primary 26A33 Secondary 34A12 34G20 

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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  • Hamid Boulares
    • 1
  • Abdelouaheb Ardjouni
    • 2
    • 3
  • Yamina Laskri
    • 4
  1. 1.Depatement of Mathematics, Faculty of SciencesGuelma UniversityGuelmaAlgeria
  2. 2.Department of Mathematics and Informatics, Faculty of Sciences and TechnologyUniversity of Souk AhrasSouk AhrasAlgeria
  3. 3.Applied Mathematics Lab, Department of Mathematics, Faculty of SciencesUniversity of AnnabaAnnabaAlgeria
  4. 4.Depatement of Mathematics, Faculty of SciencesUniversity of AnnabaAnnabaAlgeria

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