# Positive solutions for nonlinear fractional differential equations

Positive solutions for nonlinear fractional differential equations 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} C D α x t = f ( t , x ( t ) ) + C D α - 1 g t , x t , 0 < t ≤ T , x 0 = θ 1 > 0 , x ′ 0 = θ 2 > 0 , where $$1<\alpha \le 2$$ 1 < α ≤ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Positive solutions for nonlinear fractional differential equations

, Volume 21 (3) – Dec 5, 2016
12 pages
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Publisher
Springer International Publishing
Copyright
Copyright © 2016 by Springer International Publishing
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-016-0461-x
Publisher site
See Article on Publisher Site

### 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} C D α x t = f ( t , x ( t ) ) + C D α - 1 g t , x t , 0 < t ≤ T , x 0 = θ 1 > 0 , x ′ 0 = θ 2 > 0 , where $$1<\alpha \le 2$$ 1 < α ≤ 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.

### Journal

PositivitySpringer Journals

Published: Dec 5, 2016

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