Positive solutions for nonlinear fractional differential equations. (English) Zbl 1377.26006

Summary: We study the existence and uniqueness of positive solutions of the nonlinear fractional differential equation \[ \begin{cases} ^{C}D^{\alpha}x\left( t\right) =f(t,x(t))+^{C}D^{\alpha -1}g\left( t,x\left( t\right) \right),\;0<t\leq T,\\ x\left( 0\right) =\theta_{1}>0,\;x'\left( 0\right) =\theta_{2}>0, \end{cases} \]
where \(1<\alpha \leq 2\). 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 M. M. Matar [Acta Math. Univ. Comen., New Ser. 84, No. 1, 51–57 (2015; Zbl 1340.34031)]. Finally, an example is given to illustrate our results.


26A33 Fractional derivatives and integrals
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34G20 Nonlinear differential equations in abstract spaces


Zbl 1340.34031
Full Text: DOI


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