Existence and uniqueness of positive solutions for first-order nonlinear Liouville-Caputo fractional differential equations. (English) Zbl 1442.34007

Summary: We study the existence and uniqueness of positive solutions of the first-order nonlinear Liouville-Caputo fractional differential equation \[\begin{cases} ^CD^\alpha( x( t) -g( t,x( t))) =f( t,x( t)) ,\quad 0<t\leq 1, \\ x( 0) =x_0>g( 0,x_0) >0, \end{cases}\] where \(0<\alpha \leq 1\). In the process we convert the given fractional differential equation into an equivalent integral equation. Then we construct appropriate mappings and employ the Krasnoselskii 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. Finally, an example is given to illustrate our results.


34A08 Fractional ordinary differential equations
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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