Positive solutions of a boundary value problem for a nonlinear fractional differential equation. (English) Zbl 1183.34007

Kaufmann, E.; Mboumi, E.

Electron. J. Qual. Theory Differ. Equ. 2008, Paper No. 3, 11 p. (2008).

Summary: We give sufficient conditions for the existence of at least one and at least three positive solutions to the nonlinear fractional boundary value problem
\[ \begin{aligned} & D^{\alpha}u + a(t) f(u) = 0, \quad 0<t<1,\quad 1<\alpha\leq 2,\\ & u(0) = 0,\;u'(1)= 0,\end{aligned} \]
where \( D^{\alpha}\) is the Riemann-Liouville differential operator of order \(\alpha , f: [0,\infty)\to [0,\infty)\) is a given continuous function and \(a\) is a positive and continuous function on \([0,1]\).

MSC:

34A08	Fractional ordinary differential equations
34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations

Keywords:

fractional derivative; nonlinear dynamic equation; positive solution