Triple positive solutions for a boundary value problem of nonlinear fractional differential equation. (English) Zbl 1183.34005

Bai, Chuanzhi

Electron. J. Qual. Theory Differ. Equ. 2008, Paper No. 24, 10 p. (2008).

Summary: We investigate the existence of three positive solutions for the nonlinear fractional boundary value problem
\[ D_{0+}^{\alpha} u(t) + a(t)f(t,u(t), u''(t))=0,\quad 0 < t < 1, \quad 3 < \alpha \leq 4, \]
\[ u(0) = u'(0) = u''(0)= u''(1)=0 , \]
where \(D_{0+}^{\alpha}\) is the standard Riemann-Liouville fractional derivative. The method involves applications of a new fixed-point theorem due to Bai and Ge. The interesting point lies in the fact that the nonlinear term is allowed to depend on the second order derivative \(u''\).

MSC:

34A08	Fractional ordinary differential equations
34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations
47N20	Applications of operator theory to differential and integral equations

Keywords:

fractional derivative; boundary value problem; positive solution; fixed point theorem