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

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
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