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Existence of positive solutions for a class of nonlinear fractional differential equations with integral boundary conditions and a parameter. (English) Zbl 1354.34049

Summary: In this paper, we study the existence of positive solutions for the following nonlinear fractional differential equations with integral boundary conditions: \[ \begin{cases} D_{0+}^{\alpha}u(t)+h(t)f(t,u(t))=0, \quad 0<t<1,\\ u(0)=u'(0)=u''(0)=0, \\ u(1)={\lambda}\int_ 0^{\eta}u(s)\mathrm{d}s,\end{cases} \] where \(3<\alpha\leq 4\), \(0<\eta\leq 1\), \(0\leqslant \frac{\lambda{\eta}^{\alpha}}{\alpha}<1\), \(D_{0+}^{\alpha}\) is the standard Riemann-Liouville derivative. \(h(t\)) is allowed to be singular at \(t=0\) and \(t=1\). By using the properties of the Green function, \(u_0\)-bounded function and the fixed point index theory under some conditions concerning the first eigenvalue with respect to the relevant linear operator, we obtain some existence results of positive solution.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34A08 Fractional ordinary differential equations
34B08 Parameter dependent boundary value problems for ordinary differential equations
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