Existence of positive solutions for a fractional high-order three-point boundary value problem. (English) Zbl 1343.34010
Summary: In this paper, the authors consider the following fractional high-order three-point boundary value problem: \(D^{\alpha}_{0^+}u(t)+f(t,u(t))=0\), \(t\in(0,1)\), \(u(0)=u'(0)=\dots=u^{(n-2)}(0)=0\), \(D^{\alpha-1}_{0+^u}(\eta)=kD^{\alpha-1}_{0^+}u(1)\), where \(k>1\), \(\eta\in(0,1)\), \(n-1<\alpha\leq n\), \(n\geq 3\), \(D^{\alpha}_{0^+}\) is the standard Riemann-Liouville derivative of order \({\alpha}\), and \(f:[0,1]\times[0,+\infty)\to[0,+\infty)\) is continuous. By using some fixed point index theorems on a cone for differentiable operators, the authors obtain the existence of positive solutions to the above boundary value problem.
MSC:
| 34A08 | Fractional ordinary differential equations |
| 34B15 | Nonlinear boundary value problems for ordinary differential equations |
Keywords:
fractional differential equations; three-point boundary value problems; existence results; fixed point index theorem for differentiable operatorsReferences:
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