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Positive solutions of nonlinear fractional boundary value problems in ordered Banach spaces. (Chinese. English summary) Zbl 1463.34019

Summary: The existence of positive solutions for a fractional boundary value problem \[D_{0^+}^\alpha u(t) = f(t,u(t)), \quad 0 < t < 1, \quad u(0) = u(1) = u'(0) = u'(1) = \theta\] in an ordered Banach spaces \(E\) is discussed, where \(3 < \alpha \le 4\), \(D_{0^+}^\alpha\) is the standard Riemann-Liouville differentiation, \(f:[0,1] \times P \to P\) is continuous, and \(P\) is the cone of positive elements in \(E\). An existence result of positive solutions is obtained by employing a new estimate of noncompactness measure and the fixed point index theory of condensing mapping.

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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