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Existence and uniqueness of positive solutions for a singular fractional three-point boundary value problem. (English) Zbl 1246.34006
Summary: We investigate the existence and uniqueness of positive solutions for the following singular fractional three-point boundary value problem $D^\alpha_{0^+}u(t) + f(t, u(t)) = 0, 0 < t < 1, u(0) = u'(0) = u''(0) = 0, u''(1) = \beta u''(\eta)$, where $3 < \alpha \leq 4, D^\alpha_{0^+}$ is the standard Riemann-Liouville derivative and $f : (0, 1] \times [0, \infty) \rightarrow [0, \infty)$ with $\lim_{t \rightarrow 0^+} f(t, \cdot) = \infty$ (i.e., $f$ is singular at $t = 0$). Our analysis relies on a fixed point theorem in partially ordered metric spaces.

34A08Fractional differential equations
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
34B10Nonlocal and multipoint boundary value problems for ODE
Full Text: DOI
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