Existence of positive solutions for fractional differential equation with nonlocal boundary condition. (English) Zbl 1238.34012

Gao, Hongliang; Han, Xiaoling

Int. J. Differ. Equ. 2011, Article ID 328394, 10 p. (2011).

Summary: By using a fixed point theorem, existence of positive solutions for fractional differential equations with nonlocal boundary condition \[ D^\alpha_{0+} u(t) + a(t)f(t, u(t)) = 0, ~0 < t < 1, \]
\[ u(0) = 0, ~u(1) = \sum^\infty_{i=1} \alpha_i u(\zeta_i) \] is considered, where \(1 < \alpha \leq 2\) is a real number, \(D^\alpha_{0+}\) is the standard Riemann-Liouville differentiation, and \(\zeta_i \in (0, 1), ~\alpha_i \in [0, \infty)\) with \(\sum^\infty_{i=1} \alpha_i \zeta^{\alpha -1}_i < 1, ~a \in C([0, 1], [0, \infty)),~ f \in C([0, 1] \times [0, \infty), [0, \infty))\).

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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