## The positive properties of Green’s function for fractional differential equations and its applications.(English)Zbl 1274.34062

Summary: We consider the properties of Green’s function for the nonlinear fractional differential equation boundary value problem: $$\mathbf{D}^\alpha_{0+}u(t) + f(t, u(t)) + e(t) = 0, 0 < t < 1, u(0) = u'(0) = \cdots = u^{(n-2)}(0) = 0, u(1) = \beta u(\eta)$$, where $$n - 1 < \alpha \leq n, n \geq 3, 0 < \beta \leq 1, 0 \leq \eta \leq 1, \mathbf{D}^\alpha_{0+}$$ is the standard Riemann-Liouville derivative. Here our nonlinearity $$f$$ may be singular at $$u = 0$$. As applications of Green’s function, we give some multiple positive solutions for singular boundary value problems by means of Schauder fixed-point theorem.

### MSC:

 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34B27 Green’s functions for ordinary differential equations 34A08 Fractional ordinary differential equations
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### References:

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