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

Jiang, Fuquan; Xu, Xiaojie; Cao, Zhongwei

Abstr. Appl. Anal. 2013, Article ID 531038, 12 p. (2013).

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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