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The positive properties of Green’s function for three point boundary value problems of nonlinear fractional differential equations and its applications. (English) Zbl 1248.35224

Summary: We consider the properties of Green’s function for the nonlinear fractional differential equation boundary value problem \[ {\mathbf D}^\alpha_{0+} u(t)+ f(t, u(t))+ e(t)= 0,\quad 0< t< 1, \]
\[ u(0)= 0,\quad{\mathbf D}^\beta_{0+} u(1)= a{\mathbf D}^\beta_{0+} u(\xi), \] where \(1<\alpha\leq 2\), \(0<\beta\leq 1\), \(0\leq a\leq 1\), \(0<\xi< 1\), \(\alpha-\beta-1\geq 0\), \({\mathbf D}^\alpha_{0+}\) 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:

35R11 Fractional partial differential equations
34B27 Green’s functions for ordinary differential equations
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