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Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation. (English) Zbl 1178.34006
Summary: In this paper, we consider the properties of Green’s function for the nonlinear fractional differential equation boundary value problem
${\mathbf D}^\alpha_{0+}u(t)=f(t,u(t)),\quad 0<t<1,\qquad u(0)=u(1)=u'(0)=u'(1)=0,$
where $$3<\alpha\leq 4$$ is a real number, and $${\mathbf D}^\alpha_{0+}$$ is the standard Riemann-Liouville differentiation. As an application of Green’s function, we give some multiple positive solutions for singular and nonsingular boundary value problems, and we also give the uniqueness of solution for a singular problem by means of the Leray-Schauder nonlinear alternative, a fixed-point theorem on cones and a mixed monotone method.

##### MSC:
 34A08 Fractional ordinary differential equations and fractional differential inclusions 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 47N20 Applications of operator theory to differential and integral equations 34B27 Green’s functions for ordinary differential equations
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