## The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations.(English)Zbl 1221.34068

Summary: We study the existence of multiple positive solutions for the nonlinear fractional differential equation boundary value problem $\begin{cases} D^\alpha_{0^+}u(t)+f(t,u(t))=0,\quad 0<t<1,\\ u(0)=u'(0)=u'(1)=0,\end{cases}$ where $$2<\alpha\leq 3$$ is a real number and $$D^\alpha_{0^+}$$ is the Riemann-Liouville fractional derivative. Using the properties of the Green’s function, the lower and upper solution method and a fixed-point theorem, some new existence criteria for singular and nonsingular fractional differential equation boundary value problems are established. As applications, examples are presented to illustrate the main results.

### MSC:

 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34A08 Fractional ordinary differential equations 34A37 Ordinary differential equations with impulses 47N20 Applications of operator theory to differential and integral equations
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### References:

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