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Positive solutions for boundary value problems of nonlinear fractional differential equation. (English) Zbl 1185.26011
The authors study the following nonlinear fractional boundary problem
$D^{\alpha}_{0+}u(t) + f(t,u(t))= 0,\quad 0<t<1,\;3<\alpha \leq 4,$
$u(0)=u'(0)=u''(0)=u''(1)=0,$
where $$f \in C([0,1]\times [0,+\infty), (0, +\infty))$$ and $$D^{\alpha}_{0+}$$ is the Riemann-Liouville fractional derivative. By using the method of lower and upper solutions and fixed point theorems, the existence of positive solutions of the fractional boundary problem above is established. An example is provided to illustrate the obtained results.

MSC:
 26A33 Fractional derivatives and integrals 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34B27 Green’s functions for ordinary differential equations
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References:
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