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Zbl 1185.26011
Liang, Sihua; Zhang, Jihui
Positive solutions for boundary value problems of nonlinear fractional differential equation.
(English)
[J] Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No. 11, A, 5545-5550 (2009). ISSN 0362-546X

The authors study the following nonlinear fractional boundary problem $$D\sp{\alpha}\sb{0+}u(t) + f(t,u(t))= 0,\quad 0<t<1,\ 3<\alpha \le 4,$$ $$u(0)=u'(0)=u''(0)=u''(1)=0,$$ where $f \in C([0,1]\times [0,+\infty), (0, +\infty))$ and $D\sp{\alpha}\sb{0+}$ 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 2000:
*26A33 Fractional derivatives and integrals (real functions)
34B18 Positive solutions of nonlinear boundary value problems
34B27 Green functions

Keywords: fractional boundary value problem; fixed-point theorem; positive solution; lower and upper solutions

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