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Zbl 1187.34026
Bai, Zhanbing
On positive solutions of a nonlocal fractional boundary value problem. (English)
[J] Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 2, A, 916-924 (2010). ISSN 0362-546X

Summary: We investigate the existence and uniqueness of positive solutions for a nonlocal boundary value problem $$\align & D^\alpha_{0+}u(t)+f(t,u(t))=0,\quad 0<t<1,\\ & u(0)=0,\quad \beta u(\eta)=u(1),\endalign$$ where $1<\alpha\le 2$, $0 <\beta\eta^{\alpha-1}< 1.0 < \eta < 1$, $D^\alpha_{0+}$ is the standard Riemann-Liouville differentiation. The function is continuous on $[0,1]\times [0,\infty)$. Firstly, we give Green's function and prove its positivity; secondly, the uniqueness of positive solution is obtained by the use of contraction map principle and some Lipschitz-type conditions; thirdly, by means of the fixed point index theory, we obtain some existence results of positive solution. The proofs are based upon the reduction of the problem considered to the equivalent Fredholm integral equation of second kind.

MSC 2000:: *34B10 Multipoint boundary value problems
34A08
34B18 Positive solutions of nonlinear boundary value problems

Keywords: fractional differential equation; positive solution; Green's function; fixed point index

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