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On positive solutions of a nonlocal fractional boundary value problem. (English) Zbl 1187.34026
Summary: We investigate the existence and uniqueness of positive solutions for a nonlocal boundary value problem
\[ \begin{aligned} & D^\alpha_{0+}u(t)+f(t,u(t))=0,\quad 0<t<1,\\ & u(0)=0,\quad \beta u(\eta)=u(1),\end{aligned} \]
where \(1<\alpha\leq 2\), \(0 <\beta\eta^{\alpha-1}< 1.0 < \eta < 1\), \(D^\alpha_{0+}\) 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:
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34A08 Fractional ordinary differential equations and fractional differential inclusions
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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