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Positive solutions for singular \(\alpha\)-order \((2\le\alpha\le 3)\) fractional boundary value problems on the half-line. (English) Zbl 1499.34173

Summary: This article deals with existence of positive solutions to the fractional boundary value problem \[\begin{cases}D^\alpha u(t)+f(t,u(t))=0,0 \quad 0\le t<\infty\\ u(0)=D^{\alpha-2}u(0)=\lim_{t\to\infty} D^{\alpha-1}u(t)=0\end{cases}\] where \(\alpha\in[2,3]\), \(D^\alpha\) is the standard Riemann-Liouville derivative and \(f:(0,+\infty)\times (0,+\infty)\to\mathbb{R}^+\) is a continuous function and may exhibit singular at \(u=0\). The main existence result is obtained by means of Guo-Krasnoselskii’s version of expansion and compression of a cone principal in a Banach space.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34A08 Fractional ordinary differential equations
26A33 Fractional derivatives and integrals
34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B40 Boundary value problems on infinite intervals for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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