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Wang, Yongqing; Liu, Lishan; Wu, Yonghong

Positive solutions for a nonlocal fractional differential equation. (English) Zbl 1220.34006

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 11, 3599-3605 (2011).
Summary: We study the following singular boundary value problem of a nonlocal fractional differential equation
\[ \begin{cases} D^\alpha_{0^+}u(t)+q(t)f(t,u(t))=0,\quad & 0<t<1,\;n-1<\alpha\leq n,\\ u(0)=u'(0)=\cdots=u^{(n-2)}(0)=0,& u(1)=\int^1_0u(s)\,dA(s),\end{cases} \]
where \(\alpha \geq 2\), \(D^\alpha_{0^+}\) is the standard Riemann-Liouville derivative, \(\int^1_0u(s)\,dA(s)\) is given by the Riemann-Stieltjes integral with a signed measure, \(q\) may be singular at \(t=0\) and/or \(t=1,f(t,x)\) may also have a singularity at \(x=0\). Existence and multiplicity of positive solutions are obtained by means of fixed point index theory in cones.

Cited in 126 Documents

MSC:

34A08 Fractional ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations

Keywords:

fractional differential equation; positive solution; singular problem; fixed point index
PDFBibTeX XMLCite
\textit{Y. Wang} et al., Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 11, 3599--3605 (2011; Zbl 1220.34006)
Full Text: DOI

References:

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