Positive solutions for nonlinear fractional semipositone differential equation with nonlocal boundary conditions

Volume 9, Issue 6, pp 3992--4002 http://dx.doi.org/10.22436/jnsa.009.06.45

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Authors

Xinan Hao - School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, P. R. China. Lishan Liu - School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, P. R. China. - Department of Mathematics and Statistics, Curtin University, Perth, 6845WA, Australia. Yonghong Wu - Department of Mathematics and Statistics, Curtin University, Perth, 6845WA, Australia.

Abstract

In this paper, we study the existence of positive solutions to the nonlinear fractional order singular and semipositone nonlocal boundary value problem \[ \begin{cases} \mathfrak{D}^\alpha_{0^+}u(t)+f(t,u(t))=0,\,\,\,\,\, 0<t<1,\\ u(0)=u'(0)=...=u^{(n-2)}(0)=0,\,\,\,\,\, u(1)=\mu\int^1_0 u(s)ds. \end{cases} \] by using the Leray-Schauder nonlinear alternative and a fixed-point theorem on cones, where \(0 < \mu < \alpha; 2 \leq n - 1 < \alpha \leq n, \mathfrak{D}^\alpha_{0^+}\) is the standard Riemann-Liouville derivative, and f(t; u) is semipositone and may be singular at u = 0.

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Keywords

• Singular fractional differential equation
• semipositone
• positive solutions
• nonlocal boundary conditions.

MSC

•  34B10
•  34B16
•  26A33

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