Electron. J. Diff. Equ., Vol. 2012 (2012), No. 63, pp. 1-14.

Existence of positive solutions for singular fractional differential equations with integral boundary conditions

Jingfu Jin, Xiping Liu, Mei Jia

Abstract:
This article shows the existence of a positive solution for the singular fractional differential equation with integral boundary condition
$$\displaylines{ {}^C\!D^p u(t)=\lambda h(t)f(t, u(t)), \quad t\in(0, 1), \cr u(0)-au(1)=\int^1_0g_0(s)u(s)\,ds, \cr u'(0)-b\,{}^C\!D^qu(1)=\int^1_0g_1(s)u(s)\,ds, \cr u''(0)=u'''(0)=\dots =u^{(n-1)}(0)=0, }$$
where $\lambda $ is a parameter and the nonlinear term is allowed to be singular at t=0, 1 and u=0. We obtain an explicit interval for $\lambda $ such that for any $\lambda $ in this interval, existence of at least one positive solution is guaranteed. Our approach is by a fixed point theory in cones combined with linear operator theory.

Submitted January 17, 2012. Published April 19, 2012.
Math Subject Classifications: 34B16, 34B18, 26A33
Key Words: Caputo derivative; fractional differential equations; positive solutions; integral boundary conditions; singular differential equation

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Jingfu Jin
College of Science
University of Shanghai for Science and Technology
Shanghai 200093, China
email: jinjingfu2005@126.com
Xiping Liu
College of Science
University of Shanghai for Science and Technology
Shanghai 200093, China
email: xipingliu@163.com
Mei Jia
College of Science
University of Shanghai for Science and Technology
Shanghai 200093, China
email: jiamei-usst@163.com

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