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

• Main Authors: Jingfu Jin, Xiping Liu, Mei Jia
• Format: Article
• Language: English
• Published: Texas State University 2012-04-01
• Series: Electronic Journal of Differential Equations
• Subjects: Caputo derivative; fractional differential equations; positive solutions; integral boundary conditions; singular differential equation
• Online Access: http://ejde.math.txstate.edu/Volumes/2012/63/abstr.html

This article shows the existence of a positive solution for the singular fractional differential equation with integral boundary condition $$displaylines{ {}^C!D^p u(t)=lambda h(t)f(t, u(t)), quad tin(0, 1), cr u(0)-au(1)=int^1_0g_0(s)u(s),ds, cr u'(0)-b,{}^C!D^qu(1)=int^1_0g_1(s)u(s),ds, cr u''(0)=u'''(0)=dots =u^{(n-1)}(0)=0, }$$ 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.