Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 63, pp. 1-14.
Title: Existence of positive solutions for singular fractional
differential equations with integral boundary conditions
Authors: Jingfu Jin (Univ. of Shanghai for Science and Tech., China)
Xiping Liu (Univ. of Shanghai for Science and Tech., China)
Mei Jia (Univ. of Shanghai for Science and Tech., China)
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