Positivity 12 (2008), 711–724
2008 Birkh¨auser Verlag Basel/Switzerland
1385-1292/040711-14, published online October 4, 2008
Nonnegative solution for singular nonlinear
fractional differential equation with coefﬁcient
that changes sign
Abstract. In this paper, we consider the existence of nonnegative solutions of
initial value problem for singular nonlinear fractional differential equation
u),u(0) = 0, 0 <t≤ 1, 0 <α<s<1
are the standard Riemann-Liouville fractional derivatives,
f :[0, ∞) × (−∞, +∞) → [0, +∞), a :(0, 1] → R, may be change sign,
a :[0, 1] → R,0≤ r<s− α,andλ>0 is a parameter. Our analysis relies
on the Schauder ﬁxed point theorem.
Mathematics Subject Classiﬁcation (2000). 26A33; 34A12.
Keywords. Riemann-Liouville fractional derivatives and integrals, Singular
problem, Nonnegative solutions, Schauder ﬁxed point theorem.
Many papers and books on fractional calculus, fractional differential equations
have appeared recently (see [1–10], etc). About the development of the fractional
differential equations, we refer to the survey paper by Kilbas and Trujillo in .
In , by using the nonlinear alternative of Leray-Schauder type, Bai investigated
the existence of positive solution for the nonlinear fractional equation
u(t)=λa(t)f(u), 0 <t≤ 1
with initial value
u(0) = 0
where 0 <s<1, D
is the standard Riemann-Liouville fractional derivative, λ>0
is a parameter, and a(t),f(u) satisfy the following conditions;
(H1) f :[0, +∞) → [0, +∞) is continuous and f(0) > 0;