Results Math 72 (2017), 747–763
2017 Springer International Publishing
published online February 11, 2017
Results in Mathematics
Existence of Nonnegative Solutions for a
Fractional Integro-Diﬀerential Equation
Johnny Henderson and Rodica Luca
Abstract. We investigate the existence of nonnegative solutions for a frac-
tional integro-diﬀerential equation subject to multi-point boundary con-
ditions, by using the Banach contraction mapping principle and the Kras-
nosel’skii ﬁxed point theorem for the sum of two operators.
Mathematics Subject Classiﬁcation. 34A08, 34B18, 45G15.
Keywords. Fractional integro-diﬀerential equation, multi-point boundary
conditions, nonnegative solutions.
We consider the nonlinear fractional integro-diﬀerential equation
u(t)+f(t, u(t),Tu(t),Su(t)) = 0,t∈ (0, 1), (E)
with the multi-point boundary conditions
u(0) = u
(0) = ··· = u
(0) = 0,D
where α ∈ R, α ∈ (n − 1,n], n ∈ N, n ≥ 3, a
∈ R for all i =1,...,m,
(m ∈ N), 0 <ξ
< ··· <ξ
≤ 1, p, q ∈ R, p ∈ [1,n−2], q ∈ [0,p], D
the Riemann–Liouville derivative of order α, Tu(t)=
H(t, s)u(s)ds for all t ∈ [0, 1].
In this paper, we study the existence of nonnegative solutions for problem
(E)–(BC), by using the Banach contraction mapping principle and the Kras-
nosel’skii ﬁxed point theorem for the sum of two operators. By a nonnegative