Positivity 12 (2008), 725–732
2008 Birkh¨auser Verlag Basel/Switzerland
1385-1292/040725-8, published online May 27, 2008
Existence and uniqueness of solutions
for singular integral equation
Zhongwei Cao, Daqing Jiang, Chengjun Yuan and Donal O’Regan
Abstract. Using the mixed monotone method we establish existence and
uniqueness results for a singular integral equation. The theorem obtained
is very general and complements previous known results.
Mathematics Subject Classiﬁcation (2000). 34B15; 34B16.
Keywords. Mixed monotone operator, integral equation, singular, existence,
In recent years integral equation have been studied extensively in the literature
(see [1–9] and references therein). Most of the results concern single or multiple
positive solutions. However very few uniqueness results are available.
Recently some results have appeared on the uniqueness of solutions for
singular higher-order boundary value problems (see [10,11] and ). In this paper
we present a uniqueness result for the singular integral equation
K(t, s)f(s, x(s))ds, 0 ≤ t ≤ 1,λ>0. (1.1)
where f(t, x) ∈ C((0, 1)×(0, +∞), (0, +∞)),and f may be singular at t = 0 and/or
t = 1, and also may be singular at x =0.
Let P be a normal cone of a Banach space E,ande ∈ P with e≤1,e= θ.
The work was supported by the National Natural Science Foundation of China (No.10571021 and
No.10701020) and Key Laboratory for Applied Statistics of MOE(KLAS) and Subject Foundation
of Harbin University (No. HXK200714).