Chapter 1
Singularity Subtraction for Nonlinear
Weakly Singular Integral Equations
of the Second Kind
Mario Ahues, Filomena D. d’Almeida, Rosário Fernandes,
and Paulo B. Vasconcelos
1.1 Introduction
The reference Banach space is the set X := C
0
([a,b], R) with the supremum norm.
We consider the operator K defined by
K(x)(s):=
b
a
g(|s − t |)N(s,t,x(t))dt, x ∈ X, s ∈[a,b],
where g is a weakly singular function in the following sense:
lim
s →0
+
g(s) = +∞, and g ∈ C
0
(]0,b − a], R
+
) ∩ L
1
([0,b − a], R
+
).
To be consistent with [An81] and [AhEtAl01], we assume that g is a decreasing
function on ]0,b − a].
The factor N , containing the values x(t) ∈ R of the functional variable x ∈ X
for t ∈[a,b], is a continuous function
N :[a,b]×[a,b]× R → R, (s,t,u) → N(s,t,u),
with continuous partial derivative with respect to the third variable.
M. Ahues ()
Université de Lyon, Lyon, France
e-mail: mario.ahues@univ-st-etienne.fr
F. D. d’Almeida · P. B. Vasconcelos
Universidade do Porto, Porto, Portugal
e-mail: falmeida@fe.up.pt; pjv@fep.up.pt
R. Fernandes
Universidade do Minho, Braga, Portugal
e-mail: rosario@math.uminho.pt
© Springer Nature Switzerland AG 2019
C. Constanda, P. Harris (eds.), Integral Methods in Science and Engineering,
https://doi.org/10.1007/978-3-030-16077-7_1
1