In the paper, we provide the construction of a coincidence degree being a homotopy invariant detecting the existence of solutions of equations or inclusions of the form
Ax
∈
F
(
x
),
x
∈
U
, where
A
:
D
(
A
)
⊸
E
is an
m
-accretive operator in a Banach space
E
,
F
:
K
⊸
E
is a weakly upper semicontinuous set-valued map constrained to an open subset
U
of a closed set
K
⊂
E
. Two different approaches are presented. The theory is applied to show the existence of non-trivial positive solutions of some nonlinear second-order partial differential equations with discontinuities.
This article is part of the theme issue ‘Topological degree and fixed point theories in differential and difference equations’.