Nielsen coincidence theory for $(n,m)$-valued pairs

We define a Nielsen coincidence number for pairs of maps $(f,g)$ where $f$ is $n$-valued and $g$ is $m$-valued. The classical definition of the Nielsen number does not work in this setting, but we propose a theory where coincidence classes are allowed to overlap. We invoke a notion of multiplicity of a coincidence point, which indicates to how many classes it can belong, and we show that our Nielsen number is a homotopy-invariant lower bound for the number of coincidence points with multiplicity. We show a computational result in terms of single-valued lift factors, and using this, a Wecken property for self-maps of the circle.