In discrete mathematics, a discrete fixed-point is a fixed-point for functions defined on finite sets, typically subsets of the integer grid
Discrete fixed-point theorems were developed by Iimura,[1] Murota and Tamura,[2] Chen and Deng[3] and others.
Such conditions imply that the function does not change too drastically when moving between neighboring points of the integer grid.
There are various direction-preservation conditions, depending on whether neighboring points are considered points of a hypercube (HGDP), of a simplex (SGDP) etc.
Continuous fixed-point theorems often require a convex set.
, where the domain X is a nonempty subset of the Euclidean space
ch(X) denotes the convex hull of X. Iimura-Murota-Tamura theorem:[2] If X is a finite integrally-convex subset of
is a hypercubic direction-preserving (HDP) function, then f has a fixed-point.
Herings-Laan-Talman-Yang fixed-point theorem:[6] Let X be a non-empty convex compact subset of
Let f: X → X be a locally gross direction preserving (LGDP) function: at any point x that is not a fixed point of f, the direction of
The theorem is originally stated for polytopes, but Philippe Bich extends it to convex compact sets.
An LGDP function may even be neither upper nor lower semi-continuous.
Moreover, there is a constructive algorithm for approximating this fixed point.