Applications of the fixed point theorem for mappings contracting perimeters of triangles: ODEs, PDEs, integer and difference equations

Abstract

Mappings contracting perimeters of triangles are Banach contractions whenever every point of the underlying metric space is an accumulation point. Consequently, applying the perimeter-contraction theorem on a complete metric space without isolated points yields an existence result already covered by Banach's fixed point theorem. This observation applies, in particular, to operators associated with differential and integral equations acting on standard nontrivial function spaces, such as spaces of continuous functions $C[a,b]$, Lebesgue spaces $L_p$, and Sobolev spaces $W^{k,p}$, equipped with their usual norm metrics. Thus, applications extending beyond Banach's theorem for the same operator and metric require a complete invariant domain containing some number of isolated points. Such a domain may be a closed subset of a standard function space, despite the absence of isolated points in the ambient space. The main goal of this paper is to illustrate applications of the fixed point theorem for mappings contracting perimeters of triangles in settings where the associated operators are not Banach contractions. The differential-equation examples admit interpretations as stylized models combining continuous evolution, nonlocal measurement, and quantized feedback. The integer and difference-equation examples illustrate the effects of rounding on equilibria and periodic responses.