Epsilon Nielsen coincidence theory (Q5919146)

Nielsen fixed point theory deals with the estimation of the number of fixed points of self-maps up to homotopy. Clearly, two homotopic self-maps on a metric space may have a very large distance, and hence this theory does not look very suitable for application in analysis, where relation among the fixed point sets of self-maps in small distances is interesting. A so-called epsilon Nielsen fixed point theory was developed by \textit{R. F. Brown} [Fixed Point Theory Appl. 2006, No. 2, Article 29470, 10 p. (2006; Zbl 1093.55003)], concerning the estimation of the number of fixed points of self-maps up to a small perturbation. In other words, the relation between fixed point sets of two self-maps is addressed if the two self-maps have a small epsilon distance. In some nice metric spaces, such as Riemannian manifolds, two self-maps with small distance must be homotopic. Thus, the epsilon Nielsen fixed point theory is a kind of restricted version of the classical one, and also contains some aspects for the purpose of application to analysis. The author of the paper under review generalizes the idea of epsilon Nielsen fixed point theory to coincidence theory. Two maps \(f: X\to Y\) and \(f': X\to Y\) are said to be \(\varepsilon\)-homotopic if \(f\) and \(f'\) are connected by a homotopy \(\{h_t\}\) such that \(d(h_t, h_s)<\varepsilon\) for all \(t,s \in I\). Two pairs of maps \((f, g): X\to Y\) and \((f', g'): X\to Y\) are said to be \(\varepsilon\)-homotopic if \(f\) and \(f'\) are \(\varepsilon_1\)-homotopic and \(g\) and \(g'\) are \(\varepsilon_2\)-homotopic, where \(\varepsilon_1+ \varepsilon_2<\varepsilon\). The key concept in this paper is a lower bound \(N^\varepsilon(f, g)\) for the number of coincidence points of all pairs of maps which are \(\varepsilon\)-homotopic to \((f,g)\). It should be noticed that \(N^\varepsilon(f, g)\) is not invariant even under \(\varepsilon\)-homotopies. Indeed, \(\varepsilon\)-homotopy is not an equivalence relation among all pairs of maps. Because of this, \(N^\varepsilon(f, g)\) is not a parallel notation of the classical coincidence Nielsen number \(N(f,g)\), and more technical arguments are involved. Moreover, in the case of root theory, the author proves the minimum theorem, i.e. \(N^\varepsilon(f, g)\) can be realized if \(g\) is a constant map.