The epsilon Nielsen fixed point and coincidence theory

Topological fixed point(coincidence) theory deals with the estimation of the number of fixed points(coincidences) of maps.In this book, we introduce recent developments in the epsilon Nielsen fixed point and coincidence theory of map.There is well known the epsilon Nielsen theory expressing the epsilon Nielsen number is a bigger than the usual Nielsen number. We generalize these ideas to the setting of coincidence two maps. We first extend the algebraic method of computing the epsilon Nielsen coincidence number. Then, we prove the problem of minimizing the number of the epsilon coincidence points of two mappings by deforming them through the epsilon homotopies and extend the epsilon Nielsen fixed point theory of single-valued maps to n-valued maps. Finally we present a counterexample that exhibits the negative answer of the questions formulated by Fel'shtyn and Troitsky and discuss the generalization of usual the commutator subgroup, especially for some relations among Nielsen fixed point theory and the twisted commutator subgroup in Appendix. Some calculations and examples are given.