Handbook of Topological Fixed Point Theory

Fixed point theory concerns itself with a very simple, and basic, mathematical setting. For a functionf that has a setX as bothdomain and range, a ?xed point off isa pointx ofX for whichf(x)=x. Two fundamental theorems concerning ?xed points are those of Banach and of Brouwer. In Banach’s theorem, X is a complete metric space with metricd andf:X?X is required to be a contraction, that is, there must existL< 1 such thatd(f(x),f(y))?Ld(x,y) for allx,y?X. Theconclusion is thatf has a ?xed point, in fact exactly one of them. Brouwer’stheorem requiresX to betheclosed unit ball in a Euclidean space and f:X?X to be a map, that is, a continuous function. Again we can conclude that f has a ?xed point. But in this case the set of?xed points need not be a single point, in fact every closed nonempty subset of the unit ball is the ?xed point set for some map. ThemetriconX in Banach’stheorem is used in the crucialhypothesis about the function, that it is a contraction. The unit ball in Euclidean space is also metric, and the metric topology determines the continuity of the function, but the focus of Brouwer’s theorem is on topological characteristics of the unit ball, in particular that it is a contractible ?nite polyhedron. The theorems of Banach and Brouwer illustrate the di?erence between the two principal branches of ?xed point theory: metric ?xed point theory and topological ?xed point theory.

現在にいたるまで、位相幾何的不動点定理に関する書籍はいずれもこの理論の一部に紙面を割いていたに過ぎなかった.本書は、位相幾何的不動点定理における最新のトレンドをすべて網羅した初の参考文献である.不動点定理、とりわけ非線形解析、微分方程式、および動的システムに興味がある大学院生や研究者にとって本書は有用であろう.その内容はまた、位相幾何的力学を探求している理論物理学者や、数理経済学者、ならびに人口動態の専門家の興味も刺激することであろう.
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