Fixed point theory is very simple, but is based on fundamentals in Mathematics. For any continuous function $$f:Xrightarrow X$$ a fixed point of $$f$$
is a point $$xin X$$ satisfying the identity $$f(x)=x.$$ Two fundamental theorems concerning fixed points are Banach Theorem and Brouwer Theorem. Banach theorem states that if $$X$$
is a complete metric space and $$f$$
is a contraction then $$f$$
has a unique fixed point. In Brouwer theorem, $$X$$
must be the closed unit ball in a Euclidean space. Then any $$f$$
has a fixed point. But in this case, the set of fixed points is not necessarily a one-point set.
In Banach theorem, a metric on $$X$$
is used in the crucial assumption that $$f$$
is a contraction. The unit ball in a Euclidean space is also a metric space and the metric topology determines the continuity of continuous functions, however the essence of Brouwer theorem is a topological property of the unit ball, namely the unit ball is compact and contractible. Banach theorem and Brouwer theorem tell us a difference between two major branches of fixed point theory, metric space fixed point theory and topological fixed point theory. It is impossible to distinguish two fixed point theories in an exact way, and it is not easy to determine a certain topics belong to which branch. In general, the fixed point theory is regarded as a branch of topology. But due to deep influence on topics related to nonlinear analysis or dynamic systems, many parts of the fixed point theory can be considered as a branch of analysis.