<p>
The first result on fixed points of mappings was obtained in 1911. At that
time, L. Brouwer proved that: Every continuous mapping from a compact convex
set in a finite-dimensional space into itself has at least one fixed point. In 1922, S.
Banach introduced a class of contractive mappings in metric spaces and proved the
famous contraction mapping principle: Each contractive mapping from a complete
metric space (X, d) into itself has a unique fixed point. The birth of the Banach
contraction mapping principle and its application to study the existence of solutions
of differential equations marks a new development of the study of fixed point theory.
After that, many mathematicians have studied to extend the Banach contraction
mapping principle for classes of maps and different spaces. Expanding only contractive
mappings, till 1977, was summarized and compared with 25 typical formats by B.E.Rhoades.
</p>
