Abstract:
The theory of fixed point is one of the most incredible asset of present day Mathematical
Analysis. Theorem concerning the presence and properties of fixed point
are known as fixed point theorems. Fixed point theory is the beautiful mixture of
Analysis, Topology and Geometry which has many applications in various fields. In
19th century the study of fixed point theory was initiated by Poincare and in the
20th century this area developed by many mathematicians like Brouwer, Schauder,
Kakutani, Banach, Kannan, Tarski and others.
In 1922 the concept of Banach Space was introduced by Stefen Banach and
introduced a Fixed Point Theorem for contraction mapping. There are numerous
generaliztaion of Banach contraction principle to unique fixed point of the mapping.
Sehgal, Kannan, Caristi and Husain worked for some generalization of contraction
mappings and proved number of the result for contraction mapping.
As the fundamental properties of contraction mapping do not extend to nonexpansive
and mean nonexpansive mapping. Now, the study of nonexpansive and
mean nonexpansive mapping is the main feature in recent development of fixed
points. Contractive mappping, isometries and orthogonal projections are all nonexpansive
mapping. In most of the cases we have studied fixed points for different
types of mapppings. The objective of following work is to find the fixed point for
mean-nonexpansive semigroup and fundamentally nonexpansive mapping.
