Abstract
Nonexpansive mappings are the natural generalization of contraction mappings. A nonexpansive
mapping need not have a fixed point in complete metric space. However, it is
possible to guarantee the existence of fixed points for nonexpansive mappings by endowing
the space with a sufficiently rich geometric structure like uniform convexity and normal
structure with reflexivity. These mappings are important due to their connection with the
monotonicity methods and appear in applications for the initial value, variational inequality,
optimization, equilibrium, and many other problems in nonlinear analysis. In this thesis, we
study some fixed point results for some broader classes of nonexpansive mappings. Firstly,
we present some fixed point results for a general class of nonexpansive type mappings and
associated Krasnosel’ski˘ı mappings in Banach spaces. Then, we consider Halpern and Mann
type iteration procedures and present some convergence results for one parameter nonexpansive
type semigroups. We also study viscosity approximation methods for finding a common
point of the set of solutions of a variational inequality problem and the set of fixed points of
a multi-valued quasi nonexpansive mapping in a Banach space. Further, we present a new
iterative algorithm and obtain some strong convergence results to approximate a common
point of a monotone bounded mapping and the set of fixed points of a generalized nonexpansive
mapping in Banach spaces, which is a solution of a variational inequality problem.
Finally, we present some results on stability of fixed points for quasi contraction mapping.
We also present a number of applications of our results in initial value problem, split feasibility
problem, convex minimization problem, generalized mixed equilibrium problem etc.
Keywords: Nonexpansive mapping, condition (E), metric projection, uniformly convex
space, Opial property, nonexpansive semigroup, common fixed points, split feasibility problem,
variational inequality problem, viscosity approximation, quasi-nonexpansive mapping,
inverse strongly monotone mapping, common point, stability.
AMS Classification (2010): 47H09, 47H10, 54H25.