Abstract
The Banach contraction theorem is a fundamental result in metric fixed point theory. It has
numerous applications in nonlinear analysis. The relation theoretic version of this theorem
and its extensions and generalizations have received great attention in recent years. In this
thesis, we present some new fixed point results for single and multi-valued mappings in
relational metric spaces with a couple of applications.
Nonexpansive mappings can be considered as contraction mappings’ inherent generalization.
A nonexpansive mapping may not possess a fixed point in a complete metric space.
Nevertheless, it is feasible to ensure the existence of fixed points for nonexpansive mappings
by equipping the space with a sufficiently intricate geometric structure, such as uniform convexity
and normal structure with reflexivity. In this thesis, we present an existence result
for a general class of nonexpansive mappings. We also obtain some endpoint existence and
convergence results of these mappings in Banach spaces.