On coincidence point and fixed point theorems for a general class of multivalued mappings in incomplete metric spaces with an application

Abstract

In this paper, we prove existence of fixed and coincidence points
for a general class of multivalued mappings satisfying a new
generalized contractive condition in incomplete metric spaces which
generalize a number of published results in the last decades. In
addition, this article not only brings a new approaches on the
subject and but also involves several non-trivial examples which
demonstrate the significance of the motivation. Finally, the
obtained results of this paper provide a result on the convergence
of successive approximations for certain operators (not necessarily
linear) on a norm space (not necessarily a Banach space). In
particular, we conclude that the renowned Kelisky-Rivlin theorem
works on iterates of the Bernstein operators on an incomplete
subspace of $C[0,1]$.