Abstract:

This dissertation is the contribution to the Metric fixed point theory, the area
that has recently been rapidly developing. It contains five chapters.
The first chapter gives the proof of one already known lemma. This lemma is
used in the proof of Banach’s theorem for orbital complete metric spaces.
The second chapter contains the proofs of eight theorems, which generalize
some known results from the theory of fixed points in metric spaces (BoydWong’s,
´ Ciri´c
’
s, Pant’
s, and other). Some of these theorems are modifications of the known
ones, while three are completely new.
Three theorems are proven in the third chapter. They generalize the result of
fixed point of mapping defined in compact metric space given by Nemytzki, as well
as one generalization of Edelstein’s theorem. The proofs and stated corollaries of
some theorems are original.
Chapter four discusses bmetric spaces as a generalization of metric spaces.
The generalization of Zamfirescu’s theorem of bmetric spaces is presented as well
as some of its applications. A new result concerning weakly almost contractive
mappings is also determined.
Chapter five contains some new results in cone metric spaces. Two theorems are
presented as the analogue of the same theorems in the setting of standard metric
spaces. A completely new theorem is established which results in the Banach’s
theorem in cone metric spaces whereby the cone does not need to be normal. A
generalization of Fisher’s theorem in cone metric spaces over a regular cone is also
proven.
Almost all results in this dissertation are confirmed by corresponding examples,
which explain how these results differ from the already known results. 