The purpose of this paper is to prove end-point theorems for multivalued mappings satisfying comparatively a more general contractive condition in ordered complete metric spaces. Afterwards, we extend the results of previous sections and prove common end-point results for a pair of
Fixed-point theory for multivalued mappings was originally initiated by Von Neumann in the study of game theory. Fixed-point theorem for multivalued mappings is quite useful in control theory and has been frequently used in solving the problem of economics and game theory.
The theory of multivalued nonexpansive mappings is comparatively complicated as compare to the corresponding theory of single-valued nonexpansive mappings. It is therefore natural to expect that the theory of noncontinuous nonself-multivalued mappings would be much more complicated.
The study of fixed-points for multivalued contraction mappings was equally an active topic as single-valued mappings. The development of geometric fixed-point theory for multivalued was initiated with the work of Nadler Jr. [
Let
Since then, this discipline has been further developed, and many profound concepts and results have been established; for example, the work of Border [
Let
For all
A point
Let
Let
Let
A function
On the other hand, fixed-point theory has developed rapidly in metric spaces endowed with a partial ordering. The first result in this direction was given by Ran and Reurings [
The results of this paper are divided in three sections. In the first section we establish the existence of end-points for a multivalued mapping under a more general contractive condition in partially ordered metric spaces. The consequences of the main theorem are also given. The second section is devoted for common end-point results for a pair of weakly isotone increasing multivalued mappings. In the third section, we present common end-point results for a pair of weakly isotone increasing multivalued mappings satisfying weakly contractive condition.
In this section, we prove end-point theorems for a multivalued mapping in ordered complete metric space.
Let there exists for
for all comparable
By the assumption (i), there exists
Using the monotone property of
Then from (
Therefore,
Assuming that
Hence
Then by the monotone property of
Taking
Let there exists for
for all comparable
The following corollary is a special case of Theorem
Let there exists for
for all comparable
In the following theorem we replace condition (
Let there exists for
for all comparable
If we assume
Then, the continuity of
In this section, we prove common end-point theorems for a pair of
To complete the result, we need notion of
Let
Note that, in particular, for single-valued mappings
Let
Define a sequence
Since
Continuing this process we construct a monotone increasing sequence
Suppose that
Then from (
Assuming that
Hence
Then by the monotone property of
Putting
Let
In Theorem
Let
If we assume
In this section, we prove common end-point theorems for a pair of weakly isotone increasing multivalued mappings under weakly contractive condition.
To complete the result, we need notion of weakly contractive condition given by Rhoades [
Let
Let
Also suppose that
Define a sequence
Continuing this process, we conclude that
Suppose that
Similar corollaries can be derived from Theorem
The present version of the paper owes much to the precise and kind remarks of the learned referees.