CERTAIN RESULTS IN b-METRIC SPACE USING SUBCOMPATIBLE, FAINTLY COMPATIBLE MAPPINGS

In this paper we prove a common fixed point theorem in b-metric space using faintly compatible, subcompatible, occasionally weakly compatible and reciprocally continuous mappings. Furthermore appropriate examples are provided to support our conclusions.


INTRODUCTION
The fixed point theory plays very important role in analysis and it has got wide applications in many fields of mathematics. Many fixed point theorems have been developed on various areas in 8383 CERTAIN RESULTS IN b-METRIC SPCACE the current context. Introducing the concept of compatible mappings, Junck [1] generated many results in metric space. Later on many theorems were proved using weaker form compatible mappings namely weakly compatible mappings and occasionally weakly compatible mappings.
In [2], [3], [4] and [5] many results can be witnessed using the above mentioned conditions. In the recent past b-metric space emerged as one of the generalizations of metric space. Czerwik [6] introduced the concept of b-metric space. Thereafter some more fixed point theorems were extracted in b-metric space, such as [7], [8] and [9] using a variety discovered a variety of constraints. J.R. Roshan et al. [10] used compatible and continuous mappings to prove a common unique fixed point theorem in b-metric space. We improve their result in this study by utilizing certain weaker conditions, such as faintly compatible mappings, occasionally weakly compatible, subcompatible and reciprocally continuous mappings. We now discuss some examples to find the relation among the above definitions.      Therefore the pair (T,S) is sub compatible.

Example 2.4: Let
The following theorem was proved in [10].

Theorem
Let the self maps f,g,S and T be defined on a b-metric space (X,d) which is complete with the given conditions:

Theorem:
Let (X,d) be b-metric space which is complete and four self mappings f,g,S, and T are satisfying is occasionally weakly compatible (b4) the pair ( ) S f , is sub compatible and reciprocally continuous.
Then the four maps f, g, S, and T, share a unique common fixed point. Proof: for some j, then the above inequality gives Therefore the above gives ( ) ( ).
, , , , Using (1) and (2) we get From equation (3),we get take the sup limit as  → j on both the sides and by Lemma(2.6),which gives We get from above (4) and (7). .
Therefore  is the required common fixed point.
Uniqueness can be easily obtained. Now we discuss a suitable example to support our Theorem.

Example:
Introduce the self maps as so that the condition (b1) is satisfied.
are the coincidence points for the maps g ,T.
As a result, while g,T are OWC mappings, but they are not weakly compatible.
Take a sequence as p Thus the pair (f,S) and (g,T) satisfy all the conditions of the Theorem 3.1.
T S g f therefore 2 is the unique common fixed point for f,g,S and T.
We now prove another generalization Theorem of 2.5.

Theorem:
Let (X,d) be a complete b-metric space and the self mappings f,g,S, and T meet the conditions (b1) ( ) ( ) Now we assert .
Therefore  is the required common fixed point.
Uniqueness follows easily. Now we justify our Theorem with the following example.

Example:
The self maps defined as below Therefore the pair (g,T) is occasionally weakly compatible but not weakly compatible mappings.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.