WEAK CONTRACTION CONDITION FOR FAINTLY COMPATIBLE MAPPINGS INVOLVING CUBIC TERMS OF METRIC FUNCTIONS

In this paper, we obtain a generalized common fixed point theorem for four mappings using the conditions of non-compatibility and faint compatibility satisfying a generalized ∅ −weak contraction condition that involves cubic terms of d(x, y). Also, we provide an example in support of our result.


INTRODUCTION
Banach fixed point theorem is the basic tool to study fixed point theory which ensure the existence and uniqueness of a fixed point under appropriate conditions. This theorem provides a technique for solving a variety of applied problems in mathematical sciences and engineering.
In 1997, Alber and Gueree-Delabriere [2] introduced the concept of weak contraction and Rhoades [3]  In 2013, Murthy and Prasad [4] introduced a new type of inequality having cubic terms of ( , ) that extended and generalized the results of Alber and Gueree-Delabriere [2] and others cited in the literature of fixed point theory.
In 2018, Jain et al. [5] introduced a new type of inequality having cubic terms of ( , ) that extended and generalized the results of Murthy et al. [4] and others cited in the literature of fixed point theory for two pairs of compatible mappings.
In this paper, we extend and generalize the result of Jain et al. [5] for four mappings using the conditions of non-compatibility and faint compatibility satisfying a generalized ∅ −weak contraction condition that involves cubic terms of ( , ).

PRELIMINARIES
In this section, we give some basic definitions and results that are useful for proving our main results.
The notion of commutativity of mappings in fixed point theory was first used by Jungck [6] to obtain a generalization of Banach's fixed point theorem for a pair of mappings. This result was further generalized, extended and unified by using various types of minimal commutative mappings.

Definition 2.1[6]
The pair ( , ) of a metric space ( , ) are said to be commuting if = for all in .
The first ever attempt to relax the commutativity of mappings to weak commutative was initiated by Sessa [7] as follows: In 1996, Jungck and Rhoades [9] introduced the notion of weakly compatible mappings which is more general than that of compatibility. In 2010, Pant et al. [11] redefined the concept of occasionally weakly compatible mappings by introducing conditional commutativity.
Definition 2.8 [11] The pair ( , ) of a metric space ( , ) are said to be conditionally commuting if the pair commutes on a non-empty subset of the set of coincidence points whenever the set of coincidences is non empty.
Again, Pant et al. [12] gave the concept of conditional compatibility which is independent of compatibility condition and proved that in case of existence of unique common fixed point/ coincidence point; conditional compatibility cannot be reduced to the compatibility condition.
Further, they also proved that conditional compatibility need not imply Commutativity at the coincidence points.

Remark 2.11
Compatibility, weakly compatible, occasionally weakly compatible implies faint compatibility, but converse is not true in general.

Remark 2.12
Faint compatibility and non-compatibility are independent concepts.
In one of the interesting paper, Jungck [19] established a common fixed point theorem for four mappings in a complete metric space. Now, we prove our main result for the existence of common fixed point for four mappings in a non-complete metric space using the concept of faintly compatible mappings which is analogous to the result of Jungck [19].

MAIN RESULTS
In this section, we extend and generalize the result of Jain et al. [5] for four mappings using the conditions of non-compatibility and faint compatibility satisfying a generalized ∅ −weak contraction condition that involves cubic terms of ( , ). Using the condition (iii), we get  Hence, = = = = . On putting = = in (iii) and on simplification, we get ( , ) = 0 ⇒ = .

WEAK CONTRACTION CONDITION FOR FAINTLY COMPATIBLE MAPPINGS
Now, we give an example in support of our result. Taking = and = in Theorem 3.1, we obtain the following corollary. Then and have a unique common fixed point in .
Taking = = (Identity map) in Theorem 3.1, we obtain the following result as corollary: