COMMON FIXED POINT THEOREMS FOR WEAKLY COMPATIBLE MAPPINGS USING GENERALIZED ψ-WEAK CONTRACTION

In this paper, first we introduce generalized ψ −weak contraction condition that involves cubic and quadratic terms of distance function d(x, y). Secondly; we discuss common fixed point theorems for weakly compatible and weakly compatible mappings along with property (E.A.) and common limit range property. At the end, we provide an example and an application of our main theorem satisfying integral type contraction condition.


INTRODUCTION
Fixed point theory for contraction mappings and related mappings has played a fundamental role in many aspects of nonlinear functional analysis for many years. The theory has generally involved an interconnecting of geometrical and topological arguments in a Banach space setting. Fixed point theory results indicate that under certain conditions self-mapping on a set admits a fixed point. Among all the results Banach contraction principle is the most celebrated due to its simplicity and ease of application in major areas of mathematics.
Banach fixed point theorem is the basic tool to study fixed point theory and shows 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.
Most of the problems of applied mathematics reduce to inequality which in turn their solutions give rise to the fixed points of certain mappings. It was the new era of the fixed point theory literature when the notion of commutativity mappings was used by Jungck [5] to obtain a generalization of Banach's fixed point theorem for a pair of mappings. The first ever attempt to relax the commutativity to weak commutativity was initiated by Sessa [13]. Further, in 1986 Jungck [6] introduced more generalized commutativity, so called compatibility. One can notice that the notion of weak commutativity is a point property, while the notion of compatibility is an iterate of sequence. In 1996, Jungck [8] introduced the notion of weakly compatible mappings and showed that compatible maps are weakly compatible, but not converse may not be true.
In 2002, Aamri and El-Moutawakil [1] introduced the concept of property (E.A.) for the self-mappings, which also includes the notion of the class of non-compatible mappings.
Sintunavarat and Kumam [14] further generalized the notion of property (E.A.) by introducing the notion of common limit in the range property (CLR property). The significance of the CLR property and property (E.A.) have the following properties:(i)both the properties relaxes the continuity hypothesis of all the involved mappings and also relaxes the containment condition of the range subspace of the mapping into the range subspaces of the other mappings, which is generally required for constructing the sequences of joint iterates in fixed point results.(ii) the property (E.A.) replaces the completeness requirement of the space (or the range subspaces of the mappings involved) by the condition of the range subspace of the mapping to be closed, whereas (CLR) property ensures that the requirement of the completeness of the space (or range subspaces of any of the mappings involved) can be relaxed entirely and need not to be replaced by any other condition.
If k≤ 1 then T is said to be non-expansive, if 0 < < 1T is said to be a contraction.
Banach fixed point theorem states that every contraction mapping on a complete metric space has a unique fixed point.
In 1969, Boyd and Wong [3] replaced the constant in Banach contraction principle by a control function as follows: Let ( , ) be a complete metric space and ∶ [0 , ∞) → [0, ∞) be upper semi continuous from the right such that 0 ≤ ( ) < for all > 0. In connection with control function : R + → R + different authors have considered some of the following properties: converges for all t>0, ψ n is the nth iterate is lower semi continuous Here we note that (i) and (ii) implies (iii) ; (ii) and (iv) implies (iii)

(i) and (v) implies (ii)
A function satisfying (i) and (v) that is is non decreasing and lim n → ∞ ψ n (t) = 0 for all t ≥ 0 is called as a comparison function.
Several fixed point theorems and common fixed point theorems have been unified considering a general condition by an implicit function.
The notion of compatibility is an iterate of sequence. In 1996, Jungck [8] introduced the notion of weakly compatible mappings and showed that compatible maps are weakly compatible, but converse may not be true.

Definition 2.2[8]
Two self-mappings f and g on a metric space ( , ) are called weakly compatible if they commute at their coincidence point i.e., if = for some ∈ then = .

Definition 2.3[1]
Let f and g be two self-mappings of a metric space ( , ). We say that f and g satisfy (E.A) property if there exists a sequence { } in X such that = = , for some in .
Pathak et.al [11] has shown that weak compatibility and (E.A.) property are independent to each other.
In 2011, Sintunavarat and Kumam [14] coined the idea of common limit range property (called CLR) which relaxes the requirement of completeness to compute fixed point.

Definition 2.4[14]
Two self mappings and on a metric space ( , )are are said to satisfy the common limit in the range of property denoted as CLRg property if = = , for some in .
Now we introduce the generalized −weak contraction for a pair of mappings in the following way: Let , , and are self mappings on a metric space ( , ) satisfying the following conditions:
First we prove that { 2 } is non increasing sequence and converges to zero.
Case II. If n is odd, then in a similar way, one can obtain 2 +1 ≤ 2 .
It follows that the sequence { 2 } is decreasing.
On putting = ( ) and = ( ) in (C2), we get Letting → ∞, and using property of , we have which is a contradiction.
On setting = and = 2 +1 in (C2) and proceeding limit, we have } This implies that = and hence = = . Therefore, is a coincidence point of and . Since = ∈ ⊂ there exist ∈ such that = .
Next we claim that = . Now putting = 2 and = in (C2), we have Next we show that = . For this put = and = 2 +1 in (C2) For this put = and = 2 +1 in (C2) and proceeding limit as n → ∞, 3  Next we claim that = . Now put = 2 and = in (C2) This completes the proof.
If we put = in theorem 3.1, then we obtain the following results  Now suppose that BX is closed subset of X, then there exists ∈ such that = .
Subsequently, we have First we claim that = .
Now putting = and = in (C2) PAWAN KUMAR, NEERU YADAV, BALBIR SINGH We shall continue our discussion to find fixed point for the mapping satisfying weakly compatible mappings along with common limit range property. Now suppose that BX is a closed subset of X, there exists a point ∈ such that = .
Now we show that = . Putting = and = in (C2), we have

4.APPLICATION
In 2002 Branciari [1] obtained a fixed point theorem for a single mapping satisfying an analogue of a Banach contraction principle for integral type inequality. , where ∶ + → + is a "Lebesgue-integrable function" which is summable, nonnegative, and such that, for each ∈ > 0, ∫ φ(t)dt > 0.
∈ 0 Then has a unique fixed point ∈ such that, for each ∈ , lim →∞ = .
Now we prove the following theorem as an application of theorem 3.1. Remark4.1.Every contractive condition of integral type automatically includes a corresponding contractive condition not involving integrals, by setting φ (t) = 1.