Some Common Fixed Point Theorems in Menger PM Spaces

Employing the common property E.A , we prove some common fixed point theorems for weakly compatible mappings via an implicit relation in Menger PM spaces. Some results on similar lines satisfying quasicontraction condition as well as ψ-type contraction condition are also proved in Menger PM spaces. Our results substantially improve the corresponding theorems contained in Branciari, 2002 ; Rhoades, 2003 ; Vijayaraju et al., 2005 and also some others inMenger aswell as metric spaces. Some related results are also derived besides furnishing illustrative examples.


Introduction and Preliminaries
Sometimes, it is found appropriate to assign the average of several measurements as a measure to ascertain the distance between two points. Inspired from this line of thinking, Menger 1, 2 introduced the notion of Probabilistic Metric spaces in short PM spaces as a generalization of metric spaces. In fact, he replaced the distance function d : R × R → R with a distribution function F p,q : R → 0, 1 wherein for any number x, the value F p,q x describes the probability that the distance between p and q is less than x. In fact the study of such spaces received an impetus with the pioneering work of Schweizer and Sklar 3 . The theory of PM spaces is of paramount importance in Probabilistic Functional Analysis especially due to its extensive applications in random differential as well as random integral equations.
Fixed point theory is one of the most fruitful and effective tools in mathematics which has enormous applications within as well as outside mathematics. The theory of fixed points in PM spaces is a part of Probabilistic Analysis which continues to be an active area of mathematical research. By now, several authors have already established numerous fixed 2 Fixed Point Theory and Applications point and common fixed point theorems in PM spaces. For an idea of this kind of the literature, one can consult the results contained in 3-14 . In metric spaces, Jungck 15 introduced the notion of compatible mappings and utilized the same as a tool to improve commutativity conditions in common fixed point theorems. This concept has been frequently employed to prove existence theorems on common fixed points. However, the study of common fixed points of noncompatible mappings is also equally interesting which was initiated by Pant 16 . Recently, Aamri and Moutawakil 17 and Liu et al. 18 respectively, defined the property E.A and the common property E.A and proved some common fixed point theorems in metric spaces. Imdad et al. 19 extended the results of Aamri and Moutawakil 17 to semimetric spaces. Most recently, Kubiaczyk and Sharma 20 defined the property E.A in PM spaces and used it to prove results on common fixed points wherein authors claim to prove their results for strict contractions which are merely valid up to contractions.
In 2002, Branciari 21 proved a fixed point result for a mapping satisfying an integral-type inequality which is indeed an analogue of contraction mapping condition. In recent past, several authors e.g., 22-26 proved various fixed point theorems employing relatively more general integral type contractive conditions. In one of his interesting articles, Suzuki 27 pointed out that Meir-Keeler contractions of integral type are still Meir-Keeler contractions. In this paper, we prove the fixed point theorems for weakly compatible mappings via an implicit relation in Menger PM spaces satisfying the common property E.A . Our results substantially improve the corresponding theorems contained in 21, 24, 26, 28 along with some other relevant results in Menger as well as metric spaces. Some related results are also derived besides furnishing illustrative examples.
In the following lines, we collect the background material to make our presentation as self-contained as possible.
Definition 1.1 see 3 . A mapping F : R → R is called distribution function if it is nondecreasing and left continuous with inf{F t : t ∈ R} 0 and sup{F t : t ∈ R} 1.
Let L be the set of all distribution functions whereas H be the set of specific distribution function also known as Heaviside function defined by Definition 1.2 see 1 . Let X be a nonempty set. An ordered pair X, F is called a PM space if F is a mapping from X × X into L satisfying the following conditions: 3 F p,q x 1 and F q,r y 1, then F p,r x y 1, for all p, q, r ∈ X and x, y ≥ 0.
Every metric space X, d can always be realized as a PM space by considering F : X × X → L defined by F p,q x H x − d p, q for all p, q ∈ X. So PM spaces offer a wider framework than that of the metric spaces and are general enough to cover even wider statistical situations.
Example 1.4. The following are the four basic t-norms.
i The minimum t-norm: ii The product t-norm: iii The Lukasiewicz t-norm: iv The weakest t-norm, the drastic product:

1.2
In respect of above mentioned t-norms, we have the following ordering: Throughout this paper, Δ stands for an arbitrary continuous t-norm. Sx n t ∈ X, for some t ∈ X, 1.5 but for some t 0 > 0, lim n → ∞ F ASx n ,SAx n t 0 is either less than 1 or nonexistent.  This shows that the pairs A, S and B, T share the common property E.A .

Implicit Relation
Let F 6 be the set of all continuous functions Φ t 1 , t 2 , t 3 , t 4 , t 5 , t 6 : 0, 1 6 → R satisfying the following conditions: where ψ : 0, 1 → 0, 1 is increasing and continuous function such that ψ t > t for all t ∈ 0, 1 . Notice that where ψ : 0, 1 → 0, 1 is an increasing and continuous function such that ψ t > t for all t ∈ 0, 1 and φ : R → R is a Lebesgue integrable function which is summable and satisfies Observe that Fixed Point Theory and Applications where ψ : 0, 1 → 0, 1 is an increasing and continuous function such that ψ t > t for all t ∈ 0, 1 and φ : R → R is a Lebesgue integrable function which is summable and satisfies Observe that

Main Results
We begin with the following observation. ii for any p, q ∈ X, Φ ∈ F 6 and for all x > 0, iii S X (or T X is a closed subset of X. 3.7

Then the pairs A, S and B, T have a point of coincidence each. Moreover, A, B, S and T have a unique common fixed point provided that both the pairs
Suppose that S X is a closed subset of X, then t Su for some u ∈ X. If t / Au, then applying inequality 3.1 , we obtain Φ F Au,By n x , F Su,T y n x , F Su,Au x , F Ty n ,By n x , F Su,By n x , F Ty n ,Au x ≥ 0 3.8 which on making n → ∞, reduces to Φ F Au,t x , 1, F t,Au x , 1, 1, F t,Au x ≥ 0 3.9 which is a contradiction to Φ 1 . Hence Au Su t.
Since A X ⊂ T X , there exists v ∈ X such that t Au Tv. If t / Bv, then using inequality 3. St, and t is a common fixed point of A, B, S and T . The uniqueness of common fixed point is an easy consequences of inequality 3.1 .
By choosing A, B, S and T suitably, one can derive corollaries involving two or three mappings. As a sample, we deduce the following natural result for a pair of self-mappings by setting B A and T S in Theorem 3.3 .

Corollary 3.4. Let A and S be self-mappings on a Menger space X, F, Δ . Suppose that i the pair A, S enjoys the property (E.A),
ii for all p, q ∈ X, Φ ∈ F 6 and for all x > 0, iii S X is a closed subset of X.
Then A and S have a coincidence point. Moreover, if the pair A, S is weakly compatible, then A and S have a unique common fixed point. Since S X and T X are closed subsets of X, we obtain t Su Tv for some u, v ∈ X.
If t / Au, then using inequality 3.1 , we have Φ F Au,By n x , F Su,T y n x , F Su,Au x , F Ty n ,By n x , F Su,By n x , F Ty n ,Au x ≥ 0 3.17 which on making n → ∞ reduces to which is a contradiction to Φ 1 , and hence t Au Tv Su. The rest of the proof can be completed on the lines of the proof of Theorem 3.3, hence it is omitted.

Theorem 3.7. The conclusions of Theorem 3.5 remain true if condition (ii) of Theorem 3.5 is replaced by the following:
iii A X ⊂ T X and B X ⊂ S X .

Corollary 3.8. The conclusions of Theorems 3.3 and 3.5 remain true if conditions (ii) (of Theorem 3.3) and (iii)' (of Theorem 3.7) are replaced by the following:
iv A X and B X are closed subsets of X whereas A X ⊂ T X and B X ⊂ S X .
As an application of Theorem 3.3, we prove the following result for four finite families of self-mappings. While proving this result, we utilize Definition 1.13 which is a natural extension of commutativity condition to two finite families of mappings. . . , T q } be four finite families of self-mappings of a Menger PM space X, F, Δ with A A 1 A 2 · · · A m , B B 1 B 2 · · · B p , S S 1 S 2 · · · S n and T T 1 T 2 · · · T q satisfying condition 3.  By setting A 1 A 2 · · · A m G, B 1 B 2 · · · B p H, S 1 S 2 · · · S n I and T 1 T 2 · · · T q J in Theorem 3.9, we deduce the following.

10
Fixed Point Theory and Applications Corollary 3.11. Let G, H, I and J be self-mappings of a Menger space X, F, Δ such that the pairs G m , I n and H p , J q share the common property (E.A) and also satisfy the condition Φ F G m x,H p y z , F I n x,J q y z , F I n x,G m x z , F I n x,H p y z , F J q y,H p y z , F J q y,G m x z ≥ 0 3.19 for all x, y ∈ X, for all z > 0 and m, n, p and q are fixed positive integers. If I n X and J q X are closed subsets of X, then G, H, I and J have a unique common fixed point provided, GI IG and HJ JH.
Remark 3.12. Corollary 3.11 is a slight but partial generalization of Theorem 3.3 as the commutativity requirements i.e., GI IG and HJ JH in this corollary are stronger as compared to weak compatibility in Theorem 3.3. Corollary 3.11 also presents the generalized and improved form of a result according to Bryant 33 in Menger PM spaces.
Our next result involves a lower semicontinuous function ψ : 0, 1 → 0, 1 such that ψ t > t for all t ∈ 0, 1 along with ψ 0 0 and ψ 1 1. 3.21 If S X is a closed subset of X, then 3.21 . Therefore, there exists a point u ∈ X such that Su t. Now we assert that Au Su. If it is not so, then setting p u, q y n in 3.20 , we get  Remark 3.14. Theorem 3.13 generalizes the main result of Kohli and Vashistha 9 to two pairs of self-mappings as Theorem 3.13 never requires any condition on the containment of ranges amongst involved mappings besides weakening the completeness requirement of the space to closedness of suitable subspaces along with suitable commutativity requirements of the involved mappings. Here one may also notice that the function ψ is lower semicontinuous whereas all the involved mappings may be discontinuous at the same time.
Remark 3.15. Notice that results similar to Theorems 3.5 -3.9 and Corollaries 3.4-3.11 can also be outlined in respect of Theorem 3.13, but we omit the details with a view to avoid any repetition.
We conclude this paper with two illustrative examples which demonstrate the validity of the hypotheses of Theorem 3.3 and Theorem 3.13.

Fixed Point Theory and Applications
Example 3. 16. Let X, F, Δ be a Menger space, where X 0, 2 with a t-norm defined by Δ a, b min{a, b} for all a, b ∈ 0, 2 , ψ s √ s for all s ∈ 0, 1 and F p,q t t t p − q , ∀p, q ∈ X, t > 0. 3.28 Define A, B, S and T by: Ax Bx 1,

3.30
It is easy to see that for all x, y ∈ X and t > 0