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On common fixed point theorems for semi-compatible mappings in Menger space

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Bijendra Singh, Arihant Jain, Bholaram Lodha

On common fixed point theorems for semi-compatible mappings in Menger space

Abstract. In this paper, the concept of semi-compatibility and weak compatibility in Menger space has been applied to prove a common fixed point theorem for six self maps. Our result generalizes and extends the result of Pathak and Verma [6].

2000 Mathematics Subject Classification: Primary 47H10, Secondary 54H25.

Key words and phrases: Probabilistic metric space, Menger space, common fixed point, compatible maps, semi-compatible maps, weak compatibility.

1. Introduction. There have been a number of generalizations of metric space.

One such generalization is Menger space initiated by Menger [4]. It is a probabilistic generalization in which we assign to any two points x and y, a distribution function Fx,y. Schweizer and Sklar [8] studied this concept and gave some fundamental results on this space. Sehgal and Bharucha-Reid [9] obtained a generalization of Banach Contraction Principle on a complete Menger space which is a milestone in developing fixed point theory in Menger space.

Recently, Jungck and Rhoades [3] termed a pair of self maps to be coincidentally commuting or equivalently weakly compatible if they commute at their coincidence points. Sessa [10] initiated the tradition of improving commutativity in fixed point theorems by introducing the notion of weak commuting maps in metric spaces.

Jungck [2] soon enlarged this concept to compatible maps. The notion of compatible mapping in a Menger space has been introduced by Mishra [5].

Cho, Sharma and Sahu [1] introduced the concept of semicompatibility in a d-complete topological space. Popa [7] proved interesting fixed point results using implicit real functions and semi-compatibility in d-complete topological space. In the sequel, Pathak and Verma [6] proved a common fixed point theorem in Menger space using compatibility and weak compatibility.

In this paper, a fixed point theorem for six self maps has been proved using the concept of semi-compatible maps and weak compatibility which turns out be a material generalization of the result of Pathak and Verma [6].

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2. Preliminaries.

Definition 2.1 A mapping F : ℝ → ℝ+ is called adistribution if it is nondecre- asing left continuous with

inf{F(t) : t ∈ ℝ} = 0 and sup{F(t) : t ∈ ℝ} = 1.

We shall denote by L the set of all distribution functions while H will always denote the specific distribution function defined by

H(t) =

 0, t ¬ 0, 1, t > 0 .

Definition 2.2 ([5]) A mapping t : [0, 1] × [0, 1] → [0, 1] is called a t − norm if it satisfies the following conditions :

(t-1) t(a, 1) = a, t(0, 0) = 0;

(t-2) t(a, b) = t(b, a) ;

(t-3) t(c, d) ­ t(a, b) ; for c ­ a, d ­ b,

(t-4) t(t(a, b), c) = t(a, t(b, c)) for all a, b, c, d ∈ [0, 1].

Definition 2.3 ([5]) A probabilistic metric space (PM-space) is an ordered pair (X, F) consisting of a non empty set X and a function F : X × X → L, where L is the collection of all distribution functions and the value of F at (u, v) ∈ X × X is represented by Fu,v. The function Fu,vassumed to satisfy the following conditions:

(PM-1 ) Fu,v(x) = 1, for all x > 0, if and only if u = v;

(PM-2 ) Fu,v(0) = 0;

(PM-3 ) Fu,v= Fv,u;

(PM-4 ) If Fu,v(x) = 1 and Fv,w(y) = 1 then Fu,w(x + y) = 1, for all (PM-4 ) u, v, w∈ X and x, y > 0.

Definition 2.4 ([5]) AMenger space is a triplet (X, F, t) where (X, F) is a PM- space and t is a t-norm such that the inequality

(PM-5 ) Fu,w(x + y) ­ t{Fu,v(x), Fv,w(y)}, for all u, v, w ∈ X, x, y ­ 0.

Definition 2.5 ([5]) A sequence {xn} in a Menger space (X, F, t) is said to be convergent and converges to a point x in X if and only if for each ε > 0 and λ > 0, there is an integer M(ε, λ) such that Fxn,x(ε) > 1 − λ for all n ­ M(ε, λ).

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Further the sequence {xn} is said to be Cauchy sequence if for ε > 0 and λ > 0, there is an integer M(ε, λ) such that

Fxn,xm(ε) > 1 − λ for all m, n ∈ M(ε, λ).

A Menger PM-space (X, F, t) is said to be complete if every Cauchy sequence in X converges to a point in X. A complete metric space can be treated as a complete Menger space in the following way :

Proposition 2.6 ([5]) If (X, d) is a metric space then the metric d induces map- ping F : X ×X → L, defined by Fp,q(x) = H(x−d(p, q)), p, q ∈ X, where H(k) = 0, for k ¬ 0 and H(k) = 1, for k > 0.

Further if, t : [0, 1]×[0, 1] → [0, 1] is defined by t(a, b) = min{a, b}. Then (X, F, t) is a Menger space. It is complete if (X, d) is complete.

The space (X, F, t) so obtained is called the induced Menger space.

Definition 2.7 ([6]) Self mappings A and S of a Menger space (X, F, t) are said to be weak compatible if they commute at their coincidence points i.e. Ax = Sx for x∈ X implies ASx = SAx.

Definition 2.8 ([6]) Self mappings A and S of a Menger space (X, F, t) are said to be compatible if FASxn,SAxn(x) → 1 for all x > 0, whenever {xn} is a sequence in X such that Axn, Sxn→ u for some u in X, as n → ∞.

Definition 2.9 Self mappings A and S of a Menger space (X, F, t) are said to be semi-compatible if FASxn,Su(x) → 1 for all x > 0, whenever {xn} is a sequence in X such that Axn, Sxn→ u, for some u in X, as n → ∞.

Now, we give an example of pair of self maps (S, T ) which is semicompatible but not compatible. Further we observe here that the pair (T, S) is not semi-compatible though (S, T ) is semi-compatible.

Example 2.10 Let (X, d) be a metric space where X = [0, 1] and (X, F, t) be the induced Menger space with Fp,q(ε) = H(ε − d(p, q)), ∀p, q ∈ X and ∀ε > 0. Define self maps S and T as follows :

Sx =

 x, if 0 ¬ x < 12, 1, if 12 ¬ x ¬ 1 and

T x =

 1 − x, if 0 ¬ x < 12, 1, if 12 ¬ x ¬ 1 . Take xn= 121n. Now,

FSxn,1/2(ε) = H(ε − (1/n)). Therefore, limn→∞FSxn,1/2(ε) = H(ε) = 1.

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Hence, Sxn→ 1/2 as n → ∞.

Similarly, T xn → 1/2 as n → ∞.

Also

FST xn,T Sxn(ε) = H

 ε−

1 2 −

1 n



6= 1, ∀ε > 0.

Hence, the pair (S, T ) is not compatible.

Again, limn→∞FST xn,T x(ε) = limn→∞FST xn,1(ε) = H (ε − |1 − 1|) = 1, ∀ε > 0.

Thus, (S, T ) is semi-compatible.

Now, limn→∞FT Sxn,Sx(ε) 6= 1, ∀ε > 0. Thus, (T, S) is not semi-compatible.

Remark 2.11 In view of above example, it follows that the concept of semi-compatibility is more general than that of compatibility.

Lemma 2.12 ([6]) Let (X, F, ∗) be a Menger space with t-norm ∗ such that the fa- mily {∗n(x)}n∈N is equicontinuous at x = 1 and let E denote the family of all func- tions φ : ℝ+→ ℝ+ such that φ is non-decreasing with limn→∞φn(t) = +∞, ∀t > 0.

If {yn}n∈N is a sequence in X satisfying the condition Fyn,yn+1(t) ­ Fyn−1,yn(φ(t)),

for all t > 0 and α ∈ [−1, 0], then {yn}n∈N is a Cauchy sequence in X.

3. Main Result.

Theorem 3.1 Let A, B, S, T, P and Q be self maps of a complete Menger space (X, F, ∗) with ∗ = min satisfying :

(3.1.1) P (X) ⊆ ST (X), Q(X) ⊆ AB(X);

(3.1.2) AB = BA, ST = T S, P B = BP , QT = T Q;

(3.1.3) either P or AB is continuous;

(3.1.4) (P, AB)is semi-compatible and (Q, ST ) is weak compatible;

(3.1.5) [1 + αFABx,ST y(t)] ∗ FP x,Qy(t)

(3.1.5) ­ αmin{FP x,ABx(t) ∗ FQy,ST y(t), FP x,ST y(2t) ∗ FQy,ABx(2t)}

(3.1.5) +FABx,ST y(φ(t)) ∗ FP x,ABx(φ(t)) ∗ FQy,ST y(φ(t)) (3.1.5) ∗FP x,ST y(2φ(t)) ∗ FQy,ABx(2φ(t))

(3.1.5) for all x, y ∈ X, t > 0 and φ ∈ E.

Then A, B, S, T, P and Q have a unique common fixed point in X.

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Proof Suppose x0∈ X. From condition (3.1.1) ∃x1, x2∈ X such that P x0= ST x1 and Qx1= ABx2.

Inductively, we can construct sequences {xn} and {yn} in X such that y2n= P x2n = ST x2n+1 and y2n+1 = Qx2n+1= ABx2n+2

for n = 0, 1, 2, . . . . Step 1.

Let us show that Fyn+2,yn+1(t) ­ Fyn+1,yn(φ(t)).

For, putting x2n+2 for x and x2n+1 for y in (3.1.5) and then on simplification, we have

[1 + αFABx2n+2,ST x2n+1(t)] ∗ FP x2n+2,Qx2n+1(t)

­ αmin{FP x2n+2,ABx2n+2(t) ∗ FQx2n+1,ST x2n+1(t), FP x2n+2,ST x2n+1(2t) ∗ FQx2n+1,ABx2n+2(2t)}

+FABx2n+2,ST x2n+1(φ(t))

∗FP x2n+2,ABx2n+2(φ(t)) ∗ FQx2n+1,ST x2n+1(φ(t))

∗FP x2n+2,ST x2n+1(2φ(t)) ∗ FQx2n+1,ABx2n+2(2φ(t))

[1 + αFy2n+1,y2n(t)] ∗ Fy2n+2,y2n+1(t)

­ αmin{Fy2n+2,y2n+1(t) ∗ Fy2n+1,y2n(t), Fy2n+2,y2n(2t) ∗ Fy2n+1,y2n+1(2t)}

+Fy2n+1,y2n(φ(t)) ∗ Fy2n+2,y2n+1(φ(t)) ∗ Fy2n+1,y2n(φ(t))

∗Fy2n+2,y2n(2φ(t)) ∗ Fy2n+1,y2n+1(2φ(t))

Fy2n+2,y2n+1(t) + αFy2n+1,y2n(t) ∗ Fy2n+2,y2n+1(t)

­ αmin{Fy2n+2,y2n(2t), Fy2n+2,y2n(2t)}

+Fy2n+1,y2n(φ(t)) ∗ Fy2n+2,y2n+1(φ(t)) ∗ Fy2n+2,y2n(2φ(t)) ∗ 1

Fy2n+2,y2n+1(t) + αFy2n+1,y2n(t) ∗ Fy2n+2,y2n+1(t)

­ αFy2n+2,y2n(2t) + Fy2n+1,y2n(φ(t)) ∗ Fy2n+2,y2n+1(2φ(t)) Fy2n+2,y2n+1(t) + αFy2n+2,y2n(2t)

­ αFy2n+2,y2n(2t) + Fy2n+1,y2n(φ(t)) ∗ Fy2n+2,y2n+1(φ(t)) ∗ Fy2n+1,y2n(φ(t))

Fy2n+2,y2n+1(t) ­ Fy2n+1,y2n(φ(t)) ∗ Fy2n+2,y2n+1(φ(t))

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or

Fy2n+2,y2n+1(t) ­ Fy2n+1,y2n+2(φ(t)) ∗ Fy2n,y2n+1(φ(t)) or

Fy2n+2,y2n+1(t) ­ min{Fy2n+1,y2n+2(φ(t)), Fy2n,y2n+1(φ(t))}.

If Fy2n+1,y2n+2(φ(t)) is chosen ’min’ then we obtain

Fy2n+2,y2n+1(t) ­ Fy2n+2,y2n+1(φ(t)), ∀t > 0, a contradiction as φ(t) is non-decreasing function. Thus,

Fy2n+2,y2n+1(t) ­ Fy2n+1,y2n(φ(t)), ∀t > 0.

Similarly, by putting x2n+2 for x and x2n+3 for y in (3.1.5), we have Fy2n+3,y2n+2(t) ­ Fy2n+2,y2n+1(φ(t)), ∀t > 0.

Using these two, we obtain

Fyn+2,yn+1(t) ­ Fyn+1,yn(φ(t)), ∀n = 0, 1, 2, ..., t > 0.

Therefore, by lemma 2.12, {yn} is a Cauchy sequence in X, which is complete.

Hence {yn} → z ∈ X. Also its subsequences converges as follows : {P x2n} → z and {ST x2n+1} → z,

(3.1.6) {Qx2n+1} → z and {ABx2n+2} → z.

Case I. Suppose P is continuous.

As P is continuous and (P, AB) is semi-compatible, we get (3.1.7) P ABx2n+2→ P z and P ABx2n+2→ ABz.

Since the limit in Menger space is unique, we get

(3.1.8) P z = ABz.

Step 2.

We prove P z = z. Put x = z, y = x2n+1 in (3.1.5) and let P z 6= z. Then [1 + αFABz,ST x2n+1(t)] ∗ FP z,Qx2n+1(t)

­ αmin{FP z,ABz(t) ∗ FQx2n+1,ST x2n+1(t), FP z,ST x2n+1(2t) ∗ FQx2n+1,ABz(2t)}

+FABz,ST x2n+1(φ(t)) ∗ FP z,ABz(φ(t)) ∗ FQx2n+1,ST x2n+1(φ(t))

∗FP z,ST x2n+1(2φ(t)) ∗ FQx2n+1,ABz(2φ(t)) Letting n → ∞ and using (3.1.6) and (3.1.8), we get

[1 + αFP z,z(t)] ∗ FP z,z(t)

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­ αmin{FP z,P z(t) ∗ Fz,z(t), FP z,z(2t) ∗ Fz,P z(2t)}

+FP z,z(φ(t)) ∗ FP z,P z(φ(t)) ∗ Fz,z(φ(t)) ∗ FP z,z(2φ(t)) ∗ Fz,P z(2φ(t))

FP z,z(t) + α{FP z,z(t) ∗ FP z,z(t)}

­ αmin{1 ∗ 1, FP z,z(2t) ∗ FP z,z(2t)}

+FP z,z(φ(t)) ∗ 1 ∗ 1 ∗ FP z,z(2φ(t)) ∗ Fz,P z(2φ(t))

FP z,z(t) + αFP z,z(t) ­ αmin{1, FP z,z(2t)} + FP z,z(φ(t)) ∗ FP z,z(2φ(t)) FP z,z(t) + αFP z,z(t) ­ αFP z,z(2t) + FP z,z(φ(t)) ∗ FP z,z(2φ(t))

FP z,z(t) + αFP z,z(t) ­ α{FP z,z(t) ∗ Fz,z(t)} + FP z,z(φ(t)) ∗ FP z,z(φ(t)) ∗ Fz,z(φ(t)) FP z,z(t) + αFP z,z(t) ­ α{FP z,z(t) ∗ 1} + FP z,z(φ(t)) ∗ 1

FP z,z(t) + αFP z,z(t) ­ αFP z,z(t) + FP z,z(φ(t)) FP z,z(t) ­ FP z,z(φ(t))

which is a contradiction and hence, P z = z and so z = P z = ABz.

Step III. Put x = Bz and y = x2n+1 in (3.1.5), we get [1 + αFABBz,ST x2n+1(t)] ∗ FP Bz,Qx2n+1(t)

­ αmin{FP Bz,ABBz(t) ∗ FQx2n+1,ST x2n+1(t), FP Bz,ST x2n+1(2t) ∗ FQx2n+1,ABBz(2t)}

+FABBz,ST x2n+1(φ(t)) ∗ FP Bz,ABBz(φ(t)) ∗ FQx2n+1,ST x2n+1(φ(t))

∗FP Bz,ST x2n+1(2φ(t)) ∗ FQx2n+1,ABBz(2φ(t)).

As BP = P B, AB = BA so we have P (Bz) = B(P z) = Bz and AB(Bz) = B(AB)z = Bz. Letting n→ ∞ and using (3.1.6), we get

[1 + αFBz,z(t)] ∗ FBz,z(t)

­ αmin{FBz,Bz(t) ∗ Fz,z(t), FBz,z(2t) ∗ Fz,Bz(2t)}

+FBz,z(φ(t)) ∗ FBz,Bz(φ(t)) ∗ Fz,z(φ(t)) ∗ FBz,z(2φ(t)) ∗ Fz,Bz(2φ(t))

FBz,z(t) + α{FBz,z(t) ∗ FBz,z(t)}

­ αmin{1 ∗ 1, FBz,z(2t)} + FBz,z(φ(t)) ∗ 1 ∗ 1 ∗ FBz,z(2φ(t))

FBz,z(t) + αFBz,z(t) ­ αFBz,z(2t) + FBz,z(φ(t)) ∗ FBz,z(2φ(t))

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FBz,z(t) + αFBz,z(t) ­ α{FBz,z(t) ∗ Fz,z(t)} + FBz,z(φ(t)) ∗ FBz,z(φ(t)) ∗ Fz,z(φ(t)) FBz,z(t) + αFBz,z(t) ­ α{FBz,z(t) ∗ 1} + FBz,z(φ(t)) ∗ 1

FBz,z(t) + αFBz,z(t) ­ αFBz,z(t) + FBz,z(φ(t)) FBz,z(t) ­ FBz,z(φ(t))

which is a contradiction and we get Bz = z and so z = ABz = Az.

Therefore,

(3.1.9) P z = Az = Bz = z.

Step IV. Since P (X) ⊆ ST (X) there exists u ∈ X such that z = P z = ST u. Put x = x2n and y = u in (3.1.5), we get

[1 + αFABx2n,ST u(t)] ∗ FP x2n,Qu(t)

­ αmin{FP x2n,ABx2n(t) ∗ FQu,ST u(t), FP x2n,ST u(2t) ∗ FQu,ABx2n(2t)}

+FABx2n,ST u(φ(t)) ∗ FP x2n,ABx2n(φ(t)) ∗ FQu,ST u(φ(t)) ∗ FP x2n,ST u(2φ(t))

∗FQu,ABx2n(2φ(t)).

Letting n → ∞ and using (3.1.6), we get

[1 + αFz,z(t)] ∗ Fz,Qu(t)

­ αmin{Fz,z(t) ∗ FQu,z(t), Fz,z(2t) ∗ FQu,z(2t)} + Fz,z(φ(t)) ∗ Fz,z(φ(t))

∗FQu,z(φ(t)) ∗ Fz,z(2φ(t)) ∗ FQu,z(2φ(t))

Fz,Qu(t) + αFz,Qu(t) ­ αmin{FQu,z(t), FQu,z(2t)} + FQu,z(φ(t)) ∗ FQu,z(2φ(t))

FQu,z(t) + αFQu,z(t)

­ αmin{FQu,z(t), FQu,z(t) ∗ Fz,z(t)} + FQu,z(φ(t)) ∗ FQu,z(φ(t)) ∗ Fz,z(φ(t))

FQu,z(t) + αFQu,z(t) ­ αFQu,z(t) + FQu,z(φ(t)) FQu,z(t) ­ FQu,z(φ(t))

which is a contradiction by lemma 2.12 and we get Qu = z and so Qu = z = ST u.

Since (Q, ST ) is weak-compatible, we have

ST Qu = QST u i.e. ST z = Qz.

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Step V. Put x = x2n and y = z in (3.1.5), we have [1 + αFABx2n,ST z(t)] ∗ FP x2n,Qz(t)

­ αmin{FP x2n,ABx2n(t) ∗ FQz,ST z(t), FP x2n,ST z(2t) ∗ FQz,ABx2n(2t)}

+FABx2n,ST z(φ(t)) ∗ FP x2n,ABx2n(φ(t)) ∗ FQz,ST z(φ(t)) ∗ FP x2n,ST z(2φ(t))

∗FQz,ABx2n(2φ(t)).

Letting n → ∞ and using (3.1.6) and step IV, we get [1 + αFz,Qz(t)] ∗ Fz,Qz(t)

­ αmin{Fz,z(t) ∗ FQz,Qz(t), Fz,Qz(2t) ∗ FQz,z(2t)} + Fz,Qz(φ(t)) ∗ Fz,z(φ(t))

∗FQz,Qz(φ(t)) ∗ Fz,Qz(2φ(t)) ∗ FQz,Qz(2φ(t))

Fz,Qz(t) + α{Fz,Qz(t) ∗ FQz,z(t)}

­ αmin{1 ∗ Fz,Qz(2t)} + Fz,Qz(φ(t)) ∗ Fz,Qz(2φ(t))

FQz,z(t) + αFQz,z(t) ­ αFQz,z(2t) + FQz,z(φ(t)) ∗ FQz,z(φ(t)) ∗ Fz,z(φ(t)) FQz,z(t) + αFQz,z(t) ­ α{FQz,z(t) ∗ Fz,z(t)} + FQz,z(φ(t))

FQz,z(t) + αFQz,z(t) ­ αFQz,z(t) ∗ FQz,z(φ(t)) FQz,z(t) ­ FQz,z(φ(t))

which is a contradiction and we get Qz = z.

Step VI. Put x = x2n and y = T z in (3.1.5), we have [1 + αFABx2n,ST T z(t)] ∗ FP x2n,QT z(t)

­ αmin{FP x2n,ABx2n(t) ∗ FQT z,ST T z(t), FP x2n,ST T z(2t) ∗ FQT z,ABx2n(2t)}

+FABx2n,ST T z(φ(t)) ∗ FP x2n,ABx2n(φ(t)) ∗ FQT z,ST T z(φ(t))

∗FP x2n,ST T z(2φ(t)) ∗ FQT z,ABx2n(2φ(t)).

As QT = T Q and ST = T S, we have

QT z = T Qz = T z and ST (T z) = T (ST z) = T z.

Letting n → ∞, we get

[1 + αFz,T z(t)] ∗ Fz,T z(t)

­ αmin{Fz,z(t) ∗ FT z,T z(t), Fz,T z(2t) ∗ FT z,z(2t)} + Fz,T z(φ(t)) ∗ Fz,z(φ(t))

∗FT z,T z(φ(t)) ∗ Fz,T z(2φ(t)) ∗ FT z,z(2φ(t))

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Fz,T z(t)+α{Fz,T z(t)∗Fz,T z(t)} ­ α min{1∗FT z,z(2t)}+Fz,T z(φ(t))∗1∗1∗FT z,z(2φ(t)) FT z,z(t) + αFT z,z(t) ­ αFT z,z(2t) + FT z,z(φ(t)) ∗ FT z,z(2φ(t))

FT z,z(t) + αFT z,z(t) ­ α{FT z,z(t) ∗ Fz,z(t)} + FT z,z(φ(t)) ∗ FT z,z(φ(t)) ∗ Fz,z(φ(t)) FT z,z(t) + αFT z,z(t) ­ αFT z,z(t) + FT z,z(φ(t))

FT z,z(t) ­ FT z,z(φ(t)) which is a contradiction and we get T z = z.

Now, ST z = T z = z implies Sz = z. Hence,

(3.1.10) Sz = T z = Qz = z.

Combining (3.1.9) and (3.1.10), we get

Az = Bz = P z = Qz = Sz = T z = z i.e. z is a common fixed point of A, B, P , Q, S and T . Case II. Suppose AB is continuous.

Since AB is continuous and (P, AB) is semi-compatible, we get (3.1.11) (AB)2x2n→ ABz, P ABx2n→ ABz.

Now, we prove ABz = z.

Step VII. Put x = ABx2n and y = x2n+1 in (3.1.5) and assuming ABz 6= z, we get

[1 + αFABABx2n,ST x2n+1(t)] ∗ FP ABx2n,Qx2n+1(t)

­ αmin{FP ABx2n,ABABx2n(t) ∗ FQx2n+1,ST x2n+1(t), FP ABx2n,ST x2n+1(2t)

∗FQx2n+1,ABABx2n(2t)} + FABABx2n,ST x2n+1(φ(t)) ∗ FP ABx2n,ABABx2n(φ(t))

∗FQx2n+1,ST x2n+1(φ(t)) ∗ FP ABx2n,ST x2n+1(2φ(t)) ∗ FQx2n+1,ABABx2n(2φ(t)).

Letting n → ∞ and using (3.1.11), we get

[1 + αFABz,z(t)] ∗ FABz,z(t)

­ αmin{FABz,ABz(t) ∗ Fz,z(t), FABz,z(2t) ∗ Fz,ABz(2t)} + FABz,z(φ(t))

∗FABz,ABz(φ(t)) ∗ Fz,z(φ(t)) ∗ FABz,z(2φ(t)) ∗ Fz,ABz(2φ(t))

FABz,z(t) + α{FABz,z(t) ∗ FABz,z(t)}

­ αmin{1 ∗ 1, FABz,z(2t)} + FABz,z(φ(t)) ∗ 1 ∗ 1 ∗ FABz,z(2φ(t))

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FABz,z(t) + αFABz,z(t) ­ αmin{1, FABz,z(2t)} + FABz,z(φ(t)) ∗ FABz,z(2φ(t)) FABz,z(t)+αFABz,z(t) ­ α{FABz,z(t)∗Fz,z(t)}+FABz,z(φ(t))∗FABz,z(φ(t))∗Fz,z(φ(t))

FABz,z(t) + αFABz,z(t) ­ αFABz,z(t) + FABz,z(φ(t)) FABz,z(t) ­ FABz,z(φ(t))

which is a contradiction and we get ABz = z.

Step VIII. Put x = z and y = x2n+1 in (3.1.5), we get [1 + αFABz,ST x2n+1(t)] ∗ FP z,Qx2n+1(t)

­ αmin{FP z,ABz(t) ∗ FQx2n+1,ST x2n+1(t), FP z,ST x2n+1(2t) ∗ FQx2n+1,ABz(2t)}

+FABz,ST x2n+1(φ(t)) ∗ FP z,ABz(φ(t)) ∗ FQx2n+1,ST x2n+1(φ(t))

∗FP z,ST x2n+1(2φ(t)) ∗ FQx2n+1,ABz(2φ(t)).

Letting n → ∞ and using (3.1.6), we get

[1 + αFz,z(t)] ∗ FP z,z(t)

­ αmin{FP z,z(t) ∗ Fz,z(t), FP z,z(2t) ∗ Fz,z(2t)} + Fz,z(φ(t)) ∗ FP z,z(φ(t))

∗Fz,z(φ(t)) ∗ FP z,z(2φ(t)) ∗ Fz,z(2φ(t))

FP z,z(t) + αFP z,z(t)

­ αmin{FP z,z(t), FP z,z(2t)} + FP z,z(φ(t)) ∗ FP z,z(2φ(t))

FP z,z(t) + αFP z,z(t) ­ αmin{FP z,z(t), FP z,z(t) ∗ Fz,z(t)} + FP z,z(φ(t)) ∗ Fz,z(φ(t)) FP z,z(t) + αFP z,z(t) ­ αmin{FP z,z(t), FP z,z(t)} + FP z,z(φ(t))

FP z,z(t) + αFP z,z(t) ­ αFP z,z(t) + FP z,z(φ(t)) FP z,z(t) ­ FP z,z(φ(t))

which is a contradiction and hence, we get P z = z. Hence, P z = z = ABz.

Further using step III, we get Bz = z.

Thus ABz = z gives Az = z and so Az = Bz = P z = z.

Also, it follows from steps IV, V and VI that Sz = T z = Qz = z.

Hence, we get

Az = Bz = P z = Sz = T z = Qz = z

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i.e. z is a common fixed point of A, B, P , Q, S and T in this case also.

Uniqueness :

Let z1 be another common fixed point of A, B, P , Q, S and T . Then Az1= Bz1= P z1= Sz1= T z1= Qz1= z1, assuming z 6= z1

Put x = z and y = z1in (3.1.5), we get

[1 + αFABz,ST z1(t)] ∗ FP z,Qz1(t)

­ αmin{FP z,ABz(t) ∗ FQz1,ST z1(t), FP z,ST z1(2t) ∗ FQz1,ABz(2t)}

+FABz,ST z1(φ(t)) ∗ FP z,ABz(φ(t)) ∗ FQz1,ST z1(φ(t))

∗FP z,ST z1(2φ(t)) ∗ FQz1,ABz(2φ(t))

[1 + αFz,z1(t)] ∗ Fz,z1(t)

­ αmin{Fz,z(t) ∗ Fz1,z1(t), Fz,z1(2t) ∗ Fz1,z(2t)} + Fz,z1(φ(t))

∗Fz,z(φ(t)) ∗ Fz1,z1(φ(t)) ∗ Fz,z1(2φ(t)) ∗ Fz1,z(2φ(t))

Fz,z1(t) + α{Fz,z1(t) ∗ Fz,z1(t)}

­ αmin{1, Fz,z1(2t)} + Fz,z1(φ(t)) ∗ Fz,z1(2φ(t))

Fz,z1(t) + αFz,z1(t) ­ αFz,z1(2t) + Fz,z1(φ(t)) ∗ Fz,z1(φ(t)) ∗ Fz,z(φ(t)) Fz1,z(t) + αFz1,z(t) ­ α{Fz1,z(t) ∗ Fz,z(t)} + Fz1,z(φ(t)) ∗ 1

Fz1,z(t) + αFz1,z(t) ­ αFz1,z(t) + Fz1,z(φ(t)) Fz1,z(t) ­ Fz1,z(φ(t))

which is a contradiction.

Hence z = z1and so z is the unique common fixed point of A, B, S, T , P and Q.

This completes the proof.

Remark 3.2 If we take B = T = I, the identity map on X in theorem 3.1, then condition (3.1.2) is satisfied trivially and we get

Corollary 3.3 Let A, S, P and Q be self maps of a complete Menger space (X, F, ∗) with ∗ = min satisfying :

(a) P (X) ⊆ S(X), Q(X) ⊆ A(X);

(b) either P or A is continuous;

(c) (P, A) is semi-compatible and (Q, S) is weak compatible;

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(d) [1 + αFAx,Sy(t)] ∗ FP x,Qy(t) ­ αmin{FP x,Ax(t) ∗ FQy,Sy(t), FP x,Sy(2t) ∗ FQy,Ax(2t)}

+FAx,Sy(φ(t)) ∗ FP x,Ax(φ(t)) ∗ FQy,Sy(φ(t)) ∗ FP x,Sy(2φ(t)) ∗ FQy,Ax(2φ(t))

for all x, y ∈ X, t > 0 and φ ∈ E.

Then A, S, P and Q have a unique common fixed point in X.

Remark 3.4 In view of Remark 3.2, corollary 3.3 is a generalization of the result of Pathak and Verma [6] in the sense that condition of compatibility of the first pair of self maps has been restricted to semi-compatibility.

References

[1] Y.J. Cho, B.K. Sharma and R.D. Sahu, Semi-compatibility and fixed points, Math. Japon.

42 (1) (1995), 91–98.

[2] G. Jungck, Compatible mappings and common fixed points, Internat. J. Math. and Math. Sci.

9(4) (1986), 771–779.

[3] G. Jungck and B.E. Rhoades, Fixed points for set valued functions without continuity, Indian J. Pure Appl. Math.29 (1998), 227–238.

[4] K. Menger, Statistical metrics, Proc. Nat. Acad. Sci. USA.28 (1942), 535 –537.

[5] S.N. Mishra, Common fixed points of compatible mappings in PM-spaces, Math. Japon.36(2) (1991), 283–289.

[6] H.K. Pathak and R.K. Verma, Common fixed point theorems for weakly compatible mappings in Menger space and application, Int. Journal of Math. Analysis, Vol.3 No. 24 (2009), 1199–

1206.

[7] V. Popa, Fixed points for non-surjective expansion mappings satisfying an implicit relation, Bul. Stiint. Univ. Baia Mare Ser. B Fasc. Mat.-Inform.18 (1) (2002), 105-108.

[8] B. Schweizer and A. Sklar, Statistical metric spaces, Pacific J. Math.10 (1960), 313–334.

[9] V.M. Sehgal and A.T. Bharucha-Reid, Fixed points of contraction maps on probabilistic metric spaces, Math. System Theory6 (1972), 97–102.

[10] S. Sessa, On a weak commutativity condition of mappings in fixed point consideration, Publ.

Inst. Math. Beograd32(46) (1982), 146–153.

Bijendra Singh

School of Studies in Mathematics, Vikram University Ujjain (M.P.) 456010, India

Arihant Jain

Department of Applied Mathematics, S.G.S.I.T.S.

Ujjain (M.P.) 456650, India E-mail: arihant2412@gmail.com Bholaram Lodha

School of Studies in Mathematics, Vikram University Ujjain (M.P.) 456010, India

(Received: 6.05.2010)

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