Vol 3, No 3 (2012)

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Available online through

ISSN 2229 – 5046

COMMON FIXED POINT THEOREMS FOR WEAKLY COMPATIBLE MAPPINGS

IN MENGER SPACE

Krishnapal Singh Sisodia*

Department of Mathematics, BITS, Bhopal (M.P), India

E-mail:

(Received on: 13-10-11; Accepted on: 30-10-11)

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ABSTRACT

I

n this paper, we prove some common fixed point theorems for a pair of self mappings using M.S. property and satisfy certain sufficient conditions in setting of Menger space.

Keywords: Menger space, Probabilistic distance, Probabilistic metric space, Distribution function, t-norm, weakly

compatible, common fixed point, M.S. property.

2000 Mathematics Subject Classification: 47H10, 54H25.

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INTRODUCTION:

There have been a number of generalizations of Metric space. One such generalization is Menger space introduced in 1942 by Menger [2] who used distribution functions instead of non negative real numbers as values of the metric. Schweizer and Sklar [7] studied this concept and then the important development of Menger space theory was due to Sehgal and Bharucha-Reid [4].

Sessa [5] introduced weakly commuting maps in metric spaces. Jungck [1] enlarged this concept to compatible maps. The notion of compatible maps in Menger spaces has been introduced by Mishra [3]. Recently Singh and Jain [6] generalized the results of Mishra [3] using the concept of weak compatibility and compatibility of pair of self maps. In this paper, we extend the result of Sisodia, K.S. [8] to obtain a common fixed point for a pair of self mappings in Menger space.

PRELIMINARIES:

Definition1: A binary operation ∗: [0,1] × [0,1]→[0,1] is a continuous t-norm if ∗ is satisfying the following conditions:

(a) ∗ is commutative and associative; (b) ∗ is continuous;

(c) a∗1=a for all aϵ[0,1];

(d) a∗b≤c∗d whenever a≤c and b≤d and a,b,c,d ϵ[0,1].

Examples of t-norms are a∗b=max {a+b-1, 0} and a∗b= min {a, b}.

Definition2: A distribution function is a function F:[-∞,∞]→[0,1] which is left continuous on R, non-decreasing and F(-∞)=0, F(∞)=1.

We will denote ∆ by the family of all distribution functions on [-∞, ∞]. H is a special element of ∆ defined by

H (t)=�_1,0,_{𝑖𝑖𝑖𝑖}𝑖𝑖𝑖𝑖𝑡𝑡 ≤_𝑡𝑡_{> 0}0,.

If X is a non empty set, F: X×X→∆ is called a probabilistic distance on X and F(x, y) is usually denoted by Fxy.

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Definition3: The ordered pair (X, F) is called a probabilistic metric space (shortly PM-space) if X is a non empty set and F is a probabilistic distance satisfying the following conditions: for all x, y, zϵX and t, s>0;

(i) Fxy(t)=1⇔x=y; (ii) Fxy(0)=0; (iii) Fxy= Fyx;

(iv) Fxz(t) =1; Fzy(s) =1⇒ Fxy(t+s)=1.

The ordered triple (X, F,∗) is called Menger space if (X, F) is a PM-space, ∗ is a t-norm and the following condition is also satisfied: for all x, y, zϵX and t, s>0;

(v) Fxy(t+s)≥ Fxz(t)∗ Fzy(s).

Proposition1 [4]: Let (X, d) be a metric space, then the metric d induces a distribution function F defined by Fxy(t)=H(t-d(x, y)) for all x, yϵX and t>0. If t-norm ∗ is a∗b=min {a, b} for all a,b ϵ[0,1] then (X, F,∗) is a Menger space. Further, (X, F,∗) is a complete Menger space if (X, d) is complete.

Definition4: Let (X, F,∗) be a Menger space and ∗ be a continuous t-norm.

(a) A sequence {𝑥𝑥_𝑛𝑛} in X is said to be converge to a point x in X (written 𝑥𝑥_𝑛𝑛→ 𝑥𝑥) iff for every є>0 and 𝜆𝜆 ∈(0, 1), there exists an integer 𝑛𝑛₀=𝑛𝑛₀(є,𝜆𝜆) such that 𝐹𝐹_𝑥𝑥_𝑛𝑛𝑥𝑥(є)>1-λ for all𝑛𝑛 ≥ 𝑛𝑛₀.

(b) A sequence {𝑥𝑥_𝑛𝑛} in X is said to be Cauchy if for every є>0 and 𝜆𝜆 ∈(0, 1), there exists an integer 𝑛𝑛₀=𝑛𝑛₀(є,𝜆𝜆) such that 𝐹𝐹_𝑥𝑥_𝑛𝑛_𝑥𝑥_𝑛𝑛₊_𝑝𝑝(є)>1-λ for all𝑛𝑛 ≥ 𝑛𝑛₀ and p>0.

(c) A Menger space in which every Cauchy sequence is convergent is said to be complete.

Definition5: Self maps A and B of a Menger space (X, F,∗) are said to be weakly compatible (or coincidently commuting) if they commute at their coincidence points, i.e. if Ax=Bx for some xϵX then ABx=BAx.

Definition6: Self maps A and B of a Menger space (X, F,∗) are said to be compatible if 𝐹𝐹_{𝐴𝐴𝐴𝐴𝑥𝑥}_𝑛𝑛_{𝐴𝐴𝐴𝐴𝑥𝑥}_𝑛𝑛(𝑡𝑡)→1 for all t>0, whenever {𝑥𝑥_𝑛𝑛} is a sequence in X such that 𝐴𝐴𝑥𝑥𝑛𝑛,𝐴𝐴𝑥𝑥_𝑛𝑛→ 𝑥𝑥 for some x in X as n→∞.

Definition6: Self maps A and B of a Menger space (X, F,∗) are called semi compatible if 𝐹𝐹_{𝐴𝐴𝐴𝐴𝑥𝑥}_𝑛𝑛_{𝐴𝐴𝑥𝑥}(𝑡𝑡)→1 for all t>0, whenever {𝑥𝑥_𝑛𝑛} is a sequence in X such that 𝐴𝐴𝑥𝑥𝑛𝑛,𝐴𝐴𝑥𝑥_𝑛𝑛→ 𝑥𝑥 for some x in X.

Lemma1 [6]: Let {𝑥𝑥_𝑛𝑛} be a sequence in a Menger space (X, F,∗) with continuous t-norm ∗ and𝑡𝑡 ∗t≥ 𝑡𝑡. If there exists a constant k ϵ(0, 1) such that𝐹𝐹_𝑥𝑥_𝑛𝑛_𝑥𝑥_𝑛𝑛₊₁(𝑘𝑘𝑡𝑡)≥ 𝐹𝐹_𝑥𝑥_{𝑛𝑛 −}₁_𝑥𝑥_𝑛𝑛(𝑡𝑡) for all t>0 and n=1, 2… Then {𝑥𝑥_𝑛𝑛} is a Cauchy sequence in X.

Lemma2 [6]: Let (X, F,∗) be a Menger space. If there exists k ϵ(0, 1) such that𝐹𝐹_{𝑥𝑥𝑥𝑥}(𝑘𝑘𝑡𝑡)≥ 𝐹𝐹_{𝑥𝑥𝑥𝑥}(𝑡𝑡) for all 𝑥𝑥,𝑥𝑥ϵ𝑋𝑋 and t>0, then 𝑥𝑥=𝑥𝑥.

Definition7 [8]: Let A and B be two self maps of a Menger space (X, F,∗), we say that A and B satisfy M.S. property, if there exists a sequence {𝑥𝑥_𝑛𝑛} in X such that𝐴𝐴𝑥𝑥_𝑛𝑛, 𝐴𝐴𝑥𝑥_𝑛𝑛 → 𝑥𝑥₀ for some 𝑥𝑥₀∈ 𝑋𝑋 as n→∞.

i.e. lim𝑛𝑛→∞𝐹𝐹𝐴𝐴𝑥𝑥𝑛𝑛𝑥𝑥0(𝑡𝑡) = lim𝑛𝑛→∞𝐹𝐹𝐴𝐴𝑥𝑥𝑛𝑛𝑥𝑥0(𝑡𝑡) = 1 for all 𝑡𝑡 ∈(0,∞).

Example1: Let X= [0,∞). Let 𝐹𝐹_{𝑥𝑥𝑥𝑥}(𝑡𝑡)=_𝑡𝑡_+|_{𝑥𝑥−𝑥𝑥}𝑡𝑡 _| for all t>0.

Define A, B: X→[0,∞) by 𝐴𝐴𝑥𝑥=𝑥𝑥

5 and 𝐴𝐴𝑥𝑥= 2𝑥𝑥

5 for all𝑥𝑥 ∈ 𝑋𝑋. Then lim𝑛𝑛→∞𝐴𝐴𝑥𝑥𝑛𝑛 = lim𝑛𝑛→∞𝐴𝐴𝑥𝑥𝑛𝑛 = 0, where 𝑥𝑥𝑛𝑛 = 1

𝑛𝑛.

MAIN RESULTS:

Theorem1: Let A and B be two weakly compatible self mappings of a Menger space (X, F,∗) with 𝑡𝑡 ∗t≥ 𝑡𝑡 such that (i) A and B satisfy the M.S. property,

(ii) For each 𝑥𝑥 ≠ 𝑥𝑥 in X and t>0 𝐹𝐹_{𝐴𝐴𝑥𝑥𝐴𝐴𝑥𝑥}(𝑞𝑞𝑡𝑡)≥min⁡{𝐹𝐹_{𝐴𝐴𝑥𝑥𝐴𝐴𝑥𝑥}(𝑡𝑡),𝐹𝐹_{𝐴𝐴𝑥𝑥𝐴𝐴𝑥𝑥}(𝑡𝑡)∗ 𝐹𝐹_{𝐴𝐴𝑥𝑥𝐴𝐴𝑥𝑥}(𝑡𝑡),𝐹𝐹_{𝐴𝐴𝑥𝑥𝐴𝐴𝑥𝑥}(𝑡𝑡)∗ 𝐹𝐹_{𝐴𝐴𝑥𝑥𝐴𝐴𝑥𝑥}(𝑡𝑡)}, where for 0<q<1

(iii) 𝐴𝐴(𝑋𝑋)⊂ 𝐴𝐴(𝑋𝑋).

(iv) 𝐴𝐴(𝑋𝑋) or A(X) is a complete subspace of X

Then A and B have a unique common fixed point.

Proof: Since A and B satisfy the M.S. property, there exists a sequence {𝑥𝑥_𝑛𝑛} in X such that 𝑙𝑙𝑖𝑖𝑙𝑙_{𝑛𝑛→∞}𝐴𝐴𝑥𝑥_𝑛𝑛 =

lim

n→∞Bxn=x0 for some𝑥𝑥₀∈ 𝑋𝑋. Suppose that B(X) is complete, then𝑙𝑙𝑖𝑖𝑙𝑙_{𝑛𝑛→∞}𝐴𝐴𝑥𝑥_𝑛𝑛 =𝐴𝐴𝐵𝐵 for some𝐵𝐵 ∈ 𝑋𝑋.

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Condition (ii) implies that

𝐹𝐹𝐴𝐴𝑥𝑥𝑛𝑛𝐴𝐴𝐵𝐵(𝑞𝑞𝑡𝑡)≥min⁡{𝐹𝐹𝐴𝐴𝑥𝑥𝑛𝑛𝐴𝐴𝐵𝐵(𝑡𝑡), [𝐹𝐹𝐴𝐴𝑥𝑥𝑛𝑛𝐴𝐴𝑥𝑥𝑛𝑛(𝑡𝑡)∗ 𝐹𝐹𝐴𝐴𝐵𝐵𝐴𝐴𝐵𝐵(𝑡𝑡)], [𝐹𝐹𝐴𝐴𝐵𝐵𝐴𝐴 𝑥𝑥𝑛𝑛(𝑡𝑡)∗ 𝐹𝐹𝐴𝐴𝑥𝑥𝑛𝑛𝐴𝐴𝐵𝐵(𝑡𝑡)]}

Letting limit 𝑛𝑛 →∞

𝐹𝐹𝐴𝐴𝐵𝐵𝐴𝐴𝐵𝐵(𝑞𝑞𝑡𝑡)≥min⁡{1,𝐹𝐹𝐴𝐴𝐵𝐵𝐴𝐴𝐵𝐵(𝑡𝑡)∗ 𝐹𝐹𝐴𝐴𝐵𝐵𝐴𝐴𝐵𝐵(𝑡𝑡),𝐹𝐹𝐴𝐴𝐵𝐵𝐴𝐴𝐵𝐵(𝑡𝑡)∗1}

= {1,𝐹𝐹_{𝐴𝐴𝐵𝐵𝐴𝐴𝐵𝐵}(𝑡𝑡), 𝐹𝐹_{𝐴𝐴𝐵𝐵𝐴𝐴𝐵𝐵}(𝑡𝑡)}

𝐹𝐹𝐴𝐴𝐵𝐵𝐴𝐴𝐵𝐵(𝑞𝑞𝑡𝑡)≥ 𝐹𝐹𝐴𝐴𝐵𝐵𝐴𝐴𝐵𝐵(𝑡𝑡)

∴ 𝐴𝐴𝐵𝐵=𝐴𝐴𝐵𝐵.

Now we show that 𝐴𝐴𝐵𝐵 is the common fixed point of A and B.

Since A and B are weakly compatible,

𝐴𝐴𝐴𝐴𝐵𝐵=𝐴𝐴𝐴𝐴𝐵𝐵=𝐴𝐴𝐴𝐴𝐵𝐵=𝐴𝐴𝐴𝐴𝐵𝐵

𝐹𝐹𝐴𝐴𝐵𝐵𝐴𝐴𝐴𝐴𝐵𝐵(𝑞𝑞𝑡𝑡)≥min{𝐹𝐹𝐴𝐴𝐵𝐵𝐴𝐴𝐴𝐴𝐵𝐵(𝑡𝑡),𝐹𝐹𝐴𝐴𝐵𝐵𝐴𝐴𝐵𝐵(𝑡𝑡)∗ 𝐹𝐹𝐴𝐴𝐴𝐴𝐵𝐵𝐴𝐴𝐴𝐴𝐵𝐵(𝑡𝑡),𝐹𝐹𝐴𝐴𝐴𝐴𝐵𝐵𝐴𝐴𝐵𝐵(𝑡𝑡)∗ 𝐹𝐹𝐴𝐴𝐵𝐵𝐴𝐴𝐴𝐴𝐵𝐵(𝑡𝑡)}

= min{𝐹𝐹_{𝐴𝐴𝐵𝐵𝐴𝐴𝐴𝐴𝐵𝐵}(𝑡𝑡),𝐹𝐹_{𝐴𝐴𝐵𝐵𝐴𝐴𝐵𝐵}(𝑡𝑡)∗ 𝐹𝐹_{𝐴𝐴𝐴𝐴𝐵𝐵𝐴𝐴𝐴𝐴𝐵𝐵}(𝑡𝑡),𝐹𝐹_{𝐴𝐴𝐴𝐴𝐵𝐵𝐴𝐴𝐵𝐵}(𝑡𝑡)∗ 𝐹𝐹_{𝐴𝐴𝐵𝐵𝐴𝐴𝐴𝐴𝐵𝐵}(𝑡𝑡)}

= min{(𝑡𝑡), 1,𝐹𝐹_{𝐴𝐴𝐵𝐵𝐴𝐴𝐴𝐴𝐵𝐵}(𝑡𝑡)∗ 𝐹𝐹_{𝐴𝐴𝐵𝐵𝐴𝐴𝐴𝐴𝐵𝐵}(𝑡𝑡)}

𝐹𝐹𝐴𝐴𝐵𝐵𝐴𝐴𝐴𝐴𝐵𝐵(𝑞𝑞𝑡𝑡)≥ 𝐹𝐹𝐴𝐴𝐵𝐵𝐴𝐴𝐴𝐴𝐵𝐵(𝑡𝑡)

∴ 𝐴𝐴𝐴𝐴𝐵𝐵=𝐴𝐴𝐵𝐵.

Hence 𝐴𝐴𝐵𝐵 is the common fixed point of A and B.

Even, if we assume that A(X) is complete and proceed as above the result will be same.

Now it is left to prove that the fixed point is unique.

Let 𝑥𝑥₀and 𝑥𝑥₀ be two common fixed points of A and B. then

𝐹𝐹𝑥𝑥0𝑥𝑥0(𝑞𝑞𝑡𝑡) =𝐹𝐹𝐴𝐴𝑥𝑥0𝐴𝐴𝑥𝑥0(𝑞𝑞𝑡𝑡)

≥min{ 𝐹𝐹_{𝐴𝐴𝑥𝑥}₀_{𝐴𝐴𝑥𝑥}₀(𝑡𝑡), [𝐹𝐹_{𝐴𝐴𝑥𝑥}₀_{𝐴𝐴𝑥𝑥}₀(𝑡𝑡)∗ 𝐹𝐹_{𝐴𝐴𝑥𝑥}₀_{𝐴𝐴𝑥𝑥}₀(𝑡𝑡)], [𝐹𝐹_{𝐴𝐴𝑥𝑥}₀_{𝐴𝐴𝑥𝑥}₀(𝑡𝑡)∗ 𝐹𝐹_{𝐴𝐴𝑥𝑥}₀_{𝐴𝐴𝑥𝑥}₀(𝑡𝑡)]}

= min{ (𝑡𝑡), [𝐹𝐹_𝑥𝑥₀_𝑥𝑥₀(𝑡𝑡)∗ 𝐹𝐹_𝑥𝑥₀_𝑥𝑥₀(𝑡𝑡)], [𝐹𝐹_𝑥𝑥₀_𝑥𝑥₀(𝑡𝑡)∗ 𝐹𝐹_𝑥𝑥₀_𝑥𝑥₀(𝑡𝑡)]}

𝐹𝐹𝑥𝑥0𝑥𝑥0(𝑞𝑞𝑡𝑡)≥ 𝐹𝐹𝑥𝑥0𝑥𝑥0(𝑡𝑡).

∴ 𝑥𝑥0=𝑥𝑥0.

Corollary1: Let A and B be two non compatible weakly compatible self mappings of a Menger space (X, F,∗) with

𝑡𝑡 ∗ 𝑡𝑡 ≥ 𝑡𝑡 such that

(i) 𝐹𝐹_{𝐴𝐴𝑥𝑥𝐴𝐴𝑥𝑥}(𝑞𝑞𝑡𝑡)≥min{ 𝐹𝐹_{𝐴𝐴𝑥𝑥𝐴𝐴𝑥𝑥}(𝑡𝑡),𝐹𝐹_{𝐴𝐴𝑥𝑥𝐴𝐴𝑥𝑥}(𝑡𝑡)∗ 𝐹𝐹_{𝐴𝐴𝑥𝑥𝐴𝐴𝑥𝑥}(𝑡𝑡),𝐹𝐹_{𝐴𝐴𝑥𝑥𝐴𝐴𝑥𝑥}(𝑡𝑡)∗ 𝐹𝐹_{𝐴𝐴𝑥𝑥𝐴𝐴𝑥𝑥}(𝑡𝑡)} (ii) 𝐴𝐴(𝑋𝑋)⊂ 𝐴𝐴(𝑋𝑋).

If B(X) or A(X) is complete subspace of X, then A and B have a unique common fixed point.

Theorem2: Let A and B be two weakly compatible self mappings of a Menger space (X, F,∗) with 𝑡𝑡 ∗t≥ 𝑡𝑡 such that (i) A and B satisfy the M.S. property,

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(iv) 𝐴𝐴(𝑋𝑋) or A(X) is a complete subspace of X

Then A and B have a unique common fixed point.

Proof: Since A and B satisfy the M.S. property, there exists a sequence {𝑥𝑥_𝑛𝑛} in X such that 𝑙𝑙𝑖𝑖𝑙𝑙_𝑛𝑛→∞𝐴𝐴𝑥𝑥𝑛𝑛 =

lim

n→∞Bxn=x0 for some𝑥𝑥₀∈ 𝑋𝑋. Suppose that B(X) is complete, then lim_𝑛𝑛→∞𝐴𝐴𝑥𝑥_𝑛𝑛=𝐴𝐴𝐵𝐵 for some𝐵𝐵 ∈ 𝑋𝑋. ∴lim𝑛𝑛→∞𝐴𝐴𝑥𝑥𝑛𝑛=𝐴𝐴𝐵𝐵 by (i).

Now, we claim that𝐴𝐴𝐵𝐵=𝐴𝐴𝐵𝐵.

If𝐴𝐴𝐵𝐵 ≠ 𝐴𝐴𝐵𝐵, then 𝐹𝐹_{𝐴𝐴𝐵𝐵𝐴𝐴𝐵𝐵}(𝑡𝑡) < 1 for all t.

Condition (ii) implies that

𝐹𝐹𝐴𝐴𝑥𝑥𝑛𝑛𝐴𝐴𝐵𝐵(𝑞𝑞𝑡𝑡)≥min{ 𝐹𝐹𝐴𝐴𝑥𝑥𝑛𝑛𝐴𝐴𝐵𝐵(𝑡𝑡),

[𝐹𝐹_{𝐴𝐴𝑥𝑥 𝑛𝑛 𝐴𝐴𝑥𝑥𝑛𝑛}(𝑡𝑡)+𝐹𝐹𝐴𝐴𝐵𝐵𝐴𝐴𝐵𝐵(𝑡𝑡)]

2 ,

[𝐹𝐹_{𝐴𝐴𝑥𝑥 𝑛𝑛 𝐴𝐴𝐵𝐵}(𝑡𝑡)+𝐹𝐹_{𝐴𝐴𝐵𝐵𝐴𝐴 𝑥𝑥𝑛𝑛}(𝑡𝑡)]

2 }

Letting limit 𝑛𝑛 →∞

𝐹𝐹𝐴𝐴𝐵𝐵𝐴𝐴𝐵𝐵(𝑞𝑞𝑡𝑡)≥min{ 1,[1+𝐹𝐹𝐴𝐴𝐵𝐵𝐴𝐴𝐵𝐵₂ (𝑡𝑡)],[1+𝐹𝐹𝐴𝐴𝐵𝐵𝐴𝐴𝐵𝐵₂ (𝑡𝑡)]}

𝐹𝐹𝐴𝐴𝐵𝐵𝐴𝐴𝐵𝐵(𝑞𝑞𝑡𝑡)≥[1+𝐹𝐹𝐴𝐴𝐵𝐵𝐴𝐴𝐵𝐵₂ (𝑡𝑡)]

≥ 𝐹𝐹_{𝐴𝐴𝐵𝐵𝐴𝐴𝐵𝐵}(𝑡𝑡) for all t.

Because, if [1+𝐹𝐹𝐴𝐴𝐵𝐵𝐴𝐴𝐵𝐵₂ (𝑡𝑡)]<𝐹𝐹_{𝐴𝐴𝐵𝐵𝐴𝐴𝐵𝐵}(𝑡𝑡) then𝐹𝐹_{𝐴𝐴𝐵𝐵𝐴𝐴𝐵𝐵}(𝑡𝑡) > 1, which is a contradiction.

Hence𝐴𝐴𝐵𝐵=𝐴𝐴𝐵𝐵.

Let us now prove that 𝐴𝐴𝐵𝐵 is the common fixed point of A and B.

Suppose that𝐴𝐴𝐵𝐵 ≠ 𝐴𝐴𝐴𝐴𝐵𝐵. Since A and B are weakly compatible, 𝐴𝐴𝐴𝐴𝐵𝐵=𝐴𝐴𝐴𝐴𝐵𝐵 and therefore

𝐴𝐴𝐴𝐴𝐵𝐵=𝐴𝐴𝐴𝐴𝐵𝐵 and𝐴𝐴𝐴𝐴𝐵𝐵=𝐴𝐴𝐴𝐴𝐵𝐵.

Then by (ii), we have

𝐹𝐹𝐴𝐴𝐵𝐵𝐴𝐴𝐴𝐴𝐵𝐵(𝑞𝑞𝑡𝑡)≥min{𝐹𝐹𝐴𝐴𝐵𝐵𝐴𝐴𝐴𝐴𝐵𝐵(𝑡𝑡),[𝐹𝐹𝐴𝐴𝐵𝐵𝐴𝐴𝐵𝐵(𝑡𝑡)+₂𝐹𝐹𝐴𝐴𝐴𝐴𝐵𝐵𝐴𝐴𝐴𝐴𝐵𝐵 (𝑡𝑡)],[𝐹𝐹𝐴𝐴𝐵𝐵𝐴𝐴𝐴𝐴𝐵𝐵 (𝑡𝑡)+₂𝐹𝐹𝐴𝐴𝐴𝐴𝐵𝐵𝐴𝐴𝐵𝐵 (𝑡𝑡)]}

= min{𝐹𝐹_{𝐴𝐴𝐵𝐵𝐴𝐴𝐴𝐴𝐵𝐵}(𝑡𝑡),[1+𝐹𝐹𝐴𝐴𝐴𝐴𝐵𝐵𝐴𝐴𝐴𝐴𝐵𝐵₂ (𝑡𝑡)],[𝐹𝐹𝐴𝐴𝐵𝐵𝐴𝐴𝐴𝐴𝐵𝐵 (𝑡𝑡)+₂𝐹𝐹𝐴𝐴𝐴𝐴𝐵𝐵𝐴𝐴𝐵𝐵(𝑡𝑡)]}

= min{𝐹𝐹_{𝐴𝐴𝐵𝐵𝐴𝐴𝐴𝐴𝐵𝐵}(𝑡𝑡),[1+𝐹𝐹𝐴𝐴𝐴𝐴𝐵𝐵𝐴𝐴𝐴𝐴𝐵𝐵 (𝑡𝑡)]

2 ,𝐹𝐹𝐴𝐴𝐵𝐵𝐴𝐴𝐴𝐴𝐵𝐵(𝑡𝑡)}

= 𝐹𝐹_{𝐴𝐴𝐵𝐵𝐴𝐴𝐴𝐴𝐵𝐵}(𝑡𝑡)

Because𝐹𝐹_{𝐴𝐴𝐵𝐵𝐴𝐴𝐴𝐴𝐵𝐵}(𝑡𝑡) ≤ [1+𝐹𝐹𝐴𝐴𝐴𝐴𝐵𝐵𝐴𝐴𝐴𝐴𝐵𝐵 (𝑡𝑡)] 2

Thus𝐹𝐹_{𝐴𝐴𝐵𝐵𝐴𝐴𝐴𝐴𝐵𝐵}(𝑡𝑡)≥ 𝐹𝐹_{𝐴𝐴𝐵𝐵𝐴𝐴𝐴𝐴𝐵𝐵}(𝑡𝑡) for all t.

This implies that𝐴𝐴𝐴𝐴𝐵𝐵=𝐴𝐴𝐵𝐵.

Hence 𝐴𝐴𝐵𝐵 is the common fixed point of A and B.

Even, if we assume that A(X) is complete and proceed as above the result will be same.

Finally we show that the fixed point is unique.

(5)

Mar.-2012, Page: 1261-1265

𝐹𝐹𝑥𝑥0𝑥𝑥0(𝑞𝑞𝑡𝑡) =𝐹𝐹𝐴𝐴𝑥𝑥0𝐴𝐴𝑥𝑥0(𝑞𝑞𝑡𝑡)

≥min{𝐹𝐹_{𝐴𝐴𝑥𝑥}₀_{𝐴𝐴𝑥𝑥}₀(𝑡𝑡),[𝐹𝐹𝐴𝐴𝑥𝑥0𝐴𝐴𝑥𝑥0(𝑡𝑡)+₂𝐹𝐹𝐴𝐴𝑥𝑥0𝐴𝐴𝑥𝑥0(𝑡𝑡)],[𝐹𝐹𝐴𝐴𝑥𝑥0𝐴𝐴𝑥𝑥0(𝑡𝑡)+₂𝐹𝐹𝐴𝐴𝑥𝑥0𝐴𝐴𝑥𝑥0(𝑡𝑡)]}

= min{𝐹𝐹_𝑥𝑥₀_𝑥𝑥₀(𝑡𝑡),[𝐹𝐹𝑥𝑥0𝑥𝑥0(𝑡𝑡)+₂𝐹𝐹𝑥𝑥0𝑥𝑥0(𝑡𝑡)],[𝐹𝐹𝑥𝑥0𝑥𝑥0(𝑡𝑡)+₂𝐹𝐹𝑥𝑥0𝑥𝑥0(𝑡𝑡)]}

= min{𝐹𝐹_𝑥𝑥₀_𝑥𝑥₀(𝑡𝑡), 1,𝐹𝐹_𝑥𝑥₀_𝑥𝑥₀(𝑡𝑡)}

⇒ 𝐹𝐹𝑥𝑥0𝑥𝑥0(𝑞𝑞𝑡𝑡)≥ 𝐹𝐹𝑥𝑥0𝑥𝑥0(𝑡𝑡), for all t.

∴ 𝑥𝑥0=𝑥𝑥0.

Hence the theorem.

Corollary2:Let A and B be two non-compatible weakly compatible self mappings of a Menger space (X, F,∗) with

𝑡𝑡 ∗t≥ 𝑡𝑡 such that

(i) 𝐹𝐹_{𝐴𝐴𝑥𝑥𝐴𝐴𝑥𝑥}(𝑞𝑞𝑡𝑡)≥min⁡{𝐹𝐹_{𝐴𝐴𝑥𝑥𝐴𝐴𝑥𝑥}(𝑡𝑡),[𝐹𝐹𝐴𝐴𝑥𝑥𝐴𝐴𝑥𝑥(𝑡𝑡)+𝐹𝐹𝐴𝐴𝑥𝑥𝐴𝐴𝑥𝑥(𝑡𝑡)]

2 ,

[𝐹𝐹𝐴𝐴𝑥𝑥𝐴𝐴𝑥𝑥(𝑡𝑡)+𝐹𝐹𝐴𝐴𝑥𝑥𝐴𝐴𝑥𝑥(𝑡𝑡)]

2 }, where for 0<q<1. (ii) 𝐴𝐴(𝑋𝑋)⊂ 𝐴𝐴(𝑋𝑋).

If B(X) or A(X) is complete subspace of X, then A and B have a unique common fixed point.

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[4] V. M. Sehgal, A. T. Bharucha-Reid, Fixed point of contraction mapping on PM spaces, Math. Systems Theory, 6(1972), 97-100.

[5] S. Sessa, On a weak commutative condition in fixed point consideration, Publ. Inst. Math., 32(1982), 146-153.

[6] B. Singh, S. Jain, A fixed point theorem in Menger Space through weak compatibility, J. Math. Anal. Appl., 301(2005), 439-448.

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