Fixed Point Results by Altering Distances in Fuzzy Metric Spaces

We establish fixed point theorems in complete fuzzy metric space by using notion of altering distance, initiated by Khan et al. [Bull. Austral. Math. Soc. 30 (1984), 1-9]. Also, we find an affirmative answer in fuzzy metric space to the problem of Sastry [TamkangJ. Math., 31(3) (2000), 243-250].


Introduction
The concept of fuzzy sets was introduced by Zadeh.With the concept of fuzzy sets, the fuzzy metric space was introduced by Kramosil and Michalek [1].Grabiec [2] proved the contraction principle in the setting of fuzzy metric space.Also, George and Veermani [3] modified the notion of fuzzy metric space with the help of continuous t-norm.Fuzzy set theory has applications in applied sciences such as neural network theory, stability theory, mathematical programming, modelling theory, engineering sciences, medical sciences (medical genetics, nervous system), image processing, control theory and communication.
Boyd and Wong [4] introduced the notion of Φ-contractions.In 1997, Alber and Guerre-Delabriere [5] defined the ϕ-weak contraction which was a generalization of Φ-contractions.Many researchers studied the notion of weak contractions on different settings which generalized the Banach Contraction Mapping Principle.Another interesting and significant fixed point results as a generalization of Banach Contraction Principle have been established by using the notion of alerting distance function, a new notion propounded by Khan et al. [6].Altering Distance Functions are control functions which alter the distance between two points in a metric space.For more details, we refer to [6]- [12].
Sastry et al. [13] proved the following: A S and ( ) , B T be weakly commuting pairs of self mappings of a complete me- tric space ( ) In this paper, we prove common fixed point theorems which provide an affirmative answer to the above question on existence of fixed point in fuzzy metric spaces.

Preliminaries
To set up our results in the next section, we recall some basic definitions.
is continuous, for all x, y, z ∊ X and s, t > 0.
We note that ( ) , , M x y ⋅ is non-decreasing for all x, y ∊ X.

Definition 2.4 Let ( )
, , X M * be a fuzzy metric space.A sequence { } n x is said to be 1) G-Cauchy (i.e., Cauchy sequence in sense of Grabiec [5]) if = for all t > 0 and each p > 0. 2

Definition 2.5 [16]-[18]
A pair of self mappings (f, g) on fuzzy metric space ( ) for some z in X. Definition 2.6 An altering distance function or control function is a function such that the following axioms hold: 1) ψ is monotonic increasing and continuous; x y X ∈ , 0 t > and if for a number ( ) Lemma 2.2 [5].Let (X, M, *) be fuzzy metric space and { } n y be a sequence in X.If there exists a number ( ) , , , ,

Definition 2.7 [13]
A pair of self mappings ( ) , f g on fuzzy metric space ( ) for some z in X.

Main Results
Theorem 3.1 Let ( ) A S and ( ) , B T be weakly commuting pairs of self mappings of a complete fuzzy metric space ( )

= =
This can be done by virtue of (3.1).Now, we prove that { } n y is a Cauchy sequence.For ( ) , , , ,

M y y q t M y y t M y y qt
, , , , as ψ is decreasing, so we have  .AAu ASu SAu SSu = = = Also, B and T are weakly commuting, we get (3.9) .BBw BTw TBw TTw = = = Finally, we show that AAu Au  which gives that .AAu Au = Therefore Au is a common fixed point of A and S. Similarly, we can show that BBw Bw = and since , TBw BBw = we have Bw , a common fixed point of B and T. Finally, , Au Bw = we have Au as a common fixed point for A, B, S and T. The uniqueness follows from 2) and hence the theorem.Theorem 3.2 Let ( ) , A S and ( ) , B T be weakly commuting pairs of self mappings of a complete fuzzy which gives that AAz Az = .Therefore, Az is a common fixed point of A and S. Similarly, we can show that BBw Bw = and since TBw BBw = , we have Bw , a common fixed point of B and T. Finally, Az Bw = , we have Az as a common fixed point for A, B, S and T. The uniqueness follows from 2) and hence the theorem.

Definition 2.1 [14] A
fuzzy set A in X is a function with domain X and values in [0, 1].

Definition 2.2 [14]
A binary operation *: We claim that lim n SAx Az n y is a Cauchy sequence in X.Since X is complete, there is a point z in X such as n → ∞ .Now, suppose that A and S are ψ -compatible then we have (3.5)