SOME NOVEL FIXED POINT RESULTS FOR (Ω, ∆)-WEAK CONTRACTION CONDITION IN COMPLETE FUZZY METRIC SPACES

ABSTRACT In the present article, some fixed point theorems are investigated for two pairs of weakly compatible maps through (Ω, ∆)-type weak contractive maps in the framework of fuzzy metric spaces. The results studied in this workpiece are generalizations of some recent results existing in literature. Also, some illustrative examples are presented in last section to check the authenticity of our results.


INTRODUCTION
Contraction principle given by Banach (1922) is the most eminent result in the era of metrical fixed point theory.Though this principle requires the continuity of the mapping, still it works as the back-bone even for the recent results in different metric spaces.An open question on the continuity of the mapping in Banach principle is answered by many authors.In 1968, Kannan (1968) settled this problem in a robust way by introducing the following inequality: Later on, Rakotch (1962), Boyd & Wong (1969) extended the contraction inequality due to Banach (1922)  where φ : [0, ∞] → [0, ∞] is a non-decreasing continuous function such that φ (t) vanishes at t = 0.
On the other hand, Alber & Guerre-Delabriere (1997) introduced a modified contractive condition in Hilbert spaces which was further elaborated by Rhoades (2001) as follows: If a mapping T : U → U satisfies the following condition: then T possesses a fixed point.Zhang & Song (2009) proved unique common fixed point results for hybrid generalized ∆-weak contractive mappings in complete metric spaces whereas Doric ( 2009) established some related theorems using control functions.This work was an extension to the results due to Zhang & Song (2009).Then, Murthy et al. (2015) proved some results using weak contractive condition on two pairs of discontinuous weakly compatible mappings.
In 1975, the concept of fuzzy metric space is initiated by Kramosil & Michalek (1975) with the concept of t-norm.Later on, George & Veeramani (1994) extended the notion of fuzzy metric space by defining the Hausdorff topology in this framework.After that, Mihet (2008) introduced the fuzzy version of Banach's result and introduced fuzzy ψ-contractive type mapping in non-Archimedean fuzzy environment.A key distinction between a fuzzy metric and a classical metric is that the latter contains a parameter in its definition.This concept has been used successfully in engineering applications including colour picture filtering and perceived colour disparities.(For details, one can refer to the study of Camarena et al. (2008), Camarena et al. (2010), Morillas et al. (2009), Morillas et al. (2007), Morillas et al. (2005), Morillas et al. (2008a), Morillas et al. (2008b)).
It has been demonstrated, in particular, that the class of topological spaces that are fuzzy metrizable matches with the class of topological spaces that may be metrized and then some traditional metric completeness and compactness theorems have been modified for fuzzy metric spaces.
(See George & Veeramani (1995), Gregori & Romaguera (2000)).However, compared to the traditional theories of metric completion, the theory of fuzzy metric completion is significantly distinct.In actuality, some fuzzy metric spaces are non-completable (See Gregori (2002)).The example below demonstrates the existence of a fuzzy metric space that forbids fuzzy metric completion.
We firstly claim that (M, * ) is a fuzzy metric on W .
Observe that the first four characteristics are nearly evident.(for m, n ≥ 3): 4. For every u, v ∈ W ; M(u, v, .) is a continuous function on (0, ∞).
Also, a straightforward calculation reveals that, for every m, n, p ≥ 3 and s,t > 0; Finally, the relationships listed below are simple: Thus, for every u, v, w ∈ W and s,t > 0, we have Hence, (M, * ) is a fuzzy metric on W .
Next, we assert that in the fuzzy metric space (W, M, * ); {u m } ∞ m=3 is a Cauchy sequence.
For fixed ε ∈ (0, 1) and s > 0, there exists m 0 ≥ 3 such that m=3 and {v m } ∞ m=3 do not converge in W w.r.t. the topology ς M induced by (M, * ).Actually, ς M is the discrete topology on W as for every m ≥ 3 and each s > 0, we have for , s = {u m } and , s = {v m }.
To demonstrate the two prior equality claims, it is sufficient to observe that for m, n ≥ 3, with m ̸ = n, and s > 0, we have . Similarly, and for m, n ≥ 3 and s > 0, we have Hence, (W, M, * ) is not complete.
The main intent of our work is to extend and generalize (∆, Ω)-weak contraction due to Murthy et al. (2015) to fuzzy metric spaces.The authenticity of the results is further verified with some illustrative examples.
Theorem 1. [Murthy et al. (2015)] Let (U, d) be a metric space equipped with completeness, and C, D, E and T be the self mappings defined on U satisfying for all ρ, σ ∈ U, with ρ ̸ = σ and and and T have a unique common fixed point in U.
* is a continuous t-norm if it satisfies the postulates stated below: 1. * is commutative as well as associative; 2. * is a continuous binary operation; Definition 2 (George & Veeramani (1994)).
Let (U, M, * ) be a FMS.Then, 2. any sequence {ρ n } in U is named a Cauchy sequence if ∀ t > 0 and for each 3. A fuzzy metric space in which every Cauchy sequence convergent in it, is named as complete fuzzy metric space.

MAIN RESULTS
Theorem 2. Let (U, M, * ) be a complete fuzzy metric space, and let Θ, D, E and T : U → U be four mappings satisfying for all ρ, σ ∈ U, with ρ ̸ = σ and (Θ, E) and (D, T ) are weakly compatible pairs, where Ω : [0, 1] → [0, 1] is a non-decreasing and continuous function with Then Θ, D, E and T possess a unique common fixed point.
Taking limit n tends to ∞ in ( 18) and using ( 19), we get which gives, This is impossible with ∆ function, therefore Next, we claim that the sequence {σ n } is Cauchy.
For this, it is sufficient to prove that the sub-sequence {σ 2n } of the sequence {σ n } is Cauchy.Let us assume in a contrary manner that {σ 2n } is not a Cauchy sequence.Consider the sequences {2n(k)} and {2m(k)} such that 2n(k) > 2m(k) > 2k for k ∈ N and Choose 2n(k) to be the smallest index in such a way that (21) holds true. Then, , where Taking k → ∞ in ( 23), we get As ∆ is discontinuous at t = 1 where ∆(t) = 0 and ∆(t) < 1 ∀ t < 1, the last term in ( 29) vanishes, which eventually lead to a contradiction.
Thus, {σ n } is a Cauchy sequence.By the property of completeness, this sequence converges to some point ζ (say) in U. Consequently, its sub-sequences also converges to ζ in U i.e.
The result below is obtained by taking Ω = I(identity function): Corollary 1.Let (U, M, * ) be a fuzzy metric space equipped with completeness property.Let Θ, D, E and T : U → U be self-mappings holding following inequality: where ρ, σ ∈ U, ρ ̸ = σ ,  ) and ∆ is discontinuous at the point t = 1 with ∆(t) = 0.
Then Θ, D, E and T possess a unique common fixed point in U.
Then Θ and D possess a unique fixed point in U.
If the aforementioned condition another result will be deduced as follows: Theorem 4. Let (U, M, * ) be a fuzzy metric space equipped with completeness.Let Θ, D, E and T be self-mappings defined on U such that they satisfy the following inequality: where ρ, σ ∈ U, ρ ̸ = σ , Then Θ, D possess a unique common fixed point in U.