INTERPOLATIVE GENERALISED MEIR-KEELER CONTRACTION

: Introduction/purpose: The aim of this paper is to introduce the notion of an interpolative generalised Meir-Keeler contractive condition for a pair of self maps in a fuzzy metric space, which enlarges, unifies and generalizes the Meir-Keeler contraction which is for only one self map. Using this, we establish a unique common fixed point theorem for two self maps through weak compatibility. The article includes an example, which shows the validity of our results. Methods: Functional analysis methods with a Meir-Keeler contraction. Results: A unique fixed point for self maps in a fuzzy metric space is obtained. Conclusions: A fixed point of the self maps is obtained.


Introduction
In 1965 L. Zadeh (Zadeh, 1965) introduced the theory of fuzzy sets. Later on, in 1978, the concept of a fuzzy metric space was introduced by Kramosil and Michalek in (Kramosil & Michalek, 1975), which was modified by George and Veeramani (George & Veeramani, 1994) in order to obtain a Hausdorff topology for this class of fuzzy metric spaces. Then in year 1988, Grabiec (Grabiec, 1988) gave a fuzzy version of the Banach (Banach, 1922) contraction principle in the setting of a fuzzy metric space. Over the past years, various authors have tried to generalize the fixed point theorem by modifying and varying the contractive condition, see, e.g., (Gregori & Sapena, 2002), (Jain & Jain, 2021), (Mihet, 2008), (Saha et al, 2016), (Tirado, 2012) and (Wardowski, 2013) in the sense of George and Veeramani. In 2019, Zheng and Wang (Zheng & Wang, 2019) introduced a Meir-Keeler contraction in the setting of a fuzzy metric (Schweizer & Sklar, 1983) space and proved some fixed point results for a self map.
Inspired with the interpolative theory, Karapinar and Agrawal (Karapinar & Agarwal, 2019) introduced the notion of an interpolative Rus-Reich-Ćirić type contraction via the simulation function in a metric space. Motivated by this paper, we introduce an interpolative generalised Meir-Keeler contraction (Gregori & Minana, 2014) for two self maps (Rhoades, 2001) in the setting of a fuzzy metric space, which enlarges, unifies and generalizes the existing Meir-Keelar contraction in a fuzzy metric (Mihet, 2010) space through weak compatibility (Banach, 1922).
The structure of the paper is as follows: After the preliminaries, we introduce a interpolative generalised Meir-Keeler contraction in the setting of a fuzzy metric space. Then we study the Meir-Keeler contractive mapping due to Zheng and Wang (Zheng & Wang, 2019). In section 4, the existence of a unique common fixed point of an interpolative generalised Meir-Keeler contractive mapping has been established through weak compatibility followed by an example. Preliminaries DEFINITION 1. (George & Veeramani, 1994) The well-known examples of the t-norm are the minimum t-norm * m , a * m b = min{a, b} written as * m and the product t-norm * , a * b = ab.
DEFINITION 2. (George & Veeramani, 1994) A fuzzy metric space is an ordered triple (X, M, * ) such that X is a (nonempty) set, * is a continuous t-norm and M is a fuzzy set on X × X × (0, +∞) satisfying the following conditions, for all x, y, z ∈ X and t, s > 0; Note that in view of the condition (GV2) we have M (x, x, t) = 1, for all x ∈ X and t > 0 and M (x, y, t) < 1, for all x ̸ = y and t > 0. The following notion was introduced by George and Veeramani in (George & Veeramani, 1994). DEFINITION 3. (George & Veeramani, 1994) A sequence {x n } in a fuzzy metric space (X, M, * ) is said to be M-Cauchy, or simply Cauchy, if for each ϵ ∈ (0, 1) and each t > 0 there exists an n 0 ∈ N, such that M (x n , x m , t) > 1 − ϵ, for all n, m ≥ n 0 . Equivalently, {x n } is Cauchy if lim n,m→+∞ M (x n , x m , t) = 1, for all t > 0. LEMMA 1. (Grabiec, 1988) Let (X, M, * ) be a fuzzy metric space. Then M (x, y, .) is non-decreasing for all x, y ∈ X. THEOREM 1. (George & Veeramani, 1994) Let (X, M, * ) be a fuzzy metric space. A sequence {x n } n∈N in X converges to x ∈ X if and only if lim n→∞ M (x n , x, t) = 1. LEMMA 2. (Saha et al, 2016) If * is a continuous t-norm {α n }, {β n }, {γ n } are sequences such that α n → α, β n → β and γ n → γ as n → +∞ then DEFINITION 5. (Zheng & Wang, 2019) Let (X, M, * ) be a fuzzy metric space. A mapping f : X → X is said to be a fuzzy Meir-Keeler contractive mapping with respect to δ ∈ △ if the following condition holds: for all x, y ∈ X, t > 0. DEFINITION 6. (Jain et al, 2009) Two self maps f and g in a fuzzy metric space (X, M, * ) are said to be weakly compatible if they commute at their coincidence points i.e. for x ∈ X, f x = gx = y implies gy = f y.
Interpolative generalised Meir-Keeler contraction DEFINITION 7. Let (X, M, * ) be a fuzzy metric space. A pair (f, g) of self maps in X is said to be an interpolative generalised Meir-Keeler contractive if there exists α, β ∈ [0, 1) with α + β < 1 and for all x, y ∈ X, t > 0 Thus for x ̸ = y.
REMARK 2. Taking g = I, the identity map in equation (2)we obtain which is an interpolative generalised Meir-Keeler contraction, for a self map f . (5) which is precisely the Meir-Keeler contraction, for a self map given by Zheng and Wang (Zheng & Wang, 2019).

Main results
Our first new result is the next one: THEOREM 2. : Let f and g be self maps in a fuzzy metric space (X, M, * ) satisfying the following conditions: (4.11) f (X) ⊆ g(X); (4.12) The pair (f, g) is an interpolative generalised Meir-Keeler contraction; The pair (f, g) is weakly compatible.

Then f and g have a unique common fixed point in X if and only if there
Proof. : Suppose the pair (f, g) has a unique common fixed point Construct a sequence {y n }, by defining f x n = gx n+1 = y n , for n = 0, 1, 2, . . .. First we show that if the two maps f and g have a common fixed point then it is unique. Let u and v be two common fixed points of f and g. Then which is not true as the left hand quantity is less than 1.So u = v. Thus, if the pair (f, g) has a common fixed point then it is unique.
Step 1 To see the existence of a common fixed point of the self maps f and g, we consider the following cases. CASE I Suppose any two terms of the sequence {y n } are equal i. e. for some n ∈ N, y n = y n+1 . As is a point of coincidence of the pair (f, g). As the pair (f, g) is weakly compatible we have f z = gz. Now we show that f z = z. Suppose, if possible on the contrary, that f z ̸ = z so f z ̸ = f x n+1 . By using equation (3) we have , for all t > 0 implies (M (z, f z, t)) (α+β) > 1, which is not possible. Hence f z = z. Therefore, z is a common fixed point of the pair (f, g) in this case. So we can assume the consecutive terms of the sequence {y n } are distinct.
i. e. M (y n , y n+1 , t) < M (y n , y n+1 , t), which is not possible. So this case does not arise.
Taking the limit supremum on both sides of equation (11), and using the properties of M and * , and by lemma 3, we obtain Since M is bounded with the range in (0, 1], continuous and nondecreasing in the third variable t, it follows from lemma 3, that h 1 is continuous from the left. Therefore, for λ → 0 , we obtain Let Again, for all n ≥ 1, λ > 0 (10) Taking the limit infimum as n → +∞ in the above inequality, we obtain Since M is bounded with the range in (0, 1] , continuous and nondecreasing in the third variable t, it follows from lemma 3, that h 2 is continuous from the right. So letting λ → 0 , we obtain Combining the inequalities (12) and (13), we get STEP 3 In this step, we show that lim n→+∞ M (y p(n) , y q(n) , t 0 ) = 1 − η. From equation (10) we have Also for all n ≥ 1 and λ > 0 we have M (y p(n) , y q(n) , t 0 + 2λ) ≥ M (y p(n) , y p(n)−1 , λ) * M (y p(n)−1 , y q(n)−1 , t 0 ) * * M (y q(n)−1 , y q(n) , λ) Taking the limit infimum as n → +∞ in the above inequality, using (9), (14) and the properties of M and * and by lemma 2, we obtain (14) (16) Since M is bounded with the range in (0, 1], continuous and nondecreasing in the third variable t, it follows from lemma 3 that lim n→+∞ M (y p(n) , y q(n) , t 0 ) is a continuous function of t from the right.
Therefore, for λ → 0 , we obtain Combining inequalities (15) and (17), we get STEP 4 In this step, we show that the sequence {y n } is an M-Cauchy sequence.
As z ∈ g(X) there exists v ∈ X such that STEP 5 Now we show that gv = f v. Suppose, on the contrary, that f v ̸ = gv(= w). Then exists a positive integer n 0 such that gv ̸ = gx n , for all n ≥ n 0 .
For n → +∞ and using equations (9), (23) and (24) we get As the pair of self maps (f, g) is weakly compatible, we have STEP 6 Now we show that f z = z. Suppose, on the contrary that f z ̸ = z.
i. e. M (f z, z, t) (α+β) > 1 which is not possible as the left hand side is less than 1.Thus, f u = gu = u.
Taking g = I in Theorem 2, then the sequence {x n } = {x 0 , f x 0 , · · ·f n x 0 , , · · ·} becomes a Picard sequence for the self map f and we have  1+d(x,y) , then (X, M, .) is a complete stationary fuzzy metric space with the product t-norm. Define δ as follows: Then δ ∈ △.
Taking α = 0 = β. observe that for all values of x, y ∈ X, f (x), f (y) ∈ [0, 1 3 ). We show that the quadruple (X, M, δ, f ) is an interpolative Meir-Keeler contractive. For this we prove the following condition: Therefore, the inequality ϵ− < M (x, y) ≤ ϵ gives Hence Thus, the quadruple (X, M, δ, f ) is an interpolative Meir-Keeler contractive and x = 0 is the unique fixed point of the map f .