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-C´iric´ 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 MeirKeeler 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) A mapping ∗ : [0, 1] × [0, 1] → [0, 1] is called a continuous triangular norm (t-norm for short) if ∗ is continuous and satisfies the following conditions:

(i) ∗ is commutative and associative, i.e. a ∗ b = b ∗ a and a ∗ (b ∗ c) = (a ∗ b) ∗ c, for all a, b, c ∈ [0, 1];

(ii) 1 ∗ a = a, for all a ∈ [0, 1];

(iii) a ∗ c ≤ b ∗ d, for a ≤ b, c ≤ d for a, b, c, d ∈ [0, 1].

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;

(GV1) M(x, y, t) > 0;

(GV2) M(x, y, t) = 1 if and only if x = y;

(GV3) M(x, y, t) = M(y, x, t);

(GV4) M(x, z, t + s) ≥ M(x, y, t) ∗ M(y, z, s);

(GV5) M(x, y, .) : (0, +∞) → (0, 1] is continuous.

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 n₀ ∈ N, such that M(x_n, x_m, t) > 1 − ϵ, for all n, m ≥ n₀. 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.

DEFINITION 4. (George & Veeramani, 1994) (X, M, ∗) (or simply X) is called M-complete if every M-Cauchy sequence in X is convergent.

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

and

LEMMA 3. (Saha et al, 2016) Let {f(k, .) : (0, +∞) → (0, 1]; k = 0, 1, 2, ..., } be a sequence of functions such that f(k, .) is continuous and monotone increasing for each k ≥ 0. Then is a left continuous function in t and is a right continuous function in it.

Denote △ = {δ|δ : (0, 1] → (0, 1]} where δ is right continuous.

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, fx = gx = y implies gy = fy.

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

where

REMARK 1. From equation (2) for all x $\ne$ y ∈ X, t > 0 the pair (f, g) is a strict contraction i.e.

Thus for x $\ne$ y

REMARK 2. Taking g = I, the identity map in equation (2) we obtain

where

which is an interpolative generalised Meir-Keeler contraction, for a self map f.

REMARK 3. Taking α = 0, β = 0 in equation (4) then M(x, y) = M(x, y, t) and we have

which is precisely the Meir-Keeler contraction, for a self map given by Zheng and Wang (Zheng & Wang, 2019).

LEMMA 4. (Zheng & Wang, 2019) If δ ∈ △, then for t ∈ (0, 1), there exists k = k(t) ∈ N such that

Before we prove the main result, we prove the following lemma:

LEMMA 5. Let (f, g) be a pair of an interpolative Meir-Keeler contractive mapping with respect to δ ∈ △ and f(X) ⊆ g(X). Construct a sequence {y_n}, by fx_n = gx_n+1 = y_n, for n = 0, 1, 2, . . .. Then lim_n→+∞ M(y_n, y_n+1, t) = 1.

Proof. Suppose if possible on the contrary that lim_n→+∞ M(y_n, y_n+1, t) = p(< 1). For α, β ∈ [0, 1) we have

By using lemma (4), for p < 1 and δ ∈ △ we can find k = k(p) ∈ N such that

Since lim_n→+∞ M(x_n, x_n+1) = p, we can find n₀ such that when n > n₀,

Also, there exists n1 such that whenever n > n₁,

Let n > max{n₀, n₁}. Then both equations (6), and ( 7) hold for such n. Taking iin equation (2) we get

i. e. M(y_n, y_n+1, t) > p + , which contradicts the fact that lim_n→+∞ M(yn, yn+1, t) = p. Therefore, lim_n→+∞ M(y_n, y_n+1, t) = 1.

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;

(4.13) f(X) is complete;

(4.14) The pair (f, g) is weakly compatible.

Then f and g have a unique common fixed point in X if and only if there exists x0 ∈ X such that ∧ _t>0 M(x₀, f(x₀), t) > 0.

Proof. : Suppose the pair (f, g) has a unique common fixed point u then u = fu = gu. Therefore, M(u, fu, t) = 1, ∀t > 0. Hence

Conversely, suppose that there exists x₀ ∈ X such that ∧_t>0 M(x₀, f(x₀), t) > 0. Construct a sequence {y_n}, by defining fx_n = gx_n+1 = yn, 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 u = fu = gu and v = fv = gv. We show that u = v.

Suppose, on the contrary that u $\ne$ v, then fu $\ne$ fv. Now

Now

M(u, v, t) = M(fu, fv, t),

> M(u, v), using (3)

= (M(u, v, t))^{(1−α−β)}

i. e. M(u, v, t) > (M(u, v, t))^{(1−α−β)} ,

implies

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 y_n = fx_n = gx_n+1 = fx_n+1 = gx_n+2 = y_n+1 we have fx_n+1 = gx_n+1. Let fx_n+1 = gx_n+1 = z. So x_n+1 is a point of coincidence of the pair (f, g). As the pair (f, g) is weakly compatible we have fz = gz. Now we show that fz = z. Suppose, if possible on the contrary, that fz $\ne$ z so fz $\ne$ fx_n+1. By using equation (3) we have

i.e.

M(z, fz, t) > (M(z, fz, t))^{(1−α−β)} , for all t > 0 implies (M(z, fz, t))^(α+β) > 1, which is not possible. Hence fz = 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 {yn} are distinct.

Again, to see the existence of a common fixed point in other cases, we first show that all the terms of the sequence {y_n} are distinct.

CASE II Suppose y_n = y_m, for some m > (n + 1), then as all the consecutive terms of the sequence {y_n} are distinct, we claim that y_n+1 = y_m+1. Suppose if possible on the contrary y_n+1 $\ne$ y_m+1 then y_n $\ne$ y_n+1 $\ne$ y_m+1 $\ne$ y_m implies y_n $\ne$ ̸y_m which contradicts our assumption. So we have y_n+1 = y_m+1. Also and

Thus

So

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.

Thus, we conclude that for distinct n, m ∈ N, y_n y_m. Therefore, the elements of the sequence {y_n} are distinct. From equation (8) we have

for all t > 0. Thus, {M(y_n, y_n+1, t)}, for each t > 0, is a strictly increasing sequence, which is bounded above by 1. Therefore, by lemma 5, for t > 0,

Now we prove that the sequence {y_n} is M-Cauchy. Suppose if possible on the contrary that it is not true; then there exist η ∈ (0, 1), t₀ > 0 and the sequences {p(n)}, {q(n)} (p(n) being the smallest ones of the index)

STEP 2: In this step, we show that lim_n→∞ M(y_p(n)−1 , y_q(n)−1 , t₀) = 1 − η. Now for all n ≥ 1, 0 < λ < t₀/2, we obtain,

Let

Taking the limit supremum on both sides of equation (11), and using the properties of M and ∗, and by lemma 3, we obtain

using (10).

Since M is bounded with the range in (0, 1], continuous and nondecreasing in the third variable t, it follows from lemma 3, that h₁ is continuous from the left. Therefore, for λ → 0 , we obtain

Let

Again, for all n ≥ 1, λ > 0

using (10).

Taking the limit infimum as n → +∞ in the above inequality, we obtain

using (9).

Since M is bounded with the range in (0, 1] , continuous and nondecreasing in the third variable t, it follows from lemma 3, that h₂ 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(_yp(n) , y_q(n) , t₀) = 1 − η. From equation (10) we have

Also for all n ≥ 1 and λ > 0 we have

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

Since M is bounded with the range in (0, 1], continuous and nondecreasing in the third variable t, it follows from lemma 3 that 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.

Using equations (9) and (14) at t = t₀, we have

Therefore

And

For n → +∞ and using equations (20) and (21), we have

implies that (1 − η)^(α+β) ≥ 1., which is not possible as(1 − η) < 1. So,{y_n} is an M-Cauchy sequence in g(X) which is M-complete. Therefore, there exists z ∈ g(X) such that

i. e.

As z ∈ g(X) there exists v ∈ X such that

STEP 5 Now we show that gv = fv. Suppose, on the contrary, that fv gv(= w). Then exists a positive integer n₀ such that gv gx_n, for all n ≥ n₀.

Now

For n → +∞ and using equations (9), (23) and (24) we get

M(z, fv, t) ≥ [M(fv, z, t)]^β i. e. M(fv, z, t)]^1−β > 1, which is not possible if fv z. Hence fv = u and we have

As the pair of self maps (f, g) is weakly compatible, we have

STEP 6 Now we show that fz = z. Suppose, on the contrary that fz z. Then gz z.

and

i. e. M(fz, z, t)^(α+β) > 1 which is not possible as the left hand side is less than 1.Thus, fu = gu = u.

Taking g = I in Theorem 2, then the sequence {x_n} = {x₀, fx₀, · · ·fⁿx₀, , · · ·} becomes a Picard sequence for the self map f and we have

COROLLARY 1. Let f be an interpolative fuzzy Meir-Keeler contractive map on a M-complete fuzzy metric space (X, M, ∗).Then the map f has a unique fixed point in X.

REMARK 4. If we take α = 0 and β = 0 in the above corollary, we obtain Theorem 3.1 of Zheng and Wang (Zheng & Wang, 2019).

EXAMPLE 1. (of Theorem 4.1) Let X = [0, 1] . Define a self map f : X → X by f(x) = x/2 , and g(x) = x, the identity map on X. Taking M(x, y) = 1/1+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 MeirKeeler contractive. For this we prove the following condition:

Therefore, the inequality ϵ− < M(x, y) ≤ ϵ gives

which implies that x, y ∈ [0, 1]. 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.

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Author notes

vuk.stojiljkovic999@gmail.com

Additional information

FIELD: Mathematics

ARTICLE TYPЕ: Original scientific paper

EDITORIAL NOTE: The third author of this article, Stojan N. Radenović, is a current member of the Editorial Board of the Military Technical Courier. Therefore, the Editorial Team has ensured that the double blind reviewing process was even more transparent and more rigorous. The Team made additional effort to maintain the integrity of the review and to minimize any bias by having another associate editor handle the review procedure independently of the editor – author in a completely transparent process. The Editorial Team has taken special care that the referee did not recognize the author’s identity, thus avoiding the conflict of interest.

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