We are aware that there are two categories of fixed points that are under discussion [1]. The first category, known as Banach fixed point theorems, deals with contraction [2]. The second type is more complex and deals with compact mappings [3]. A crucial role is played by the metric fixed point theorem. In a variety of spaces, including Banach, Hilbert, cone, soft, and others, numerous writers have demonstrated the fixed point theorem. Zadeh first proposed the idea of fuzzy sets. Kaleva and Seikkala [4], Deng [5], Eklund and Gahler [6], and Kramosil and Michalek [7] have presented the idea of many approaches to fuzzy metric spaces. The fixed point theory has also been examined by numerous writers in these fuzzy metric spaces and for fuzzy mappings and numerous other individuals recently initiated studies on probabilistic 2-metric spaces.

Given the abstract qualities implied by the area function in Euclidean spaces, we know that 2-metric space is a real valued function of a point triple on a set X. Now, the volume function suggests 3-metric space, which is what is expected. Naturally, this is introduced in a different way than 2-metric space theory. A common fixed point theorem for three mappings in fuzzy metric space is proved in this study. The fuzzy 2-metric and fuzzy 3-metric spaces are also covered by this solution [8–10].

Definition 1. An operation in binary *: [0, 1] × [0, 1] → [0, 1] is referred to as a t-norm in ([0,1], *) if all of the following conditions are satisfied for every a, b, c, and d that are contained in [0,1]:

I. a*1 = a.

II. a*(b*c) = (a*b)*c.

III. a*b = b*a.

IV. a*b ≤ c*d whenever a ≤ c and b ≤ d.

Definition 2. If X is an arbitrary set, * is a continuous t-Norm, and M is a fuzzy set in X × [0, ∞], and all x, y, and z ∈X satisfy the following conditions, then the 3-tuple (X, ℳ, *) is referred to as a fuzzy metric space (FM space).

FM-1 For all t >0 if and only if x= y, ℳ(x, y, t) = 1, FM-2 ℳ(x, y, t) = ℳ(y, x, t) (symmetry), FM-3 ℳ(x, y, t) ∗ ℳ(y, z, s) ≤ ℳ(x, z, t + s), FM-4 ℳ(x, y, 0) = 0, FM-5 lim ℳ(x, y, t) = 1, FM-6 ℳ(x, y, . ): [0, 1] → [0, 1] is left continuous, Definition 3. Let (X, ℳ, *) is a fuzzy metric space.

I. A sequence {xn} in X can be represented by lim xn = x and is said to converge to a point x∈X. If lim ℳ(xn , x, t) = 1, for all t > 0.

II. If for all t > 0 and p > 0, then lim ℳ(xn+p , xn, t) = 1 is a Cauchy sequence -a sequence {xn} in X.

III. A fuzzy metric space is complete if all of its Cauchy sequences converge.

Definition 4. If X is an arbitrary set, * is a continuous t-Norm, and ℳ is a fuzzy set in X × [0, ∞], and all x, y, z, and u∈ X satisfy the following conditions for all t1, t2, and t3> 0, then the 3-tuple (X, ℳ, *) is referred to as a fuzzy 2-metric space.

FM’-1 For every t >0 if and only if x= y, ℳ(x, y, z, t) = 1, FM’-2 ℳ(x, y, z, t) = ℳ(x, z, y, t) = ℳ(y, z, x, t) (about three variables symmetric), FM’-3 ℳ(x, y, u, t1) ∗ ℳ(x, u, z, t2) ∗ ℳ(x, u, z, t3) ≤ ℳ(x, y, z, t1 + t2 + t3),(Tetrahedron inequality in 2- metric space is equivalent to this).

The probability that the area of the triangle is less than t can be deduced from the function value ℳ(x, y, z, t), FM’-4 ℳ(x, y, z, 0) = 0, FM’-5 ℳ(x, y, z, . ): [0, 1) → [0, 1] is left continuous.

Definition 5. Assume that (X, ℳ, *) is a fuzzy 2-metric space:

If lim ℳ(xn, a, t) = 1 , for any a in X and t > 0, then a sequence {xn} in fuzzy 2 metric space X is said to converge to a point ∈ .

If lim ℳ(xn+p, xn, a, t) = 1 , for any a in X and t> 0, p > 0, then a sequence {xn} in fuzzy 2-metric space X is referred to as a Cauchy sequence.

Every Cauchy sequence that converges in a fuzzy two-metric space is considered complete.

Definition 6. If X is an arbitrary set, * is a continuous t-Norm, and ℳ is a fuzzy set in × [0, ∞], and all x, y, z, u, and w ∈ X and t1, t2, t3 and t4> 0, then the 3-tuple (X, ℳ, *) is referred to as a fuzzy 3-metric space.

FM”-1 For every t > 0 if and only if x= y, ℳ(x, y, z, w, t) = 1.

FM”-2 ℳ(x, y, z, w, t) = ℳ(x, w, z, y, t) = ℳ(y, z, w, x, t) = ℳ(z, w, x, y, t) = ⋯ (about three variables symmetric).

FM”-3 ℳ(x, y, z, u, t1) ∗ ℳ(x, y, u, w, t2) ∗ ℳ(x, u, z, w, t3) ∗ ℳ(u, y, z, w, t4) ≤ ℳ(x, y, z, w, t1 + t2 + t3 +

t4).

FM”-4 ℳ(x, y, z, w, 0) = 0.

FM”-5 ℳ(x, y, z, w, . ): [0, 1) → [0, 1] is left continuous.

Definition 7. Assume that (X, ℳ, *) is a fuzzy 3-metric space:

If lim ℳ(xn, x, a, b, t) = 1 , for any a,b in X and t > 0, then a sequence {xn} in fuzzy 3 metric space X is said to converge to a point ∈ .

If lim ℳ(xn+p, xn, a, b, t) = 1 , for any a,b in X and t> 0, p > 0, then a sequence {xn} in fuzzy 3-metric space X is referred to as a Cauchy sequence.

Every Cauchy sequence that converges in a fuzzy 3-metric space is considered complete.

Theorem 1. Let (X, ℳ, *) be a fuzzy metric space that is complete and has the condition (FM-5). If S & are continuous mapping of X in X if there is a continuous mapping A of X into (X)⋂ (X) that commutes with & , then and have a common fixed point in X and ℳ(Ax, Ay, qt) ≥

min {ℳ( x,Ay,t),ℳ( x,Ax,t),ℳ( y,Ax,t) }

for all x,y,z ∈X, t>0 and 0<q<1. A, , and then have a unique

min {ℳ( x,Ay,t),ℳ( y,Ax,t) }

common fixed point.

min{ℳ( x2n,Ax2n+1,t),ℳ( x2n,Ax2n,t),ℳ( x2n+1,Ax2n,t)}

ℳ(Ax2n, Ax2n+1, qt) ≥

min{ℳ( x2n,Ax2n+1,t),ℳ( x2n+1,Ax2n,t)} min{ℳ(Ax2n+1,Ax2n+1,t),ℳ(Ax2n−1,Ax2n,t),ℳ(Ax2n,Ax2n,t)}

≥ ≥ ℳ(Ax2n−1, Ax2n, t).

min{ℳ(Ax2n+1,Ax2n+1,t),ℳ(Ax2n,Ax2n,t)}

[since, ℳ(Ax2n+1, Ax2n+1, t) = ℳ(Ax2n, Ax2n, t) = 1] ≥ ℳ (Ax2n−1, Ax2n, ).

Consequently, ℳ(Ax2n, Ax2n+1, qt) ≥ ℳ (Ax2n−1, Ax2n, ) Using induction ℳ(Ax2k, Ax2m+1, qt) ≥ ℳ (Ax2k−1, Ax2m, ) ℳ(Ax2k, Ax2m+1, qt) ≥ M (Ax2k−1, Ax2m, ) … ≥ ℳ (Ax0, Ax2m+1−2k, ).

q q2k

If 2k > 2m + 1 then ℳ(Ax2k, Ax2m+1, qt) ≥ ℳ (Ax2k−1, Ax2m, ) … ≥ ℳ (Ax2k−(2m+1), Ax0, ),

q q2m+1

Simple induction yields ℳ(Axn, Axn+p, qt) ≥ ℳ (Ax0, Axp, ),

in the case of n = 2k, p = 2m + 1 and by (FM-3) ℳ(Axn, Axn+p, qt) ≥ ℳ (Ax0, Ax1, ) ∗ ℳ (Ax0, Axp, ), 2q 2q for any positive integer p and n in N, ifn = 2k, p = 2m or n = 2k + 1, p = 2m, By observing that ℳ (Ax0, Axp, ) → 1 as n → ∞ for each positive integer p & n in N.

the space X is complete. As a result, Az = z = z and ℳ(Az, A z, qt) ≥ ℳ(Az, AAz, qt) ≥ min{ℳ( z, AAz, t), ℳ( z, Az, t), ℳ( Az, Az, t)} ≥ min{ℳ( z, A z, t), ℳ(Az, Az, t), ℳ(A z, Az, t)} ≥ min{ℳ( z, A z, t), ℳ(Az, Az, t), ℳ(A z, z, t)} ≥ ℳ( z, A z, t) ≥

2 2 t

ℳ( z, AAz, t) ≥ ℳ(Az, A z, t) … ≥ ℳ (Az, A z, ).

2 t 2

Given that lim ℳ (Az, A z, ) = 1 Consequently, Az = A z Z is therefore the common fixed point of A, , and . Let w (w ≠ z) be another common fixed point of , , and A for uniqueness. Then, we write min{ℳ( z, Aw, t), ℳ( z, Az, t), ℳ( w, Az, t)} ℳ(Az, Aw, qt) ≥ .

min{ℳ( z, Aw, t), ℳ( w, Az, t)} This indicates ℳ(z, w, qt) ≥ ℳ(z, w, t).

Thus, we write z = w using the lemma. The proof of Theorem 1 is now complete. Theorem for fuzzy 2-metric spaces is now proved.

Theorem 2. Let (X, ℳ, *) be a fuzzy 2-metric space that is complete and has the condition. If S & are continuous mapping of X in X if there is a continuous mapping A of X into (X)⋂ (X) that commutes with & , then and have a common fixed point in X and ℳ(Ax, Ay, a, qt) ≥

min{ℳ( x,Ay,a,t),ℳ( x,Ax,a,t),ℳ( y,Ax,a,t)}

forall ,y,a, ∈X, t>0 and 0<q<1. limℳ(x, y, z, t) = 1 for all x, y, z ∈

min{ℳ( x,Ay,a,t),ℳ( y,Ax,a,t)} t→∞

X. A, , and then have a unique common fixed point.

min{ℳ( x2n,Ax2n+1,a,t),ℳ( x2n,Ax2n,a,t),ℳ( x2n+1,Ax2n,a,t)}

ℳ(Ax2n, Ax2n+1, a, qt) ≥

min{ℳ( x2n,Ax2n+1,a,t),ℳ( x2n+1,Ax2n,a,t)} min{ℳ(Ax2n+1,Ax2n+1,a,t),ℳ(Ax2n−1,Ax2n,a,t),ℳ(Ax2n,Ax2n,a,t)}

≥ ≥ ℳ(Ax2n−1, Ax2n, a, t)

min{ℳ(Ax2n+1,Ax2n+1,a,t),ℳ(Ax2n,Ax2n,a,t)}

[since, ℳ(Ax2n+1, Ax2n+1, a, t) = ℳ(Ax2n, Ax2n, a, t) = 1] ≥ ℳ (Ax2n−1, Ax2n, a, ).

Consequently, ℳ(Ax2n, Ax2n+1, a, qt) ≥ ℳ (Ax2n−1, Ax2n, a, ).

Using induction ℳ(Ax2k, Ax2m+1, a, qt) ≥ ℳ (Ax2k−1, Ax2m, a, ).

ℳ(Ax2k, Ax2m+1, a, qt) ≥ ℳ (Ax2k−1, Ax2m, a, ) … ≥ ℳ (Ax0, Ax2m+1−2k, )

q q2k

If 2k > 2m + 1, then ℳ(Ax2k, Ax2m+1, a, qt) ≥ ℳ (Ax2k−1, Ax2m, a, ) … ≥ ℳ (Ax2k−(2m+1), Ax0, a, ).

q q2m+1

Simple induction yields ℳ(Axn, Axn+p, a, qt) ≥ ℳ (Ax0, Axp, a, ).

In the case of n = 2k, p = 2m + 1 and by (FM-4).

t t t

ℳ(Axn, Axn+p, a, qt) ≥ ℳ (Ax0, Ax1, a, ) ∗ ℳ (Ax0, Axp, a, ) ∗ ℳ (Ax1, Axp, a, ).

n n n

3q 3q 3q

For any positive integer p & n in N, ifn = 2k, p = 2m or n = 2k + 1, p = 2m, By observing that ℳ (Ax0, Axp, a, ) → 1 as n → ∞ for each positive integer p & n in N.

the space X is complete. As a result, Az = z = z and ℳ(Az, A z, a, qt) ≥ ℳ(Az, AAz, a, qt) ≥ min{ℳ( z, AAz, a, t), ℳ( z, Az, a, t), ℳ( Az, Az, a, t)} ≥ min{ℳ( z, A z, a, t), ℳ(Az, Az, a, t), ℳ(A z, Az, a, t)} ≥ min{ℳ( z, A z, a, t), ℳ(Az, Az, a, t), ℳ(A z, z, a, t)}

2 2 t

≥ ℳ( z, A z, a, t) ≥ ℳ( z, AAz, a, t) ≥ ℳ(Az, A z, a, t) … ≥ ℳ (Az, A z, a, ).

2 t 2

Given that lim ℳ (Az, A z, a, ) = 1 Consequently, Az = A z.

Z is therefore the common fixed point of A, , and .

Let w (w ≠ z) be another common fixed point of , , and A for uniqueness. Then, using (5), we write

min{ℳ( z, Aw, a, t), ℳ( z, Az, a, t), ℳ( w, Az, a, t)} ℳ(Az, Aw, a, qt) ≥ .

min{ℳ( z, Aw, a, t), ℳ( w, Az, a, t)}

This indicates ℳ(z, w, a, qt) ≥ ℳ(z, w, a, t).

Thus, we write z = w using the lemma. The proof of Theorem 2 is now complete. Theorem for fuzzy 3-metric spaces is now proved.

Theorem 3. Let (X, ℳ, *) be a fuzzy 3-metric space that is complete and has the condition . If S & are continuous mapping of X in X if there is a continuous mapping A of X into (X)⋂ (X) that commutes with & , then and have a common fixed point in X and ℳ(Ax, Ay, a, b, qt) ≥

min {ℳ( x,Ay,a,b,t),ℳ( x,Ax,a,b,t),ℳ( y,Ax,a,b,t) }

For all x,y,a,b ∈ , t>0 and 0<q<1. limℳ(x, y, z, w, t) =

min {ℳ( x,Ay,a,b,t),ℳ( y,Ax,a,b,t) } t→∞

1 for all x, y, z ∈ X A, , and then have a unique common fixed point.

ℳ(Ax2n, Ax2n+1, a, b, qt) min{ℳ( x2n, Ax2n+1, a, b, t), ℳ( x2n, Ax2n, a, b, t), ℳ(Tx2n+1, Ax2n, a, b, t)} min{ℳ( x2n, Ax2n+1, a, b, t), ℳ( x2n+1, Ax2n, a, b, t)} min{ℳ(Ax2n+1, Ax2n+1, a, b, t), ℳ(Ax2n−1, Ax2n, a, b, t), ℳ(Ax2n, Ax2n, a, b, t)} min{ℳ(Ax2n+1, Ax2n+1, a, b, t), ℳ(Ax2n, Ax2n, a, b, t)} ≥ ℳ(Ax2n−1, Ax2n, a, b, t) [since, ℳ(Ax2n+1, Ax2n+1, a, b, t) = ℳ(Ax2n, Ax2n, a, b, t) = 1] ≥ ℳ (Ax2n−1, Ax2n, a, b, ).

Consequently, ℳ(Ax2n, Ax2n+1, a, qt) ≥ ℳ (Ax2n−1, Ax2n, a, b, ).

Using induction ℳ(Ax2k, Ax2m+1, a, b, qt) ≥ ℳ (Ax2k−1, Ax2m, a, b, ).

ℳ(Ax2k, Ax2m+1, a, b, qt) ≥ ℳ (Ax2k−1, Ax2m, a, b, ) … ≥ ℳ (Ax0, Ax2m+1−2k, a, b, ) If 2k > 2m + 1, then ℳ(Ax2k, Ax2m+1, a, b, qt) ≥ ℳ (Ax2k−1, Ax2m, a, b, ) … ≥ ℳ (Ax2k−(2m+1), Ax0, a, b, ).

q2m+1

Simple induction yields ℳ(Axn, Axn+p, a, b, qt) ≥ ℳ (Ax0, Axp, a, b, ).

In the case of n = 2k, p = 2m + 1 and by (FM-3) ℳ(Axn, Axn+p, a, b, qt) ≥ ℳ (Ax0, Axp, a, Ax1, ) ∗ ℳ (Ax0, Axp, Ax1, b, ) ∗

4qn 4qn

ℳ (Ax0, Ax1, a, b, ) ∗ ℳ (Ax1, Axp, a, b, ).

4qn 4qn

For any positive integer p and n in N, if n = 2k, p = 2m or n = 2k + 1, p = 2m, By observing that ℳ (Ax0, Axp, a, b, ) → 1 as n → ∞ for each positive integer p and n in N.

ℳ(Az, A z, a, b, qt) ≥ ℳ(Az, AAz, a, b, qt) ≥ min{ℳ( z, AAz, a, b, t), ℳ( z, Az, a, b, t), ℳ( Az, Az, a, b, t)} ≥ min{ℳ( z, A z, a, b, t), ℳ(Az, Az, a, b, t), ℳ(A z, Az, a, b, t)} ≥ min{ℳ( z, A z, a, t), ℳ(Az, Az, a, t), ℳ(A z, Sz, a, t)} ≥ ℳ( z, A z, a, b, t) ≥ ℳ( z, AAz, a, t) ≥ ℳ(Az, A z, a t) … ≥ ℳ (Az, A z, a, b, ).

the space X is complete. As a result, Az = z = z and Given that limℳ (Az, A z, a, b, ) = 1 Consequently, Az = A z. Z is therefore the common fixed point of A, , and .

Let w (w ≠ z) be another common fixed point of , , and A for uniqueness. Then, using (5), we write

min{ℳ( z,Aw,a,b,t),ℳ( z,Az,a,b,t),ℳ( w,Az,a,b,t)}

ℳ(Az, Aw, a, b, qt) ≥ .

min{ℳ( z,Aw,a,b,t),ℳ( w,Az,a,b,t)}

This indicates ℳ(z, w, a, b, qt) ≥ ℳ(z, w, a, b, t).

Thus, we write z = w using the lemma. The proof of Theorem 3 is now complete.

The main objective of this study was to introduce a new kind of contractive condition in order to expand the theory of fixed points in fuzzy 3-metric spaces. In contrast to previous fixed point results that depend on similarity-type or classical contraction criteria, our method adapts the condition to represent the characteristics of interactions between three elements in a fuzzy environment.

Using this contractive mapping, we demonstrate the existence and uniqueness of the fixed point under the suggested condition, thus proving a new fixed point theorem. The method of proof is based on building a convergent sequence with the fuzzy 3-metric and showing that it satisfies the fixed point requirements.

The authors would like to express their sincere appreciation to all individuals whose valuable comments and support contributed to the successful completion of this study.

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

The data supporting the findings of this study are available from the corresponding author upon reasonable request.