Modular Fuzzy Metric-Like Spaces

Modular Fuzzy Metric-Like Spaces Nizamettin Ufuk Bostan1, Banu Pazar Varol1 1Department of Mathematics, Kocaeli University, 41001, Kocaeli - TÜRKİYE Corresponding author: [email protected] ORCID IDs: First Author: 0000-0003-4708-8663 Second Author: 0000-0002-8627-7910 DOI : 10.5281/zenodo.18033595 Abstract The aim of this paper is to define the concept of modular fuzzy metric-like space by combining of modular-like metric space and fuzzy metric-like space. We give examples and propositions to approve the importance of this new concept. Fundamental notions containing convergence, Cauchy sequence and completeness are carefully defined. We also demonstrate that, in a modular fuzzy metric-like space, the limit of the convergent sequence may not be unique. The fixed point theorem is proved, expanding the classical findings to this space. The applicability of fixed point theorem is shown through an example, expressing both theoretical potency and practical benefit of the proposed approach. Keywords: metric-like spaces, fuzzy metric spaces, fuzzy metric-like spaces, modular-like metric spaces, modular fuzzy metric-like spaces, fixed point theory, contraction principle 1 Introduction and Motivation The basis of metric space theory were laid more than a century ago by Fréchet [ 13 ] and Hausdorff [ 16 ], centered on the notion of distance between any two points in a set. In 1922, Banach [ 2 ] introduced what is now known as the Banach’s fixed point theorem or the Banach Contraction Principle, establihed within the setting of metric spaces. Later, Amini-Harandi [ 1 ] gave the concept of metric-like space and proved corresponding fixed point theorems, supported by illustrative examples that expressed the structure. Further developments were made by Hosseini and Fošner [ 17 ], who introduced several related notions for metric-like spaces, including equal-like points, cluster points, completely separate points, as well as definitions of distance between a point and a subset and between two subsets in a metric-like space. Fuzzy set theory, introduced by Zadeh [ 26 ], has since become a widely studied area across various scientific and applied disciplines. The definition of fuzzy metric space was first given by Kramosil and Michálek [ 19 ]. Later, George and Veeramani [ 14 , 15 ] modified this definition to gain the result that the formed structure generates a Hausdorff topology, thereby strengthening and expanding the practicality of fuzzy metrics. Based on these developments, Shukla and Abbas [ 25 ] introduced the concept of fuzzy metric-like space generalizing the concept of fuzzy metric space of George and Veeramani, and proved several fixed point theorems in this more comprehensive space. M35-1 KOSC-2025 Proceedings Nakano [ 21 ] first introduced the definition of modular, a concept later further developed by Orlicz [ 22 ]. The concepts of metric modular and modular metric space were presented by Chistyakov [ 4 , 5 ] in the course of building the theory of these structures. According to Chistyakov [ 7 ], while a metric on a set measures non-negative finite distances between points, a metric modular is given to describe non-negative, possibly infinite-valued, velocities. Some of his results appear in [ 6 ], and his fixed point theorems together with their applications are introduced in detail in [ 7 ]-[ 10 ]. An exhaustive compilation of Chistyakov’s contributions to metric modulars and modular metric spaces was later provided in [ 11 ]. Mongkolkeha et al. [ 20 ] established and proved existence theorems of fixed points for contraction mappings in modular metric spaces. Rasham et al. [ 23 ] presented the concept of modular-like metric space and proved some common fixed point theorems for two families of set-valued mappings satisfying contraction condition in this space. In the same work [ 23 ], they also obtained new results in graph theory concerning multigraph-dominated contractions within modular-like metric spaces. Additional fixed point results in modular-like metric spaces can be found in [12]. Kerim et al. [ 18 ] introduced a new space termed the modular fuzzy metric space in the sense of Kramosil-Michálek. They explored its main properties and illustrated the structure of space with several examples. Based on this foundation, they obtained existence and uniqueness results for fixed points of continuous mappings in this space and showed the applicability of their findings. Then Bostan and Pazar Varol [ 3 ] introduced the concept of modular fuzzy metric space in the sense of George-Veeramani in 2023 In this study, we define the concept of modular fuzzy metric-like space by combining modularlike metric space and fuzzy metric-like space. We examine some properties of this space and give various examples that provide a better understanding of the structure of space. We give the notions of modular fuzzy metric-like space such as contraction, convergence, Cauchy sequence, completeness and show that in a modular fuzzy metric-like space, the limit of the convergent sequence may not be unique. Then we establish and prove fixed point theorem in the modular fuzzy metric-like space. We also demonstrate the power of the structure we established and of the results we gained in this structure by giving an example for the fixed point theorem we proved. Our aim here is to contribute to fixed point theory by being interested in fixed point theorem in modular fuzzy metric-like space, which is a generalization of modular fuzzy metric space in the sence of George-Veeramani. We aim to carry the applications of fixed point theory in mathematics and engineering to modular fuzzy metric-like space by modulating the concept of fuzzy metric-like space. The new mathematical structure introduced in this paper may enable researchers working in fixed point theory or its applications to establish various fixed point theorems under different contraction conditions and to explore their applicability in broader contexts. 2 Mathematical Preliminaries In this section, we introduce several essential definitions an concepts that will be used throughout the paper. Throughout the text, we denote by IR the set of all real numbers and by IN the set of all positive integers. Definition 2.1. [ 1 ] Let W = ∅ . A mapping σ : W×W→ [0 ,∞ )is called metric-like on W M35-2 2nd Kocaeli Science Congress, November 19-21, 2025 if the following three conditions hold for all w, y, z ∈W: (ML1) σ(w, z)=0⇒w=z, (ML2) σ(w, z) = σ(z, w), (ML3) σ(w, z)≤σ(w, y)+σ(y, z). The pair (W, σ)is called a metric-like space. Example 2.1. [ 1 ] Let W = [0 ,∞ ). Define the function σ : W×W→ [0 ,∞ )by σ ( w, z ) = max{w, z}. Then (W, σ)is a metric-like space. Definition 2.2. [ 26 ] A fuzzy set Q in W is characterized by a membership function fQ ( w ) which associates each point in W with a real number in the interval [0 , 1]. The value of fQ ( w )at wrepresent the grade of membership of win W. Definition 2.3. [ 24 ] A binary operation ∗ : [0 , 1] × [0 , 1] → [0 , 1] is called a continuous t-norm if ∗satisfies the following conditions for all o, u, j, l ∈[0,1]: (1) o∗1 = o (2) o∗u=u∗oand o∗(u∗j) = (o∗u)∗j (3) If o≤jand u≤l, then o∗u≤j∗l (4) ∗is continuous. Example 2.2. [24] The binary operations defined as follows are the continuous t-norms: (1) o∗u=ou (2) o∗u= max{0, o +u−1} (3) o∗u= min{o, u} Definition 2.4. [ 14 ] A triplet ( W, Q, ∗ )is called fuzzy metric space if W is an arbitrary set, ∗ is a continuous t-norm and Q is a fuzzy set on W2× (0 ,∞ )satisfying the following conditions, for all w, y, z ∈Wand t, s > 0: (FM1) Q(w, y, t)>0, (FM2) Q(w, y, t)=1⇔w=y, (FM3) Q(w, y, t)=Q(y, w, t), (FM4) Q(w, y, t)∗Q(y, z, s)≤Q(w, z, t +s), (FM5) Q(w, y, .) : (0,∞)→[0,1] is continuous. Example 2.3. [ 14 ] Let W = IR . Define o∗u = ou and Q ( w, y, t ) = e−|w−y| t for all w, y ∈Q=IR and t>0. Then (W, Q, ∗)is a fuzzy metric space. Example 2.4. [ 14 ] Let ( W, δ )be a metric space. Define o∗u = ou and Qδ ( w, y, t ) = ktn ktn+mδ(w,y)k, m, n ∈IR+. Then (W, Q, ∗)is a fuzzy metric space. Remark 2.1. [ 14 ] In the above example by taking k = m = n = 1, we get Qδ ( w, y, t ) = t t+δ(w,y). This fuzzy metric induced by a metric δis called the standard fuzzy metric. Definition 2.5. [ 25 ] A triplet ( W, Q, ∗ )is called fuzzy metric-like space if W is an arbitrary set, ∗ is a continuous t-norm and Q is a fuzzy set on W2× (0 ,∞ )satisfying the following conditions, for all w, y ∈Wand t, s > 0: (FML1) Q(w, y, t)>0, (FML2) Q(w, y, t)=1⇒w=y, (FML3) Q(w, y, t)=Q(y, w, t), (FML4) Q(w, y, t)∗Q(y, z, s)≤Q(w, z, t +s), (FML5) Q(w, t, .) : (0,∞)→[0,1] is continuous. 2nd Kocaeli Science Congress, November 19-21, 2025 M35-3 KOSC-2025 Proceedings Example 2.5. [ 25 ] Let W = [0 , 1]. Define o∗u = ou and Q ( w, y, t ) = t t+max{w,y} for all w, y ∈Wand t > 0. Then (W, Q, ∗)is a fuzzy metric-like space. Definition 2.6. [ 4 ] Let W = ∅ . A mapping ζ : (0 ,∞ ) ×W×W→ [0 ,∞ ]is called metric modular on Wif the following hold for each w, y, z ∈W: (MM1) ζ(w, y, λ)=0⇔w=y, for all λ > 0, (MM2) ζ(w, y, λ)=ζ(y, w, λ), for all λ > 0, (MM3) ζ(w, y, λ +γ)≤ζ(w, z, λ)+ζ(z, y, γ), for all λ, γ > 0. For simplicity, the notation ζλ ( w, y )will be used instead of ζ ( w, y, λ )for modular structures throughout the remainder of the paper. Example 2.6. [ 7 ] Let ( W, δ )be metric space. Define the function ζ : (0 ,∞ ) ×X×X→ [0 ,∞ ] by ζλ(w, y) = δ(w,y) λfor all λ > 0and w, y ∈W. Then ζis a metric modular on W. Definition 2.7. [ 23 ] Let W = ∅ . A function φ : (0 ,∞ ) ×W×W→ [0 ,∞ )is called modular-like metric on W, if it satisfies the following three conditions for each w, y, z ∈W: (MLM1) φλ(w, y)=0⇒w=y, for all λ > 0, (MLM2) φλ(w, y)=φλ(y, w), for all λ>0, (MLM3) φλ+γ(w, y)≤φλ(w, z)+φγ(z, y), for all λ, γ > 0. Then the triplet (W, φ)is called modular-like metric space. Definition 2.8. [ 18 ] A triplet ( W, Qβ,∗ )is called modular fuzzy metric space if W is an arbitrary set, ∗ is a continuous t-norm and Qβ is fuzzy set on W2× (0 ,∞ )satisfying the following conditions where β > 0, for all w, y, z ∈Wand t, s > 0: (MFM1) Qβ(w, y, 0) = 0,Qβ(w, y, t)>0, for all β > 0, (MFM2) Qβ(w, y, t)=1⇔w=y, for all β > 0, (MFM3) Qβ(w, y, t)=Qβ(y, w, t), for all β > 0, (MFM4) Qβ(w, y, t)∗Qµ(y, z, s)≤Qβ+µ(w, z, t +s), for all β, µ > 0, (MFM5) Qβ(w, y, .) : (0,∞)→[0,1] is left continuous. Here Qβis called a modular fuzzy metric. Example 2.7. [ 18 ] Let ζ be a metric modular on W and o∗u = ou for all o, u ∈ [0 , 1]. Define fuzzy set Qβon W2×(0,∞)as follows: Qβ(w, y, t)=e −ζβ(w,y) tfor all w, y ∈Wand β, t > 0. Then (W, Qβ,∗)is a modular fuzzy metric space. Example 2.8. [ 18 ] Let ζ be a metric modular on W and o∗u = ou for all o, u ∈ [0 , 1]. Define fuzzy set Qβon W2×(0,∞)as follows: Qβ(w, y, t) = t t+ζβ(w,y)for all w, y ∈Ωand β, t > 0. Then (W, Qβ,∗)is a modular fuzzy metric space. Definition 2.9. [ 3 ] A triplet ( W, Qβ,∗ )is called modular fuzzy metric space (in the sence of George-Veeramani) if W is an arbitrary set, ∗ is a continuous t-norm and Qβ is fuzzy set on W2×(0,∞)satisfying the following conditions where β > 0, for all w, y, z ∈Wand t, s > 0: (MFMGV1) Qβ(w, y, t)>0, for all β > 0, (MFMGV2) Qβ(w, y, t)=1⇔w=y, for all β > 0, (MFMGV3) Qβ(w, y, t)=Qβ(y, w, t), for all β > 0, (MFMGV4) Qβ(w, y, t)∗Qµ(y, z, s)≤Qβ+µ(w, z, t +s), for all β, µ > 0, (MFMGV5) Qβ(w, y, .) : (0,∞)→[0,1] is continuous. M35-4 2nd Kocaeli Science Congress, November 19-21, 2025 Here Qβis called modular fuzzy metric (in the sence of George-Veeramani). Example 2.9. [ 3 ] Let ζ be a metric modular on W such that ζ : (0 ,∞ ) ×W2→ [0 ,∞ ). Define o∗u=ou for all o, u ∈[0,1] and fuzzy set Qβon W2×(0,∞)as follows: Qβ(w, y, t)=e −ζβ(w,y) tfor all w, y ∈Wand β, t > 0. Then (W, Qβ,∗)is a modular fuzzy metric space. Example 2.10. [ 3 ] Let ζ be a metric modular on W such that ζ : (0 ,∞ ) ×W2→ [0 ,∞ ). Define o∗u=ou for all o, u ∈[0,1] and fuzzy set Qβon W2×(0,∞)as follows: Qβ(w, y, t) = t t+ζβ(w,y)for all w, y ∈Ωand β, t > 0. Then (W, Qβ,∗)is a modular fuzzy metric space. 3 Results We begin this section by introducing the definition of modular fuzzy metric-like space and investigating its main properties. We then provide illustrative examples, propositions and present the notions of convergence, Cauchy sequences, and completeness within this mathematical structure. Finally, we give and prove fixed point theorem in this space. Definition 3.1. The triplet ( W, Qβ,∗ )is called a modular fuzzy metric-like space if W is an arbitrary nonempty set, ∗ is a continuous t norm and Qβ is a fuzzy set on W×W× (0 ,∞ ) satisfying the following conditions for all w, y, z ∈Wand t, s > 0: (MFML1) Qβ(w, y, t)>0, for all β > 0, (MFML2) Qβ(w, y, t)=1⇒w=y, for all β > 0, (MFML3) Qβ(w, y, t)=Qβ(y, w, t), for all β > 0, (MFML4) Qβ(w, y, t)∗Qµ(y, z, s)≤Qβ+µ(w, z, t +s), for all β, µ > 0 (MFML5) Qβ(w, y, .) : (0,∞)→[0,1] is continuous for all β > 0. Example 3.1. Let W = [0 ,∞ )and o∗u = ou for all o, u ∈ [0 , 1]. Define the fuzzy set Qβ in W2×(0,∞)as follows: Qβ ( w, y, t ) = e−(max{w,y} β) t for all w, y ∈W and β > 0and t∈ (0 ,∞ ). Then ( W, Qβ,∗ )is a modular fuzzy metric-like space. (MFML1)-(MFML3) and (MFML5) are obvious. Now, we prove (MFML4). Since max{w, y} ≤ max{w, z}+max{z, y}, we have max{w,y} β+µ≤max{w,z} β+max{z,y} µ. ⇒max{w,y} β+µ≤(t+s) t max{w,z} β+(t+s) t max{z,y} µ ⇒max{w,y} (β+µ)(t+s)≤max{w,z} βt +max{z,y} µs ⇒max{w,y} (β+µ)(t+s)≤max{w,z} βt +max{z,y} µs ⇒e max{w,y} (β+µ)(t+s)≤e max{w,z} βt +max{z,y} µs =e max{w,z} βt e max{z,y} µs ⇒e −max{w,y} (β+µ)(t+s)≥e −max{w,z} βt e −max{z,y} µs ⇒Qβ+µ(w, y, t +s)≥Qβ(w, z, t)∗Qµ(z, y, s)for all β, µ > 0. Hence (MFML4) holds. Example 3.2. Let W = [0 ,∞ )and o∗u = ou for all o, u ∈ [0 , 1]. Define the fuzzy set Qβ in W2×(0,∞)as follows: Qβ ( w, y, t ) = t t+max{w,y} β for all w, y ∈W, β > 0and t > 0. Then ( W, Qβ,∗ )is a modular fuzzy metric-like space. 2nd Kocaeli Science Congress, November 19-21, 2025 M35-5 KOSC-2025 Proceedings (MFML1)-(MFML3) and (MFML5) are obvious. Now, we prove (MFML4). We know max{w, y}≤max{w, z}+max{z, y}. Then max{w,y} β+µ≤max{w,z}+max{wz,y} β+µ ≤max{w,z} β+µ+max{z,y} β+µ ≤max{w,z} β+max{z,y} µ Thus, max{w,y} β+µ≤max{w,z} β+max{z,y} µ. ⇒ max{w,y} β+µ t+s≤ max{w,z} β+max{z,y} µ t+s = max{w,z} β t+s+ max{z,y} µ t+s ≤ max{w,z} β t+ max{z,y} µ s =smax{w,z} β+tmax{z,y} µ ts ⇒1 + max{w,y} β+µ t+s≤1 + smax{w,z} β+tmax{z,y} µ ts Since max{w,z} β.max{z,y} µ≥0, ⇒t+s+max{w,y} β+µ t+s≤ts+smax{w,z} β+tmax{z,y} µ+max{w,z} β.max{z,y} µ ts ⇒ts ts+smax{w,z} β+tmax{z,y} µ+max{w,z} β.max{z,y} µ ≤t+s t+s+max{w,y} β+µ ⇒t t+max{w,z} β .s s+max{z,y} µ ≤t+s t+s+max{w,y} β+µ ⇒Qβ(w, z, t)∗Qµ(z, y, s)≤Qβ+µ(w, y, t +s)for all β, µ > 0. Hence, (MFML4) holds. Proposition 3.1. Let ( W, σ )be a metric-like space and o∗u = ou for all o, u ∈ [0 , 1]. Define fuzzy set Qβon W2×(0,∞)as follows: Qβ ( w, y, t ) = e −(σ(w,y) β) t for all w, y ∈W and β, t > 0. Then ( W, Qβ,∗ )is a modular fuzzy metric-like space. Proof. (MFML1)- (MFML3) and (MFLM5) are obvious. Now, we prove (MFML4). Since σ is metric-like, we have σ(w, y)≤σ(w, z) + σ(z, y)for w, z, y ∈W. ⇒σ(w,y) (β+µ)(t+s)≤σ(w,z)+σ(z,y) (β+µ)(t+s)≤σ(w,z) βt +σ(z,y) µs ⇒e σ(w,y) (β+µ)(t+s)≤e σ(w,z) βt +σ(z,y) µs =e σ(w,z) βt e σ(z,y) µs ⇒ ⇒ e−σ(w,y) (β+µ)(t+s)≥e−σ(w,z) βt e−σ(z,y) µs ⇒Qβ(w, z, t)∗Qµ(z, y, s)≤Qβ+µ(w, y, t +s). Hence (MFML4) holds. Proposition 3.2. Let ( W, σ )be a metric-like space and o∗u = ou for all o, u ∈ [0 , 1]. Define fuzzy set Qβon W2×(0,∞)as follows: Qβ ( w, y, t ) = t t+σ(w,y) β for all w, y ∈W and β, t > 0. Then, ( W, Qβ,∗ )is modular fuzzy metric-like space. Proof. (MFML1)- (MFML3) and (MFLM5) are obvious. Now, we prove (MFML4). Since σ is metric-like, we have σ(w, y)≤σ(w, z) + σ(z, y)for w, z, y ∈W. ⇒σ(w,y) (β+µ)(t+s)≤σ(w,z)+σ(z,y) (β+µ)(t+s)≤σ(w,z) βt +σ(z,y) µs =µsσ(w,z)+βtσ(z,y) βtµs ⇒1 + σ(w,y) (β+µ)(t+s)≤1 + µsσ(w,z)+βtσ(z,y) βtµs ⇒(β+µ)(t+s)+σ(w,y) (β+µ)(t+s)≤βtµs+µsσ(w,z)+βtσ(z,y) βtµs ≤βtµs+µsσ(w,z)+βtσ(z,y)+σ(w,z)σ(z,y) βtµs ⇒βtµs βtµs+µsσ(w,z)+βtσ(z,y)+σ(w,z)σ(z,y)≤(β+µ)(t+s) (β+µ)(t+s)+σ(w,y) ⇒βt βt+σ(w,z).µs µs+σ(z,y)≤(β+µ)(t+s) (β+µ)(t+s)+σ(w,y) M35-6 2nd Kocaeli Science Congress, November 19-21, 2025 ⇒Qβ(w, z, t)∗Qµ(z, y, s)≤Qβ+µ(w, y, t +s). Hence (MFML4) holds. Proposition 3.3. Let φ be a modular-like metric on W . Define o∗u = ou for all o, u ∈ [0 , 1] and Qβ : W×W× (0 ,∞ ) → [0 , 1] by Qβ ( w, y, t ) = e −φβ(w,y) t for all w, y ∈W and β, t > 0. Then (W, Qβ,∗)is a modular fuzzy metric-like space. Proof. (MFML1)- (MFML3) and (MFLM5) are obvious. Now, we prove (MFML4). Since φβ+µ(w, z)≤φβ(w, y) + φβ(y, z),∀β, µ > 0, we have φβ+µ(w, z)≤(t+s t)φβ(w, y) + (t+s s)φβ(y, z) ⇒φβ+µ(w,z) t+s≤φβ(w,y) t+φβ(y,z) s ⇒e φβ+µ(w,z) t+s≤e(φβ(w,y) t+φβ(y,z) s)=e φβ(w,y) t.e φβ(y,z) t ⇒e−φβ+µ(w,z) t+s≥e−φβ(w,y) t.e−φβ(y,z) t ⇒Qβ(w, z, t)∗Qµ(z, y, s)≤Qβ+µ(w, y, t +s). Hence (MFML4) holds. Proposition 3.4. Let φ be a modular-like metric on W . Define o∗u = ou for all o, u ∈ [0 , 1] and Qβ : W×W× (0 ,∞ ) → [0 , 1] by Qβ ( w, y, t ) = t t+φβ(w,y) for all w, y ∈W and β, t > 0. Then (W, Qβ,∗)is a modular fuzzy metric-like space. Proof. (MFML1)- (MFML3) and (MFLM5) are obvious. Now, we prove (MFML4). Since φβ+µ(w, z)≤φβ(w, y) + φβ(y, z),∀β, µ > 0, we have ⇒φβ+µ(w,z) t+s≤φβ(w,y) t+s+φβ(y,z) t+s≤φβ(w,y) t+φβ(y,z) s=sφβ(w,y)+tφµ(y,z) ts ⇒1 + φβ+µ(w,z) t+s≤1 + sφβ(w,y)+tφµ(y,z) ts Since φβ(w, y).φµ(y, z)≥0, we get t+s+φβ+µ(w,z) t+s≤ts+sεβ(w,y)+tφµ(y,z)+φβ(w,y)φµ(y,z) ts ⇒t+s t+s+φβ+µ(w,z)≥ts ts+sφβ(w,y)+tφµ(y,z)+φβ(w,y)φµ(y,z)=t t+φβ(w,y).s φµ(y,z). Hence, Qβ(w, y, t)∗Qµ(y, z, s)≤Qβ+µ(w, z, t +s)and (MFML4) holds. Proposition 3.5. Let ( W, Q, ∗ )be a fuzzy metric space. Define fuzzy set Qβ on W2× (0 ,∞ ) as follows; Qβ ( w, y, t ) = Q ( w, y, βt )for all w, y ∈W and β, t > 0. Then, ( W, Qβ,∗ )is modular fuzzy metric-like space. Proof. (MFML1)- (MFML3) and (MFLM5) are obvious. Now, we prove (MFML4). Qβ+µ(w, y, t +s) = Q(w, y, (β+µ)(t+s)) =Q(w, y, βt +βs +µt +µs) ≥Q(w, y, (βt +µs)) ≥Q(w, z, βt)∗Q(z, y, µs) =Qβ(w, z, t)∗Qµ(z, y, s) ⇒Qβ+µ(w, y, t +s)≥Qβ(w, z, t)∗Qµ(z, y, s). Definition 3.2. Let (W, Qβ,∗)be a modular fuzzy metric-like space. (a) Sequence {wn}n∈N in W is called convergent to w∈W if limn→∞ Qβ ( wn, w, t ) = Qβ(w, w, t)for all β, t > 0. (b) Sequence {wn}n∈N in W is called a Cauchy sequence if limn→∞ Qβ ( wn+p, wn, t )exists and is finite for all β, t > 0, p ≥1. (c) A modular fuzzy metric-like space is called complete if every Cauchy sequence {wn}n∈N in Wconverges to some w∈Wsuch that 2nd Kocaeli Science Congress, November 19-21, 2025 M35-7 KOSC-2025 Proceedings limn→∞ Qβ(wn, w, t)=Qβ(w, w, t) = limn→∞ Qβ(wn+p, wn, t)for all β, t > 0, p ≥1. Remark 3.1. In a modular fuzzy metric-like space, the limit of a convergent sequence may not be unique. Consider Example 3.2. Define the sequence {wn}n∈N in W by {wn} = { 1 + 1 n} for all n∈IN. If w≥2, then limn→∞ Qβ ( wn, w, t ) = limn→∞ βt βt+max{wn,w} = limn→∞ βt βt+w = βt βt+w = βt βt+max{w,w} = Qβ(w, w, t). Hence, the sequence {wn}n∈Nconverges to all w∈Wwith w≥2. Definition 3.3. Let ( W, Qβ,∗ )be a modular fuzzy metric-like space. We will say that the mapping T : W→W is a modular fuzzy contractive mapping if there exists k∈ (0 , 1) such that 1 Qβ(T(w),T (y),t)−1≤k[1 Qβ(w,y,t)−1] for all β, t > 0and w, y ∈W. Theorem 3.1. Let ( W, Qβ, ⋆ )be a complete modular fuzzy metric-like space and T : W→W be a modular fuzzy contractive mapping with contractive constant k . Then T has a unique fixed point w∈Wand Qβ(w, w, t)=1for all β, t > 0. Proof. Let ( W, Qβ, ⋆ )be a complete modular fuzzy metric-like space. For an arbitrary w0∈W , define a sequence {wn} in W by wn = T ( wn−1 )for all n∈IN . If wn = wn−1 for some n∈IN , then wn is a fixed point of T since wn = T ( wn−1 ) = T ( wn ). Therefore, we assume that wn = wn−1 for all n∈IN . For any n∈IN and β, t > 0, from Definition 3.3 we obtain the following, 1 Qβ(wn,wn+1,t)−1 = 1 Qβ(T(wn−1),T (wn),t)−1≤k[1 Qβ(wn−1,wn,t)−1] = k Qβ(wn−1,wn,t)−k. Let Qβ(wn, wn+1, t) = Q(n) β(t)and 1−k=h. Since 1−k=h∈(0,1),1 Q(n) β(t)−1≤k Qβ(wn−1,wn,t)−k=k Q(n−1) β(t)−k. Then, 1 Q(n) β(t)≤k Q(n−1) β(t)+h. Let this approach be maintained; then 1 Qβ(wn−1,wn,t)−1 = 1 Qβ(T(wn−2),T (wn−1),t)−1≤k[1 Qβ(wn−2,wn−1,t)−1] = k Qβ(wn−2,wn−1,t)−k ⇒1 Q(n−1) β(t)−1≤k Q(n−2) β(t)−k ⇒1 Q(n−1) β(t)≤k Q(n−2) β(t)−k+ 1 = k Q(n−2) β(t)+h. The others can be shown similarly. Then, 1 Q(n) β(t)≤k Q(n−1) β(t)+h =k1 Q(n−1) β(t))+h ≤k[k Q(n−2) β(t)+h] + h =k21 Q(n−2) β(t)+kh +h ≤k2[k Q(n−3) β(t)+h] + kh +h =k3[k Q(n−3) β(t)+h] + k2h+kh +h If we continue like this, we get that M35-8 2nd Kocaeli Science Congress, November 19-21, 2025 1 Q(n) β(t)≤kn1 Q(0) β(t)+kn−1h+kn−2h+... +kh +h =kn Q(0) β(t)+ (kn−1+kn−2+... +k+ 1)h =kn Q(0) β(t)+h(1−kn) 1−k =kn Q(0) β(t)+(1−kn) Hence, we have 1 kn Q(0) β(t) +1−kn≤Q(n) β(t), for all β, t > 0, n ∈IN. (1) Now, p≥1, n ∈IN, for all β, t > 0, Qβ(wn, wn+p, t)≥Qβ p (wn, wn+1,t p)∗Qpβ−β p (wn+1, wn+p,pt−t p) ≥Qβ p (wn, wn+1,t p)∗Qβ p (wn+1, wn+2,t p)∗Qpβ−2β p (wn+2, wn+p,pt−2t p) ≥Qβ p (wn, xn+1,t p)∗Qβ p (wn+1, wn+2,t p)∗Qβ p (wn+2, wn+3,t p) ∗Qpβ−3β p (wn+3, wn+p,pt−3t p) ≥Qβ p (wn, wn+1,t p)∗Qβ p (wn+1, wn+2,t p)∗Qβ p (wn+2, wn+3,t p)∗... ∗Qpβ−(p−2)β p (wn+p−2, wn+p,pt−(p−2)t p) ≥Qβ p (wn, wn+1,t p)∗Qβ p (wn+1, wn+2,t p)∗... ∗Qβ p (wn+p−2, wn+p−1,t p) ∗Qpβ−(p−2)β−β p (wn+p−1, wn+p,pt−(p−2)t−t p) ≥Qβ p (wn, wn+1,t p)∗Qβ p (wn+1, wn+2,t p)∗... ∗Qβ p (wn+p−2, wn+p−1,t p) ∗Qβ p (wn+p−1, wn+p,t p) =Q(n) β p (t p)∗Q(n+1) β p (t p)∗... ∗Q(n+p−2) β p (t p)∗Q(n+p−1) β p (t p). From (1), Qβ(wn, wn+p, t)≥1 kn Q(0) β p (t p) +1−kn∗1 kn+1 Q(0) β p (t p) +1−kn+1 ∗... ∗1 kn+p−1 Q(0) β p (t p) +1−kn+p−1 ≥1 kn Q(0) β p (t p) +1 ∗1 kn+1 Q(0) β p (t p) +1 ∗... ∗1 kn+p−1 Q(0) β p (t p) +1 Taking the limit as n→ ∞, since k∈(0,1),limn→∞ kn= 0 and we get limn→∞ Qβ ( wn+p, wn, t ) ≥ 1 ∗ 1 ∗... ∗ 1=1. Moreover, 1 ≥limn→∞ Qβ ( wn+p,wn,t ) ≥ 1and then limn→∞ Qβ(wn+p, xn, t) = 1, for all β, t > 0and p≥1. Hence {wn} is a Cauchy sequence on ( W, Qβ,∗ ). Since ( W, Qβ,∗ )is complete, there exists w∈Wsuch that limn→∞ Qβ(wn, w, t)=Qβ(w, w, t) = limn→∞ Qβ(wn+p, wn, t). (2) Now, let me show you that wis a fixed point of T: We know that 1 Qβ(T(wn),T (w),t)−1≤k[1 Qβ(wn,w,t)−1] = k Qβ(wn, w, t)−k, 1 k Qβ(wn,w,t)+1−k≤Qβ(T(wn), T (w), t). Using the above inequalities, we get Qβ(w, T (w), t)≥Qβ 2 (w, wn+1,t 2)=Qβ 2 (w, wn+1,t 2)∗Qβ 2 (T(wn), T(w),t 2) ≥Qβ 2 (w, wn+1,t 2)∗1 k Qβ 2 (wn,w, t 2)+1−k Taking the limit as n→ ∞, from (2), limn→∞ Qβ(wn, w, t)=1. Then, 1 ≥limn→∞ Qβ ( w, T ( w ) , t ) ≥ 1 ∗ 1and limn→∞ Qβ ( w, T ( w ) , t ) = Qβ ( w, T ( w ) , t ) = 1, for all β, t > 0. 2nd Kocaeli Science Congress, November 19-21, 2025 M35-9