Definition 2.1 ([13, 14]). A mapping η : ℝ⟶[0, 1] is called a fuzzy real number or fuzzy interval if it meets the following two conditions, with its α-level set defined as [η]_α = {δ ∈ ℝ : η(δ) ≥ α},

• 1.

  ∃δ₀ ∈ ℝ s.t. η(δ₀) = 1.

• 2.

  Each α ∈ (0, 1] defines a closed real interval [η]_α = [λ_α, ρ_α], bounded by endpoints satisfying −∞<λ_α ≤ ρ_α < +∞.

The set $$ encompasses all fuzzy real numbers defined as above. When η(δ) = 0 holds for every δ < 0, the element $$ is classified as nonnegative, with $$ designating the subset of all such non-negative fuzzy real numbers.

Characterized by $$ and $$ whenever δ ≠ 0, the fuzzy number $$ is evidently contained in $$. Furthermore, ℝ can be mapped into $$ through the correspondence $$, where $$ is defined as $$ for each a ∈ ℝ.

Definition 2.2 ([8]). Consider a nonempty set $$ with b ≥ 1, and let $$ be a function mapping $$ to $$. Assume $$ and $$ are symmetric, nondecreasing functions from [0, 1] × [0, 1] to [0, 1], satisfying $$ and $$. For any α ∈ (0, 1] and elements $$, the mapping is defined as follows:

Then, $$ is defined as a fuzzy b-metric, and the five-tuple $$ is known as KS-FbMS with the coefficient b, and if

(BM1) $$⇐x = y;

(BM2) $$, $$;

(BM3) $$:

(BM$$) $$, if μ ≤ λ₁(x, z), ν ≤ λ₁(z, y) and b(μ + ν) ≤ λ₁(x, y);

(BM$$) $$, if μ ≥ λ₁(x, z), ν ≥ λ₁(z, y) and b(μ + ν) ≥ λ₁(x, y).

For illustrative examples of KS-FbMS, one may consult [8, 9].

The following propositions enumerate several properties (a detailed exposition is provided in [8]).

Proposition 2.1 ([8]). Let $$ be a KS - FbMS, and suppose that

($$) $$;

$$ for each $$, $$ s.t. $$, $$;

Then, ($$1) ⇒ ($$2) ⇒ for each $$, $$ s.t. $$

Definition 2.3 ([8]). Let $$ be a KS - FbMS.

• 1.

  A sequence {x_n} in $$ converges to $$ (denoted $$) if, $$, $$;

• 2.

  A sequence {x_n} in $$ is a Cauchy sequence if, ∀ε > 0 and $$, $$ s.t. $$, whenever n, m ≥ N;

• 3.

  $$ is complete if every Cauchy sequence in $$ converges.

Lemma 2.1 ([11]). Let ϕ : ℝ⁺⟶ℝ⁺ satisfy ϕ⁻¹({0}) = {0}.

• 1.

  For a nondecreasing ϕ, if ϕⁿ(p)⟶0 as n⟶∞ holds for all p > 0, then $$ and ϕ(p) < p, ∀p > 0.

• 2.

  If ϕ(p) < p and $$, ∀p > 0, then $$, ∀p > 0.

Lemma 2.2 ([11]). Let ψ : ℝ⁺⟶ℝ⁺ satisfy ψ⁻¹({0}) = {0}.

• 1.

  For a nondecreasing ψ, if ψⁿ(p)⟶+∞ as n⟶∞ holds for all p > 0, then p < ψ(p), ∀p > 0.

• 2.

  If p < ψ(p) and $$, ∀p > 0, then $$, ∀p > 0.

Theorem 3.1. Let $$ be a complete KS-FbMS with ($$) and the coefficient b ≥ 1. Suppose that $$ satisfies the following:

Proof 3.1. First from Lemma 2.1 (1), we have

From (1) and (2), we have T, which is continuous. Now, let $$ and ε > 0 be arbitrary. Choose n₀ ∈ ℕ such that

Put $$, and for each m ∈ ℕ, set x_m = fx_m−1 = f^mx₀; it follows from (1) that

so $$, $$. The next step shows that {x_m} is a Cauchy sequence. Now, choose m₀ ∈ ℕ s.t. $$, $$. By (1), (3), and Proposition 2.1, then for each $$$$ s.t.

Assume that $$, $$, then

By induction, for any k ∈ ℕ and $$, we obtain that

For any m > m₀ and k ∈ ℕ, by (1) and (4), we have

Passing to the as m⟶∞ in the above inequality, we obtain that

Remark 3.1. Theorem 3.1 quantifies the independence of the Banach contraction principle from the space coefficient b.

Corollary 3.1. Let $$ be a complete KS-FMS with ($$). Assume that $$ satisfies

Remark 3.2. Notice that Corollary 3.1 is a generalization of [11, Theorem 3.1(b)]. Indeed the condition $$(p > 0) of Corollary 3.1 is weaker than the condition $$(p > 0) of [11, Theorem 3.1(b)]. For example, ϕ(p) = p/1 + p (p > 0) satisfies the hypothesis of ϕ in Corollary 3.1 and does not satisfy the hypothesis of ϕ in [11, Theorem 3.1(b)].

Corollary 3.2. ([12], Theorem 12.2). Let $$ be a complete b - MS with constant b ≥ 1. Suppose that $$ satisfies

Corollary 3.3. ([15], Theorem 2.1). Let $$ be a complete b - MS with constant b ≥ 1. Suppose that $$ satisfies

Proof 3.2. Define ϕ(p) = λp for all p ≥ 0. It is easy to get that all assumptions of Corollary 3.2 are satisfied. Then, T has a unique fixed point $$ and $$.

Remark 3.3. In numerous established fixed point theorems for Banach-type contractions within b-metric spaces (where b ≥ 1), the contraction constants λ are predominantly confined to the interval [0, 1/b), as demonstrated in works such as [16, 17, 18]. Do the above conclusions remain true for any λ ∈ [1/b, 1)? In [15, Theorem 2.1], Dung and Hang successfully addressed this problem and extended the Banach contraction principle from metric spaces to b-metric spaces, utilizing a contraction constant λ within the range [0, 1). Our Corollary 3.3 is exactly the same as [15, Theorem 2.1].

Lemma 4.1. Let [α, β] be a closed interval with α > 0 and ϕ ∈ Φ, then

• i.

  $$;

• ii.

  if {u_n} is a nonnegative real number sequence by u_n+1 ≤ ϕ(u_n), for u₀ < β, there exists a finite positive integer K such that u_k < α for all k ≥ K.

Proof 4.1.

• i.

  Suppose not, then we have $$. It follows that there exists a sequence {p_k} ⊂ [α, β] such that p_k − ϕ(p_k)⟶0 as k⟶∞. Since {p_k} is bounded, there exists a subsequence $$ of {p_k} satisfying $$ as i⟶∞, p₀ ∈ [α, β]. Owing to $$ as i⟶∞, we have that

• ii.

  By means of (i), for each p ∈ [α, β], we denote

For u₀ < β, from (8), it follows that

If u₁ < α, then we take K = 1, and the desired result follows immediately. If u₁ ∈ [α, β), then by (9), we have

Continuing this process, for i ∈ ℕ, if u_i < α, then we take K = i, and the desired result follows immediately. If u_i ∈ [α, β), we have u_i+1 < β − (i + 1)σ. Hence, there exists a finite positive integer K such that u_k < α for all k ≥ K with u₀ < β.

Theorem 4.1. Let $$ be a complete KS-FbMS with ($$) and the coefficient b ≥ 1. Suppose that $$ is a mapping such that

where ϕ ∈ Φ. Then, T has a unique fixed point $$.

Proof 4.2. Since ϕ⁻¹({0}) = {0} and ϕ(p) < p, ∀p > 0, we have ϕ(p) ≤ p, ∀p ∈ ℝ⁺. Given $$, let $$ be a sequence by x_n = Tx_n−1 = Tⁿx₀, ∀n ∈ ℕ. Suppose that $$ and $$. By (11), we have

It follows from (12) that $$ is a nonnegative and nonincreasing sequence, and hence,

In fact, if $$, then,

For closed interval [1/2b, 1], it follows from Lemma 4.1 that for all $$ and $$ with $$, there exists a finite positive integer k such that $$. Notice that $$. Thus, there exists a natural number N, s.t.

By induction, for any m > N, we have $$. Let $$.

The next step shows that {x_n} is a Cauchy sequence in $$. For any ε > 0, according to Lemma 4.1, there exists finite positive integer k such that $$ and $$ and $$. Furthermore, for any m, n ∈ ℕ with m > n ≥ k, due to $$, it follows from Lemma 4.1 that $$. Thus, {x_n} is a Cauchy sequence.

Lemma 4.2. Let [α, β] be a closed interval with α > 0 and ψ ∈ Ψ, then $$.

Proof 4.3. Suppose not, then we have $$. It follows that there exists a sequence {p_k} ⊂ [α, β] such that ψ(p_k) − p_k⟶0 as k⟶∞. Since {p_k} is bounded, there exists a subsequence $$ of {p_k} satisfying $$ as i⟶∞, p₀ ∈ [α, β]. Owing to $$ as i⟶∞, we have that

Lemma 4.3. Let $$ be a KS-FbMS with ($$) and the coefficient b ≥ 1. Let [α, β] be a closed interval with 0 < α < β ≤ ε. Suppose that $$ satisfies

Proof 4.4. By means of Lemma 4.2, for each p ∈ [α, β], we denote

For all $$ and $$ with $$, without loss of generality, we assume that $$ From (14) and (15), it follows that

Moreover, $$. It follows from (16) that

If $$, then we take n₀ = 1, and the desired result follows immediately; if $$, then by (17), we have

Continuing this process, for k ∈ ℕ, if $$, then we take n₀ = k; if $$, we have $$ and $$ with $$. Hence, there exists a finite positive integer n₀ such that $$ and $$ with $$.

Theorem 4.2. Let $$ be a complete KS - FbMS with ($$) and the coefficient b ≥ 1. Suppose that $$ is a mapping such that

Proof 4.5. Since ψ⁻¹({0}) = {0} and p < ψ(p), ∀p > 0, we have p ≤ ψ(p), ∀p ∈ ℝ⁺. Given $$, let $$ be a sequence by x_n = Tx_n−1 = Tⁿx₀, ∀n ∈ ℕ. Suppose that $$ and $$. By (19), we have

Moreover, $$ is a nonnegative nonincreasing sequence; and hence,

In fact, if $$, then,

Without loss of generality, we can assume that ε₀ < ε/2b². For the interval [ε₀, b²ε₀ + ε/3], according to Lemma 4.3, there exists a finite positive integer i such that $$ and $$ with $$. It follows Proposition 2.1 that $$ s.t.

Since $$ and $$ converge to 0 as k⟶∞, there exists K ∈ ℕ s.t.

Hence, we get

This shows that $$. Moreover, we have Tx^∗ = x^∗. If $$ with Ty^∗ = y^∗ and x^∗ ≠ y^∗, then there exists $$ s.t. $$. From the hypothesis on ψ and (19), it follows that

We obtain the following some corollaries from Theorem 4.1, since KS-FbMS is a generalization of KS-FMS and b-MS.

Corollary 4.1. ([11], Theorem 3.1(a)). Let $$ be a complete KS-FMS with ($$). Suppose that $$ satisfies

The Alber–Guerre Delabriere contraction (see [19, 20]) represents a notable extension of Banach’s contraction principle. Building on this generalization, Corollary 4.1 enables the derivation of a fixed-point theorem within the KS-FMS framework.

Corollary 4.2. ([11], Theorem 4.1). Let $$ be a complete KS-FMS with ($$) and $$ be a mapping. If there exists a nondecreasing function ϕ : ℝ⁺⟶ℝ⁺ with ϕ⁻¹({0}) = {0} such that

Then, T has a unique fixed point $$.

Proof 4.6. Let λ(p) = p − ϕ(p), (p ≥ 0). The next proves that λ(p) satisfies the hypothesis of Corollary 4.1. It is clear that λ(0) = 0. Since ϕ⁻¹({0}) = {0}, we obtain ϕ(p) > 0, (p > 0), which deduces that λ(p) < p, (p > 0). For ξ ∈ (p/2, 3p/2), the monotonicity of ϕ(p) yields

Then, we obtain

Therefore, we have λ(p) satisfies the hypothesis of Corollary 4.1. Using (21), we have

Finally, by Corollary 4.1, we obtain the desired result.

Remark 4.1. Notice that we can also obtain the fixed point theorem of Alber–Guerre Delabriere type in KS-FbMS. The result can be established using a similar approach to that in the proof of Corollary 4.2.

Corollary 4.3. Let $$ be a complete b - MS with b ≥ 1. Suppose that $$ is a mapping s.t.

Corollary 4.4. Let $$ be a complete b - MS with b ≥ 1 and $$ be a mapping. If there exists a nondecreasing function ϕ : ℝ⁺⟶ℝ⁺ with ϕ⁻¹({0}) = {0} s.t.

Then, T has a unique fixed point $$.

Proof 4.7. The proof is analogous to the proof of Corollary 4.2.

Corollary 4.5. ([11], Theorem 3.2). Let $$ be a complete KS - FMS with ($$). Suppose that $$ is a mapping s.t.

Corollary 4.6. Let $$ be a complete b - MS with b ≥ 1. Suppose that $$ is a mapping s.t.

Remark 4.2. Note that the conclusion is up to date in b -MS.