Common Fixed Point Theorems in a New Fuzzy Metric Space

Definition 2.1 (see [6].)The 3-tuple (X, M, *) is said to be a fuzzy metric space if X is an arbitrary set, * is a continuous t-norm, and M is a fuzzy set on X² × (0, ∞) satisfying the following conditions, for all x, y, z ∈ X,   t, s > 0:

• (i)

  M(x, y, t) > 0;

• (ii)

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

• (iii)

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

• (iv)

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

• (v)

  M(x, y, −):(0, ∞)→[0,1] is continuous.

If (X, M, *) is a fuzzy metric space, it will be said that (M, *) is a fuzzy metric on X.

Lemma 2.2 (see [6].)Let (X, M, *) be a fuzzy metric space and let τ be the topology induced by the fuzzy metric. Then, for a sequence {x_n} _n∈ℕ in X, x_n → x if and only if M(x_n, x, t) → 1 as n → ∞ for all t > 0.

Definition 2.3 (see [6].)A sequence {x_n} _n∈ℕ in a fuzzy metric space (X, M, *) is called a Cauchy sequence if and only if for each 1 > ε > 0,   t > 0, there exists n₀ ∈ ℕ such that M(x_n, x_m, t) > 1 − ε for all n, m ≥ n₀. A fuzzy metric space is said to be complete if and only if every Cauchy sequence is convergent.

Definition 2.4 (see [13].)Let A be a nonempty subset of a fuzzy metric space (X, M, *). For a ∈ X and t > 0, M(a, A, t) = sup {M(a, y, t)∣y ∈ A,   t > 0}.

Lemma 2.5 (see [28].)Let G be a set and let {G_α : α ∈ [0,1]} be a family of subsets of G such that

• (1)

  G₀ = G;

• (2)

  α ≤ β implies G_β⊆G_α;

• (3)

  α₁ ≤ α₂ ≤ ⋯, lim _n→∞ α_n = α implies $$.

Then, the function φ : G → [0,1] defined by φ(x) = sup {α ∈ [0,1] : x ∈ G_α} has the property that {x ∈ G : φ(x) ≥ α} = G_α  for every α ∈ [0,1].

Lemma 2.6 (see [26].)Let (X, M, *) be a fuzzy metric space. Then, (C(X), H_M, *) is complete if and only if (X, M, *) is complete.

Lemma 2.7 (see [26].)Let (X, d) be a metric space. Then, the Hausdorff fuzzy metric $$ of the standard fuzzy metric (M_d, ·) coincides with standard fuzzy metric $$ of the Hausdorff metric H_d on C(X).

Definition 3.1. Let (X, M, *) be a fuzzy metric space. The M_∞-fuzzy metric between two fuzzy sets is induced by the Hausdorff fuzzy metric H_M as

where μ₁, μ₂ ∈ 𝒞(X),   t > 0, and is the fuzzy separation of μ₁ from μ₂.

Lemma 3.2. Let (X, M, *) be a fuzzy metric space, μ₁, μ₂, μ₃ ∈ 𝒞(X), s, t > 0. Then one has

• (1)

  M_∞(μ₁, μ₂, t)∈(0,1],

• (2)

  M_∞(μ₁, μ₂, t) = M_∞(μ₂, μ₁, t),

• (3)

  ρ_∞(μ₁, μ₂, t) = 1 if and only if μ₁⊆μ₂,

• (4)

  M_∞(μ₁, μ₂, t) = 1 if and only if μ₁ = μ₂,

• (5)

  if μ₁⊆μ₂, then ρ_∞(μ₁, μ₃, t + s) ≥ M_∞(μ₂, μ₃, t),

• (6)

  ρ_∞(μ₁, μ₃, t + s) ≥ M_∞(μ₁, μ₂, t)*ρ_∞(μ₂, μ₃, s),

• (7)

  M_∞(μ₁, μ₃, t + s) ≥ M_∞(μ₁, μ₂, t)*M_∞(μ₂, μ₃, s),

• (8)

  M_∞(μ₁, μ₂, −):(0, ∞)→[0,1] is continuous.

Proof. For (1), by the definition of the λ-cut [μ₁] _λ, for every λ ∈ I, [μ₁] _λ is nonempty compact in X. By the theorem of nested intervals, there exists a point a₀ in [μ₁] _λ for every λ ∈ I, likewise, there exists a points b₀ in [μ₂] _λ for every λ ∈ I. Thus, M_∞(μ₁, μ₂, t) > 0. Moreover, it is clear that A = B⇔H_M(A, B, t) = 1⇔M_∞(μ₁, μ₂, t) = 1.

For (2), it is clear that M_∞(μ₁, μ₂, t) = M_∞(μ₂, μ₁, t).

For (3), since ρ_∞(μ₁, μ₂, t) = 1 if and only if ρ([μ₁] _λ, [μ₂] _λ, t) = 1 for all λ ∈ I, which implies [μ₁] _λ⊆[μ₂] _λ for all λ ∈ I, we have that ρ_∞(μ₁, μ₂, t) = 1 if and only if μ₁⊆μ₂.

For (4), it follows from (3).

For (5), for every λ ∈ I, any x ∈ [μ₁] _λ, y ∈ [μ₂] _λ and z ∈ [μ₃] _λ, by the proof of Theorem  1 in [26], we have

with all x, y, z ∈ X, which implies for all x ∈ [μ₁] _λ and all y ∈ [μ₂] _λ. Since μ₁⊆μ₂, then ρ([μ₁] _λ, [μ₂] _λ, s) = 1. By (iv) of Definition 2.1 and the arbitrariness of x and y, we have which implies Consequently, ρ_∞(μ₁, μ₃, t + s) ≥ M_∞(μ₂, μ₃, t).

For (6), for every λ ∈ I, by the proof of (5) and (iv) of Definition 2.1, we have

Consequently, ρ_∞(μ₁, μ₃, t + s) ≥ M_∞(μ₁, μ₂, t)*ρ_∞(μ₂, μ₃, s).

For (7), for every λ ∈ I, by the proof of (6), we have

Similarly, it can be shown that Hence, M_∞(μ₁, μ₃, t + s) ≥ M_∞(μ₁, μ₂, t)*M_∞(μ₂, μ₃, s).

For (8), by the continuity on (0, ∞) of the function t ↦ H_M(A, B, t), it is clear that M_∞(μ₁, μ₂, −):(0, ∞)→[0,1] is continuous.

Theorem 3.3. Let (X, M, *) be a fuzzy metric space. Then, (𝒞(X), M_∞, *) is a fuzzy metric space, where M_∞ is a fuzzy set on the 𝒞(X) × 𝒞(X)×(0, +∞).

Proof. It is easily proved by Lemma 3.2.

Example 3.4. Let d be the Euclidean metric on ℝ, and let A = [a₁, a₂] and let B = [b₁, b₂] be two compact intervals. Then, H_d(A, B) = max {|a₁ − b₁ | , |a₂ − b₂|}. Let (ℝ, M_d, *) be a fuzzy metric space, where a*b the usual multiplication for all a, b ∈ [0,1], and M_d is defined on ℝ × ℝ × (0, ∞) by

Denote by 𝒞(ℝ) the totality of fuzzy sets μ : ℝ → [0,1] which satisfy that for each λ ∈ I, the λ-cut of μ[μ] _λ = {x ∈ ℝ : μ(x) ≥ λ} is a nonempty compact interval. For any λ-cuts of fuzzy sets μ₁, μ₂ ∈ 𝒞(ℝ) and for all t > 0, by a simple calculation, we have So by Definition 3.1, we get

Definition 4.1. Let (𝒞(X), M_∞, *) be a fuzzy metric space. For t ∈ (0, +∞), define B(μ, r, t) with center a fuzzy set μ ∈ 𝒞(X) and radius r, 0 < r < 1, t > 0 as

Proposition 4.2. Every B(μ, r, t) is an open set.

Proof. It is identical with the proof in [6].

Proposition 4.3. Let (𝒞(X), M_∞, *) be a fuzzy metric space. Define $$.

Then, $$ is a topology on 𝒞(X).

Proof. It is identical with the proof in [6].

Definition 4.4. A sequence {μ_n} in a fuzzy metric space (𝒞(X), M_∞, *) is a Cauchy sequence if and only if for each ε > 0, t > 0, there exists n₀ ∈ ℕ such that M_∞(μ_n, μ_m, t) > 1 − ε for all n, m ≥ n₀.

Lemma 4.5. Let (𝒞(X), M_∞, *) be a fuzzy metric space on fuzzy metric M_∞ and let τ be the topology induced by the fuzzy metric M_∞. Then, for a sequence {μ_n} in 𝒞(X), μ_n → μ if and only if M_∞(μ, μ_n, t) → 1 as n → ∞.

Proof. It is identical with the proof of Theorem  3.11 in [6].

Theorem 4.6. The fuzzy metric space (𝒞(X), M_∞, *) is complete provided (X, M, *) is complete.

Proof. Let (X, M, *) be a complete fuzzy metric space and let a sequence {μ_n, n ≥ 1} be a Cauchy sequence in 𝒞(X). Consider a fixed 0 < λ < 1. Then, {[μ_n] _λ, n ≥ 1} is a Cauchy sequence in (C(X), H_M, *), where C(X) denotes all nonempty compact subsets of (X, M, *).

Since (C(X), H_M, *) is complete by Lemma 2.6, it follows that [μ_n] _λ → μ_λ ∈ C(X). Actually, from the definition of M_∞ and the continuity of H_M, it is easy to see that [μ_n] _λ → μ_λ, uniformly in λ ∈ [0,1].

Now, consider the family {μ_λ : λ ∈ [0,1]}, where μ₀ = X. Take λ ≤ β, we have

Since [μ_n] _β⊆[μ_n] _λ, it follows that ρ([μ_n] _β, [μ_n] _λ, t/3) = 1. Thus, for each 0 < ε < 1, ρ(μ_β, μ_λ, t) ≥ ρ(μ_β, [μ_n] _β, t/3)*ρ([μ_n] _λ, μ_λ, t/3) if n is large enough. Hence, ρ(μ_β, μ_λ, t) = 1, and by Lemma 3.2, we have μ_β⊆μ_λ.

Now, take λ_n↑ and lim _n→∞λ_n = λ. We have to show that $$. It is clear that

On the other hand, we have for fixed j. However, Consequently, for every 0 < ε < 1, there exists 0 < ε₀ < ε < 1 such that (1 − ε₀)*(1 − ε₀)*(1 − ε₀) > 1 − ε. For given ε₀, since [μ_j] _λ → μ_λ, there exists $$ such that for $$. Now, for any p ≥ 1. Since $$, we obtain Now, $$ for j ≥ j₀ and all t > 0. Note that (since the convergence [μ_j] _λ → μ_λ is uniform in λ) j₀ does not depend on p. Since $$ decreases to $$, if follows that $$ for some p₀ (depending on j).

Thus, $$, if j is large.

Finally, by taking j large enough, we obtain

that is, From (4.3) and (4.9), it yields $$. Thus, Lemma 2.5 is applicable and there exists μ ∈ 𝒞(X) for every λ ∈ [0,1] such that [μ_n] _λ → μ_λ. It remains to show that μ_n → μ in (𝒞(X), M_∞, *).

Let ε > 0. Then, since {μ_n} is a Cauchy sequence, there exists n_ε such that n, m > n_ε implies M_∞(μ_n, μ_m, t) > 1 − ε.

Let n(>n_ε) be fixed. Then,

Thus, μ_n → μ in the M_∞-fuzzy metric. The proof is completed.

Lemma 4.7. Let (X, M, *) be a compact fuzzy metric space and compact subsets A, B ∈ C(X). Then, for each x ∈ A and t > 0, there exists a y ∈ B such that M(x, y, t) ≥ H_M(A, B, t).

Proof. Suppose there exists a x₀ ∈ A such that M(x₀, y, t) < H_M(A, B, t) for any y ∈ B and t > 0. Then,

that is, So, This is a contradiction with x ∈ A.

Lemma 4.8. Let (X, M, *) be a compact fuzzy metric space, t > 0 and A, B ∈ C(X). Then, for any compact set A₁⊆A, there exists a compact set B₁⊆B such that H_M(A₁, B₁, t) ≥ H_M(A, B, t).

Proof. Let C = {y: there exists a x ∈ A₁ such that M(x, y, t) ≥ H_M(A, B, t)} and let B₁ = C⋂B. For any x ∈ A₁⊆A,   t > 0, by Lemma 4.7, there exists a y ∈ B such that

Thus, B₁ ≠ ∅, moreover, B₁ is compact since it is closed in X and B₁⊆B.

Now, for any x ∈ A₁, t > 0, there exists a y ∈ B₁ such that

Thus, we have M(x, B₁, t) ≥ H_M(A, B, t), which implies that Similarly, it can be shown that ρ(A₁, B₁, t) ≥ H_M(A, B, t).

Hence, H_M(A₁, B₁, t) ≥ H_M(A, B, t). This completes the proof.

Theorem 4.9. Let (X, M, *) be a compact fuzzy metric space and μ₁, μ₂ ∈ 𝒞(X), t > 0. Then, for any μ₃ ∈ 𝒞(X) satisfying μ₃⊆μ₁, there exists a μ₄ ∈ 𝒞(X) such that μ₄⊆μ₂ and

Proof. Since μ₁, μ₂, and μ₃ are normal, we have ∅ ≠ [μ₃] _λ⊆[μ₁] _λ and ∅ ≠ [μ₂] _λ for all λ ∈ I. Let

and let B_λ = C_λ⋂[μ₂] _λ. For any x ∈ [μ₃] _λ⊆[μ₁] _λ, by Lemma 4.7, there exists a y ∈ [μ₂] _λ such that Thus, B_λ is nonempty compact in X, moreover, B_λ⊆Bγ if 0 ≤ γ ≤ λ ≤ 1.

From the proof of Lemma 4.8, we have

By Lemma  3.1 in [28], there exists a fuzzy set μ₄ with the property that [μ₄] _λ = B_λ for λ ∈ I. Since B_λ are nonempty compact for all λ ∈ I, we have μ₄ ∈ 𝒞(X). Consequently, This completes the proof.

Definition 4.10 (see [24].)Let X, Y be any fuzzy metric space. ℱ is said to be a fuzzy mapping if and only if ℱ is a mapping from the space 𝒞(X) into 𝒞(Y), that is, ℱ(μ) ∈ 𝒞(Y) for each μ ∈ 𝒞(X).

Theorem 5.1. Let (X, M, *) be a compact fuzzy metric space and let $$ be a sequence of fuzzy self-mappings of 𝒞(X). Let ϕ : [0,1]→[0,1] be a nondecreasing function satisfying the following condition: ϕ is continuous from the left and

where ϕⁿ denote the nth iterative function of ϕ. Suppose that for each μ₁, μ₂ ∈ 𝒞(X), and for arbitrary positive integers i and j,   i ≠ j,   t > 0, then there exists μ_* ∈ 𝒞(X) such that μ_*⊆ℱ_i(μ_*) for all i ∈ Z₊.

Proof. Let μ₀ ∈ 𝒞(X) and μ₁⊆ℱ₁(μ₀). By Theorem 4.9, for any t > 0, there exists μ₂ ∈ 𝒞(X) such that μ₂⊆ℱ₂(μ₁) and

Again by Theorem 4.9, for any t > 0, we can find μ₃ ∈ 𝒞(X) such that μ₃⊆ℱ₃(μ₂) and By induction, we produce a sequence {μ_n} of points of 𝒞(X) such that Now, we prove that {μ_n} is a Cauchy sequence in 𝒞(X). In fact, for arbitrary positive integer n, by the inequality (5.2), Lemma 3.2, and the formula (5.5), we have where μ_n⊆ℱ_n(μ_n−1) implies ρ_∞(μ_n, ℱ_n(μ_n−1), 2t) = 1, by (3) of Lemma 3.2. In addition, it is easy to get that ϕ(h) > h for all h ∈ (0,1). In fact, suppose that there exists some t₀ ∈ (0,1) such that ϕ(h₀) ≤ h₀. Since ϕ is nondecreasing, we have Since ϕ(h)*ϕ²(h)* ⋯ *ϕⁿ(h) → 1 as n → ∞, for all h ∈ (0,1), then we have ϕⁿ(h₀) → 1 as n → ∞. From the inequality (5.7), we have 1 ≤ h₀. This is a contradiction which implies ϕ(h) > h for all h ∈ (0,1). We can prove that M_∞(μ_n−1, μ_n, t) ≤ M_∞(μ_n, μ_n+1, t). In fact, if M_∞(μ_n−1, μ_n, t) > M_∞(μ_n, μ_n+1, t), then from the inequality (5.6), we get which is a contradiction. Thus, from the inequality (5.6), we have Furthermore, for arbitrary positive integers m and k, we have and ϕ(h)*ϕ²(h)* ⋯ *ϕⁿ(h) → 1 as n → ∞, for all h ∈ (0,1), it follows that is convergent, which implies that {μ_n} is a Cauchy sequence in 𝒞(X). Since X is a compact fuzzy metric space, it follows X is complete. By Theorem 4.6, 𝒞(X) is complete. Let μ_n → μ_*. Next, we show that μ_*⊆ℱ_i(μ_*) for all i ∈ Z₊. In fact, for arbitrary positive integers i and j, i ≠ j, by Theorem 4.9, we have where μ_j⊆ℱ_j(μ_j−1) implies ρ_∞(μ_j, ℱ_j(μ_j−1), t) = 1. Letting n → ∞, M_∞(μ_n, μ_*, t) = 1, and using the left continuity of ϕ, we have which implies ρ_∞(μ_*, ℱ_i(μ_*), t) = 1. Hence, by Lemma 3.2, it follows that μ_*⊆ℱ_i(μ_*). Then, the proof is completed.

Theorem 5.2. Let (X, M, *) be a compact fuzzy metric space and let $$ be a sequence of fuzzy self-mappings of 𝒞(X). Suppose that for each μ₁, μ₂ ∈ 𝒞(X), and for arbitrary positive integers i and j, i ≠ j, t > 0,

where ϕ(h₁, h₂, h₃, h₄, h₅):(0,1] ⁵ → [0,1] is nondecreasing and continuous from the left for each variable. Denote γ(h) = ϕ(h, h, h, a, b), where (a, b)∈{(h*h, 1), (1, h*h)}. If where γⁿ denote the nth iterative function of γ, then there exists μ_* ∈ 𝒞(X) such that μ_*⊆ℱ_i(μ_*) for all i ∈ Z₊.

Proof. Let μ₀ ∈ 𝒞(X) and μ₁⊆ℱ₁(μ₀). By Theorem 4.9, for any t > 0, there exists μ₂ ∈ 𝒞(X) such that μ₂⊆ℱ₂(μ₁) and

Again by Theorem 4.9, for any t > 0, we can find μ₃ ∈ 𝒞(X) such that μ₃⊆ℱ₃(μ₂) and By induction, we produce a sequence {μ_n} of points of 𝒞(X) such that Now, we prove that {μ_n} is a Cauchy sequence in 𝒞(X). In fact, for arbitrary positive integer n, by the inequality (5.14), Lemma 3.2, and the formula (5.18), we have where μ_n⊆ℱ_n(μ_n−1) implies ρ_∞(μ_n, ℱ_n(μ_n−1), 2t) = 1 by (3) in Lemma 3.2 Likewise, we have γ(h) > h for all h ∈ (0,1), t > 0. If M_∞(μ_n−1, μ_n, t) > M_∞(μ_n, μ_n+1, t), then from the inequality (5.19), we obtain which is a contradiction. Thus, from the inequality (5.19), we have Furthermore, for arbitrary positive integers m and k, we have Furthermore, for arbitrary positive integers m and k, we have Since ϕ(h)*ϕ²(h)* ⋯ *ϕⁿ(h) → 1 as n → ∞, for all h ∈ (0,1), it follows that is convergent, this implies that {μ_n} is a Cauchy sequence in 𝒞(X). Since X is a compact fuzzy metric space, it follows that X is complete. By Theorem 4.6, 𝒞(X) is complete. Let μ_n → μ_*. Now, we show that μ_*⊆ℱ_i(μ_*) for all i ∈ Z₊. In fact, for arbitrary positive integers i and j, i ≠ j, by Theorem 4.9, we have where μ_j⊆ℱ_j(μ_j−1) implies ρ_∞(μ_j, ℱ_j(μ_j−1), t) = 1. Letting n → ∞, M_∞(μ_n, μ_*, t) = 1, and using the left continuity of ϕ, we have which implies ρ_∞(μ_*, ℱ_i(μ_*), t) = 1. Hence, by Lemma 3.2, it follows that μ_*⊆ℱ_i(μ_*), then the proof is completed.

Example 5.3. Let (𝒞(X), M_∞, *) be a fuzzy metric space, where X = [−1,1], M_d, H_M, and M_∞ are the same as in Example 3.4. Then, (𝒞(X), M_∞, *) is a compact metric space.

Now, define ϕ : [0,1]→[0,1] as $$, and define $$ a sequence of fuzzy self-mappings of 𝒞(X) as

For arbitrary positive integers i and j, without loss of generality, suppose i < j. For each μ₁, μ₂ ∈ 𝒞(X), by a routine calculation, we have

Therefore, by Theorem 5.1, we assert that the sequence of fuzzy self-mappings $$ has a common fixed point μ_* in 𝒞(X). In fact, it is easy to check that