On Common Fixed Point Theorems in the Stationary Fuzzy Metric Space of the Bounded Closed Sets

Definition 1 (see [39].)A triangular norm (or t-norm for short) is a binary operation * on the unit interval [0,1], that is, a function * : [0,1] ² → [0,1], such that for all a, b, c, d ∈ [0,1] the following four axioms are satisfied:

• (1)

  a*1 = a (boundary condition);

• (2)

  a*b ≤ c*d whenever a ≤ c and b ≤ d (monotonicity);

• (3)

  a*b = b*a (commutativity);

• (4)

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

A t-norm * is said to be continuous if it is a continuous function in [0,1] ²; a t-norm * is said to be positive if a*b > 0 whenever a, b ∈ (0,1]. The following are examples of t-norms: a*_Pb = a · b; a∧b = min (a, b), where a · b denotes the usual multiplication for all a, b ∈ [0,1].

Definition 2 (see [40].)A stationary fuzzy metric space is an ordered triple (X, M, *) such that X is an arbitrary nonempty set, * is a continuous t-norm, and M is a fuzzy set of X × X satisfying the following conditions, for all x, y, z ∈ X:

• (1)

  M(x, y) > 0,

• (2)

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

• (3)

  M(x, y) = M(y, x),

• (4)

  M(x, y) ≥ M(x, z)*M(z, y).

Example 3 (see [14].)Let (X, d) be a metric space. Denote by a · b the usual multiplication for all a, b ∈ [0,1], and define M_d on X × X by

for all x, y ∈ X. Then (M_d, ·) is a stationary fuzzy metric on X which will be called a standard stationary fuzzy metric.

Definition 4 (see [41].)A stationary fuzzy pseudometric space is an ordered triple (X, M, *) such that X is an arbitrary nonempty set, * is a continuous t-norm, and M is a fuzzy set of X × X satisfying the following conditions, for all x, y, z ∈ X:

• (1)

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

• (2)

  M(x, y) = M(y, x),

• (3)

  M(x, y) ≥ M(x, z)*M(z, y).

Definition 5 (see [42].)Let (X, M, *) be a stationary fuzzy metric space and A ⊂ X. If, for all ε ∈ (0,1), B_M(x, ε)⋂ (A − {x}) ≠ ∅, then x is an accumulation point of A; the set of all accumulation points of A is called the derived set of A, denoted by d(A); the union of A and d(A) is called the closure of A, denoted by $$. If d(A) ⊂ A, then A is a closed set of X.

Definition 6 (see [42].)Let (X, M, *) be a stationary fuzzy metric space, and A ⊂ X. If there exists r ∈ (0,1) such that for all x, y ∈ A we have M (x, y) > 1 − r, then we say that A is a bounded subset of X; if X itself is a bounded set we will say that (X, M, *) is a bounded stationary fuzzy metric space.

Definition 7 (see [43].)Let (X, M, *) be a stationary fuzzy metric space. For all A, B ∈ 𝒫(X), we define a function H_M on 𝒫(X) × 𝒫(X) by

where ρ(A, B) = inf _x∈A M (x, B).

Definition 8. Let (X, M, *) be any stationary metric space. A₀ ∈ 𝒞ℬ(X) is said to be a fixed point of a self-mapping f of 𝒞ℬ(X) if and only if A₀⊆f(A₀).

Proposition 9. Let (X, M, *) be a stationary fuzzy metric space. Then, for all A, B, C ∈ 𝒫(X),

• (1)

  ρ(A, B) = 1 if and only if $$ if and only if   $$ if and only if $$,

• (2)

  for all x, x^′ ∈ X, M(x, B) ≥ M(x, x^′)*M(x^′, B),

• (3)

  $$,

• (4)

  ρ(A, C) ≥ ρ(A, B)*ρ(B, C),

• (5)

  if $$, then ρ(A, C) ≥ H_M (B, C),

• (6)

  ρ(A, C) ≥ H_M (A, B)*ρ(B, C),

• (7)

  H_M (A, B) = 1 if and only if $$.

Proof. In fact, we can prove this proposition by a similar proof of Proposition 1 in [42].

Theorem 10. Let (X, M, *) be a stationary fuzzy metric space, and then (𝒫(X), H_M, *) is a stationary fuzzy pseudometric space.

Proof. Let A, B, C ∈ 𝒫(X); by the definition of H_M, (1) of Proposition 9, and the commutativity of *, it is clear that H_M (A, B) = 1 if and only if A = B and H_M (A, B) = H_M (B, A).

In addition, by (4) of Proposition 9 and the commutativity of *, we obtain

Consequently, by the definition of H_M, we get We conclude that (𝒫(X), H_M, *) is a stationary fuzzy pseudometric space.

Proposition 11 (see [42].)Let (X, M, *) be a stationary fuzzy metric space. If A, B ⊂ X are any two bounded subsets of X, then A ∪ B is a bounded subset of X.

Theorem 12. Let (X, M, *) be a stationary fuzzy metric space. Then (𝒞ℬ(X), H_M, *) is a stationary fuzzy metric space.

Proof. Let A, B, C ∈ 𝒞ℬ(X). By Proposition 11, we have A ∪ B ∈ 𝒞ℬ(X), which means there exists r ∈ (0,1) such that M (x, y) > 1 − r, for all x ∈ A, y ∈ B. Hence, for any x ∈ A, we can get that

Thus we obtain Similarly, we can get Consequently, by the positivity of *, we have H_M (A, B) = ρ(A, B)*ρ(B, A) > 0.

By the definition of H_M, (7) of Proposition 9, and the commutativity of *, it is clear that H_M (A, B) = 1 if and only if A = B and H_M (A, B) = H_M (B, A). In addition, by (4) of Proposition 9 and the commutativity of *, we obtain

Consequently, by the definition of H_M, we get We conclude that (𝒞ℬ(X), H_M, *) is a stationary fuzzy metric space.

Example 13. Let (X, d) be a metric space. Then the Hausdorff stationary fuzzy metric $$ of the standard fuzzy metric (M_d, ·) coincides with the standard fuzzy metric $$ of the Hausdorff fuzzy metric H_d on 𝒞ℬ(X).

In fact, let A, B ∈ 𝒞ℬ(X); for each x₀ ∈ A, we have

Consequently, we obtain We conclude that $$ on 𝒞ℬ(X).

Theorem 14. Let (X, M, *) be a stationary fuzzy metric space. Then the Hausdorff-Bourbaki uniformity $$ coincides with the uniformity $$ on 𝒞ℬ(X).

Proof. If (A, B) ∈ {(A, B) ∈ 𝒞ℬ(X) × 𝒞ℬ(X) : B⊆U_n+1 (A),   A⊆U_n+1 (B)}, then, for any y ∈ B⊆U_n+1 (A), there exists x_y ∈ A⊆U_n+1 (B), such that M(x_y, y) > 1 − (1/(n + 1)). Thus we obtain

for all y ∈ B⊆U_n+1 (A). Consequently we have Similarly, we can get for all A⊆U_n+1 (B). Thus we conclude that

If (A, B) ∈ {(A, B) ∈ 𝒞ℬ(X) × 𝒞ℬ(X) : H_M (A, B) > 1 − (1/n)}, then, for each x ∈ A, we have

Let α = M (x, B), and then, for each x ∈ A, there exists x_y ∈ B such that Thus we obtain A⊆U_n(B).

Similarly, by

we can get B⊆U_n(A). It follows that Hence we obtain the following relations:

We conclude that $$ on 𝒞ℬ(X).

Theorem 15. Let (X, M, *) be a stationary fuzzy metric space. Then (𝒞ℬ(X), H_M, *) is complete if and only if (X, M, *) is complete.

Proof. In fact, we can prove it by a similar proof of Theorem 3 in [20].

Lemma 16. Let (X, M, *) be a stationary fuzzy metric space and A, B ∈ 𝒞ℬ(X). Then

• (1)

  for arbitrarily ε ∈ (0,1) and any x ∈ A, there exists y ∈ B such that M (x, y) ≥ H_M (A, B) − ε;

• (2)

  for any x ∈ A and any β ∈ [0,1), there exists y ∈ B such that M (x, y) ≥ βH_M (A, B).

Proof. We only prove (1) since it is equivalent to (2).

For each x ∈ A, there exists y ∈ B such that, for any ε ∈ (0,1),

This completes the proof.

Lemma 17. Let (X, M, *) be a stationary fuzzy metric space and A, B ∈ 𝒞ℬ(X). Then

• (1)

  for arbitrarily ε ∈ (0,1) and any closed subset A₁⊆A, there exists closed subset B₁⊆B such that H_M (A₁, B₁) ≥ H_M (A, B) − ε;

• (2)

  for any closed subset A₁⊆A and any β ∈ [0,1), there exists closed subset B₁⊆B such that H_M (A₁, B₁) ≥ βH_M (A, B).

Proof. We only prove (1) since it is equivalent to (2). Let ε ∈ (0,1) and let

Assume $$. For any δ > 0, there exists a y ∈ C₀ such that M (z, y) > 1 − δ. By (2) of Proposition 9, we have By the arbitrariness of δ, we have M (z, A₁) ≥ H_M (A, B) − ε. Then we get

Conversely, suppose z ∈ D. Take a descending positive number sequence $$ such that δ_n → 0 as n → ∞. For each δ_n, there exists a x ∈ A₁ such that

Let $$. Then we have $$, and $$. Hence we can get

Thus, we obtain $$. Let B₁ = B ∩ D. For each x ∈ A₁⊆A, by Lemma 16, there exists a y ∈ B such that Consequently, B₁ is a nonempty closed subset of B.

For any x ∈ A₁, there exists y ∈ B₁ such that M (x, y) ≥ H_M (A, B) − ε, which implies that

Thus we obtain

By the definition of B₁, we can get

for all y ∈ B₁. Thus we obtain Consequently, we easily obtain the following inequality:

This completes the proof.

Lemma 18 (see [29].)Let ϕ : [0,1] → [0,1] be a nondecreasing function satisfying the following conditions:

• (i)

  ϕ is continuous from the left,

• (ii)

  ϕⁿ(h) → 1    (n → ∞)  for  all  h ∈ (0,1],

where ϕⁿ denote the nth iterative function of ϕ. Then

• (1)

  for each h ∈ (0,1), such that ϕ(h) > h,

• (2)

  ϕ(1) = 1.

Theorem 19. Let (X, M, *) be a complete stationary fuzzy metric space and let $$ be a sequence of self-mappings of 𝒞ℬ(X). If there exists a constant q ∈ (1, +∞), such that, for each A₁, A₂ ∈ 𝒞ℬ(X), and for arbitrary positive integers i and j, i ≠ j,

where ϕ satisfies the conditions of Lemma 18. Then there exists an A^* ∈ 𝒞ℬ(X) such that A^*⊆f_i(A^*), for all i ∈ ℕ⁺.

Proof. Let A₀, A₁ ∈ 𝒞ℬ(X) and A₁⊆f₁(A₀), and β = (1/q) ∈ (0,1). By Lemma 17, there exists A₂ ∈ 𝒞ℬ(X), such that A₂⊆f₂(A₁) and

Again by Lemma 17, we can find A₃ ∈ 𝒞ℬ(X) such that A₃⊆f₃(A₂) and By induction, we produce a sequence {A_n} of points of 𝒞ℬ(X) such that Now we prove that {A_n} is a Cauchy sequence in 𝒞ℬ(X). In fact, for arbitrary positive integer n, by inequality (38) and formula (41), we have where A_n⊆f_n (A_n−1), which implies that ρ(A_n, f_n (A_n−1)) = 1.

If H_M (A_n−1, A_n) ∧ H_M (A_n, A_n+1) = H_M (A_n, A_n+1), then

From A_n+1⊆f_n+1 (A_n) = {y : there  exists  x ∈ A_n  such  that  f_n+1 (x) = y}, it follows that H_M (A_n, A_n+1) ∈ (0,1]. Hence, there are two cases.

Case 1. If H_M (A_n, A_n+1) = 1, by (2) of Lemma 18 we can get

that is, H_M (A_n, A_n+1) ≥ ϕ(H_M (A_n−1, A_n)).

Case 2. If H_M (A_n, A_n+1) ∈ (0,1), by (1) of Lemma 18, we can get

Obviously, (43) and (45) are contradictory. Hence, we have that is, H_M (A_n, A_n+1) ≥ ϕ(H_M (A_n−1, A_n)).

Consequently, we easily obtain the following inequalities:

Thus, for arbitrary positive integer p, we have

Since ϕⁿ(h) → 1 (n → ∞), for all h ∈ (0,1], we get

which implies that lim _n→∞ H_M (A_n+p, A_n) = 1. Hence, {A_n} is a Cauchy sequence. In addition, since (X, M, *) is a complete stationary fuzzy metric space, by Theorem 15, we get that (𝒞ℬ(X), H_M, *) is complete. Thus there exists an A^* ∈ 𝒞ℬ(X) such that A_n → A^* as n → ∞; that is, lim _n→∞ H_M (A_n, A^*) = 1.

Next, we show that A^*⊆f_i (A^*), that is, ρ(A^*, f_i (A^*)) = 1, for all i ∈ ℕ⁺. In fact, for arbitrary positive integers i and j, i ≠ j, by Proposition 9, we have

Moreover, we have Since ϕ is continuous from the left and * is a continuous positive t-norm, we can obtain Consequently, we conclude that that is, ρ(A^*, f_i (A^*)) = 1. By (1) of Proposition 9, we obtain A^*⊆f_i (A^*), for all i ∈ ℕ⁺.

Corollary 20. Let (X, M, *) be a complete stationary fuzzy metric space and let f be a self-mapping of 𝒞ℬ(X). If there exists a constant q ∈ (1, +∞), such that, for each A₁, A₂ ∈ 𝒞ℬ(X),

where ϕ satisfies the conditions of Lemma 18, then there exists an A^* ∈ 𝒞ℬ(X) such that A^*⊆f(A^*).

Proof. In fact, we can define a sequence of fuzzy self-mappings of 𝒞𝒞(X) as f_i = f, for i = 1,2, …. Thus, this result is a special case of Theorem 19.

Theorem 21. Let (X, M, *) be a complete stationary fuzzy metric space and let $$ be a sequence of self-mappings of 𝒞ℬ(X). If there exists a constant q ∈ (1, +∞), such that, for each A₁, A₂ ∈ 𝒞ℬ(X), and for arbitrary positive integers i and j, i ≠ j,

where ϕ(h₁, h₂, h₃, h₄, h₅) : (0,1]⁵ → [0,1] is nondecreasing and continuous from the left for each variable, let rⁿ(h) = ϕ(h, h, h, a, b), where (a, b) ∈ {(h*h, 1), (1, h*h)}. If where rⁿ denotes the nth iterative function of r, then there exists an A^* ∈ 𝒞ℬ(X) such that A^*⊆f_i(A^*), for all i ∈ ℕ⁺.

Proof. Let A₀, A₁ ∈ 𝒞ℬ(X) and A₁⊆f₁(A₀), and β = (1/q) ∈ (0,1). By Lemma 17, there exists A₂ ∈ 𝒞ℬ(X), such that A₂⊆f₂(A₁) and

Again by Lemma 17, we can find A₃ ∈ 𝒞ℬ(X) such that A₃⊆f₃(A₂) and By induction, we produce a sequence {A_n} of points of 𝒞ℬ(X) such that Now we prove that {A_n} is a Cauchy sequence in 𝒞ℬ(X). In fact, for arbitrary positive integer n, by inequality (55) and formula (59), we have where A_n⊆f_n (A_n−1), which implies that ρ(A_n, f_n (A_n−1)) = 1.

If H_M (A_n−1, A_n)∧ H_M (A_n, A_n+1) = H_M (A_n, A_n+1), then

From we get H_M (A_n, A_n+1) ∈ (0,1]. Hence, there are two cases.

Case 1. If H_M (A_n, A_n+1) = 1, by (2) of Lemma 18, we can get

that is, H_M (A_n, A_n+1) ≥ r (H_M (A_n−1, A_n)).

Case 2. If H_M (A_n, A_n+1) ∈ (0,1), by (1) of Lemma 18, we can get

Obviously, (61) and (64) are contradictory. Hence, we have that is, H_M (A_n, A_n+1) ≥ r (H_M (A_n−1, A_n)).

Consequently, we easily obtain the following relations:

Thus, for arbitrary positive integer p, we have

By rⁿ(h) → 1  (n → ∞), for all h ∈ (0,1], and continuity of *, we can get

which implies that lim _n→∞ H_M (A_n+p, A_n) = 1. Hence, we get {A_n}⊆𝒞ℬ(X) that is a Cauchy sequence. In addition, since (X, M, *) is a complete stationary fuzzy metric space, by Theorem 15, we have that (𝒞ℬ(X), H_M, *) is complete. Thus there exists an A^* ∈ 𝒞ℬ(X) such that A_n → A^* as n → ∞; that is, lim _n→∞ H_M (A_n, A^*) = 1.

Next, we show that A^*⊆f_i (A^*), that is, ρ(A^*, f_i (A^*)) = 1, for all i ∈ ℕ⁺. In fact, for arbitrary positive integers i and j, i ≠ j, by Proposition 9, we have

Moreover, we have Since ϕ is continuous from the left and * is a continuous positive t-norm, hence, we can obtain Consequently, we conclude that that is, ρ(A^*, f_i (A^*)) = 1. By (1) of Proposition 9, we obtain A^*⊆f_i (A^*), for all i ∈ ℕ⁺.

Corollary 22. Let (X, M, *) be a complete stationary fuzzy metric space and let f be a self-mapping of 𝒞ℬ(X). If there exists a constant q ∈ (1, +∞), such that, for each A₁, A₂ ∈ 𝒞ℬ(X),

where ϕ(h₁, h₂, h₃, h₄, h₅) : (0,1]⁵ → [0,1] is nondecreasing and continuous from the left for each variable, let rⁿ(h) = ϕ(h, h, h, a, b), where (a, b) ∈ {(h*h, 1), (1, h*h)}. If where rⁿ denotes the nth iterative function of r, then there exists an A^* ∈ 𝒞ℬ(X) such that A^*⊆f(A^*).

Proof. In fact, we can define a sequence of fuzzy self-mappings of 𝒞𝒞(X) as f_i = f, for i = 1,2, …. Thus, this result is a special case of Theorem 21.

Theorem 23. Let (X, M, *) be a complete stationary fuzzy metric space and let $$ be a sequence of self-mappings of 𝒞ℬ(X). If there exists a constant q ∈ (1, +∞), such that, for each A₁, A₂ ∈ 𝒞ℬ(X), and, for arbitrary positive integers i and j, i ≠ j,

where ϕ satisfies the conditions of Lemma 18, then there exists an A^* ∈ 𝒞ℬ(X) such that A^*⊆f_i (A^*), for all i ∈ ℕ⁺.

Proof. Let A₀, A₁ ∈ 𝒞ℬ(X) and A₁⊆f₁ (A₀), and β = (1/q) ∈ (0,1). By Lemma 17, there exists A₂ ∈ 𝒞ℬ(X), such that A₂⊆f₂ (A₁) and

Again by Lemma 17, we can find A₃ ∈ 𝒞ℬ(X) such that A₃⊆f₃ (A₂) and By induction, we produce a sequence {A_n} of points of 𝒞ℬ(X) such that Now we prove that {A_n} is a Cauchy sequence in 𝒞ℬ(X). In fact, for arbitrary positive integer n, by inequality (75) and formula (78), we have Thus, from inequality (79), we easily obtain the following relations: Furthermore, for arbitrary positive integers n and p, we get that Since for arbitrary h ∈ (0,1), ϕⁿ(h) → 1    (n → ∞), and by continuity of *, we have that is, {A_n} is a Cauchy sequence in 𝒞ℬ(X). By Theorem 15, 𝒞ℬ(X) is complete since X is complete. Consequently, there exists A^* ∈ 𝒞ℬ(X) such that A_n → A^*    (n → ∞); that is, lim _n→∞ H_M (A_n, A^*) = 1.

Next, we show that A^*⊆f_i (A^*); that is, ρ(A^*, f_i (A^*)) = 1, for all i ∈ ℕ⁺. In fact, for arbitrary positive integers i and j, i ≠ j, by Proposition 9 we have

Moreover, we have Consequently, we get Since ϕ is continuous from the left and * is a continuous positive t-norm, hence, we can obtain that is, ρ(A^*, f_i (A^*)) = 1. By (1) of Proposition 9, we obtain A^*⊆f_i (A^*), for all i ∈ ℕ⁺.

Corollary 24. Let (X, M, *) be a complete stationary fuzzy metric space and let f be a self-mapping of 𝒞ℬ(X). If there exists a constant q ∈ (1, +∞), such that, for each A₁, A₂ ∈ 𝒞ℬ(X),

where ϕ satisfies the conditions of Lemma 18, then there exists an A^* ∈ 𝒞ℬ(X) such that A^*⊆f(A^*).

Proof. In fact, we can define a sequence of fuzzy self-mappings of 𝒞ℬ(X) as f_i = f, for i = 1,2, …. Thus, this result is a special case of Theorem 23.

Example 25. Let $$ be a stationary fuzzy metric space, where X = [−1,1], and $$ is the same as in Example 13. Then, $$ is a complete stationary fuzzy metric space.

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

For arbitrary positive integers i and j, without loss of generality, suppose i < j. For each A₁, A₂ ∈ 𝒞ℬ(X), by a routine calculation, we have Therefore, by Theorem 23, we assert that the sequence of self-mappings $$ has a common fixed point A^* in 𝒞ℬ(X). In fact, it is easy to check that A^* = {0}.