Some Equivalences between Cone b-Metric Spaces and b-Metric Spaces

Definition 1 (see [1].)Let X be a nonempty set and d : X × X → [0, +∞). Then, d is called a b-metric on X if

• (1)

  d(x, y) = 0 if and only if x = y;

• (2)

  d(x, y) = d(y, x) for all x, y ∈ X;

• (3)

  there exists s ≥ 1 such that d(x, z) ≤ s[d(x, y) + d(y, z)] for all x, y, z ∈ X.

The pair (X, d) is called a b-metric space. A sequence {x_n} is called convergent to x in X, written lim _n→∞x_n = x, if lim _n→∞d(x_n, x) = 0. A sequence {x_n} is called a Cauchy sequence if lim _n,m→∞d(x_n, x_m) = 0. The b-metric space (X, d) is called complete if every Cauchy sequence in X is a convergent sequence.

Remark 2. On a b-metric space (X, d), we consider a topology induced by its convergence. For results concerning b-metric spaces, readers are invited to consult papers [1, 2].

Remark 3. Let (X, d) be a b-metric space. For each r > 0 and x ∈ X, we set

In [3], Akkouchi claimed that the topology 𝒯(d) on X associated with d is given by setting U ∈ 𝒯(d) if and only if, for each x ∈ U, there exists some r > 0 such that B(x, r) ⊂ U and the convergence of {x_n} _n in the b-metric space (X, d) and that in the topological space (X, 𝒯(d)) are equivalent. Unfortunately, this claim is not true in general; see Example 13. Note that; on a b-metric space, we always consider the topology induced by its convergence. Most of concepts and results obtained for metric spaces can be extended to the case of b-metric spaces. For results concerning b-metric spaces, readers are invited to consult papers [1, 2].

Definition 4 (see [11].)P is called a cone if and only if

• (1)

  P is closed and nonempty and P ≠ {θ};

• (2)

  a, b ∈ ℝ;   a, b ≥ 0;   x, y ∈ P imply that ax + by ∈ P;

• (3)

  P∩(−P) = {θ}.

The cone P is called normal if there exists K ≥ 1 such that, for all x, y ∈ E, we have θ ≤ x ≤ y implies ∥x∥ ≤ K∥y∥. The least positive number K satisfying the above is called the normal constant of P.

Definition 5 (see [11], Definition 1.)Let X be a nonempty set and d : X × X → E satisfy

• (1)

  θ ≤ d(x, y) for all x, y ∈ X and d(x, y) = θ if and only if x = y;

• (2)

  d(x, y) = d(y, x) for all x, y ∈ X;

• (3)

  d(x, y) ≤ d(x, z) + d(z, y) for all x, y, z ∈ X.

Then d is called a cone metric on X, and (X, d) is called a cone metric space.

Definition 6 (see [8], Definition 2.1.)Let X be a nonempty set and d : X × X → P satisfy

• (1)

  θ ≤ d(x, y) for all x, y ∈ X and d(x, y) = θ if and only if x = y;

• (2)

  d(x, y) = d(y, x) for all x, y ∈ X;

• (3)

  d(x, y) ≤ s[d(x, z) + d(z, y)] for some s ≥ 1 and all x, y, z ∈ X.

Then d is called a cone b-metric with coefficient s on X and (X, d) is called a cone b-metric space with coefficient s.

Definition 7 (see [8], Definition 2.4.)Let (X, d) be a cone b-metric space and {x_n} a sequence in X.

• (1)

  {x_n} is called convergent to x, written lim _n→∞x_n = x, if for each c ∈ E with θ ≪ c, there exists n₀ such that d(x_n, x) ≪ c for all n ≥ n₀.

• (2)

  {x_n} is called a Cauchy sequence if for each c ∈ E with θ ≪ c there exists n₀ such that d(x_n, x_m) ≪ c for all n, m ≥ n₀.

• (3)

  (X, d) is called complete if every Cauchy sequence in X is a convergent sequence.

Lemma 8 (see [8], Proposition 2.5.)Let (X, d) be a cone b-metric space, P a normal cone with normal constant K, x ∈ X, and {x_n} a sequence in X. Then one has the following.

• (1)

  lim _n→∞x_n = x if and only if  lim _n→∞d(x_n, x) = θ.

• (2)

  The limit point of a convergent sequence is unique.

• (3)

  Every convergent sequence is a Cauchy sequence.

• (4)

  {x_n} is a Cauchy sequence if  lim _n,m→∞d(x_n, x_m) = θ.

Lemma 9 (see [8], Remark 2.6.)Let (X, d) be a cone b-metric space over an ordered real Banach space E with a cone P. Then one has the following.

• (1)

  If a ≤ b and b ≪ c, then a ≪ c.

• (2)

  If a ≪ b and b ≪ c, then a ≪ c.

• (3)

  If θ ≤ u ≪ c for all c ∈ int P, then u = θ.

• (4)

  If c ∈ int P, θ ≤ a_n for all n ∈ ℕ and lim _n→∞a_n = θ, then there exists n₀ such that a_n ≪ c for all n ≥ n₀.

• (5)

  If θ ≪ c, θ ≤ d(x_n, x) ≤ b_n for all n ∈ ℕ and lim _n→∞b_n = θ, then d(x_n, x) ≪ c eventually.

• (6)

  If θ ≤ a_n ≤ b_n for all n ∈ ℕ and lim _n→∞a_n = a, lim _n→∞b_n = b, then a ≤ b.

• (7)

  If a ∈ P, 0 ≤ λ < 1, and a ≤ λ · a, then a = θ.

• (8)

  For each α > 0, one has α · int P ⊂ int P.

• (9)

  For each δ > 0 and x ∈ int P, there exists 0 < γ < 1 such that ∥γ · x∥ < δ.

• (10)

  For each θ ≪ c₁ and c₂ ∈ P, there exists θ ≪ d such that c₁ ≪ d and c₂ ≪ d.

• (11)

  For each θ ≪ c₁ and θ ≪ c₂, there exists θ ≪ e such that e ≪ c₁ and e ≪ c₂.

Remark 10 (see [10], Remark 1.3.)Every cone metric space is a cone b-metric space. Moreover, cone b-metric spaces generalize cone metric spaces, b-metric spaces, and metric spaces.

Example 11 (see [10], Example 2.2.)Let

and d(x, y)(t) = |x − y|²e^t for all x, y ∈ X and t ∈ [0,1]. Then (X, d) is a cone b-metric space with coefficient s = 2, but it is not a cone metric space.

Example 12 (see [10], Example 2.3.)Let X be the set of Lebesgue measurable functions on [0,1] such that $$, E = C_ℝ[0,1], P = {φ ∈ E : φ ≥ 0}. Define d : X × X → E as

for all u, v ∈ X and t ∈ [0,1]. Then (X, d) is a cone b-metric space with coefficient s = 2, but it is not a cone metric space.

Example 13. Let X = {0,1, 1/2, …, 1/n, …} and

Then we have the following.

• (1)

  d is a b-metric on X with coefficient s = 8/3.

• (2)

  0 ∈ B(1,2) but B(0, r)⊄B(1,2) for all r > 0.

Proof. (1) For all x, y ∈ X, we have d(x, y) ≥ 0, d(x, y) = 0 if and only if x = y and d(x, y) = d(y, x).

If d(x, y) = d(0,1) = 1, then

If d(x, y) = d(0, 1/2n) = 1/2n, then If d(x, y) = d(1/2k, 1/2n) = |1/2k − 1/2n|, then If d(x, y) = d(1/2k, 1/(2n + 1)) = 4 with 1/(2n + 1) ≠ 1, then If d(x, y) = d(1/(2k + 1), 1/(2n + 1)) = 4 with 1/(2k + 1) ≠ 1 and 1/(2n + 1) ≠ 1, then If d(x, y) = d(1/2k, 1) = 4, then If d(x, y) = d(1/(2k + 1), 1) = 4, then If d(x, y) = d(1/(2k + 1), 0) = 4, then

By the previous calculations, we get d(x, y)≤(8/3)[d(x, z) + d(z, y)] for all x, y, z ∈ X. This proves that d is a b-metric on X with s = 8/3.

(2) We have B(1,2) = {x ∈ X : d(x, 1) < 2} = {1,0}. Then 0 ∈ B(1,2).

For each r > 0, since d(0, 1/2n) = 1/2n, we have 1/2n ∈ B(0, r) for n  being large enough. Note that d(1, 1/2n) = 4, so  1/2n ∉ B(1,2) for all n ∈ ℕ. This proves that B(0, r)⊄B(1,2).

Theorem 14. Let (X, d) be a cone b-metric space with coefficient s and

for all x, y ∈ X. Then one has the following.

• (1)

  D is a b-metric on X.

• (2)

  lim _n→∞x_n = x in the cone b-metric space (X, d) if and only if lim _n→∞x_n = x in the b-metric space (X, D).

• (3)

  {x_n} is a Cauchy sequence in the cone b-metric space (X, d) if and only if {x_n} is a Cauchy sequence in the b-metric space (X, D).

• (4)

  The cone b-metric space (X, d) is complete if and only if the b-metric space (X, D) is complete.

Proof. (1) For all x, y ∈ X, it is obvious that D(x, y) ≥ 0 and D(x, y) = D(y, x).

If x = y, then D(x, y) = inf {∥u∥ : u ∈ P,   u ≥ θ} = 0.

If D(x, y) = inf {∥u∥ : u ∈ P,   u ≥ (1/s)d(x, y)} = 0, then, for each n ∈ ℕ, there exists u_n ∈ P such that u_n ≥ (1/s)d(x, y) and ∥u_n∥ < 1/n. Then lim _n→∞u_n = θ, and by Lemma 9(6), we have d(x, y) ≤ θ. It implies that d(x, y) ∈ P∩(−P). Therefore, d(x, y) = θ; that is, x = y.

For each x, y, z ∈ X, we have

Since u₂, u₃ ∈ P and u₂ ≥ (1/s)d(x, y),  u₃ ≥ (1/s)d(y, z), we have Then we have It implies that Now, we have By the previously metioned, D is a b-metric on X.

(2) Necessity. Let lim _n→∞x_n = x in the cone b-metric space (X, d). For each ε > 0, by Lemma 9(8), if θ ≪ c, then θ ≪ s · ε · (c/∥c∥). Then, for each c ∈ E with θ ≪ c, there exists n₀ such that d(x_n, x) ≪ s · ε · (c/∥c∥) for all n ≥ n₀. Using Lemma 9(8) again, we get (1/s)d(x_n, x) ≪ ε · (c/∥c∥). It implies that

for all n ≥ n₀. This proves that lim _n→∞D(x_n, x) = 0; that is, lim _n→∞x_n = x in the b-metric space (X, D).

Sufficiency. Let lim _n→∞x_n = x in the b-metric space (X, D). For each θ ≪ c, there exists ε > 0 such that c + B(0, ε) ⊂ P. For this ε, there exists n₀ such that

Then, there exist v ∈ P  and  d(x_n, x) ≤ v such that ∥v∥ ≤ ε/2. So −v ∈ B(0, ε), and we have c − v ∈ int P. Therefore, d(x_n, x) ≤ v ≪ c for all n ≥ n₀. By Lemma 9(1), we get d(x_n, x) ≪ c for all n ≥ n₀. This proves that lim _n→∞x_n = x in the cone b-metric space (X, d).

(3) Necessity. Let {x_n} be a Cauchy sequence in the cone b-metric space (X, d). For each ε > 0, by Lemma 9(6), if θ ≪ c, then θ ≪ s · ε · (c/∥c∥). Then for each c ∈ E with θ ≪ c, there exists n₀ such that d(x_n, x_m) ≪ s · ε · (c/∥c∥) for all n, m ≥ n₀. Using Lemma 9(6) again, we get (1/s)d(x_n, x_m) ≪ ε · (c/∥c∥). It implies that

for all n, m ≥ n₀. This proves that {x_n} is a Cauchy sequence in the b-metric space (X, D).

Sufficiency. Let {x_n} be a Cauchy sequence in the b-metric space (X, D). Then lim _n,m→∞D(x_n, x_m) = 0. For each θ ≪ c, there exists ε > 0 such that c + B(0, ε) ⊂ P. For this ε, there exists n₀ such that

for all n, m ≥ n₀. Then, there exists v ∈ P, d(x_n, x_m) ≤ v such that ∥v∥ ≤ ε/2. So −v ∈ B(0, ε), and we have c − v ∈ int P. Therefore, d(x_n, x_m) ≤ v ≪ c for all n, m ≥ n₀. By Lemma 9(1), we get d(x_n, x_m) ≪ c for all n, m ≥ n₀. This proves that {x_n} is a Cauchy sequence in the cone b-metric space (X, d).

(4) It is a direct consequence of (2) and (3).

Corollary 15 (see [13], Lemma 2.1.)Let (X, d) be a cone metric space. Then

for all x, y ∈ X is a metric on X.

Corollary 16 (see [10], Theorem 2.2.)Let (X, d) be a cone metric space and

for all x, y ∈ X. Then the metric space (X, D) is complete if and only if the cone metric space (X, d) is complete.

Example 17. Let (X, d) be a cone b-metric space as in Example 11. We have

It implies that Then D is not a metric on X. This proves that Corollaries 15 and 16 are not applicable to given cone b-metric space (X, d).

Example 18. Let (X, d) be a cone b-metric space as in Example 12. We have

For u(s) = 0, v(s) = 1, and w(s) = 2 for all s ∈ [0,1], we have Then D is not a metric on X. This proves that Corollaries 15 and 16 are not applicable to given cone b-metric space (X, d).

Corollary 19. Let (X, d) be a cone b-metric space with coefficient s, let T : X → X be a map, and let D be defined as in Theorem 14. Then the following statements hold.

• (1)

  If d(Tx, Ty) ≤ kd(x, y) for some k ∈ [0,1) and all x, y ∈ X, then

  $$ ()
  for all x, y ∈ X.

• (2)

  If d(Tx, Ty) ≤ λ₁d(x, Tx) + λ₂d(y, Ty) + λ₃d(x, Ty) + λ₄d(y, Tx) for some λ₁, λ₂, λ₃, λ₄ ∈ [0,1) with λ₁ + λ₂ + s(λ₃ + λ₄) < min {1, 2/s} and all x, y ∈ X, then

  $$ ()
  for all x, y ∈ X.

Proof. (1) For each x, y ∈ X and v ∈ P with v ≥ (1/s)d(x, y), it follows from Lemma 9(8) that

Thus, {kv : v ∈ P,   v ≥ (1/s)d(x, y)}⊂{u : u ∈ P,   u ≥ (1/s)d(Tx, Ty)}. Then we have It implies that D(Tx, Ty) ≤ kD(x, y).

(2) Let x, y ∈ X and v₁, v₂, v₃, v₄ ∈ P satisfy

From Lemma 9(8), we have

It implies that Then we have This proves that D(Tx, Ty) ≤ λ₁D(x, Tx) + λ₂D(y, Ty) + λ₃D(x, Ty) + λ₄D(y, Tx).

Corollary 20. Let (X, d) be a complete cone b-metric space with coefficient s, and let T : X → X be a map. Then the following statements hold.

• (1)

  (see [9, Theorem 2.1]) If d(Tx, Ty) ≤ kd(x, y) for all x, y ∈ X, then T has a unique fixed point.

• (2)

  (see [9, Theorem 2.3]) If d(Tx, Ty) ≤ λ₁d(x, Tx) + λ₂d(y, Ty) + λ₃d(x, Ty) + λ₄d(y, Tx) for some λ₁, λ₂, λ₃, λ₄ ∈ [0,1) with λ₁ + λ₂ + s(λ₃ + λ₄) < min {1, 2/s} and all x, y ∈ X, then T has a unique fixed point.

Proof. Let D be defined as in Theorem 14. It follows from Theorem 14(4) that (X, D) is a complete b-metric space.

• (1)

  By Corollary 19 (1), we see that T satisfies all assumptions of [5, Theorem 2]. Then T has a unique fixed point.

• (2)

  By Corollary 19 (2), we see that T satisfies all assumptions in [6, Theorem 3.7], where K = s, f = T, g is the identity, and a₁ = 0,   a₂ = λ₁,   a₃ = λ₂, and a₄ = λ₃,   a₅ = λ₄. Note that condition (3.10) in [6, Theorem 3.7] was used to prove (3.16) and K(a₂ + a₃ + a₄ + a₅) < 2 at line 3, page 7 in the proof of [6, Theorem 3.7]. These claims also hold if a₁ = 0 and λ₁ + λ₂ + s(λ₃ + λ₄) < min {1, 2/s}. Then T has a unique fixed point.

Remark 21. By similar arguments as in Corollaries 19 and 20, we may get fixed point theorems on cone b-metric spaces in [8, 10] from preceding ones on b-metric spaces in [3, 5].