An Ulam stability result on quasi-b-metric-like spaces

Let X be a nonempty set and let γ : X × X → [0, + ∞) be a mapping satisfying the following

conditions for all x, y, z ∈ X:

(ML1) γ(x, y) = 0 ⇒ x= y;

(ML2) γ(x, y)= γ(y, x);

(ML3) γ(x, y) ≤ γ(x, z) + γ(z, y).

Then the mapping γ is called a metric-like and the pair (X, γ) is called a metric-like space (dislocated space).

Clearly, the conditions of a metric-like on X coincide with the conditions of a metric except that γ(x, x)does not need to be 0 in general. Therefore, a metric space is always a metric-like space but a metric-like space is not metric space in general. Alghamdi et al. [10] combined the concepts of b-metric space and metric-like space recently and introduced the so-called b-metric-like spaces.

[10]). Let X be a nonempty set and let γ_b : X ×X → [0, + ∞) be a function such that for all

x, y, z ∈X and a constant s ≥ 1, the following conditions are satisfied:

(BML1) γ_b(x, y) = 0 ⇒ x= y;

(BML2) γ_b(x, y) = γ_b(y, x);

(BML3) γ_b(x, y)⇒ s[γ_b(x, z) + γ_b(z, y)].

The map γ_bis called a b-metric-like and the pair (X, γ_b) is called a b-metric-like space with constant s.

Let (X, γ_b)be a b-metric-like space. For any x ∈ X and r > 0, the set B(x, r) = {y ∈X : |γ_b(x, y) — γ_b(x, x)| < r}is called an open ball centered at x of radius r > 0.

Also very recently, Zhu et al.[11] defined the concept of quasi-metric-like spaces and proved some fixed point theorems on quasi-metric-like spaces. The definition of quasi-metric-like is given next.

([11]). Let X be a nonempty set. A mapping ρ : X ×X → [0, + ∞) is called a quasi-metric-like if

for all x, y, z ∈X,

(QML1) ρ(x, y) = 0 ⇒ x = y;

(QML2) ρ(x, z) ≤ρ(x, y) + ρ(y, z).

The pair (X, ρ) is called a quasi-metric-like space.

Since the symmetry condition is not required for quasi-metric-like spaces, whenever ρ(x, y) = 0 we may have ρ(y, x) ≠0. Therefore, instead of(QML1) we use the condition given in [12],

After their results on quasi-metric-like spaces, Zhu et al. [13] combined b-metric space and quasi-metric-like space and introduced the concept of quasi-b-metric-like spaces. The authors also presented some fixed point results on quasi-b-metric-like spaces. On the other hand, also very recently Klin-eam and Suanoom [12] studied cyclic Banach and Kannan type contraction mappings on quasi-b-metric-like. We give next the definition of quasi-b-metric-like spaces.

([12]). A quasi-b-metric-like on a nonempty set X is a function ρ_b : X × X →[0, + ∞) such that

for all x, y, z ∈ X and a constant s > 1,

(QML1) ρ_b(x, y) = ρ_b(y, x) = 0 ⇒ x = y;

(QML2) ρ_b(x, z)≤ s[ρ_b(x, z) + ρ_b(z, y].

The pair (X ρ, _b) is called a quasi-b-metric-like space with constant s.

Inspired by the work of Klin-eam and Suanoom [12] we present the following examples of quasi-b-metric-like.

Example 1.6. The function ρ_b : ℝ × ℝ [0, ∞) defined as

for all x, y ∈ ℝ is a quasi-b-metric-like on ℝ with constant s = 2.

The function : ρ_b : ℝ x ℝ → [0, ∞) defined as

for all x, y ∈ℝ is a quasi-b-metric-like on ℝ with constants = 8.

Some topological concepts on quasi-b-metric-like spaces are defined next.

([13]). Let (X, ρ_b) be a quasi-b-metric-like space.

1. A sequence {ξ_n}⊂ X is said to converge to a point ξ ∈ X if and only if

2. A sequence {ξ_n}⊂ Χ is said to be a Cauchy sequence if lim_{m,n→ ∞}ρ_b(ξ_n,ξ_m) and lim_{m,n→ ∞}ρ_b(ξ_n,ξ_m) exist and are finite.

3. (X, ρ_b) is said to be a complete quasi-b-metric-like space if and only if for every Cauchy sequence {ξ_n} in X there exists some ξ∈ X such that

4. A sequence {ξ_η}C X is said to be a 0-Cauchy sequence if

5. (X, ρ_b) is said to be a 0-complete quasi-b-metric-like space if for every 0-Cauchy sequence {ξ_η} in X there exists a ξ ∈ X such that

6. A mapping ƒ : Χ → X is said to be continuous at ξ₀ ϵ X if for every ε > 0, there exists δ > 0 such that f(B(ξ_ο, δ))⊂ Β(ξ₀),ε).

The above definition makes it clear that in the quasi-b-metric like space (X, ρ_b), every 0-Cauchy sequence is a Cauchy sequence, and that very complete quasi-b-metric-like space is also 0-complete. Note, however, that the converses of these statements may not be true.

Due to the fact that interchange symmetry condition is not necessary on quasi-metric type spaces, we need to recall right- and left-Cauchy sequences and right- and left-completeness definitions.

Let (X, ρ_b) be a quasi-b-metric-like space.

(i) A sequence {ξ_n} in X is called a left-Cauchy sequence if and only if for every ε > 0 there exists a positive integer Ν = Ν(ε) such that ρ_b(ξ_n, ξ_m) < ε for all n ≥ m>N;

(ii) A sequence {ξ_n} in X is called a right-Cauchy sequence if and only if for every ε > 0 there exists a positive integer Ν = Ν(ε) such that ρ_b(ξ_n, ξ_m) < ε for all m ≥ n> N;

(iii) A quasi-b-metric-like space is said to be left-complete if every left-Cauchy sequence {ξ_n} in X is convergent,

(iv) A quasi-b-metric-like space is said to be right-complete if every right-Cauchy sequence {ξ_n} in X is convergent,

(v) A quasi-b-metric-like space is complete if and only if every Cauchy sequence in X is convergent.

It is obvious from the Definition 1.9 that a sequence {ξ_n} in a quasi-b-metric-like space is Cauchy if and only if it is both left-Cauchy and right-Cauchy, and a quasi-b-metric-like space is complete if and only if it is left-complete and right-complete.

In the following discussion we will employ the so-called comparison functions and their variants. These functions have been introduced by Berinde [14] and Rus [15]. We give next the definitions of comparison, (c)-comparison and (b)-comparison functions. It should be mentioned that Berinde [14] introduced the concept of (c)-comparison functions in order to investigate convergence properties of iterative sequences generated by fixed point methods, and (b)-comparison functions to be used on b-metric space (see [16, 17] for details).

CF (See [14, 15]). Comparison function is an increasing mapping φ : [0,+∞) → [0,+∞) satisfying φⁿ(t)→ 0, n → ∞ for any t ∈ [0, ∞).

CCF(See [14]) A (c)-comparison function is a function φ_c : [0, +∞) → [0, +∞) satisfying

(c₁)φ_c is increasing,

(c₂)there exist k₀ ∈ ℕ, a ∈ (0, 1) and a convergent series of nonnegative terms ∑k=1∞vk suchthat φck+1t≤aφckt+vk for k ≥ k₀ and any t ∈ [0, ∞). BCF(See [16, 17]) For a real number s ≥ 1 (b)-comparison function is a function φ_b : [0,+∞) → [0,+∞) satisfying the conditions

(b₁)φ_b, is increasing,

(b₂)there exist k₀ ∈ ℕ, a ∈ (0, 1) and aconvergent series of nonnegative terms ∑k=1∞vk such thatsk+1φbk+1t≤askφbkt+vk,fork≥k0and any t ∈ [0, ∞).

In the sequel, we denote the class of comparison functions by Φ, the class of (c)-comparison functions by Φ_c and the class of (b)-comparison functions by Φ_b. Obviously, a (b)-comparison function reduces to (c)-comparison function whenever s = 1. In addition, every (c)-comparison function is a comparison function.

We will need the following essential properties in our further discussion.

(Berinde [14], Rus [15]). For a comparison function φ : [0, +∞) →[0, +∞) the following hold:

(1) each iterate φ^k of φ k ≥ 1, is also a comparison function;

(2) φ is continuous at 0;

(3) φ(t) < t, for any t > 0.

([17]). tor a (b)-comparison function φ_b : [0, +∞) → [0, +∞) the following hold:

(1) the series ∑k=0∞skφbkt (0 converges for any t ∈ [0, +∞);

(2) the function b_s : [0, +∞) → [0, +∞) defined bybst=∑k=0∞skφbkt,t∈[0,∞)isincreasing and continuous at 0.

For more details on comparison functions and examples we refer the reader to [14, 15].

Finally, we recall the concept of α-admissible mappings introduced by Samet [5].

A mapping Τ : X → X is called α-admissible if for all ξ, n ∈X we have

where α : Χ ×X→ [0, ∞) is a given function.

For recent results related with α-admissible mappings, see [18, 19]. In their work Samet et.al [5], studied α — ψ-contractive mappings and their fixed points.

Recently, Gülyaz [8] defined α — ψ-contractive mapping in the framework of quasi-b-metric-like spaces.

Let (X, ρ_b) be a quasi-b-metric-like space and Τ : X → X be a given mapping. We say that Τ is an α — ψ-contractive mapping if there exist two functions a : Χ × X → [0, ∞) and ψ ∈Φ_b such that for all ξ, n∈ X, we have

Gülyaz [8] also proved the existence of fixed points for α — ψ-contractive mapping on quasi-b-metric-like spaces.

Let (X, ρ_b) be a 0-complete quasi-b-metric-like space with a constant s ≥ 1. Suppose that T : X → X is an α — ψ-contractive mapping. Suppose also that

(i) Τ is α-admissible;

(ii) there exists ξ₀ ∈ X such that α(Tξ₀, ξ₀) ≥ 1 and α(ξ₀,Τ ξ₀) ≥ 1;

(iii) T is continuous,

Then Τ has a fixed point.

Let (X, ρ_b) be a 0-complete quasi-b-metric-like space with constant s ≥ 1. Suppose that T : X → X is an α — ψ-contractive mapping. Suppose also that

(i) Τ is α-admissible;

(ii) there exists ξ₀ ∈ X such that α(Τ ξ₀, ξ₀) ≥ 1 and α (ξ₀, Τ ξ₀)≥ 1;

(iii) If {ξ_n} is a sequence in Χ which converges to ξ and satisfies α(ξ_n+1, ξ_n)≥ 1 and α(ξ_n, ξ_n+1) ≥ 1 for all n

then, there exists a subsequence {ξ_n(k)} of {ξ_n} such that α(ξ, ξ_n(k)))≥ 1 and α (ξ_n(k), ξ)≥ 1 for all k.

Then Τ has a fixed point.

Let (X, ρ_b) be a quasi-b-metric-like space with a constant s ≥ 1 and let α : Χ × X → [0, ∞) and

φ_b ∈ Φ_bbe two functions.

(i) An α — φ_b contractive mapping Τ : X → X is of type (A) if

where

(ii) An α — φ_b contractive mapping Τ : X → X is of type (B) if

where

(iii) An α —φ_b contractive mapping Τ : X → X is of type (C) if

where

Note that ρ_b(ξ, n)≤ Κ(ξ, n)≤ Ν(ξ, η)≤ Μ(ξ, n) for all ξ,n ∈ Χ.

Our first theorem gives conditions for the existence of a fixed point for maps in class (A).

Let (X, ρ_b) be a 0-complete quasi-b-metric-like space with a constant s ≥ 1. Suppose that Τ : X → X is an α — φ_b contractive mapping of type (A) satisfying the following:

(i) T is α-admissible;

(ii) there exists ξ₀ ∈ X such that α(Τ ξ₀,ξ₀)≥ 1 and α(ξ₀, Τ ξ₀)≥ 1;

(iii) T is continuous;

Then Τ has a fixed point.

As usual, we take ξ₀ ∈ X such that α (Τ ξ₀, ξ₀) ≥ 1 and α (ξ₀, T ξ₀) ≥ 1 and construct the sequence {ξ_n} as

Notice that if for some n₀ ≥ 0 we have ξn0=ξn0+1 then the proof is done, i.e., ξn0 is a fixed point of Τ. Assume that ξ_n ≠ξ_n+1for all n ≥ 0. Because of the lack of symmetry condition in quasi-b-metric-like spaces, we will show that the sequence {ξ_n} is both left- and right-Cauchy.

From the conditions (i) and (ii), we have

and

or, in general

Regarding (8), the contractive condition (3) with ξ = ξ_n+1and n = ξ_n becomes

where

Observe that for the last term in Μ(ξ_n, ξ_n-1), by the triangle inequality we have

and hence, either Μ(ξ_n, ξ_n-1) = max{ρ_b(ξ_n+1,ξ_n), ρ_b(ξ_n, ξ_n-1)}or Μ(ξ_n, ξ_n-1)≤ ρ_b(ξ_n, ξ_n+1). Now, we will examine all three cases.

Case 1. Suppose that Μ(ξ_n, ξ_n-1) = ρ_b(ξ_n+1,ξ_n)for some n ≥ 1. Since ρ_b(ξ_n+1,ξ_n) > 0, from (9), we have

which is a contradiction. Hence, for all n ≥ 1 either M(ξ_n, ξ_n-1) = ρ_b(ξ_n, ξ_n-1) or M(ξ_n, ξ_n-1) ≤ ρ_b(ξ_n, ξ_n+1).

Case 2. Suppose that Μ(ξ_n, ξ_n-1) = ρ_b(ξ_n, ξ_n-1) for some n ≥ 1. Regarding the properties of φ_b ∈ Φ_b, and (9), we get

for all n≥ 1. Inductively, we obtain

Applying repeatedly triangle inequality (QBML₂)and regarding (11), for all k ≥ 1, we get

Define

We obtain

Due to the assumption ξ_n ≠ξ_n+1 for all n ∈ ℕ and Lemma 1.13, we conclude that the series ∑p=0∞spφbp(ρb(ξ1,ξ0)) is convergent to some S ≥ 0. Thus, limn→∞ρb(ξn+k,ξn)=0 or, in other words, for m > n

Case 3. Suppose that Μ(ξ_n, ξ_n-1)≤ ρ_b(ξ_n, ξ_n+1)for some n ≥ 1. Using the fact that φ_b ∈Φ_b and the inequality (9), we get

for all n ≥ 1. On the other hand, if we put ξ = ξ_n and η = ξ_n+1 in (3), taking into account (8), we find

where

for all n ≥ 1. Observe that, applying triangle inequality (QBML₂)to the last term in Μ(ξ_n-1, ξ_n)we have

since we are in Case 3. On the other hand, from (16) we see that

Therefore, either M(ξ_n-1, ξ_n) ≤ ρ_b(ξ_n, ξ_n+1)or M(ξ_n-1, ξ_n) = ρ_b(ξ_n-1, ξ_n).

If Μ(ξ_n-1, ξ_n) ≤ ρ_b(ξ_n, ξ_n+1)for some n ∈ ℕ, since ρ_b(ξ_n+1, ξ_n)> 0, the inequality (17) implies

which is a contradiction.

Thus, we should have M(ξ_n-1, ξ_n) = ρ_b (ξ_n-1, ξ_n) for all n ≥ 1. The inequality (17) becomes

for all n ≥ 1, using the fact that φ_b ∈Φ_b.

By induction, we get

Hence, combining (16) and (19) we deduce

Due to (QBML₂), together with (20), for all k ≤ 1, we get

Define

Then, employing (21), we get

Using the fact that the series ∑p=0∞spφbp(ρb(ξ0,ξ1)) converges to some 𝒬 ≤ 0, we deduce, lim_n→∞ρ_b(ξ_n+k, ξ_n) = 0 which means that for m > n,

Therefore, in all possible cases, we conclude that the sequence {ξ_n}is a left Cauchy sequence.

To show that it is also right Cauchy, we proceed as follows.

Let ξ = ξ_n and η = ξ_n+1 in (3). Using (8), we get

where

Regarding the inequality (18), we have

and hence, either

or Μ(ξ_n-1, ξ_n)≤ ρ_b(ξ_n, ξ_n+1)for all n ≥1. We will discuss separately these four cases.

Case I. Suppose that M(ξ_n-1, ξ_n)≤ ρ_b(ξ_n, ξ_n+1) for some n ≥ 1. Then the inequality (23) implies

which is a contradiction because ρ_b(ξ_n, ξ_n+1) > 0.

Case II. If for some n ≥ 1 we have M(ξ_n-1, ξ_n) = ρ_b(ξ_n+1, ξ_n), then the inequality (23) becomes

Recalling (9) and (10), that is,

where

and since by the assumption

then, inequalities (26) and (27) yield

due to the properties of φ_b. This is clearly impossible since ρ_b(ξ_n+1, ξ_n) > 0 and we end up with a contradiction.

Case III. Assume that Μ(ξ_n-1, ξ_n) = ρ_b(ξ_n-1, ξ_n)for some n ≥ 1. Since ρ_b(ξ_n-1, ξ_n) > 0, from (9), we get

for all n ≥ 1. Recursively, we derive

Applying repeatedly triangle inequality (QBML₂)and regarding (28), we get for all k > 0,

Recalling (21), we see that

As a result, we have, lim_n→∞ρ_b(ξ_n, ξ_n+k)= 0 or, in other words for m > n,

Case IV. Suppose that M(ξ_n-1, ξ_n) = ρ_b(ξ_n- ξ_n-1)for some n ≥ 1. Then, (23) gives

for all n ≥ 1. Now, rewriting (9) and (10) for n — 1 we have

where

where obviously, the maximum can be either ρ_b(ξ_n, ξ_n−1)or ρ_b(ξ_n−1, ξ_n−2).

The first possibility, that is, if M(ξ_n−1, ξ_n−2)≤ ρ_b(ξ_n, ξ_n−1)for some n ≥ 1, results in

which is a contradiction. Therefore, we should have Μ(ξ_n−1, ξ_n−2)≤ ρ_b(ξ_n−1, ξ_n−2) for all n ≥ 1. Then the inequality (31) yields