Abstract
In this paper a class of general type α-admissible contraction mappings on quasi-b-metric-like spaces are defined. Existence and uniqueness of fixed points for this class of mappings is discussed and the results are applied to Ulam stability problems. Various consequences of the main results are obtained and illustrative examples are presented.
1 Introduction and preliminaries
The problem of stability of functional equations was motivated by a question of S.M. Ulam asked in 1940: “When is it true that the solution of an equation differing slightly from a given one, must of necessity be close to the solution of the given equation?” The answer of D.H. Hyers, published in [1], initiates the theory of Ulam stability theory. In this paper, we investigate the Ulam stability results by using the fixed point techniques. For further information and results on the subject we refer the reader to [2-4].
First of all, we give a set of the tools in the context of fixed point theory that will be used for Ulam stability results.
The concept of α-admissible mapping introduced by Samet et al. [5] in 2012 has been one of the most important generalizations in fixed point theory in the recent years. It is a very general concept and can be used in various abstract spaces. On the other hand, b-metric spaces received considerable interest recently and combined with αadmissibility they also proved to be one of the attractive studies in fixed point theory. In their recent study, Bota et al. [6] investigated α–ψ-contractive mapping of type-(b) in the framework of b-metric spaces. In another work, Bilgili et al. [7] discussed α —ψ contractive mappings in the setting of quasi-metric spaces and proved the existence and uniqueness of a fixed point of such mappings. Also very recently, Gülyaz [8] presented some results about α-admissible mappings on quasi-b-metric like spaces.
In this work, a class of α-admissible contractive mappings of general types are considered in the context of a quasi-b-metric like space and the existence and uniqueness of their fixed points are discussed.
First, we define some basic notions and concepts to be used in the sequel.
Metric-like spaces or, also known as dislocated spaces have been introduced by Amini-Harandi in [9].
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 γb : 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) γb(x, y) = 0 ⇒ x= y;
(BML2) γb(x, y) = γb(y, x);
(BML3) γb(x, y)⇒ s[γb(x, z) + γb(z, y)].
The map γbis called a b-metric-like and the pair (X, γb) is called a b-metric-like space with constant s.
Let (X, γb)be a b-metric-like space. For any x ∈ X and r > 0, the set B(x, r) = {y ∈X : |γb(x, y) — γb(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 ρb : X × X →[0, + ∞) such that
for all x, y, z ∈ X and a constant s > 1,
(QML1) ρb(x, y) = ρb(y, x) = 0 ⇒ x = y;
(QML2) ρb(x, z)≤ s[ρb(x, z) + ρb(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 ρb : ℝ × ℝ [0, ∞) defined as
for all x, y ∈ ℝ is a quasi-b-metric-like on ℝ with constant s = 2.
The function : ρb : 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, ρb) be a quasi-b-metric-like space.
1. A sequence {ξn}⊂ X is said to converge to a point ξ ∈ X if and only if
2. A sequence {ξn}⊂ Χ is said to be a Cauchy sequence if limm,n→ ∞ρb(ξn,ξm) and limm,n→ ∞ρb(ξn,ξm) exist and are finite.
3. (X, ρb) is said to be a complete quasi-b-metric-like space if and only if for every Cauchy sequence {ξn} in X there exists some ξ∈ X such that
4. A sequence {ξη}C X is said to be a 0-Cauchy sequence if
5. (X, ρb) is said to be a 0-complete quasi-b-metric-like space if for every 0-Cauchy sequence {ξη} 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(ξο, δ))⊂ Β(ξ0),ε).
The above definition makes it clear that in the quasi-b-metric like space (X, ρb), 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, ρb) be a quasi-b-metric-like space.
(i) A sequence {ξn} 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 ρb(ξn, ξm) < ε for all n ≥ m>N;
(ii) A sequence {ξn} 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 ρb(ξn, ξm) < ε for all m ≥ n> N;
(iii) A quasi-b-metric-like space is said to be left-complete if every left-Cauchy sequence {ξn} in X is convergent,
(iv) A quasi-b-metric-like space is said to be right-complete if every right-Cauchy sequence {ξn} 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 {ξn} 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 φc : [0, +∞) → [0, +∞) satisfying
(c1)φc is increasing,
(c2)there exist k0 ∈ ℕ, a ∈ (0, 1) and a convergent series of nonnegative terms
(b1)φb, is increasing,
(b2)there exist k0 ∈ ℕ, a ∈ (0, 1) and aconvergent series of nonnegative terms
In the sequel, we denote the class of comparison functions by Φ, the class of (c)-comparison functions by Φc and the class of (b)-comparison functions by Φb. 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 φk 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 φb : [0, +∞) → [0, +∞) the following hold:
(1) the series
(2) the function bs : [0, +∞) → [0, +∞) defined by
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, ρb) 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 ∈Φb 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, ρb) 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, ρb) 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 {ξn} is a sequence in Χ which converges to ξ and satisfies α(ξn+1, ξn)≥ 1 and α(ξn, ξn+1) ≥ 1 for all n
then, there exists a subsequence {ξn(k)} of {ξn} such that α(ξ, ξn(k)))≥ 1 and α (ξn(k), ξ)≥ 1 for all k.
Then Τ has a fixed point.
2 Existence and uniqueness theorems on complete quasi-b-metric-like spaces
In this section we present our main results. We concentrate on existence and uniqueness of fixed points for a general class of α-admissible contractive mappings.
Let (X, ρb) be a quasi-b-metric-like space with a constant s ≥ 1 and let α : Χ × X → [0, ∞) and
φb ∈ Φbbe two functions.
(i) An α — φb contractive mapping Τ : X → X is of type (A) if
where
(ii) An α — φb contractive mapping Τ : X → X is of type (B) if
where
(iii) An α —φb contractive mapping Τ : X → X is of type (C) if
where
Note that ρb(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, ρb) be a 0-complete quasi-b-metric-like space with a constant s ≥ 1. Suppose that Τ : X → X is an α — φb 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 {ξn} as
Notice that if for some n0 ≥ 0 we have
From the conditions (i) and (ii), we have
and
or, in general
Regarding (8), the contractive condition (3) with ξ = ξn+1and n = ξn becomes
where
Observe that for the last term in Μ(ξn, ξn-1), by the triangle inequality we have
and hence, either Μ(ξn, ξn-1) = max{ρb(ξn+1,ξn), ρb(ξn, ξn-1)}or Μ(ξn, ξn-1)≤ ρb(ξn, ξn+1). Now, we will examine all three cases.
Case 1. Suppose that Μ(ξn, ξn-1) = ρb(ξn+1,ξn)for some n ≥ 1. Since ρb(ξn+1,ξn) > 0, from (9), we have
which is a contradiction. Hence, for all n ≥ 1 either M(ξn, ξn-1) = ρb(ξn, ξn-1) or M(ξn, ξn-1) ≤ ρb(ξn, ξn+1).
Case 2. Suppose that Μ(ξn, ξn-1) = ρb(ξn, ξn-1) for some n ≥ 1. Regarding the properties of φb ∈ Φb, 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 ξn ≠ξn+1 for all n ∈ ℕ and Lemma 1.13, we conclude that the series
Case 3. Suppose that Μ(ξn, ξn-1)≤ ρb(ξn, ξn+1)for some n ≥ 1. Using the fact that φb ∈Φb and the inequality (9), we get
for all n ≥ 1. On the other hand, if we put ξ = ξn and η = ξn+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(ξn-1, ξn)we have
since we are in Case 3. On the other hand, from (16) we see that
Therefore, either M(ξn-1, ξn) ≤ ρb(ξn, ξn+1)or M(ξn-1, ξn) = ρb(ξn-1, ξn).
If Μ(ξn-1, ξn) ≤ ρb(ξn, ξn+1)for some n ∈ ℕ, since ρb(ξn+1, ξn)> 0, the inequality (17) implies
which is a contradiction.
Thus, we should have M(ξn-1, ξn) = ρb (ξn-1, ξn) for all n ≥ 1. The inequality (17) becomes
for all n ≥ 1, using the fact that φb ∈Φb.
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
Therefore, in all possible cases, we conclude that the sequence {ξn}is a left Cauchy sequence.
To show that it is also right Cauchy, we proceed as follows.
Let ξ = ξn and η = ξn+1 in (3). Using (8), we get
where
Regarding the inequality (18), we have
and hence, either
or Μ(ξn-1, ξn)≤ ρb(ξn, ξn+1)for all n ≥1. We will discuss separately these four cases.
Case I. Suppose that M(ξn-1, ξn)≤ ρb(ξn, ξn+1) for some n ≥ 1. Then the inequality (23) implies
which is a contradiction because ρb(ξn, ξn+1) > 0.
Case II. If for some n ≥ 1 we have M(ξn-1, ξn) = ρb(ξn+1, ξn), 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 φb. This is clearly impossible since ρb(ξn+1, ξn) > 0 and we end up with a contradiction.
Case III. Assume that Μ(ξn-1, ξn) = ρb(ξn-1, ξn)for some n ≥ 1. Since ρb(ξn-1, ξn) > 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, limn→∞ρb(ξn, ξn+k)= 0 or, in other words for m > n,
Case IV. Suppose that M(ξn-1, ξn) = ρb(ξn- ξn-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 ρb(ξn, ξn−1)or ρb(ξn−1, ξn−2).
The first possibility, that is, if M(ξn−1, ξn−2)≤ ρb(ξn, ξn−1)for some n ≥ 1, results in
which is a contradiction. Therefore, we should have Μ(ξn−1, ξn−2)≤ ρb(ξn−1, ξn−2) for all n ≥ 1. Then the inequality (31) yields
for all n ≥ 1. Thus, we deduce
If we combine the inequalities (30) with (33), we derive
for all n ≥ 1. As done in Case 2, when applying triangle inequality repeatedly for every k ≥ 1, we get
where Sn is defined in (13). Therefore, we end up with limn→∞ρb(ξn, ξn+k) = 0, or, equivalently, for m > n,
We conclude that {ξn} is a right-Cauchy sequence, and hence, a Cauchy sequence. Moreover, it is a 0-Cauchy sequence in 0-complete quasi b-metric like space (X, ρb). Thus, there exists ρb ∈ X such that for m > n we have
By the continuity of T, we obtain
that is, ξis a fixed point of Τ.
Adding the condition
(U) For every pair ξ and η of fixed points of T, α(ξ, η)≥ 1 and α(η, ξ) ≥ 1.
to theStatement of Theorem 2.3 we obtain the uniqueness of the fixed point.
By Theorem 2.3 we know that the mapping Τ has at least one fixed point. To show the uniqueness, we assume that η is another fixed point of T, such that ξ ≠ η. From the condition (U) we have, α(ξ, η)≥ 1, hence, the contractive condition for the fixed points ξ and η, that is,
where,
This yields
which implies ρb(ξn, η) = 0. In a similar way, by changing the roles of ξ and η in the contractive condition we get
where,
and thus,
Therefore, we deduceρb(η, ξ) = 0 and hence, ξ = η. This completes the uniqueness proof of the Theorem.
Regarding Remark 2.2, we can state the following result as an immediate consequence.
Let(X, ρb) be a 0-complete quasi-b-metric-like space with a constant s ≥ 1. Suppose that T : X → X is an α — φb contractive mapping of type (B) or type (C) satisfying the following:
(i) Τ is α-admissible;
(ii) there exists ξ0 ∈ X such that α(Τ ξ0, ξ0)≥ 1 and α(ξ0, Τξ0)≥ 1;
(iii) T is continuous;
(iv) For every pair ξ and η of fixed points of T, α(ξ, η)≥ 1 and α(η, ξ)≥ 1. Then Τ has a unique fixed point.
We also state another consequence of the main result obtained by taking the function α(x, y) = 1.
Let (X, ρb) be a 0-complete quasi-b-metric-like space with a constant s ≥ 1. Suppose that Τ : X → X is a continuous mapping satisfying
where
Then Τ has a unique fixed point.
Let (X, ρb) be a 0-complete quasi-b-metric-like space with a constant s ≥ 1. Suppose that T : X → X is a continuous mapping satisfying
where
or
where
Then Τ has a unique fixed point.
We also point out that the main result of Gülyaz [8], that is the Theorem 1.16, follows from our main Theorem 2.3. In addition, by taking α(x, y) = 1, we deduce another corollary.
Let (X, ρb) be a 0-complete quasi-b-metric-like space with a constant s ≥ 1. Suppose that T : X → X is a continuous mapping satisfying
where φb ∈Φb. Then Τ has a unique fixed point.
It has been shown in some recent studies that by a suitable choice of the function α, the fixed point results on partially ordered spaces and fixed point results of cyclic contraction mappings can be concluded from the fixed point results of α-admissible mappings (see [19] for details). Using the idea of these studies we present some theorems and their consequences below.
We recall the definition of cyclic contraction mappings on quasi b-metric-like spaces introduced originally by Kirket.al. [20].
Let X be nonempty set and let A and Β be nonempty subsets of X. A mapping T : AUB → AUB is cyclic if Τ(A)C Β and T(B)⊂ A.
We define generalized cyclic contraction of type as follows.
Let (X, ρb) be a quasi-b-metric-like space with a constant s ≥ 1 and let φb ∈ Φb be a given function. A generalized cyclic contractive mapping Τ : A U Β → A U Β is a cyclic mapping satisfying
where
A fixed point theorem for the generalized cyclic contractions defined above is given next.
Let (X, ρb) be a 0-complete quasi-b-metric-like space with a constant s ≥ 1. Suppose that Τ : A U B → A U B is a generalized cyclic contractive mapping. Then Τ has a unique fixed point in Α ⋂ Β.
Define α : X → X as follows.
Then all conditions of Corollary 2.5 are satisfied and Τ has a unique fixed point.
Finally, we observe that the main result (Theorem 2.9) in [12] is a consequence of Theorem 2.11. Indeed, by choosing
We end this section with illustrative examples.
Let X = [–1, 1] and ρb(ξ, η) = |ξ – η|2 + |2ξ + η|2. Then ρb is a quasi-b-metric like on X with a constant s = 2. Define
where φb(t) = kt with
fixed point which is ξ = 0.
Let X = [0, 1] and ρb(ξ, η) = |ξ – η|2 + 2|ξ|2 + |η|2. Then ρbis a quasi-b-metric like on X with a constant s = 2. Define
Choosing a (b)-comparison function φb (t) = kt where
3 Ulam-Hyers stability
In this section we apply our main results to Ulam-Hyers stability problems. We first recall the classical definition of Ulam-Hyers stability on metric space.
Let (X, d) be a metric space and Τ : X → X be an operator. The fixed point equation
is called generalized Ulam-Hyers stable if and only if there exists an increasing function φb : [0, ∞) → [0, ∞), continuous at 0 and φb(0) = 0 such that for each ε > 0 and for each ε-solution of the fixed point equation (40), that is, for each ξ* ∈ X, satisfying the inequality
there exists a solution x* ∈ X of (40) such that
lf φb(t):= ct for all t ∈ [0, ∞), where c > 0, the fixed point equation (40) is said to be Ulam-Hyers stable.
We refer the reader to Bota-Boriceanu, Petrusel [21], Lazăr [22], and Rus [23],[24] for some Ulam-Hyers stability results in the case of fixed point problems.
In the framework of quasi-b-metric like spaces the above definition can be stated as follows.
Let (X, φb) be a quasi-b-metric-like space and Τ : X → X be an operator. The fixed point equation
is called generalized Ulam-Hyers stable if and only if there exists an increasing function φb : [0, ∞) → [0, ∞), continuous at 0 and φb (0) = 0 such that for each ε > 0 and for each ε-solution of the fixed point equation (43), that is, ξ* ∈ X, satisfying the inequalities
there exists a solution x* ∈ X of (43) such that
If φb(t) : = ct for all t ∈ [0, ∞), where c > 0, the fixed pointequation (43)is said to be Ulam- Hyers stable.
In fact, the nature of the quasi-b-metric like space makes it possible to introduce two types of stability, namely, right and left stability. We define these new concepts in the following.
Let (X, ρb) be a quasi-b-metric-like space and Τ : X → X be an operator. The fixed pointequation (43)is called generalized right Ulam-Hyers stable if and only if there exists an increasing function φb : [0, ∞) → [0, φb), continuous at 0 and φb(0) = 0 such that for each ε > 0 and for each ξ*, satisfying the inequality
there exists a solution x* ∈ X of (43) such that
If φb(t) : = ct for all t ∈ [0, ∞), where c > 0, the fixed pointequation (43)is said to be right Ulam- Hyers stable.
Similarly, the fixed pointequation (43)is called generalized left Ulam-Hyers stable if and only if there exists an increasing function φb : [0, ∞) → [0, ∞), continuous at 0 and φb (0) = 0 such that for each ε > 0 and for each ξ*, satisfying the inequality
there exists a solution x* ∈ X of (43) such that
If φb(t):= ct for all t ∈ [0, ∞), where c > 0, the fixed pointequation (43)is said to be left Ulam- Hyers stable.
Clearly, if the fixed point equation (43) is both generalized right and left Ulam-Hyers stable, then it is generalized Ulam-Hyers stable.
We also recall the definition of well posed fixed point problems and apply it to quasi-b-metric-like spaces.
Let (X, ρb) be a quasi-b-metric-like space and Τ : X → X be a mapping. The fixed point problem (43) for Τ is said to be well-posed if it satisfies the following conditions:
(i) Τ has a unique fixed point x* in X;
(ii) for any sequence {ξn} in X such that
one has
Inspired by the Ulam-Hyers stability problem and the ideas given in Petra et al. [25] we state and prove the following results.
Let (X, ρb) be a 0-complete quasi-b-metric-like space with a constant s ≥ 1 and T : X → X be a mapping. Let the function β : [0, ∞) → [0, ∞), ß(r) := r — sφb(r) be strictly increasing and onto. Suppose that all the hypotheses of Corollary 2.8 are satisfied. Then the fixed pointequation (43)is generalized right Ulam-Hyers stable.
According to the Corollary 2.8, there is a unique x* ∈ X such that x* = T x*, that is, x* ∈ X is a solution of the fixed point equation (43). Let ε > 0 and let ξ* ∈ X satisfy
Using the contractive condition (38) with x = ξ* and y = x*, and triangle inequality we obtain
Then recalling the definition of β we have
or,
and since β is continuous, strictly increasing and onto, then, β–1 is also increasing and continuous with β–1 (0) = 0. Hence, the fixed equation (43) is generalized left Ulam-Hyers stable.
The following Lemma regarding right Ulam-Hyers stability can be proved in a similar way, so we give the statement only without proof.
Let (X, ρb) be a 0-complete quasi-b-metric-like space with a constant s ≥ 1 and T : X → X be a mapping. Let the function β : [0, ∞) → [0, ∞), ß(r) : = r — sφb(r) be strictly increasing and onto. Suppose that all the hypotheses of Corollary 2.8 are satisfied. Then the fixed pointequation (43)is generalized left Ulam-Hyers stable.
Using the above lemmas, we can prove the following theorem.
Let (X, ρb) be a 0-complete quasi-b-metric-like space with a constant s > 1 and Τ : X → X be a mapping. Let the function β : [0, ∞) → [0, ∞), ß(r) : = r — sφb(r) be strictly increasing and onto. Suppose that all the hypotheses of Corollary 2.8 are satisfied. Then the following hold.
(a) The fixed pointequation (43)is generalized Ulam-Hyers stable.
(b) If {ξn} is a sequence in X such that = = 0 and x* is a fixed point of Τ, then the fixed point problem (43) is well-posed.
(c) If G : X → X is a mapping such that there exists η > 0 with
then for any fixed point y* of G we have
(a) By Lemma 3.5 and Lemma 3.6 the fixed point equation (43) is generalized right and left Ulam-Hyers stable. Therefore, it is generalized Ulam-Hyers stable.
(b) Let {ξn} be a sequence in X such that limn→∞ρb(ξn, T ξn) = limn→∞ρb(T ξn, ξn) = 0 and let x* be the unique fixed point of T. Employing the contractive condition and triangle inequality we get
for all n ∈ℕ. Thus, we have,
for all n ∈ℕ. Since limn→∞ρb(ξn, T ξn)= 0, we obtain
Therefore
On the other hand, the contractive condition with ξ = x* and η = ξn and triangle inequality yields
for all n ∈ ℕ. Thus,
for all and regarding limn→∞ρb(T ξn, ξn) = 0 , we get
Consequently,
As a result, the limits in (52) and (53) show that the fixed point problem (43) is well-posed.
(c) To show the last part of the Theorem we assume that G : X → X is a mapping such that there exists η > 0
with
and let y * be a fixed point of G. Then, by the triangle inequality, we have
Then we have
and hence
In a similar way, by changing the roles of x* and y * in the discussion above we obtain
which completes the proof.
4 Conclusion
The main contribution of this study to fixed point theory is the existence-uniqueness result given in Theorem 2.3. This theorem provides the existence and uniqueness conditions for a very large class of contractive mappings on various abstract spaces. In fact, the quasi-b-metric-like spaces cover theb-metric spaces, quasi-b-metric spaces, b-metric-like spaces as well as b-metric, usual quasi-metric and metric spaces.
However, the most significant part of the study is the definition of right and left Ulam-Hyers stability to get some new Ulam-Hyers stability results. These results have applications in many areas of applied mathematics, especially in differential and integral equations. A study related particularly to the applications of the present results in the theory of ordinary differential equations is currently in progress.
Competing interests
The authors declare that they have no competing interests.
Authors’ contribution
All authors contributed equally to this work. All authors read and approved the final manuscript.
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