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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 14, Issue 1 (Jan 2016)

Issues

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

Hamed H. Alsulami
  • Corresponding author
  • Nonlinear Analysis and Applied Mathematics Research Group (NAAM), King Abdulaziz University, Jeddah, Saudi Arabia
  • Email:
/ Selma Gülyaz
  • Department of Mathematics, Cumhuriyet University, Sivas, Turkey
  • Email:
/ Erdal Karapınar
  • Department of Mathematics, Atılım University, Anakara, Turkey and NAAM, King AbdulAziz University, Jeddah, Saudi Arabia
  • Email:
/ İnci M. Erhan
  • Department of Mathematics, Atılım University, Anakara, Turkey
  • Email:
Published Online: 2016-12-28 | DOI: https://doi.org/10.1515/math-2016-0097

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.

Keywords: Fixed point; Quasi-b-metric like; α-admissible contraction mappings; Ulam-Hyers stability

MSC 2010: 47H10; 54C60; 54H25; 55M20

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 αψ-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, zX:

(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 γb is 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 xX 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], (QML1)ρ(x,y)=ρ(y,x)=0x=y.

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, zX 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 ∈is a quasi-b-metric-like onwith constant s = 2. ρb(x,y)=|xy|2+|2x+y|2

The function : ρb : ℝ x ℝ → [0, ∞) defined as ρb(x,y)=|xy|4+|x|2+2|y|2

for all x, y ∈ℝ is a quasi-b-metric-like onwith constant s = 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 sequencen}⊂ X is said to converge to a point ξ ∈ X if and only if limnρb(ξ,ξn)=limnρb(ξn,ξ)=ρb(ξ,ξ).

2. A sequencen}⊂ Χ is said to be a Cauchy sequence if limm,n → ∞ ρbnm) and limm,n → ∞ ρbnm) 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 sequencen} in X there exists some ξ∈ X such that limnρb(ξ,ξn)=limnρb(ξn,ξ)=ρb(ξ,ξ)=limm,nρb(ξm,ξn)=limm,nρb(ξn,ξm).

4. A sequenceη}C X is said to be a 0-Cauchy sequence if limm,nρb(ξn,ξm)=limm,mρb(ξm,ξn)=0.

5. (X, ρb) is said to be a 0-complete quasi-b-metric-like space if for every 0-Cauchy sequenceη} in X there exists a ξ ∈ X such that limnρb(ξ,ξn)=limnρb(ξn,ξ)=ρb(ξ,ξ)=limm,nρb(ξm,ξn)=limm,nρb(ξn,ξm).

6. A mapping ƒ : ΧX is said to be continuous at ξ0 ϵ X if for every ε > 0, there exists δ > 0 such that f(Bο, δ))⊂ Β0)).

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 sequencen} 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 ρbn, ξm) < ε for all nm>N;

(ii) A sequencen} 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 ρbn, ξm) < ε for all m ≥ n> N;

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

(iv) A quasi-b-metric-like space is said to be right-complete if every right-Cauchy sequencen} 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 sequencen} 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 k=1vk suchthat φck+1taφckt+vk for kk0 and any t ∈ [0, ∞). BCF(See [16, 17]) For a real number s ≥ 1 (b)-comparison function is a function φb : [0,+∞) → [0,+∞) satisfying the conditions

(b1)φb, is increasing,

(b2)there exist k0 ∈ ℕ, a ∈ (0, 1) and a convergent series of nonnegative terms k=1vk such that sk+1φbk+1taskφbkt+vk,forkk0 and any t ∈ [0, ∞).

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

(2) the function bs : [0, +∞) → [0, +∞) defined by bst=k=0skφbkt,t[0,) is increasing 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 : XX is called α-admissible if for all ξ, n ∈ X we have α(ξ,η)1α(Tξ,Tη)1,(1)

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 : XX 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 α(ξ,η)ρb(Tξ,Tη)ψ(ρb(ξ,η))(2)

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 : XX is an α — ψ-contractive mapping. Suppose also that

(i) Τ is α-admissible;

(ii) there exists ξ0X such that α(Tξ0, ξ0) ≥ 1 and α(ξ0 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 : XX is an α — ψ-contractive mapping. Suppose also that

(i) Τ is α-admissible;

(ii) there exists ξ0X 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 ∈ Φb be two functions.

(i) An α — φb contractive mapping Τ : XX is of type (A) if α(ξ,η)ρb(Tξ,Tη)φb(M(ξ,η)),forallξ,ηX,(3)

where M(ξ,η)=max{ρb(ξ,η),ρb(Tξ,ξ),ρb(Tη,η),14s[ρb(Tξ,η)+ρb(Tη,ξ)]}.

(ii) An α — φb contractive mapping Τ : XX is of type (B) if α(ξ,η)ρb(Tξ,Tη)φb(N(ξ,η)),forallξ,ηX,(4)

where N(ξ,η)=max{ρb(ξ,η),12s[ρb(Tξ,ξ)+ρb(Tη,η)],14s[ρb(Tξ,η)+ρb(Tη,ξ)]}.

(iii) An α —φb contractive mapping Τ : X X is of type (C) if α(ξ,η)ρb(Tξ,Tη)φb(K(ξ,η)),forallξ,ηX,(5)

where K(ξ,η)=max{ρb(ξ,η),ρb(Tξ,ξ),ρb(Tη,η)}

Note that ρb, n)≤ Κ, n)≤ Ν, η)≤ Μ, 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 : XX is an α — φb contractive mapping of type (A) satisfying the following:

(i) T is α-admissible;

(ii) there exists ξ0X such that α(Τ ξ00)≥ 1 and α0, Τ ξ0)≥ 1;

(iii) T is continuous;

Then Τ has a fixed point.

As usual, we take ξ0X such that α (Τ ξ0, ξ0) ≥ 1 and α (ξ0, T ξ0) ≥ 1 and construct the sequence {ξn} as ξn+1=Tξn for nN.

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 ξn ≠ξn+1for all n ≥ 0. Because of the lack of symmetry condition in quasi-b-metric-like spaces, we will show that the sequence {ξn} is both left- and right-Cauchy.

From the conditions (i) and (ii), we have α(ξ1,ξ0)=α(Tξ0,ξ0)1α(Tξ1,Tξ0)=α(ξ2,ξ1)1,(6)

and α(ξ0,ξ1)=α(ξ0,Tξ0)1α(Tξ0,Tξ1)=α(ξ1,ξ2)1,(7)

or, in general α(ξn+1,ξn)1andα(ξn,ξn+1)1nN.(8)

Regarding (8), the contractive condition (3) with ξ = ξn+1and n = ξn becomes ρb(ξn+1,ξn)=ρb(Tξn,Tξn1)α(ξn,ξn1)ρb(Tξn,Tξn1)φb(M(ξn,ξn1)),(9)

where M(ξn,ξn1)=max{ρb(ξn,ξn1),ρb(Tξn,ξn),ρb(Tξn1,ξn1),14s[ρb(Tξn,ξn1)+ρb(Tξn1,ξn)]}=max{ρb(ξn,ξn1),ρb(ξn+1,ξn),ρb(ξn,ξn1),14s[ρb(ξn+1,ξn1)+ρb(ξn,ξn)]}

Observe that for the last term in Μn, ξn-1), by the triangle inequality we have 14s[ρb(ξn+1,ξn1)+ρb(ξn,ξn)]14[ρb(ξn+1,ξn)+ρb(ξn1,ξn)+ρb(ξn,ξn+1)ρb(ξn+1,ξn)]14[ρb(ξn+1,ξn)+ρb(ξn,ξn1)+ρb(ξn,ξn+1)+ρb(ξn+1,ξn)]=14[2ρb(ξn+1,ξn)+ρb(ξn,ξn1)+ρb(ξn,ξn+1)]max{ρb(ξn+1,ξn),ρb(ξn,ξn1),ρb(ξn,ξn+1)},(10)

and hence, either Μn, ξn-1) = max{ρbn+1n), ρbn, ξn-1)}or Μn, ξn-1)≤ ρbn, ξn+1). Now, we will examine all three cases.

Case 1. Suppose that Μn, ξn-1) = ρbn+1n)for some n ≥ 1. Since ρbn+1n) > 0, from (9), we have ρb(ξn+1,ξn)φb(M(ξn,ξn1))=φb(ρb(ξn+1,ξn))<ρb(ξn+1,ξn)

which is a contradiction. Hence, for all n ≥ 1 either Mn, ξn-1) = ρbn, ξn-1) or Mn, ξn-1) ≤ ρbn, ξn+1).

Case 2. Suppose that Μn, ξn-1) = ρbn, ξn-1) for some n ≥ 1. Regarding the properties of φb ∈ Φb, and (9), we get ρb(ξn+1,ξn)φb(M(ξn,ξn1))φb(ρb(ξn,ξn1))<ρb(ξn,ξn1)

for all n≥ 1. Inductively, we obtain ρb(ξn+1,ξn)φbn(ρb(ξ1,ξ0)),foralln1.(11)

Applying repeatedly triangle inequality (QBML2)and regarding (11), for all k ≥ 1, we get ρb(ξn+k,ξn)s[ρb(ξn+k,ξn+1)+ρb(ξn+1,ξn)]s2[ρb(ξn+k,ξn+2)+ρb(ξn+2,ξn+1)]+sρb(ξn+1,ξn)skρb(ξn+k,ξn+k1)+sk1ρb(ξn+k1,ξn+k2)++s2ρb(ξn+2,ξn+1)+sρb(ξn+1,ξn)skφbn+k1(ρb(ξ1,ξ0))+sk1φbn+k2(ρb(ξ1,ξ0))++sφbn(ρb(ξ1,ξ0))=1sn1sn+k1φbn+k1(ρb(ξ1,ξ0))+sn+k2φbn+k2(ρb(ξ1,ξ0))++snφbn(ρb(ξ1,ξ0))(12)

Define Sn=p=0nspφbp(ρb(ξ1,ξ0))forn1.(13)

We obtain ρb(ξn+k,ξn)1sn1[Sn+k1Sn1],n1,k1.(14)

Due to the assumption ξn ≠ξn+1 for all n ∈ ℕ and Lemma 1.13, we conclude that the series p=0spφbp(ρb(ξ1,ξ0)) is convergent to some S ≥ 0. Thus, limnρb(ξn+k,ξn)=0 or, in other words, for m > n limm,nρb(ξm,ξn)=0.(15)

Case 3. Suppose that Μn, ξn-1)≤ ρbn, ξn+1)for some n ≥ 1. Using the fact that φb ∈Φb and the inequality (9), we get ρb(ξn+1,ξn)φb(M(ξn,ξn1))φb(ρb(ξn,ξn+1))<ρb(ξn,ξn+1)(16)

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 ρb(ξn,ξn+1)=ρb(Tξn1,Tξn)α(ξn1,ξn)ρb(Tξn1,Tξn)φb(M(ξn1,ξn)),(17)

where M(ξn1,ξn)=maxρb(ξn1,ξn),ρb(Tξn1,ξn1),ρb(Tξn,ξn),14s[ρb(Tξn1,ξn)+ρb(Tξn,ξn1)]=maxρb(ξn1,ξn),ρb(ξn,ξn1),ρb(ξn+1,ξn),14s[ρb(ξn,ξn)+ρb(ξn+1,ξn1)]

for all n ≥ 1. Observe that, applying triangle inequality (QBML2)to the last term in Μn-1, ξn)we have 14s[ρb(ξn,ξn)+ρb(ξn+1,ξn1)]14s[ρb(ξn,ξn+1)+ρb(ξn+1,ξn)+ρb(ξn+1,ξn)+ρb(ξn,ξn1)]=14[ρb(ξn,ξn+1)+2ρb(ξn+1,ξn)+ρb(ξn,ξn1)]max{ρb(ξn,ξn+1),ρb(ξn+1,ξn),ρb(ξn,ξn1)}=ρb(ξn,ξn+1),(18)

since we are in Case 3. On the other hand, from (16) we see that ρb(ξn,ξn1)ρb(ξn1,ξn),andρb(ξn+1,ξn)ρb(ξn,ξn+1).

Therefore, either Mn-1, ξn) ≤ ρbn, ξn+1)or Mn-1, ξn) = ρbn-1, ξn).

If Μn-1, ξn) ≤ ρbn, ξn+1)for some n ∈ ℕ, since ρbn+1, ξn)> 0, the inequality (17) implies ρb(ξn,ξn+1)φb(M(ξn1,ξn))φb(ρb(ξn,ξn+1))<ρb(ξn,ξn+1),

which is a contradiction.

Thus, we should have Mn-1, ξn) = ρbn-1, ξn) for all n ≥ 1. The inequality (17) becomes ρb(ξn,ξn+1)φb(M(ξn1,ξn))φb(ρb(ξn1,ξn))<ρb(ξn1,ξn)

for all n ≥ 1, using the fact that φb ∈Φb.

By induction, we get ρb(ξn,ξn+1)φbn(ρb(ξ0,ξ1)),for alln1.(19)

Hence, combining (16) and (19) we deduce ρb(ξn+1,ξn)φbn(ρb(ξ0,ξ1)),for alln1.(20)

Due to (QBML2), together with (20), for all k ≤ 1, we get ρb(ξn+k,ξn)s[ρb(ξn+k,ξn+1)+ρb(ξn+1,ξn)]s2[ρb(ξn+k,ξn+2)+ρb(ξn+2,ξn+1)]+sρb(ξn+1,ξn)skρb(ξn+k,ξn+k1)+sk1ρb(ξn+k1,ξn+k2)++s2ρb(ξn+2,ξn+1)+sρb(ξn+1,ξn)skφbn+k1(ρb(ξ0,ξ1))+sk1φbn+k2(ρb(ξ0,ξ1))++sφbn(ρb(ξ0,ξ1))=1sn1sn+k1φbn+k1(ρb(ξ0,ξ1))+sn+k2φbn+k2(ρb(ξ0,ξ1))++snφbn(ρb(ξ0,ξ1))

Define Qn=p=0nspφbp(ρb(ξ0,ξ1))forn1.(21)

Then, employing (21), we get ρb(ξn+k,ξn)1sn1[Qn+k1Qn1],n1,k1.

Using the fact that the series p=0spφbp(ρb(ξ0,ξ1)) converges to some 𝒬 ≤ 0, we deduce, limn→∞ ρbn+k, ξn) = 0 which means that for m > n, limm,nρb(ξm,ξn)=0.(22)

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 ρb(ξn,ξn+1)=ρb(Tξn1,Tξn)α(ξn1,ξn)ρb(Tξn1,Tξn)φb(M(ξn1,ξn))(23)

where M(ξn1,ξn)=max{ρb(ξn1,ξn),ρb(Tξn1,ξn1),ρb(Tξn,ξn),14s[ρb(Tξn1,ξn)+ρb(Tξn,ξn1)]maxρb(ξn1,ξn),ρb(ξn,ξn1),ρb(ξn+1,ξn),14s[ρb(ξn,ξn)+ρb(ξn+1,ξn1)].

Regarding the inequality (18), we have 14s[ρb(ξn,ξn)+ρb(ξn+1,ξn1)]max{ρb(ξn,ξn+1),ρb(ξn+1,ξn),ρb(ξn,ξn1)},

and hence, either M(ξn1,ξn)=max{ρb(ξn1,ξn),ρb(ξn,ξn1),ρb(ξn+1,ξn)},(24)

or Μn-1, ξn)≤ ρbn, ξn+1)for all n ≥1. We will discuss separately these four cases.

Case I. Suppose that Mn-1, ξn)≤ ρbn, ξn+1) for some n ≥ 1. Then the inequality (23) implies ρb(ξn,ξn+1)φb(M(ξn1,ξn))φb(ρb(ξn+1,ξn))<ρb(ξn,ξn+1)(25)

which is a contradiction because ρbn, ξn+1) > 0.

Case II. If for some n ≥ 1 we have Mn-1, ξn) = ρbn+1, ξn), then the inequality (23) becomes ρb(ξn,ξn+1)φb(M(ξn1,ξn))φb(ρb(ξn+1,ξn))<ρb(ξn+1,ξn).(26)

Recalling (9) and (10), that is, ρb(ξn+1,ξn)φb(M(ξn,ξn1))(27)

where M(ξn,ξn1)max{ρb(ξn,ξn1),ρb(ξn+1,ξn),ρb(ξn,ξn+1)},

and since by the assumption max{ρb(ξn1,ξn),ρb(ξn,ξn1),ρb(ξn+1,ξn)}=ρb(ξn+1,ξn)ρb(ξn,ξn+1),

then, inequalities (26) and (27) yield ρb(ξn,ξn+1)ρb(ξn+1,ξn)φb(M(ξn,ξn1))φb(ρb(ξn+1,ξn))<ρb(ξn+1,ξn)

due to the properties of φb. This is clearly impossible since ρbn+1, ξn) > 0 and we end up with a contradiction.

Case III. Assume that Μn-1, ξn) = ρbn-1, ξn)for some n ≥ 1. Since ρbn-1, ξn) > 0, from (9), we get ρb(ξn,ξn+1)φb(M(ξn1,ξn))φb(ρb(ξn1,ξn))<ρb(ξn1,ξn)

for all n ≥ 1. Recursively, we derive ρb(ξn,ξn+1)φbn(ρb(ξ0,ξ1)),n1.(28)

Applying repeatedly triangle inequality (QBML2)and regarding (28), we get for all k > 0, ρb(ξn,ξn+k)s[ρb(ξn,ξn+1)+ρb(ξn+1,ξn+k)]sφbn(ρb(ξ0,ξ1))+s2φbn+1(ρb(ξ0,ξ1))+++sk1φbn+k2(ρb(ξ0,ξ1))+skφbn+k1(ρb(ξ0,ξ1))=1sn1[snφbn(ρb(ξ0,ξ1))+++sn+k2φbn+k2(ρb(ξ0,ξ1))+sn+k1φbn+k1(ρb(ξ0,ξ1))(29)

Recalling (21), we see that ρb(ξn,ξn+k)1sn1[Qn+k1Qn1],n1,k1.

As a result, we have, limn→∞ ρbn, ξn+k)= 0 or, in other words for m > n, limm,nρb(ξn,ξm)=0.

Case IV. Suppose that Mn-1, ξn) = ρbn- ξn-1)for some n ≥ 1. Then, (23) gives ρb(ξn,ξn+1)φb(M(ξn1,ξn))φb(ρb(ξn,ξn1))<ρb(ξn,ξn1)(30)

for all n ≥ 1. Now, rewriting (9) and (10) for n — 1 we have ρb(ξn,ξn1)=ρb(Tξn1,Tξn2)α(ξn1,ξn2)ρb(Tξn1,Tξn2)φb(M(ξn1,ξn2))(31)

where M(ξn1,ξn2)max{ρb(ξn,ξn1),ρb(ξn1,ξn2),ρb(ξn1,ξn)},

where obviously, the maximum can be either ρbn, ξn−1)or ρbn−1, ξn−2).

The first possibility, that is, if Mn−1, ξn−2)≤ ρbn, ξn−1)for some n ≥ 1, results in ρb(ξn,ξn1)φb(M(ξn1,ξn2)φb(ρb(ξn,ξn1))<ρb(ξn,ξn1),

which is a contradiction. Therefore, we should have Μn−1, ξn−2)≤ ρbn−1, ξn−2) for all n ≥ 1. Then the inequality (31) yields ρb(ξn,ξn1)φb(M(ξn1,ξn2)φb(ρb(ξn1,ξn2))<ρb(ξn1,ξn2),(32)

for all n ≥ 1. Thus, we deduce ρb(ξn,ξn1)φbn1(ρb(ξ1,ξ0))for alln1.(33)

If we combine the inequalities (30) with (33), we derive ρb(ξn,ξn+1)<ρb(ξn,ξn1)φbn1(ρb(ξ1,ξ0)),(34)

for all n ≥ 1. As done in Case 2, when applying triangle inequality repeatedly for every k ≥ 1, we get ρb(ξn,ξn+k)s[ρb(ξn,ξn+1)+ρb(ξn+1,ξn+k)]sρb(ξn,ξn+1)+s2[ρb(ξn+1,ξn+2)+ρb(ξn+2,ξn+k)]sρb(ξn,ξn+1)+s2ρb(ξn+1,ξn+2)+++sk1ρb(ξn+k2,ξn+k1)+skρb(ξn+k1,ξn+k)sφbn1(ρb(ξ1,ξ0))+s2φbn(ρb(ξ1,ξ0))+++sk1φbn+k3(ρb(ξ1,ξ0))+skφbn+k2(ρb(ξ1,ξ0))1sn2sn1φbn1(ρb(ξ1,ξ0))++sn+k3φbn+k3(ρb(ξ1,ξ0))+sn+k2φbn+k2(ρb(ξ1,ξ0))1sn2[Sn+k2Sn2],n1,k1,

where Sn is defined in (13). Therefore, we end up with limn→∞ ρbn, ξn+k) = 0, or, equivalently, for m > n, limm,nρb(ξn,ξm)=0.

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 ρbX such that for m > n we have limnρb(ξn,ξ)=limnρb(ξ,ξn)=ρb(ξ,ξ)=0=limm,nρb(ξm,ξn)=limm,nρb(ξn,ξm).

By the continuity of T, we obtain ξ=limnξn+1=limnTξn=Tlimnξn=Tξ,

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 the Statement 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, ρb(ξ,η)=ρ(Tξ,Tη)α(ξ,η)ρb(Tξ,Tη)φb(M(ξ,η)),

where, M(ξ,η)=max{ρb(ξ,η),ρb(Tξ,ξ),ρb(Tη,η),14s[ρb(Tξ,η)+ρb(Tη,ξ)]}=ρb(ξ,η).

This yields ρb(ξ,η)φb(ρb(ξ,η))<ρb(ξ,η),

which implies ρbn, η) = 0. In a similar way, by changing the roles of ξ and η in the contractive condition we get ρb(η,ξ)=ρ(Tη,Tξ)α(η,ξ)ρb(Tη,Tξ)φb(M(η,ξ)),

where, M(η,ξ)=max{ρb(η,ξ),ρb(Tη,η),ρb(Tξ,ξ),14s[ρb(Tη,ξ)+ρb(Tξ,η)]}=ρb(η,ξ),

and thus, ρb(η,ξ)φb(ρb(η,ξ))<ρb(η,ξ).

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 : XX is an α — φb contractive mapping of type (B) or type (C) satisfying the following:

(i) Τ is α-admissible;

(ii) there exists ξ0X 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 : XX is a continuous mapping satisfying ρb(Tξ,Tη)φb(M(ξ,η)),forallξ,ηX,(35)

where M(ξ,η)=max{ρb(ξ,η),ρb(Tξ,ξ),ρb(Tη,η),14s[ρb(Tξ,η)+ρb(Tη,ξ)]}.

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 : XX is a continuous mapping satisfying ρb(Tξ,Tη)φb(N(ξ,η)),forallξ,ηX(36)

where N(ξ,η)=max{ρb(ξ,η),12s[ρb(Tξ,ξ)+ρb(Tη,η)],14s[ρb(Tξ,η)+ρb(Tη,ξ)]}.

or ρb(Tξ,Tη)φb(K(ξ,η)),forallξ,ηX,(37)

where K(ξ,η)=max{ρb(ξ,η),ρb(Tξ,ξ),ρb(Tη,η)}.

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 : XX is a continuous mapping satisfying ρb(Tξ,Tη)φb(ρb(ξ,η)),forallξ,ηX,(38)

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 : AUBAUB 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 ρb(Tξ,Tη)φb(K(ξ,η)),forallξA,ηB,(39)

where K(ξ,η)=max{ρb(ξ,η),ρb(Tξ,ξ),ρb(Tη,η)}.

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. α(ξ,η)=1ifξA,ηB0otherwise.

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 φb(t)=kst and noticing that ρb, η)≤ Μ, η), the main result of Klin-eam and Suanoom [12] follows immediately from Theorem 2.11.

We end this section with illustrative examples.

Let X = [–1, 1] and ρb, η) = |ξ – η|2 + |2ξ + η|2. Then ρb is a quasi-b-metric like on X with a constant s = 2. Define Tξ=ξ4.. Then we see that ρb(Tξ,Tη)=ξ41η42+2ξ4+η42=116ξη2+116ξ+η2=116(ρb(ξ,η)φb(pb(ξ,η)),

where φb(t) = kt with 116k<12 is clearly a (b)-comparison function. By the Corollary 2.8 Τ has a unique

fixed point which is ξ = 0.

Let X = [0, 1] and ρb, η) = |ξ – η|2 + 2|ξ|2 + |η|2. Then ρb is a quasi-b-metric like on X with a constant s = 2. Define Tξ=18ξ2eξ2. Without loss of generality assume that ξ η. Then we have eξ2eη21in[0,1]. ρb(Tξ,Tη)=18ξ2eξ218η2eη22+218ξ2eξ22+18η2eη22164|ξ2eη2η2eη2|2+132|ξ2eξ2|2+164|η2eη2|2=164|ξ2η2|2|eη2|2+132|ξ2|2|eξ2|2+164|η|2|2|eη2|2164|ξη|2|ξ+η|2+132|ξ|2|ξ|2+164|η|2|η|2464|ξη|2+132|ξ|2+164|η|2116(|ξη|2+2|ξ|2+|η|2)=116ρb(ξ,η).

Choosing a (b)-comparison function φb (t) = kt where 116k<12 we see that the mapping Τ satisfies the conditions of the Corollary 2.8. Hence, Τ has a unique fixed point which is ξ = 0.

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 : XX be an operator. The fixed point equation x=Tx,xX(40)

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 d(Tξ,ξ)ε,(41)

there exists a solution x*X of (40) such that d(x,ξ)φb(ε),(42)

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 : XX be an operator. The fixed point equation x=Tx,xX(43)

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 ρb(Tξ,ξ)εandρb(ξ,Tξ)ε(44)

there exists a solution x* X of (43) such that ρb(x,ξ)φb(ε)andρb(ξ,x)φb(ε)(45)

If φb(t) : = ct for all t ∈ [0, ∞), where c > 0, the fixed point equation (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 : XX be an operator. The fixed point equation (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 ρb(ξ,Tξ)ε(46)

there exists a solution x*X of (43) such that ρb(ξ,x)φb(ε).(47)

If φb(t) : = ct for all t ∈ [0, ∞), where c > 0, the fixed point equation (43) is said to be right Ulam- Hyers stable.

Similarly, the fixed point equation (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 ρb(Tξ,ξ)ε(48)

there exists a solution x*X of (43) such that ρb(x,ξ)φb(ε).(49)

If φb(t):= ct for all t ∈ [0, ∞), where c > 0, the fixed point equation (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 : XX 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 sequencen} in X such that limnρb(ξn,Tξn)=limnρb(Tξn,ξn)=0,

one has limnρb(ξn,x)=limnρb(x,ξn)=0.

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 : XX 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 point equation (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 ρb(ξ,Tξ)ε.

Using the contractive condition (38) with x = ξ* and y = x*, and triangle inequality we obtain ρb(ξ,x)s[ρb(ξ,Tξ)+ρb(Tξ,x)]=s[ρb(ξ,Tξ)+ρb(Tξ,Tx)]s[ε+φb(ρ(ξ,x))].

Then recalling the definition of β we have ρb(ξ,x)sφb(ρ(ξ,x))=β(ρb(ξ,x))sε,

or, ρb(ξ,x)β1(sε),

and since β is continuous, strictly increasing and onto, then, β1 is also increasing and continuous with β1 (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 : XX 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 point equation (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 : XX 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 point equation (43) is generalized Ulam-Hyers stable.

(b) Ifn} 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 : XX is a mapping such that there exists η > 0 with ρb(Tξ,Gξ)η,andρb(Gξ,Tξ)ηξX,(50)

then for any fixed point y* of G we have ρb(x,y)β1(sη).(51)

(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→∞ ρbn, 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 ρb(ξn,x)s[ρb(ξn,Tξn)+ρb(Tξn,x)]=s[ρb(ξn,Tξn)+ρb(Tξn,Tx)]s[ρ(ξn,Tξn)+,φb(ρb(ξn,x))],

for all n ∈ℕ. Thus, we have, β(ρb(ξn,x))=ρb(ξn,x)sφb(ρb(ξn,x))sρb(ξn,Tξn)

for all n ∈ℕ. Since limn→∞ ρbn, T ξn)= 0, we obtain limnβ(ρb(ξn,x))=0.

Therefore limnρb(ξn,x)=0.(52)

On the other hand, the contractive condition with ξ = x* and η = ξn and triangle inequality yields ρb(x,ξn)s[ρb(x,Tξn)+ρb(Tξn,ξn)]=s[ρb(Tx,Tξn)+ρb(Tξn,ξn)]s[φb(ρb(x,ξn))+ρb(Tξn,ξn)],

for all n ∈ ℕ. Thus, β(ρb(x,ξn))=ρb(x,ξn)sφb(ρb(x,ξn))sρb(Tξn,ξn),

for all and regarding limn→∞ ρb(T ξn, ξn) = 0 , we get limnβ(ρb(x,ξn))=0.

Consequently, limnρb(x,ξn)=0.(53)

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 ρb(Tξ,Gξ)η,andρb(Gξ,Tξ)ηξX,(54)

and let y * be a fixed point of G. Then, by the triangle inequality, we have ρb(x,y)=ρb(Tx,Gy)s[ρb(Tx,Ty)+ρb(Ty,Gy)]s[φb(ρb(x,y))+ρb(Ty,Gy)].

Then we have β(ρb(x,y))=ρb(xy)sφb(ρb(x,y))sρb(Ty,Gy)sη,

and hence ρb(x,y)β1(sη).(55)

In a similar way, by changing the roles of x* and y * in the discussion above we obtain ρb(y,x)β1(sη),(56)

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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Received: 2016-08-08

Accepted: 2016-10-10

Published Online: 2016-12-28

Published in Print: 2016-01-01


Citation Information: Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2016-0097.

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© 2016 Alsulami et al., published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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