**Contents**


## **About the Special Issue Editors**

**Andreea Fulgais** a lecturer at the Department of Mathematics and Computer Science at Transylvania University in Brasov, Romania. She received her Ph.D. in mathematics (2007) from Transylvania University of Brasov, with the thesis "Limit Theorems, Convergence Problems and Applications". Her research interests include functional analysis, operator theory, and fixed-point theory. She is the author of five books and over 50 publications in journals, book chapters, or conference proceedings and acts as a reviewer for various journals. She is a member of the Mathematical Sciences Society of Romania, has won certain national competitions, and has participated as a member in some research projects.

**Erdal Karapinar** is currently a visiting professor at China Medical University, Taichung, Taiwan. He received his Ph.D. in mathematics, in particular, functional analysis, at Middle East Technical University in 2014 under the supervision of Professor V. P. Zakharyuta (Sabancı University) and Prof. M. H. Yurdakul (Middle East Technical University). He was a post-doc researcher at Sabanci University from 2014 to 2015. After his military service, he worked in Izmir University of Economics, from 2015 to 2017. Later, he worked at Atilim University from 2017 to 2019. He has been a full professor in mathematics since December 2011. From January 2019 to the present, he has been a visiting professor at China Medical University. He has published more than 350 papers and a monograph. He is currently acting as an associate editor of more than 10 journals. He is also a founder and Editor-in-Chief of two journals, *Advances in the Theory of Nonlinear Analysis and its Applications* (*ATNAA*) and *Results in Nonlinear Analysis* (*RNA*). He is a highly cited researcher in mathematics, according to the Clarivate Analysis, from 2014 to 2019. He was also selected as a top reviewer in mathematics by *Publons* (which belongs to Clarivate Analysis).

#### **Preface to "Recent Advances on Quasi-Metric Spaces"**

If we were to say that fixed-point theory appeared in Liouville's article on solutions of differential equations (1837) in the second quarter of the 18th century, it would not be wrong. This approach was further developed by Picard in 1890 and entered the literature as a method of successive approximations. This method was abstracted and extracted as a separate fixed-point theorem in the setting of complete normed space by Banach in 1922.

For this reason, usually, it is said that fixed-point theory was founded by Banach. In its earlier iteration, this first fixed-point theorem was known as the Picard–Banach theorem. Later, the analog of that theorem was proved in the framework of complete metric spaces by Caccioppoli in 1931. In some literature, the Banach–Caccioppoli theorem is indicated as a first fixed-point theorem in the setting of a complete metric space.

As we mentioned above, fixed-point theory can be considered as a theory that was derived from applied mathematics. On the other hand, the techniques belong to functional analysis and topology.

In particular, this theory, and its potential application, has been investigated and focused on by a grea<sup>t</sup> number of researchers. It should be underlined that this theory has been applied in physics, economics, engineering, computer science, and so on. Indeed, an application for fixed-point theorem can be found in all fields of quantitative science.

In this Special Issue, we focused on fixed-point results in the setting of quasi-metric spaces and applications but were not restricted to it. The selected papers express our aims in this regard.

> **Andreea Fulga, Erdal Karapinar** *Special Issue Editors*

### *Article* **A Proposal for Revisiting Banach and Caristi Type Theorems in** *b***-Metric Spaces**

#### **Erdal Karapınar 1,\*, Farshid Khojasteh 2 and Zoran D. Mitrovi´c 3**


Received: 26 February 2019; Accepted: 20 March 2019; Published: 27 March 2019

**Abstract:** In this paper, we revisit the renowned fixed point theorems belongs to Caristi and Banach. We propose a new fixed point theorem which is inspired from both Caristi and Banach. We also consider an example to illustrate our result.

**Keywords:** *b*-metric; Banach fixed point theorem; Caristi fixed point theorem

**MSC:** 46T99; 47H10; 54H25

#### **1. Introduction and Preliminaries**

In fixed point theory, the approaches of the renowned results of Caristi [1] and Banach [2] are quite different and the structures of the corresponding proofs varies. In this short note, we propose a new fixed point theorem that is inspired from these two famous results.

We aim to present our results in the largest framework, *b*-metric space, instead of standard metric space. The concept of *b*-metric has been discovered several times by different authors with distinct names, such as quasi-metric, generalized metric and so on. On the other hand, this concept became popular after the interesting papers of Bakhtin [3] and Czerwik [4]. For more details in b-metric space and advances in fixed point theory in the setting of *b*-metric spaces, we refer e.g., [5–17].

**Definition 1.** *Let X be a nonempty set and s* ≥ 1 *be a real number. We say that d* : *X* × *X* → [0, 1) *is a b-metric with coefficient s when, for each x*, *y*, *z* ∈ *X,*

*(b1) d*(*<sup>x</sup>*, *y*) = *d*(*y*, *x*)*;*


*In this case, the triple* (*<sup>X</sup>*, *d*,*<sup>s</sup>*) *is called a b-metric space with coefficient s.*

The classical examples and crucial examples of *b*-metric spaces are *lp*(R) and *Lp*[0, 1], *p* ∈ (0, <sup>1</sup>). The topological notions (such as, convergence, Cauchy criteria, completeness, and so on) are defined by verbatim of the corresponding notions for standard metric. On the other hand, we should underlinethefactthat*b*-metricdoesneedtobecontinuous,forcertaindetails,seee.g.,[3,4].

 We recollect the following basic observations here. **Lemma 1.** *[14] For a sequence* (*<sup>θ</sup>n*)*n*∈<sup>N</sup> *in a b-metric space* (*<sup>X</sup>*, *d*,*<sup>s</sup>*)*, there exists a constant γ* ∈ [0, 1) *such that*

$$d(\theta\_{n+1}, \theta\_n) \le \gamma d(\theta\_{n}, \theta\_{n-1}), \text{ for all } n \in \mathbb{N}.$$

*Then, the sequence* (*<sup>θ</sup>n*)*n*∈<sup>N</sup> *is fundamental (Cauchy).*

The aim of this paper is to correlate the Banach type fixed point result with Caristi type fixed point results in *b*-metric spaces.

#### **2. Main Result**

**Theorem 1.** *Let* (*<sup>X</sup>*, *d*,*<sup>s</sup>*) *be a complete metric space and T* : *X* → *X be a map. Suppose that there exists a function ϕ* : *X* → R *with*


*Then, T has at least one fixed point in X.*

**Proof.** Let *θ*0 ∈ *X*. If *Tθ*0 = *θ*0, the proof is completed. Herewith, we assume *d*(*<sup>θ</sup>*0, *<sup>T</sup>θ*0) > 0. Without loss of generality, keeping the same argumen<sup>t</sup> in mind, we assume that *θn*+<sup>1</sup> = *Tθn* and hence

$$d(\theta\_n, \theta\_{n+1}) = d(\theta\_n, T\theta\_n) > 0. \tag{1}$$

For that sake of convenience, suppose that *an* = *d*(*<sup>θ</sup><sup>n</sup>*, *<sup>θ</sup><sup>n</sup>*−<sup>1</sup>). From (ii), we derive that

$$\begin{array}{rcl} a\_{\mathfrak{n}+1} &=& d(\theta\_{\mathfrak{n}\prime}\theta\_{\mathfrak{n}+1}) = d(T\theta\_{\mathfrak{n}-1\prime}T\theta\_{\mathfrak{n}})\\ &\leq& (\mathfrak{q}(\theta\_{\mathfrak{n}-1}) - \mathfrak{q}(T\theta\_{\mathfrak{n}-1}))d(\theta\_{\mathfrak{n}-1\prime}\theta\_{\mathfrak{n}})\\ &=& (\mathfrak{q}(\theta\_{\mathfrak{n}-1}) - \mathfrak{q}(\theta\_{\mathfrak{n}}))a\_{\mathfrak{n}}.\end{array}$$

So we have,

$$0 < \frac{a\_{n+1}}{a\_n} \le \varphi(\theta\_{n-1}) - \varphi(\theta\_n) \text{ for each } n \in \mathbb{N}.$$

Thus the sequence {*ϕ*(*<sup>θ</sup>n*)} is necessarily positive and non-increasing. Hence, it converges to some *r* ≥ 0. On the other hand, for each *n* ∈ N, we have

$$\begin{aligned} \sum\_{k=1}^{n} \frac{a\_{k+1}}{a\_k} &\le \sum\_{k=1}^{n} \left( \varphi(\theta\_{k-1}) - \varphi(\theta\_k) \right) \\ &= \left( \varphi(\theta\_0) - \varphi(\theta\_1) \right) + \left( \varphi(\theta\_1) - \varphi(\theta\_2) \right) + \dots + \left( \varphi(\theta\_{n-1}) - \varphi(\theta\_n) \right) \\ &= \left( \varphi(\theta\_0) - \varphi(\theta\_n) \to \varphi(\theta\_0) - r < \infty, \text{ as } n \to \infty. \end{aligned}$$

It means that

$$\sum\_{n=1}^{\infty} \frac{a\_{n+1}}{a\_n} < \infty.$$

Accordingly, we have

$$\lim\_{n \to \infty} \frac{a\_{n+1}}{a\_n} = 0.\tag{2}$$

On account of (2), for *γ* ∈ (0, <sup>1</sup>), there exists *n*0 ∈ N such that

$$\frac{a\_{n+1}}{a\_n} \le \gamma\_{\prime} \tag{3}$$

for all *n* ≥ *n*0. It yields that

$$d(\theta\_{n+1}, \theta\_n) \le \gamma d(\theta\_n, \theta\_{n-1}),\tag{4}$$

for all *n* ≥ *n*0. Now using Lemma 1 we obtain that the sequence {*<sup>θ</sup>n*} converges to some *ω* ∈ *X*. We claim that *ω* is the fixed point of *T*. Employing assumption (ii) of the theorem, we find that

$$\begin{aligned} d(\omega, T\omega) &\leq \quad \text{s} [d(\omega, \theta\_{n+1}) + d(\theta\_{n+1}, T\omega)] \\ &\leq \quad \text{s} [d(\omega, \theta\_{n+1}) + (\varrho(\theta\_n) - \varrho(\omega))d(\theta\_n, \omega)] \to 0 \text{ as } n \to \infty. \end{aligned}$$

Consequently, we obtain *d*(*<sup>ω</sup>*, *<sup>T</sup>ω*) = 0, that is, *Tω* = *ω*.

From Theorem 1, we ge<sup>t</sup> the corresponding result for complete metric spaces. The following example shows that the Theorem 1 is not a consequence of Banach's contraction principle.

**Example 1.** *Let X* = {0, 1, 2} *endowed with the following metric:*

$$d(0,1) = 1,\\d(2,0) = 1,\\d(1,2) = \frac{3}{2} \text{ and } d(a,a) = 0,\\\text{ for all } a \in \mathcal{X}, \ d(a,b) = d(b,a),\\\text{ for all } a, b \in \mathcal{X}.$$

*Let T*(0) = 0, *T*(1) = 2, *T*(2) = 0*. Define ϕ* : *X* → [0, ∞) *as ϕ*(2) = 2, *ϕ*(0) = 0, *ϕ*(1) = 4*. Thus for all x* ∈ *X such that d*(*<sup>x</sup>*, *Tx*) > 0, *(in this example, x* = 0*), we have*

> *d*(*T*1, *T*2) ≤ (*ϕ*(1) − *ϕ*(*T*(1)))*d*(2, <sup>1</sup>), *d*(*T*2, *T*1) ≤ (*ϕ*(2) − *ϕ*(*T*(2)))*d*(2, <sup>1</sup>), *d*(*T*1, *T*0) ≤ (*ϕ*(1) − *ϕ*(*T*(1)))*d*(1, <sup>0</sup>), *d*(*T*2, *T*0) ≤ (*ϕ*(2) − *ϕ*(*T*(2)))*d*(2, <sup>0</sup>).

*Thus the mapping T satisfies our condition and also has a fixed point. Note that d*(*T*1, *T*0) = *d*(1, <sup>0</sup>)*. Thus, it does not satisfy the Banach contraction principle.*
