*Article* **Common Fixed Point Results for Almost** *Rg***-Geraghty Type Contraction Mappings in** *b***2-Metric Spaces with an Application to Integral Equations**

**Samera M. Saleh 1, Salvatore Sessa 2,\*, Waleed M. Alfaqih <sup>3</sup> and Fawzia Shaddad <sup>4</sup>**


**Abstract:** In this paper, we define almost R*g*-Geraghty type contractions and utilize the same to establish some coincidence and common fixed point results in the setting of *b*2-metric spaces endowed with binary relations. As consequences of our newly proved results, we deduce some coincidence and common fixed point results for almost *g*-*α*-*η* Geraghty type contraction mappings in *b*2-metric spaces. In addition, we derive some coincidence and common fixed point results in partially ordered *b*2-metric spaces. Moreover, to show the utility of our main results, we provide an example and an application to non-linear integral equations.

**Keywords:** *b*2-metric space; fixed point; binary relation; almost R*g*-Geraghty type contraction

**MSC:** 47H10; 54H25

### **1. Introduction**

The extension of fixed point theory to generalized structures, such as cone metric spaces, partial metric spaces, *b*-metric spaces and 2-metric spaces has received much attention. 2-metric space is a generalized metric space which was introduced by Gähler in [1]. Unlike the ordinary metric, the 2-metric is not a continuous function. The topology induced by 2-metric space is called 2-metric topology which is generated by the set of all open spheres with two centers. It is easy to observe that 2-metric space is not topologically equivalent to an ordinary metric. Hence, there is not any relationship between the results obtained in 2-metric spaces and the correspondence results in metric spaces. For fixed point results in the setting of 2-metric spaces, the readers may refer to [2–5] and references therein.

The concept of *b*-metric spaces was introduced by Czerwik [6,7] which is a generalization of the usual metric spaces and 2-metric spaces as well. Several papers have dealt with fixed point theory for single-valued and multi-valued operators in *b*-metric spaces have been obtained (see, e.g., [8–10]).

In 2014, Mustafa et al. [11] introduced the notion of *b*2-metric spaces, as a generalization of both 2-metric and *b*-metric spaces.

On the other hand, the branch of related metric (metric space endowed with a binary relation) fixed point theory is a relatively new area was initiated by Turinici [12]. Recently, this direction of research is undertaken by several researchers such as: Bhaskar and Lakshmikantham [13], Samet and Turinici [14], Ben-El-Mechaiekh [15], Imdad et al. [16,17] and some others.

The aims of this paper are as follows:

• to define almost R*g*-Geraghty type contractions;

**Citation:** Saleh, S.M.; Sessa, S.; Alfaqih, W.M.; Shaddad, F. Common Fixed Point Results for Almost R*g*-Geraghty Type Contraction Mappings in *b*2-Metric Spaces with an Application to Integral Equations. *Axioms* **2021**, *10*, 101. https://doi. org/10.3390/axioms10020101

Academic Editor: Erdal Karapinar

Received: 19 March 2021 Accepted: 17 May 2021 Published: 24 May 2021


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<sup>1</sup> Department of Mathematics, Faculty of Science, Taiz University, Taiz 6803, Yemen; samirasaleh2007@yahoo.com


#### **2. Preliminaries**

**Definition 1** ([11])**.** *Let <sup>X</sup> be a non-empty set, <sup>s</sup>* <sup>≥</sup> <sup>1</sup> *a given real number and <sup>d</sup>* : *<sup>X</sup>*<sup>3</sup> <sup>→</sup> <sup>R</sup> *be a map satisfying the following conditions:*


*Then d is called a b*2*-metric on X and* (*X*, *d*) *is called a b*2*-metric space with parameter s*.

Obviously, for *s* = 1, *b*2-metric reduces to 2-metric.

**Example 1.** *Let* (*X*, *<sup>d</sup>*) *be a* <sup>2</sup>*-metric space and <sup>ρ</sup>*(*x*, *<sup>y</sup>*, *<sup>w</sup>*)=(*d*(*x*, *<sup>y</sup>*, *<sup>w</sup>*))*p*, *where <sup>p</sup>* <sup>≥</sup> <sup>1</sup> *is a real number. We see that ρ is a b*2*-metric with s* = 3*p*−1. *In view of the convexity of f*(*x*) = *xp*, *on* [0, ∞) *for p* ≥ 1 *and Jensen inequality, we have*

$$(a+b+c)^p \le \mathfrak{P}^{p-1}(a^p+b^p+c^p).$$

*Therefore, condition (iv) of Definition 1 is satisfied and ρ is a b*2*-metric on X*.

**Definition 2** ([11])**.** *Let* {*xn*} *be a sequence in a b*2*-metric space* (*X*, *d*). *Then*


**Definition 3** ([11])**.** *Let* (*X*, *<sup>d</sup>*) *and* (*X*¯ , ¯*d*) *be two <sup>b</sup>*2*-metric spaces and let <sup>f</sup>* : *<sup>X</sup>* <sup>→</sup> *<sup>X</sup>*¯ *be a mapping. Then f is said to be b*2*-continuous at a point z* ∈ *X if for a given ε* > 0, *there exists <sup>δ</sup>* <sup>&</sup>gt; <sup>0</sup> *such that x* <sup>∈</sup> *X and d*(*z*, *<sup>x</sup>*, *<sup>a</sup>*) <sup>&</sup>lt; *<sup>δ</sup> for all a* <sup>∈</sup> *X imply that* ¯*d*(*f z*, *f x*, *<sup>a</sup>*) <sup>&</sup>lt; *<sup>ε</sup>*. *The mapping f is b*2*-continuous on X if it is b*2*-continuous at all z* ∈ *X*.

**Proposition 1** ([11])**.** *Let* (*X*, *<sup>d</sup>*) *and* (*X*¯ , ¯*d*) *be two <sup>b</sup>*2*-metric spaces. Then a mapping <sup>f</sup>* : *<sup>X</sup>* <sup>→</sup> *<sup>X</sup>*¯ *is b*2*-continuous at a point x* ∈ *X if it is b*2*-sequentially continuous at x*, *that is, whenever* {*xn*} *is b*2*-convergent to x,* { *f*(*xn*)} *is b*2*-convergent to f*(*x*).

**Lemma 1** ([11])**.** *Let* (*X*, *d*) *be a b*2*-metric space. Suppose that* {*xn*} *and* {*yn*} *are b*2*-converge to x and y, respectively. Then, we have*

$$\frac{1}{s^2}d(\mathbf{x}, y, a) \le \liminf\_{n \to \infty} d(\mathbf{x}\_n, y\_n, a) \le \limsup\_{n \to \infty} d(\mathbf{x}\_n, y\_n, a) \le s^2 d(\mathbf{x}, y, a) \qquad \text{for all } a \in \mathcal{X}.$$

*In particular, if yn* = *y*, *is constant, then*

$$\frac{1}{n}d(\mathbf{x}, y, a) \le \liminf\_{n \to \infty} d(\mathbf{x}\_n, y, a) \le \limsup\_{n \to \infty} d(\mathbf{x}\_n, y, a) \le \text{sd}(\mathbf{x}, y, a) \qquad \text{for all } a \in \mathcal{X}.$$

**Definition 4.** *Let f and g be two self mappings on a non-empty set X. If w* = *f x* = *gx for some x* ∈ *X*, *then x is called a coincidence point of f and g and w is called a point of coincidence of f and g.*

**Definition 5** ([18])**.** *Two self mappings f and g are said to be weakly compatible if they commute at their coincidence points, that is, f x* = *gx implies that f gx* = *gfx*.

**Lemma 2** ([19])**.** *Let f and g be weakly compatible self mappings of a non-empty set X. If f and g have a unique point of coincidence w* = *f x* = *gx*, *then w is the unique common fixed point of f and g*.

A non-empty subset R of *X* × *X* is said to be a binary relation on *X*. Trivially, *X* × *X* is a binary relation on *X* known as the universal relation. For simplicity, we will write *x*R*y* whenever (*x*, *<sup>y</sup>*) ∈ R and write *<sup>x</sup>*R*<sup>y</sup>* whenever *<sup>x</sup>*R*<sup>y</sup>* and *<sup>x</sup>* <sup>=</sup> *<sup>y</sup>*. Observe that <sup>R</sup> is also a binary relation on *<sup>X</sup>* and <sup>R</sup> ⊆ R. The elements *<sup>x</sup>* and *<sup>y</sup>* of *<sup>X</sup>* are said to be <sup>R</sup>-comparable if *x*R*y* or *y*R*x*, this is denoted by [*x*, *y*] ∈ R.

**Definition 6.** *A binary relation* R *on X is said to be:*


Let *X* be a nonempty set, R a binary relation on *X* and *Y* ⊆ *X*. Then the restriction of <sup>R</sup> to *<sup>Y</sup>* is denoted by R|*<sup>Y</sup>* and is defined by R ∩ *<sup>Y</sup>*2. The inverse of <sup>R</sup> is denoted by <sup>R</sup>−<sup>1</sup> and is defined by <sup>R</sup>−<sup>1</sup> <sup>=</sup> {(*x*, *<sup>y</sup>*) <sup>∈</sup> *<sup>X</sup>* <sup>×</sup> *<sup>X</sup>* : (*y*, *<sup>x</sup>*) ∈ R} and <sup>R</sup>*<sup>s</sup>* <sup>=</sup> R∪R<sup>−</sup>1.

**Definition 7** ([20])**.** *Let X be a non-empty set and* R *a binary relation on X*. *A sequence* {*xn*} ⊆ *X is said to be an* R*-preserving sequence if xn*R*xn*+<sup>1</sup> *for all n* ∈ N0.

**Definition 8** ([20])**.** *Let X be a non-empty set and f* : *X* → *X*. *A binary relation* R *on X is said to be f -closed if for all x*, *y* ∈ *X*, *x*R*y implies f x*R *f y*.

**Definition 9** ([20])**.** *Let X be a non-empty set and f* , *g* : *X* → *X*. *A binary relation* R *on X is said to be* (*f* , *g*)*-closed if for all x*, *y* ∈ *X*, *gx*R*gy implies f x*R *f y*.

**Definition 10** ([20])**.** *Let* (*X*, *d*) *be a metric space and* R *a binary relation on X*. *We say that X is* R*-complete if every* R*-preserving Cauchy sequence in X converges to a limit in X*.

**Remark 1.** *Every complete metric space is* R*-complete, whatever the binary relation* R. *Particularly, under the universal relation, the notion of* R*-completeness coincides with the usual completeness.*

**Definition 11** ([21])**.** *Let* (*X*, *d*) *be a metric space,* R *a binary relation on X*, *f* : *X* → *X and x* ∈ *X*. *We say that f is* R*-continuous at x if, for any* R*-preserving sequence* {*xn*} ⊆ *X such that xn* → *x*, *we have f xn* → *f x*. *Moreover, f is called* R*-continuous if it is* R*-continuous at each point of X.*

**Remark 2.** *Every continuous mapping is* R*-continuous, whatever the binary relation* R. *Particularly, under the universal relation, the notion of* R*-continuity coincides with the usual continuity.*

**Definition 12** ([21])**.** *Let* (*X*, *d*) *be a metric space,* R *a binary relation on X, f* , *g* : *X* → *X and x* ∈ *X*. *We say that f is* (*g*, R)*-continuous at x if, for any sequence* {*xn*} ⊆ *M such that* {*gxn*} *is* R*-preserving and gxn* → *gx*, *we have f xn* → *f x*. *Moreover, f is called* (*g*, R)*-continuous if it is* (*g*, R)*-continuous at each point of X*.

Observe that on setting *g* = *I*, Definition 12 reduces to Definition 11.

**Remark 3.** *Every g-continuous mapping is* (*g*, R)*-continuous, whatever the binary relation* R. *Particularly, under the universal relation, the notion of* (*g*, R)*-continuity coincides with the usual g-continuity.*

**Definition 13** ([21])**.** *Let* (*X*, *d*) *be a metric space,* R *be a binary relation on X and f* , *g* : *X* → *X*. *We say that the pair* (*f* , *g*) *is* R*-compatible if for any sequence* {*xn*} ⊆ *X such that* { *f xn*} *and* {*gxn*} *are* R*-preserving and* lim*n*→<sup>∞</sup> *gxn* = lim*n*→<sup>∞</sup> *f xn* = *x* ∈ *X*, *we have* lim*n*→<sup>∞</sup> *d*(*gfxn*, *f gxn*) = 0.

**Remark 4.** *Every compatible pair is* R*-compatible, whatever the binary relation* R. *Particularly, under the universal relation, the notion of* R*-compatibility coincides with the usual compatibility.*

**Definition 14** ([20])**.** *Let* (*X*, *d*) *be a metric space. A binary relation* R *on X is said to be d-selfclosed if for any* R*-preserving sequence* {*xn*} ⊆ *X such that xn* → *x*, *there exists a subsequence* {*xnk*} *of* {*xn*} *such that* [*xnk* , *x*] ∈ R *for all k* ∈ N0.
