**4. Results for Almost** *g***-***α***-***η* **Geraghty Type Contraction Mappings in** *b***2-Metric Spaces**

Fathollahi et al. [4] introduced the concepts of triangular 2-*α*-*η*-admissible mappings as follows.

**Definition 17** ([4])**.** *Let* (*X*, *<sup>d</sup>*) *be a* <sup>2</sup>*-metric space, <sup>f</sup>* : *<sup>X</sup>* <sup>→</sup> *<sup>X</sup> and <sup>α</sup>*, *<sup>η</sup>* : *<sup>X</sup>*<sup>3</sup> <sup>→</sup> [0, <sup>∞</sup>)*. We say that f is a triangular* 2*-α-η-admissible mapping if for all a* ∈ *X*,


*If we take η*(*x*, *y*, *a*) = 1, *then we say that f is a triangular* 2*-α-admissible mapping. In addition, if we take α*(*x*, *y*, *a*) = 1, *then we say that f is a triangular* 2*-η-subadmissible mapping.*

Motivated by Fathollahi [4], we define the following concepts.

**Definition 18.** *Let* (*X*, *<sup>d</sup>*) *be a <sup>b</sup>*2*-metric space, <sup>f</sup>* , *<sup>g</sup>* : *<sup>X</sup>* <sup>→</sup> *<sup>X</sup> and <sup>α</sup>*, *<sup>η</sup>* : *<sup>X</sup>*<sup>3</sup> <sup>→</sup> [0, <sup>∞</sup>)*. We say that f is a triangular g-b*2*-α-η-admissible mapping if for all a* ∈ *X*,

*(i) α*(*gx*, *gy*, *a*) ≥ *η*(*gx*, *gy*, *a*) *implies α*(*f x*, *f y*, *a*) ≥ *η*(*f x*, *f y*, *a*), *x*, *y* ∈ *X*,

$$\begin{array}{llll} \text{(ii)} & \begin{cases} \text{a}(\text{gx}, \text{gy}, a) \ge \eta(\text{gx}, \text{gy}, a), \\ \text{a}(\text{gy}, \text{gz}, a) \ge \eta(\text{gy}, \text{gz}, a), \end{cases} & \text{implies } \text{a}(\text{gx}, \text{gz}, a) \ge \eta(\text{gx}, \text{gz}, a), \quad \text{x}, \text{y}, z \in X. \end{array}$$

*When η*(*gx*, *gy*, *a*) = 1, *we say that f is a triangular g-b*2*-α-admissible mapping. In addition, when α*(*gx*, *gy*, *a*) = 1, *we say that f is a triangular g-b*2*-η-subadmissible mapping.*

**Definition 19.** *Let* (*X*, *<sup>d</sup>*) *be a <sup>b</sup>*2*-metric space with <sup>s</sup>* <sup>≥</sup> <sup>1</sup> *and <sup>f</sup>* , *<sup>g</sup>* : *<sup>X</sup>* <sup>→</sup> *<sup>X</sup>*, *<sup>α</sup>*, *<sup>η</sup>* : *<sup>X</sup>*<sup>3</sup> <sup>→</sup> [0, <sup>∞</sup>)*. Suppose for all x*, *y*, *a* ∈ *X*,

$$M(\mathbf{x}, y, a) = \max\left\{d(\mathbf{g}\mathbf{x}, \mathbf{g}y, a), d(\mathbf{g}\mathbf{x}, f\mathbf{x}, a), d(\mathbf{g}y, fy, a), \frac{d(\mathbf{g}\mathbf{x}, fy, a) + d(\mathbf{g}y, f\mathbf{x}, a)}{2\mathbf{s}}\right\},$$

*and*

$$N(\mathbf{x}, y, a) = \min\{d(\mathbf{g}\mathbf{x}, f\mathbf{x}, a), d(\mathbf{g}y, fy, a), d(\mathbf{g}\mathbf{x}, fy, a), d(\mathbf{g}y, f\mathbf{x}, a)\}.$$

*We say that f is almost g-α-η Geraghty type contraction mapping if there exist L* ≥ 0 *and β<sup>s</sup>* ∈ Ω *such that*

$$\begin{aligned} \forall \mathbf{x}, \mathbf{y} \in X, \; a(\operatorname{gx}\_{\mathbf{}} \operatorname{gy}\_{\mathbf{}} a) \ge \eta(\operatorname{gx}\_{\mathbf{}} \operatorname{gy}\_{\mathbf{}} a) \\ \Rightarrow d(f \mathbf{x}, f \mathbf{y}, a) \le \beta\_{\mathbf{s}}(M(\mathbf{x}, \mathbf{y}, a))M(\mathbf{x}, \mathbf{y}, a) + LN(\mathbf{x}, \mathbf{y}, a), \end{aligned} \tag{36}$$

*for all a* ∈ *X*.

Now, we state the following corollaries

**Corollary 2.** *Let* (*X*, *d*) *be a complete b*2*-metric space and f* , *g* : *X* → *X*, *such that f*(*X*) ⊆ *g*(*X*), *g*(*X*) *is a b*2*-complete subspace of X*. *Assume that f is almost g-α-η Geraghty type contraction mapping and the following conditions hold:*


*Then f and g have a coincidence point in X*. *Moreover, suppose that for all coincidence points u*, *v of f and g*, *there exists w* ∈ *X such that α*(*gu*, *gw*, *a*) ≥ *η*(*gu*, *gw*, *a*) *and α*(*gv*, *gw*, *a*) ≥ *η*(*gv*, *gw*, *a*) *for all a* ∈ *X and f* , *g are weakly compatible. Then f and g have a unique common fixed point.*

**Proof.** Define R on *X* as

$$
\pi \mathcal{R} y \Longleftrightarrow \mathfrak{a}(\mathfrak{x}, y, a) \ge \eta(\mathfrak{x}, y, a).
$$

We note the following:


Finally, if for all coincidence points *u*, *v* of *f* and *g*, there exists *w* ∈ *X* such that *α*(*gu*, *gw*, *a*) ≥ *η*(*gu*, *gw*, *a*) and *α*(*gv*, *gw*, *a*) ≥ *η*(*gv*, *gw*, *a*), then *gu*R*gw* and *gv*R*gw*. That is, all hypotheses of Theorem 1 are satisfied. Therefore, *f* and *g* have a unique common fixed point.

By taking *g* = *I* in Definitions 18 and 19, we say that *f* is a triangular *b*2-*α*-*η*-admissible mapping and *f* is almost *α*-*η* Geraghty type contraction mapping.

Now, we have the following corollary.

**Corollary 3.** *Let* (*X*, *d*) *be a complete b*2*-metric space and f* : *X* → *X*. *Assume that f is almost α-η Geraghty type contraction mapping and the following conditions hold:*


*Then f has a fixed point in X*. *Moreover, if for u*, *v* ∈ *Fix*(*f*) *there exists w* ∈ *X such that α*(*u*, *w*, *a*) ≥ *η*(*u*, *w*, *a*) *and α*(*v*, *w*, *a*) ≥ *η*(*v*, *w*, *a*) *for all a* ∈ *X, then f has a unique fixed point.*

### **5. Fixed Point Results in Partially Ordered** *b***2-Metric Spaces**

Fixed point theorems for monotone operators in ordered metric spaces are widely investigated and have found various applications in differential and integral equations. This trend was started by Turinici [12] in 1986. Ran and Reurings in [24] extended the Banach contraction principle in partially ordered sets with some applications to matrix equations. The obtained result in [24] was extended and refined by many authors (see, e.g., [25–27] and references therein). The aim of this section is to deduce our results in the context of partially ordered *b*2-metric spaces. At first, we need to recall some concepts. Let *X* be a nonempty set. Then (*X*, , *d*) is called a partially ordered *b*2-metric space with *s* ≥ 1 if (*X*, *d*) is a *b*2-metric space and (*X*, ) is a partially ordered set.

**Definition 20.** *Let* (*X*, ) *be a partially ordered set and x*, *y* ∈ *X. Then x and y are called comparable if x y or y x holds.*

**Definition 21.** *Let* (*X*, ) *be a partially ordered set. A mapping f on X is said to be monotone non-decreasing if for all x*, *y* ∈ *X*, *x y implies f x f y*.

**Definition 22.** *Let* (*X*, ) *be a partially ordered set and f* , *g* : *X* → *X. One says f is g-nondecreasing if for x*, *y* ∈ *X*,

$$
\lg(\mathbf{x}) \preceq \lg(\mathbf{y}) \quad \text{implies} \quad f(\mathbf{x}) \preceq f(\mathbf{y}) .
$$

By putting R = in Theorems 1 and 2, we get the following results.

**Corollary 4.** *Let* (*X*, *d*, ) *be a complete partially ordered b*2*-metric space. Assume that f* , *g* : *X* → *X*, *are two mappings such that f*(*X*) ⊆ *g*(*X*), *g*(*X*) *is a b*2*-complete subspace of X and f is a g-non-decreasing mapping. Suppose that there exists a function β<sup>s</sup>* ∈ Ω *and L* ≥ 0 *such that*

$$d(f\mathbf{x}, f\mathbf{y}, a) \le \beta\_s(M(\mathbf{x}, \mathbf{y}, a))M(\mathbf{x}, \mathbf{y}, a) + LN(\mathbf{x}, \mathbf{y}, a),\tag{37}$$

*where*

$$M(\mathbf{x}, y, a) = \max \left\{ d(\mathbf{g}\mathbf{x}, \mathbf{g}y, a), d(\mathbf{g}\mathbf{x}, f\mathbf{x}, a), d(\mathbf{g}y, fy, a), \frac{d(\mathbf{g}\mathbf{x}, f\mathbf{y}, a) + d(\mathbf{g}y, f\mathbf{x}, a)}{2\mathbf{s}} \right\},$$

*and*

$$N(\mathbf{x}, y, a) = \min\{d(\mathbf{g}\mathbf{x}, f\mathbf{x}, a), d(\mathbf{g}y, fy, a), d(\mathbf{g}\mathbf{x}, fy, a), d(\mathbf{g}y, f\mathbf{x}, a)\},$$

*for all x*, *y*, *a* ∈ *X with gx gy*. *In addition, suppose that the following conditions hold:*


*Then f and g have a coincidence point in X*. *Moreover, suppose that for all coincidence points u*, *v of f and g*, *there exists w* ∈ *X such that gu gw or gv gw and f* , *g are weakly compatible. Then f and g have a unique common fixed point.*

By taking *g* = *I* in Corollary 4, we obtain the following corollary.

**Corollary 5.** *Let* (*X*, *d*, ) *be a complete partially ordered b*2*-metric space. Assume that f* : *X* → *X is a mapping satisfying the following conditions*


$$d(f\mathbf{x}, f\mathbf{y}, a) \le \beta\_s(M(\mathbf{x}, \mathbf{y}, a))M(\mathbf{x}, \mathbf{y}, a) + LN(\mathbf{x}, \mathbf{y}, a),\tag{38}$$

*where*

$$M(\mathbf{x}, \mathbf{y}, a) = \max \left\{ d(\mathbf{x}, \mathbf{y}, a), d(\mathbf{x}, f \mathbf{x}, a), d(\mathbf{y}, f \mathbf{y}, a), \frac{d(\mathbf{x}, f \mathbf{y}, a) + d(\mathbf{y}, f \mathbf{x}, a)}{2s} \right\},$$

*and*

$$N(\mathbf{x}, y, a) = \min\{d(\mathbf{x}, f\mathbf{x}, a), d(y, fy, a), d(\mathbf{x}, fy, a), d(y, f\mathbf{x}, a)\},$$

*for all x*, *y*, *a* ∈ *X with x y;*


*Then f has a fixed point. Moreover, if u*, *v* ∈ *Fix*(*f*) *such that there exists w* ∈ *X with u w and v w*, *then f has a unique fixed point. Then f has a fixed point. Moreover, if for every pair* (*u*, *v*) *of fixed points of f such that there exists w* ∈ *X with u w and v w*, *then f has a unique fixed point.*

#### **6. Application to Integral Equations**

In this section, we study the existence of a solution for an integral equation using the results proved in Section 3. Let *X* = (*C*[*a*, *b*], *R*) be the space of all real continuous functions on [*a*, *<sup>b</sup>*] and *<sup>ρ</sup>* : *<sup>X</sup>* <sup>×</sup> *<sup>X</sup>* <sup>→</sup> *<sup>R</sup>*<sup>+</sup> defined by

$$\rho(\mathbf{x}, y) = \max\_{t \in [a, b]} |\mathbf{x}(t) - y(t)|, \qquad \forall \mathbf{x}, y \in X.$$

Equip *<sup>X</sup>* with the 2-metric given by *<sup>σ</sup>* : *<sup>X</sup>*<sup>3</sup> <sup>→</sup> *<sup>R</sup>*<sup>+</sup> which is defined by

$$
\sigma(\mathbf{x}, y, a) = \min \{ \rho(\mathbf{x}, y), \rho(y, a), \rho(a, \mathbf{x}) \}, \qquad \forall \mathbf{x}, y, a \in X.
$$

As (*X*, *ρ*) is a complete metric space, (*X*, *σ*) is a complete 2-metric space, according to Example 1, we define a *b*2-metric on *X* by

$$d(\mathfrak{x}, \mathfrak{y}, a) = (\sigma(\mathfrak{x}, \mathfrak{y}, a))^2, \qquad \forall \mathfrak{x}, \mathfrak{y}, a \in X.$$

It follows that (*X*, *d*) is a complete *b*2-metric space with *s* = 3. Define a binary relation R on *X* by

$$\mathcal{R} = \{(\mathbf{x}, y) \in X^2 : \mathbf{x}(t) \le y(t) \text{ for all } t \in [a, \infty)\}. \tag{39}$$

Now, consider the integral equation:

$$\mathbf{x}(t) = q(t) + \int\_{a}^{b} h(t, s) A\left(s, \mathbf{x}(s)\right) ds,\tag{40}$$

where *<sup>t</sup>* <sup>∈</sup> [*a*, *<sup>b</sup>*] <sup>⊆</sup> *<sup>R</sup>*+. A solution of the Equation (40) is a function *<sup>x</sup>* <sup>∈</sup> *<sup>X</sup>* <sup>=</sup> *<sup>C</sup>*[*a*, *<sup>b</sup>*]. Assume that

(i) *h* : [*a*, *b*] × [*a*, *b*] → [0, ∞), *q* : [*a*, *b*] → *R* and *A* : [*a*, *b*] × *R* → *R* are continuous functions on [*a*, *b*];

$$\text{(ii)}\quad \int\_{a}^{b} h(t,s)dt \le r \le 1;$$

(iii) there exists *x*<sup>0</sup> ∈ *X* such that

$$\mathbf{x}\_0(t) \le q(t) + \int\_a^b h(t, \mathbf{s}) A\left(\mathbf{s}, \mathbf{x}\_0(\mathbf{s})\right) d\mathbf{s}.$$

(iv) *A* is nondecreasing in the second variable and for all *x*, *y*, *a* ∈ *X*, *s* ∈ [*a*, *b*] there exists 0 < *k* < <sup>√</sup><sup>1</sup> <sup>3</sup> such that

$$\begin{aligned} &\min \quad \left\{ \left| A\left(s, \mathbf{x}(s)\right) - A\left(s, y(s)\right) \right| \left| A\left(s, \mathbf{x}(s)\right) - a(s) \right| \left| A\left(s, y(s)\right) - a(s) \right| \right\} \\ &\le \quad \left| A\left(s, \mathbf{x}(s)\right) - A\left(s, y(s)\right) \right| \\ &\le \quad \left\| s e^{-M\left(\mathbf{x}, y, a\right)} \min \{ \left| \mathbf{x}(s) - y(s) \right|, \left| \mathbf{x}(s) - a(s) \right|, \left| y(s) - a(s) \right| \}, \end{aligned}$$

where

$$M(\mathbf{x}, y, a) = \max \left\{ d(\mathbf{x}, y, a), d(\mathbf{x}, f \mathbf{x}, a), d(y, fy, a), \frac{d(\mathbf{x}, fy, a) + d(y, f \mathbf{x}, a)}{2s} \right\}.$$

Now, we are equipped to state and prove our main result in this section.

**Theorem 3.** *Under the assumptions (i)–(iv), the integral Equation (40) has a solution in X*.

**Proof.** Define *f* : *X* → *X* by

$$f\mathfrak{x}(t) := \left. \begin{array}{c} q(t) \end{array} \right| + \left. \int\_{a}^{b} h(t,s)A\left(s,\mathfrak{x}(s)\right)ds \right|.$$

Observe that *x* is a solution for (40) if and only if *x* is a fixed point of *f* . Let *x*, *y*, *a* ∈ *X* such that *x*R*y* for all *t* ∈ [*a*, *b*]. Since *A* is nondecreasing in the second variable, we have

$$\begin{aligned} f\mathbf{x}(t) &= \, ^c q(t) + \int\_a^b h(t, \mathbf{s}) A\left(\mathbf{s}, \mathbf{x}(s)\right) ds \\ &\le \, ^c q(t) + \int\_a^b h(t, \mathbf{s}) A\left(\mathbf{s}, \mathbf{y}(s)\right) ds \\ &= \, ^c f\mathbf{y}(t) \end{aligned}$$

Hence, *f x*R *f y* and R is *f*-closed. From Condition (iii), we conclude that *x*<sup>0</sup> ≤ *f x*<sup>0</sup> for all *<sup>t</sup>* <sup>∈</sup> [*a*, *<sup>b</sup>*], then *<sup>x</sup>*0<sup>R</sup> *f x*0. Now, for any *<sup>x</sup>*, *<sup>y</sup>*, *<sup>a</sup>* <sup>∈</sup> *<sup>X</sup>* such that *f x*R *f y* we get


Therefore,

$$
\sigma(f \ge fy, a) \le \max\_{t \in [a, b]} |f \mathbf{x}(t) - fy(t)| \le r k e^{-M(\mathbf{x}, y, a)} \sigma(\mathbf{x}, y, a).
$$

It follows that

$$d(f\mathbf{x}, f\mathbf{y}, a) \le r^2 k^2 e^{-2M(\mathbf{x}, \mathbf{y}, a)} d(\mathbf{x}, \mathbf{y}, a) \le r^2 k^2 e^{-2M(\mathbf{x}, \mathbf{y}, a)} M(\mathbf{x}, \mathbf{y}, a) \le \frac{e^{-2M(\mathbf{x}, \mathbf{y}, a)}}{3} M(\mathbf{x}, \mathbf{y}, a).$$

Thus,

$$d(f\mathbf{x}, f\mathbf{y}, a) \le \frac{e^{-2M(\mathbf{x}, \mathbf{y}, a)}}{3} M(\mathbf{x}, \mathbf{y}, a) + LN(\mathbf{x}, \mathbf{y}, a)\nu$$

where

$$M(\mathbf{x}, \mathbf{y}, a) = \max \left\{ d(\mathbf{x}, \mathbf{y}, a), d(\mathbf{x}, f \mathbf{x}, a), d(\mathbf{y}, f \mathbf{y}, a), \frac{d(\mathbf{x}, f \mathbf{y}, a) + d(\mathbf{y}, f \mathbf{x}, a)}{2 \mathbf{s}} \right\},$$

and

$$N(\mathbf{x}, \mathbf{y}, a) = \min \{ d(\mathbf{x}, f\mathbf{x}, a), d(\mathbf{y}, f\mathbf{y}, a), d(\mathbf{x}, f\mathbf{y}, a), d(\mathbf{y}, f\mathbf{x}, a) \},$$

with *<sup>β</sup>s*(*t*) = *<sup>e</sup>*−2*<sup>t</sup>* <sup>3</sup> and *L* ≥ 0. Then *f* is almost a R-Geraghty type contraction. In addition, if {*xn*} ∈ *X* is an R-preserving sequence such that lim*n*→<sup>∞</sup> *xn* = *x* ∈ *X*, then *xn* ≤ *x* for all *n*. Hence, *xn*R*x*, for all *n*. Therefore, all the hypotheses of Corollary 1 are satisfied. Hence, *f* has a fixed point which is a solution for the integral Equation (40) in *X* = *C*([*a*, *b*], *R*).

**Author Contributions:** All the authors contributed equally and significantly in writing this article. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** All the authors are grateful to the anonymous referees for their excellent suggestions, which greatly improved the presentation of the paper.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


MDPI St. Alban-Anlage 66 4052 Basel Switzerland Tel. +41 61 683 77 34 Fax +41 61 302 89 18 www.mdpi.com

*Axioms* Editorial Office E-mail: axioms@mdpi.com www.mdpi.com/journal/axioms

MDPI St. Alban-Anlage 66 4052 Basel Switzerland

Tel: +41 61 683 77 34 Fax: +41 61 302 89 18

www.mdpi.com