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Extraction new results of common fixed point theorems for \(({T}, {\alpha }_{{s}}, {F})\)-contraction of six mappings in a tripled b-metric space with an application of integral equations

Abstract

The aim of this work is to usher in tripled b-metric spaces, triple weakly \(\alpha _{s}\)-admissible, triangular partially triple weakly \(\alpha _{s}\)-admissible and their properties for the first time. Also, we prove some theorems about coincidence and common fixed point for six self-mappings. On the other hand, we present a new model, talk over an application of our results to establish the existence of common solution of the system of Volterra-type integral equations in a triple b-metric space. Also, we give some example to illustrate our theorems in the section of main results. Finally, we show an application of primary results.

Introduction and preliminaries

The Banach contraction principle plays a central part in metric fixed point theory, and a great number of researchers revealed many fruitful generalizations of this resolution in diverse ways. In 1989, Bakhtin investigated the concept of b-metric space [1]. However, Czerwik initiated the study of fixed point of self-mappings in a b-metric space and proved an analogue of Banach’s fixed point theorem [2]. Since then, numerous research articles have been published comprising fixed point theorems for several classes of single-valued and multi-valued operators in b-metric spaces (for example, consider [36]). In 2012, the concept of F-contraction, which is one of these generalizations, was introduced by Wardowski [7]. He presented that every F-contraction defined in a complete metric space has a unique fixed point. Subsequently, the subject of F-contraction proved to be a milestone in the fixed point theory, and numerous research papers on F-contraction have been published (for instance, see [4, 819]). In the same year, Samet et al. investigated the idea of (\(\alpha , \psi \))-contractive and α-admissible mappings and established some significant fixed point solutions for such a variety of functions defined on a complete metric space (for more details, see [20]). Some authors such as Salimi, Latif, Hussain et al. improved the concept of α-admissibility and proved some important (common) fixed point theorems as well (for more information, see [2124].

Recently, Cosentino and Vetro established a fixed point result for Hardy–Rogers-type F-contraction [25]. Also, Minak, Helvaci, and Altun presented a fixed point result for Ćirić-type generalized F-contraction [26]. In 2018, Nazam, Muhammad, and Postolache investigated some common fixed point results for four self-mappings satisfying such kind of contractions on the \(\alpha _{s}\)-complete b-metric space and applied their conclusion to infer several new and old results, based on the idea of Ćirić-type and Hardy–Rogers-type (\(\alpha _{s}, F \))-contractions [27].

In this study, motivated by [27] and among these achievements, we are working to stretch out the Ćirić-type and Hardy–Rogers-type (\(\alpha _{s}, F \))-contractions based on six self-mappings defined on a b-metric space. Also, some common fixed point results for six self-mappings satisfying such kind of contractions are shown in the (\(T, \alpha _{s}, F \))-complete tripled b-metric space. Consequently, we discuss an application of the main result to show the existence of common solution of the system of Volterra-type integral equations.

Let X be a nonempty set, \(\mathbb{R}^{+} =(0, \infty )\), \(\mathbb{R}_{0}^{+} = [ 0, \infty )\), and \(s >1\) be a real constant. Suppose that \(d_{b}\) maps \(X\times X\times X\) into \(\mathbb{R}_{0}^{+}\) somehow that for all x, y, z, and \(a_{i}\) with \(i\in \{1,2,3,4\}\) belong to X satisfying the following conditions [9]:

  • \(d_{b} ( x, y, z )=0\) if and only if \(x = y = z\).

  • \(d_{b} ( x, y, z )>0\) if and only if \(x\neq y\) or \(x\neq z\) or \(y\neq z\).

  • \(d_{b} ( x, y, z )= d_{b} ( x, z, y )= d_{b} ( z, y, x )= d_{b} ( y, x, z )= d_{b} ( z, x, y )= d_{b} ( y, z, x )\).

  • \(d_{b} ( x, x, y )= d_{b} ( x, y, y )\).

  • \(d_{b} ( x, x, y ) \leq d_{b} ( x, y, z )\), \(d_{b} ( x, x, z ) \leq d_{b} ( x, y, z )\), \(d_{b} ( y, y, z ) \leq d_{b} ( x, y, z )\).

  • \(d_{b} ( x, y, z ) \leq s [ d_{b} ( x, a_{1}, a_{2} ) + d_{b} ( y, a_{3}, a_{4} ) + d_{b} ( z, a_{2}, a_{3} )]\).

We say that (\(X, d_{b}, s \)) is a tripled b-metric space.

Example 1.1

Let \(X= \mathbb{R}_{0}^{+}\). We define \(d_{b}: X^{3} \rightarrow \mathbb{R}_{0}^{+}\) as follows:

$$ d_{b} ( x, y, z )=\max \bigl\{ \vert x-y \vert ^{2}, \vert x-z \vert ^{2}, \vert y-z \vert ^{2} \bigr\} . $$

Then (\(X, d_{b}, s \)) is a tripled b-metric space with \(s =2\).

We bring back into reader’s mind some definitions and properties of b-metric.

Definition 1.2

(see [2])

Let A be a nonempty set, and let \(s>1\) be a real number. A mapping \(d^{*}: A^{2} \rightarrow \mathbb{R}_{0}^{+}\) is said to be a b-metric if, for all a, b, and \(c\in A\), we have:

  • \(a = b\) if and only if \(d^{*} ( a, b )=0\);

  • \(d^{*} ( a, b )= d^{*} ( b, a )\);

  • \(d^{*} ( a, b ) \leq s [ d^{*} ( a, c ) + d^{*} ( c, b )]\).

In this case, the triple (\(A, d^{*}, s \)) is called a b-metric space (with coefficient s).

Remark 1.3

Definition 1.2 allows us to remark that b-metric space is effectually more general than metric space as a b-metric is a metric when \(s=1\). It is worth to mention that the b-metric structure produces some differences to the classical case of metric spaces: the b-metric on a nonempty set M need not be continuous, open balls in such spaces need not be open sets, and so on. The following example describes the significance of a b-metric.

For the notions like convergence, completeness, Cauchy sequence in the setting of b-metric spaces, the reader is referred to Aghajani et al. [28], Czerwik [2], Amini-Harandi [29], Huang et al. [3], Khamsi and Hussain [5]. In line with Wardowski [7], Cosentino et al. [30] investigated a nonlinear function \(F: \mathbb{R}^{+} \rightarrow \mathbb{R}\) complying with the following axioms:

  • F is strictly increasing;

  • \(\lim_{r\rightarrow \infty } r_{n} =0\) if and only if \(\lim_{n\rightarrow \infty } F ( r_{n} )= -\infty \);

  • \(\lim_{r\rightarrow \infty } r_{n} =0\) there exists \(a\in (0,1)\) such that \(\lim_{r_{n} \rightarrow 0^{+}} ( r_{n} )^{a} F ( r_{n} )=0\);

  • \(\tau +F ( s r_{n} ) \leq F ( r_{n- 1} )\) implies \(\tau +F ( s^{n} r_{n} ) \leq F ( s^{n- 1} r_{n- 1} )\) for each \(n\in \mathbb{N}\) and some \(\tau >0\)

for all sequence \(\{ r_{n} \}\) of positive numbers. We denote the set of all functions satisfying the conditions (\(F_{1}\)), (\(F_{2}\)), (\(F_{3}\)), and (\(F_{4}\)) by \(\mathcal{F}_{s}\).

Example 1.4

(see [30])

Let \(F: \mathbb{R}^{+} \rightarrow \mathbb{R}\) be defined by \(F(r)=\ln r\) or \(F(r)=r+\ln r\). Then F satisfies in the conditions.

Theorem 1.5

(see [31])

Let (\(X,d\)) be a complete metric space and \(T:X\rightarrow X\) be a bijective (\(\xi ,\alpha ,\eta \))-expansive mapping of type B satisfying the following conditions:

  • \(T^{- 1}\) is α-admissible with respect to η;

  • There exists \(x_{0} \in X\) such that \(\alpha ( x_{0}, T^{- 1} x_{0} ) \geq \eta ( x_{0}, T^{- 1} x_{0} )\);

  • T is continuous.

Then T has a fixed point.

Definition 1.6

(see [32])

Let (\(X, p_{b} \)) be a partial b-metric space with the coefficient \(s\geq 1\). A mapping \(T:X\rightarrow X\) is said to be a generalized \(\alpha -\eta -\psi \)-Geraghty contractive type mapping if there exist \(\psi \in \Psi \), \(\alpha ,\eta :X\times X\rightarrow [ 0,\infty )\), and \(\beta \in \mathcal{F}\) such that

$$ \alpha ( x, y ) \geq \eta ( x, y ) \quad \text{implies}\quad \psi \bigl( s p_{b} ( Tx, Ty )\bigr) \leq \beta \bigl( \psi \bigl( M_{s}^{T} ( x, y )\bigr)\bigr) \psi \bigl( M_{s}^{T} ( x, y )\bigr) $$
(1.1)

for all \(x, y\in X\), where

$$ M_{s}^{T} ( x, y )=\max \biggl\{ p_{b} ( x, y ), p_{b} ( x, Tx ), p_{b} ( y, Ty ), \frac{p_{b} ( x, Ty ) + p_{b} ( y, Tx )}{2 s} \biggr\} . $$

Theorem 1.7

(see [32])

Let (\(X, p_{b} \)) be a \(p_{b}\)-complete partial b-metric space with the coefficient \(s\geq 1\). Let \(T:X\rightarrow X\) be a generalized \(\alpha -\eta -\psi \)-Geraghty contractive type mapping. Suppose that the following conditions hold:

  • T is a triangular α-orbital admissible mapping with respect to η;

  • There exists \(x_{1} \in X\) such that \(\alpha ( x_{1}, T x_{1} ) \geq \eta ( x_{1}, T x_{1} )\);

  • \(\{ x_{n} \}\) is α-regular with respect to η.

Then T has a fixed point.

Example 1.8

(see [32])

Let \(X=[0,\infty )\) and with the partial b-metric \(p_{b}:X\times X\rightarrow [0,\infty )\) defined by \(p_{b} (x,y)=\max \{x,y \}^{2}\) for all \(x,y\in X\). Obviously, (\(X, p_{b} \)) is a partial b-metric space with \(s=2\). Define the mapping \(T:X\rightarrow X\) given by

$$ Tx = \textstyle\begin{cases} \frac{x}{9} & \text{if } x\in [0,1];\\ \ln x+ 3 & \text{if } x\in (1, \infty ). \end{cases} $$

Define \(\psi :[0, \infty ) \rightarrow [0, \infty )\) and \(\beta :[0, \infty ) \rightarrow [0,1)\) by \(\psi ( t )= t\) and

$$ \beta ( t )= \textstyle\begin{cases} \frac{e^{-t}}{1 +t} & \text{if } x\in (0, \infty );\\ \frac{1}{2} & \text{if } t =0. \end{cases} $$

Let \(\alpha , \eta : X\times X\rightarrow [0, \infty )\) be defined by

$$ \alpha ( x, y )= \textstyle\begin{cases} 6 & \text{if } x\in [0,1];\\ 0 & \text{if } x\in (1, \infty ), \end{cases} $$

and

$$ \eta ( x, y )= \textstyle\begin{cases} 2 & \text{if } x\in [0,1];\\ 1 & \text{if } x\in (1, \infty ). \end{cases} $$

Let \(\alpha ( x, Tx ) \geq \eta ( x, Tx )\). Thus \(x, Tx\in [0,1]\) and so \(T^{2} x = T ( Tx ) \in [0,1]\), which implies that \(\alpha ( Tx, T^{2} x ) \geq \eta ( Tx, T^{2} x )\), that is, T is α-orbital admissible with respect to η. Now, let \(\alpha ( x, y ) \geq \eta ( x, y )\) and \(\alpha ( y, Ty ) \geq \eta ( y, Ty )\), we get that \(x, y, Ty\in [0,1]\) and so \(\alpha ( x, Ty ) \geq \eta ( x, Ty )\). Therefore T is triangular α-orbital admissible with respect to η. Let \(\{ x_{n} \}\) be a sequence such that \(\{ x_{n} \}\) is \(p_{b}\)-convergent to z and \(\alpha ( x_{n}, x_{n+ 1} ) \geq \eta ( x_{n}, x_{n+ 1} )\) for all \(n\in \mathbb{N}\). Then \(\{ x_{n} \} \subseteq [0,1]\) for any \(n\in \mathbb{N}\) and so \(z\in [ 0,1 ]\), from which we have \(\alpha ( x_{n}, z ) \geq \eta ( x_{n}, z )\). That is, \(\{ x_{n} \}\) is α-regular with respect to η. The condition (ii) of Theorem 1.7 is satisfied with \(x_{1} =1 \in X\) since (\(\alpha (1, T 1)=2 \geq 2= \eta (1, T 1)\). We next prove that T is a generalized α-η-ψ-Geraghty contraction type mapping. Let \(x, y\in X\) with \(\alpha ( x, y ) \geq \eta ( x, y )\). Thus \(x, y\in [0,1]\). Without loss of generality, we may assume that \(0 \leq y\leq x\leq 1\). Therefore

$$ p_{b} ( Tx, Ty )= \biggl[ \max \biggl\{ \frac{x}{9}, \frac{y}{9} \biggr\} \biggr]^{2} = \frac{x^{2}}{81} $$

and

$$ M_{s}^{T} ( x, y )=\max \biggl\{ x^{2}, x^{2}, y^{2}, \frac{x^{2} + [ \max \{ y, \frac{x}{9} \} ]^{2}}{4} \biggr\} = x^{2}. $$

Since \(\frac{2}{81} \leq \frac{1}{2 e} \leq \frac{e^{- x^{2}}}{1 + x^{2}}\), we obtain that

$$\begin{aligned} \psi \bigl( s p_{b} ( Tx, Ty )\bigr)&= \psi \biggl(2 \frac{x^{2}}{81} \biggr)= \frac{2 x^{2}}{81} \leq \frac{e^{- x^{2}}}{1 + x^{2}} \cdot x^{2} \\ &\leq \beta \bigl( \psi \bigl( x^{2} \bigr)\bigr) \psi \bigl( x^{2} \bigr) \\ &\leq \beta \bigl( \psi \bigl( M_{s}^{T} ( x, y )\bigr) \bigr) \psi \bigl( M_{s}^{T} ( x, y )\bigr). \end{aligned}$$

Thus T is a generalized α-η-ψ-Geraghty contraction type mapping. Hence all the assumptions in Theorem 1.7 are satisfied and thus T has a fixed point which is \(x =0\).

Definition 1.9

(see [27])

Let (\(M, d^{*},s\)) be a b-metric space, \(S:M\rightarrow M\) and \(\alpha _{s}:M\times M\rightarrow \mathbb{R}_{0}^{+}\) be two mappings. The mapping S is said to be \(\alpha _{s}\)-admissible if

$$ \alpha _{s} ( r_{1}, r_{2} ) \geq s^{2} \Rightarrow \alpha _{s} \bigl( S ( r_{1} ), S ( r_{2} )\bigr) \geq s^{2} \quad \text{for all } r_{1}, r_{2} \in M. $$

Theorem 1.10

(see [27])

Let M be a nonempty set and \(\alpha _{s}\) be as defined in Definition 1.9. Let f, g, S, T be \(\alpha _{s} -b\)-continuous self-mappings defined on an \(\alpha _{s} \)-complete b-metric space (\(M, d^{*},s\)) such that \(f(M)\subseteq T(M)\), \(g(M)\subseteq S(M)\). Suppose that, for all \(( r_{1}, r_{2} )\in \gamma _{f,g, \alpha _{s}}\), there exist \(F\in \mathcal{F}_{s}\) and \(\tau >0\) such that

$$ \tau +F \bigl( s d^{*} \bigl( f ( r_{1} ), g ( r_{2} )\bigr)\bigr) \leq F \bigl( \mathcal{M}_{1} ( r_{1}, r_{2} )\bigr). $$
(1.2)

Assume that the pairs (\(f, S \)), (\(g, T \)) are \(\alpha _{s}\)-compatible and the pairs (\(f, g \)) and (\(g, f \)) are triangular partially weakly \(\alpha _{s}\)-admissible with respect to T and S, respectively. Then the pairs (\(f, S \)), (\(g, T \)) have the coincidence point (say) v in M. Moreover, if \(\alpha _{s} ( Sv, Tv ) \geq s^{2}\), then v is a common fixed point of f, g, S, T.

Remark 1.11

(see [27])

If we suppose that \(\alpha _{s} (v,w)\geq s^{2}\) for each pair of common fixed point of f, g, S, T, then v is unique. Indeed, if w is another fixed point of f, g, S, T and assuming on the contrary that \(d^{*} (fv,gw)>0\), then from (1.2) we have

$$ F \bigl( s d^{*} ( v, w )\bigr)= F \bigl( s d^{*} \bigl( S ( v ), T ( w )\bigr)\bigr) \leq F \bigl( \mathcal{M}_{1} ( v, w )\bigr) -\tau , $$
(1.3)

where

$$ \begin{aligned} \mathcal{M}_{1} ( v, w )&=\max \biggl\{ d^{*} \bigl( S ( v ), T ( w )\bigr), d^{*} \bigl( f ( v ), S ( v )\bigr),\\ &\quad d^{*} \bigl( g ( w ), T ( w )\bigr) \frac{d^{*} ( S ( v ), g ( w )) + d^{*} ( f ( v ), T ( w ))}{2 s} \biggr\} . \end{aligned} $$

Thus, by (1.3), we have

$$ F \bigl( s d^{*} ( v, w )\bigr)< F \bigl( d^{*} ( v, w ) \bigr), $$

which is a contradiction. Hence, \(v = w\) and v is a unique common fixed point of self-mappings f, g, S, T.

Theorem 1.12

(see [27])

Let f, g, S, T be self-mappings defined on an \(\alpha _{s}\)-regular and \(\alpha _{s}\)-complete metric space (\(M, d^{*},s\)) such that \(f(M)\subseteq T(M)\), \(g(M)\subseteq S(M)\), and \(T(M)\) and \(S(M)\) are closed subsets of M. Suppose that, for all \(( r_{1}, r_{2} )\in \gamma _{f,g, \alpha _{s}}\), there exist \(F\in \mathcal{F}_{s}\) and \(\tau >0\) such that

$$ \tau +F ( s d^{*} \bigl( f ( r_{1} ), g ( r_{2} ) \bigr) \leq F \bigl( \mathcal{M}_{1} ( r_{1}, r_{2} )\bigr). $$
(1.4)

Assume that the pairs (\(f, S \)), (\(g, T \)) are weakly compatible and the pairs (\(f, g \)) and (\(g, f \)) are triangular partially weakly \(\alpha _{s}\)-admissible with respect to T and S, respectively. Then the pairs (\(f, S \)), (\(g, T \)) have the coincidence point v in M. Moreover, if \(\alpha _{s} ( Sv, Tv ) \geq s^{2}\), then v is a coincidence point of f, g, S, T.

Theorem 1.13

(see [27])

Let f, g, S, T be \(\alpha _{s}\)-continuous self-mappings defined on an \(\alpha _{s}\)-complete b-metric space (\(M, d^{*},s\)) such that \(f(M)\subseteq T(M)\), \(g(M)\subseteq S(M)\). Suppose that, for all \(( r_{1}, r_{2} )\in \gamma _{f,g, \alpha _{s}}\), there exist \(F\in \mathcal{F}_{s}\) and \(\tau >0\) such that

$$ \tau +F ( s d^{*} \bigl( f ( r_{1} ), g ( r_{2} ) \bigr) \leq F \bigl( \mathcal{M}_{i} ( r_{1}, r_{2} )\bigr) $$
(1.5)

holds for one of \(i =2,3,4,5,6\), where

$$\begin{aligned} \mathcal{M}_{2} ( r_{1}, r_{2} )&= a_{1} d^{*} \bigl( S ( r_{1} ), T ( r_{2} )\bigr) + a_{2} d^{*} \bigl( f ( r_{1} ), S ( r_{1} )\bigr) + a_{3} d^{*} \bigl( g ( r_{2} ), T ( r_{2} )\bigr) \\ &\quad {}+ a_{4} \bigl[ d^{*} \bigl( S ( r_{1} ), g ( r_{2} )\bigr) + d^{*} \bigl( f ( r_{1} ), T ( r_{2} )\bigr)\bigr] \end{aligned}$$

with \(a_{i} \geq 0\), \(i =1,2,3,4\), such that \(a_{1} + a_{2} + a_{3} + 2 s a_{4} =1\);

$$ \mathcal{M}_{3} ( r_{1}, r_{2} )= a_{1} d^{*} \bigl( S ( r_{1} ), T ( r_{2} )\bigr) + a_{2} d^{*} \bigl( f ( r_{1} ), S ( r_{1} )\bigr) + a_{3} d^{*} \bigl( g ( r_{2} ), T ( r_{2} )\bigr), $$

with \(a_{1} + a_{2} + a_{3} =1\);

$$\begin{aligned}& \mathcal{M}_{4} ( r_{1}, r_{2} )= k \max \bigl\{ d^{*} \bigl( f ( r_{1} ), S ( r_{1} )\bigr), d^{*} \bigl( g ( r_{2} ), T ( r_{2} )\bigr)\bigr\} \quad \textit{with } k\in [0,1); \\& \begin{aligned} \mathcal{M}_{5} ( r_{1}, r_{2} )&= a_{1} ( r_{1}, r_{2} ) d^{*} \bigl( S ( r_{1} ), T ( r_{2} )\bigr) + a_{2} ( r_{1}, r_{2} ) d^{*} \bigl( f ( r_{1} ), S ( r_{1} )\bigr)\\ &\quad {} + a_{3} ( r_{1}, r_{2} ) d^{*} \bigl( g ( r_{2} ), T ( r_{2} )\bigr) \\ &\quad {}+ a_{4} ( r_{1}, r_{2} )\bigl[ d^{*} \bigl( S ( r_{1} ), g ( r_{2} )\bigr)) + d^{*} \bigl( f ( r_{1} ), T ( r_{2} )\bigr)\bigr] \end{aligned} \end{aligned}$$

with \(a_{i} ( r_{1}, r_{2} )\), \(i =1,2,3,4\) are nonnegative functions such that

$$\begin{aligned}& \sup_{r_{1}, r_{2} \in M} \bigl\{ a_{1} ( r_{1}, r_{2} ) + a_{2} ( r_{1}, r_{2} ) + a_{3} ( r_{1}, r_{2} ) + 2 s a_{4} ( r_{1}, r_{2} )\bigr\} =1; \\& \begin{aligned} \mathcal{M}_{6} ( r_{1}, r_{2} )&= a_{1} d^{*} \bigl( S ( r_{1} ), T ( r_{2} )\bigr) + \frac{a_{2} + a_{3}}{2} \bigl[ d^{*} \bigl( f ( r_{1} ), S ( r_{1} )\bigr) + d^{*} \bigl( g ( r_{2} ), T ( r_{2} )\bigr)\bigr] \\ &\quad {}+ \frac{a_{4} + a_{5}}{2 s} \bigl[ d^{*} \bigl( S ( r_{1} ), g ( r_{2} )\bigr) + d^{*} \bigl( f ( r_{1} ), T ( r_{2} )\bigr)\bigr] \end{aligned} \end{aligned}$$

with \(a_{1} + a_{2} + a_{3} + a_{4} + a_{5} =1\).

Assume that the pairs (\(f, S \)), (\(g, T \)) are \(\alpha _{s}\)-compatible and the pairs (\(f, g \)) and (\(g, f \)) are triangular partially weakly \(\alpha _{s}\)-admissible pairs of mappings with respect to T and S, respectively. Then the pairs (\(f, S \)), (\(g, T \)) have the coincidence point v in M. Moreover, if \(\alpha _{s} ( Sv, Tv ) \geq s^{2}\), then v is a common point of f, g, S, T.

Main results

In this section, first we introduce some definitions in a tripled b-metric space (\(X, d_{b} \)) and present several examples.

Definition 2.1

Let (\(X, d_{b},s\)) be a tripled b-metric space, \(T:X\rightarrow X\) and \(\alpha _{s}: X^{3} \rightarrow \mathbb{R}_{0}^{+}\) be two mappings. The mapping T is said to be \(\alpha _{s}\)-admissible if \(\alpha _{s} ( x,y,z ) \geq s^{2}\), then \(\alpha _{s} (Tx,Ty,Tz)\geq s^{2}\) for all \(x,y,z\in X\).

Definition 2.2

Let (\(X, d_{b},s\)) be a tripled b-metric space, \(T:X\rightarrow X\) and \(\alpha _{s}: X^{3} \rightarrow \mathbb{R}_{0}^{+}\) be two mappings. The mapping T is said to be triangular \(\alpha _{s}\)-admissible if

  • \(\alpha _{s} ( x, y, z ) \geq s^{2}\) implies that \(\alpha _{s} ( Tx, Ty, Tz ) \geq s^{2}\) for all \(x, y, z\in X\);

  • \(\alpha _{s} ( x, y, z ) \geq s^{2}\) and \(\alpha _{s} ( y, z, w ) \geq s^{2}\) imply \(\alpha _{s} ( x, z, w ) \geq s^{2}\)for all \(x, y, z, w\in X\).

Definition 2.3

Let (\(X, d_{b},s\)) be a tripled b-metric space, \(f,g,h:X\rightarrow X\) and \(\alpha _{s}: X^{3} \rightarrow \mathbb{R}_{0}^{+}\) be four mappings. The tripled (\(f,g,h\)) is said to be

  • triple weakly \(\alpha _{s}\)-admissible if \(\alpha _{s} ( f ( x ), gf ( x ), hgf ( x )) \geq s^{2}\), \(\alpha _{s} ( g ( x ), hg ( x ), fhg ( x )) \geq s^{2}\), and \(\alpha _{s} ( h ( x ), fh ( x ), gfh ( x )) \geq s^{2}\) for all \(x\in X\);

  • partially weakly \(\alpha _{s}\)-admissible if \(\alpha _{s} ( f ( x ), gf ( x ), hgf ( x )) \geq s^{2}\) for all \(x\in X\).

Definition 2.4

Let (\(X, d_{b},s\)) be a tripled b-metric space and \(f,g,h,\phi :X\rightarrow X\) be four mappings such that \(f(X)\cup g(X)\cup h(X)\subseteq \phi (X)\). The triple of mappings (\(f,g,h\)) is said to be

  • triple weakly \(\alpha _{s}\)-admissible with respect to ϕ if and only if \(\alpha _{s} ( f ( x ), g ( y ), h ( z )) \geq s^{2}\) for all \(x\in X\), for all \(y\in \phi ^{- 1} gf ( x )\), for all \(z\in \phi ^{- 1} hgf ( x )\) and \(\alpha _{s} ( h ( x ), g ( y ), f ( z )) \geq s^{2}\) for all \(x\in X\), for all \(y\in \phi ^{- 1} gh ( x )\), for all \(z\in \phi ^{- 1} fgh ( x )\) and \(\alpha _{s} ( g ( x ), f ( y ), h ( z )) \geq s^{2}\) for all \(x\in X\), for all \(y\in \phi ^{- 1} fg ( x )\), for all \(z\in \phi ^{- 1} hfg ( x )\);

  • partially triple weakly \(\alpha _{s}\)-admissible with respect to ϕ if and only if

    $$ \alpha _{s} \bigl( f ( x ), g ( y ), h ( z )\bigr) \geq s^{2} $$

    for all \(x\in X\), \(y\in \phi ^{- 1} gf ( x )\), and \(z\in \phi ^{- 1} hgf ( x )\).

Definition 2.5

Let (\(X, d_{b},s\)) be a tripled b-metric space and \(f,g,h,\phi :X\rightarrow X\) be four mappings such that \(f(X)\cup g(X)\cup h(X)\subseteq \phi (X)\). The triple of mappings (\(f,g,h\)) is said to be triangular triple weakly \(\alpha _{s}\)-admissible with respect to ϕ if

  • \(\alpha _{s} ( h ( x ), g ( y ), f ( z )) \geq s^{2}\) for all \(x\in X\), for all \(y\in \phi ^{- 1} gf ( x )\), \(z\in \phi ^{- 1} hgf ( x )\), and

    $$ \alpha _{s} \bigl( h ( x ), g ( y ), f ( z )\bigr) \geq s^{2} $$

    for all \(x\in X\), for all \(y\in \phi ^{- 1} gh ( x )\), for all \(z\in \phi ^{- 1} fgh ( x )\), and \(\alpha _{s} ( g ( x ), f ( y ), h ( z )) \geq s^{2}\) for all \(x\in X\), for all \(y\in \phi ^{- 1} fg ( x )\), for all \(z\in \phi ^{- 1} hfg ( x )\);

  • \(\alpha _{s} ( x, y, z ) \geq s^{2}\) and \(\alpha _{s} ( y, z, w ) \geq s^{2}\) imply \(\alpha _{s} ( x, z, w ) \geq s^{2}\) for all \(x, y, z, w\in X\).

Example 2.6

Let \(X= \mathbb{R}_{0}^{+}\) and

$$ d_{b} ( x, y, z )=\max \bigl\{ \vert x-y \vert ^{2}, \vert x-z \vert ^{2}, \vert y-z \vert ^{2} \bigr\} $$

for all \(x, y, z\in X\). Then (\(X, d_{b}, s \)) is a tripled b-metric with \(s =2\). We define \(f ( x )= x\), \(g ( x )= x^{\frac{1}{2}}\), \(h ( x )= x^{\frac{1}{4}}\), and \(S ( x )= x^{4}\) if \(x\in [0,1)\) and \(f ( x )= g ( x )= h ( x )= S ( x )=1\), whenever \(x\in [1, \infty )\) and \(\alpha _{s}: X^{3} \rightarrow \mathbb{R}_{0}^{+}\) as follows:

$$ \alpha ( x, y, z )= \textstyle\begin{cases} \max \{ 4 +y-x,4 +z-x,4 +z-x \} , & x, y, z\in [0,1),\\ 0, & \text{otherwise}. \end{cases} $$

Then, for all \(x\in [0,1)\), \(y\in S^{- 1} ( g ( f ( x )))\), \(z\in S^{- 1} ( h ( g ( x )))\), we have \(y = x^{\frac{1}{8}}\), \(z = x^{\frac{1}{32}}\),

$$ \alpha _{s} \bigl( x, g \bigl( x^{\frac{1}{8}} \bigr), h \bigl( x^{\frac{1}{32}} \bigr) \bigr) = \alpha _{s} \bigl( x, x^{\frac{1}{16}}, x^{\frac{1}{32 \times 4}} \bigr) \geq s^{2}. $$

Thus the triple of mappings (\(f, g, h \)) is triangular weakly \(\alpha _{s}\)-admissible with respect to S. Indeed, if \(\alpha _{s} ( x, y, z ) \geq s^{2}\) and \(\alpha _{s} ( y, z, w ) \geq s^{2}\), then \(\alpha _{s} ( x, z, w ) \geq s^{2}\). Since \(y-x\geq 0\) or \(z-x\geq 0\) or \(z-y\geq 0\) and \(z-y\geq 0\) or \(w-z\geq 0\) or \(w-y\geq 0\). Thus \(w-x\geq 0\) or \(w-z\geq 0\) or \(z-x\geq 0\).

Definition 2.7

Let \(f,g,h,\phi :X\rightarrow X\) be four self-mappings defined on a tripled b-metric space such that \(f(X)\cup g(X)\cup h(X)\subseteq \phi (X)\). The triple of mappings (\(f,g,h\)) is said to be triangular triple partially weakly \(\alpha _{s}\)-admissiblewith respect to ϕ if

  • \(\alpha _{s} ( f ( x ), g ( y ), h ( z ) ) \geq s^{2}\) for all \(x\in X\), \(y\in \phi ^{- 1} ( g ( f ( x )))\), \(z\in \phi ^{- 1} ( hg ( f ( x )))\),

  • \(\alpha _{s} ( x, y, z ) \geq s^{2}\), \(\alpha _{s} ( y, z, w ) \geq s^{2}\) imply \(\alpha _{s} ( x, z, w ) \geq s^{2}\) for all \(x, y, z\in X\).

Definition 2.8

Let (\(X, d_{b},s\)) be a tripled b-metric space. The tripled b-metric space X is said to be \(\alpha _{s}\)-complete if and only if every Cauchy sequence \(\{ x_{n} \}\) in X such that \(\alpha _{s} ( x_{n}, x_{n+1}, x_{n+2} ) \geq s^{2}\) for all \(n \in \mathbb{N}\) converges in X. That is,

$$ \lim_{n\rightarrow \infty } d_{b} ( x_{n}, x, x )= \lim _{n\rightarrow \infty } d_{b} ( x_{n}, x_{n}, x )=0. $$

If X is a complete tripled metric space, then X is also an \(\alpha _{s}\)-complete tripled metric space, but the converse is not true. The following example explains this fact.

Example 2.9

Let \(X= \mathbb{R}^{+}\) and \(d_{b}: X^{3} \rightarrow \mathbb{R}_{0}^{+}\) be the tripled b-metric. Define \(\alpha _{s}: X^{3} \rightarrow \mathbb{R}_{0}^{+}\),

$$ \alpha ( x, y, z )= \textstyle\begin{cases} 4\max \{ e^{| x-y | }, e^{| y-z | }, e^{| x-z | } \} , & x, y, z\in [0, \frac{5}{2} ),\\ 0, & \text{otherwise}. \end{cases} $$

It is easy to see that (\(X, d_{b}, S \)) in not a complete tripled b-metric space, but (\(X, d_{b}, s \)) is an \(\alpha _{s}\)-complete tripled b-metric.

Definition 2.10

Let (\(X, d_{b},s\)) be a tripled b-metric space. We say that the self-mapping T is an \(\alpha _{s}\)-continuous mapping on (\(X, d_{b},s\)) if, for given \(x\in X\) and sequence \(\{ x_{n} \}\),

$$ \lim_{n\rightarrow \infty } d_{b} ( x_{n}, x, x )= \lim _{n\rightarrow \infty } d_{b} ( x_{n}, x_{n}, x )=0, $$

and \(\alpha ( x_{n}, x_{n+ 1}, x_{n+ 2} ) \geq s^{2}\) for all \(n\in \mathbb{N}\) implies

$$ \lim_{n\rightarrow \infty } d_{b} ( T x_{n}, Tx, Tx )= \lim_{n\rightarrow \infty } d_{b} ( T x_{n}, T x_{n}, Tx )=0. $$

Example 2.11

Let \(X= \mathbb{R}_{0}^{+}\) and \(d_{b}: X^{3} \rightarrow \mathbb{R}_{0}^{+}\) for all \(x,y,z\in X\), define by \(d_{b} (x,y,z)=\max \{ | x-y | ^{2},| x-z | ^{2},| y-z | ^{2} \} \) and

$$\begin{aligned}& T ( x )= \textstyle\begin{cases} \sin \pi x, & x\in [0,1],\\ \cos \pi x+ 2, & x\in (1, \infty ), \end{cases}\displaystyle \\& \alpha _{s} ( x, y, z )= \textstyle\begin{cases} x^{2} + y^{2} + 4, & x, y, z\in [0,1],\\ 0, & \text{otherwise}. \end{cases}\displaystyle \end{aligned}$$

Then T is not continuous on X; however, T is \(\alpha _{s}\)-continuous.

Definition 2.12

Let (\(X, d_{b},s\)) be a tripled b-metric space. The pairs of self-mappings (\(f,g\)), (\(g,h\)), and (\(f,h\)) are said to be \(\alpha _{s}\)-compatible if

$$\begin{aligned}& \lim_{n\rightarrow \infty } d_{b} \bigl( gh ( x_{n} ), hg ( x_{n} ), g ( x_{n} ) \bigr) =0, \\& \lim_{n\rightarrow \infty } d_{b} \bigl( fg ( x_{n} ), gf ( x_{n} ), f ( x_{n} ) \bigr) =0, \\& \lim_{n\rightarrow \infty } d_{b} \bigl( hf ( x_{n} ), fh ( x_{n} ), h ( x_{n} ) \bigr) =0, \end{aligned}$$

or \(\lim_{n\rightarrow \infty } d_{b} ( gh ( x_{n} ), hg ( x_{n} ), h ( x_{n} ) ) =0\) or \(\lim_{n\rightarrow \infty } d_{b} ( fg ( x_{n} ), gf ( x_{n} ), g ( x_{n} ) ) =0\) or

$$ \lim_{n\rightarrow \infty } d_{b} \bigl( hf ( x_{n} ), fh ( x_{n} ), f ( x_{n} ) \bigr) =0, $$

whenever \(\{ x_{n} \}\) is a sequence in X such that \(\alpha ( x_{n}, x_{n+ 1}, x_{n+ 1} ) \geq s^{2}\), and

$$ \lim_{n\rightarrow \infty } f ( x_{n} )= \lim_{n\rightarrow \infty } g ( x_{n} )= \lim_{n\rightarrow \infty } h ( x_{n} )= t $$

for some \(t\in X\).

Example 2.13

Let \(X=[1,\infty )\) and \(d_{b}:X\times X\times X\rightarrow \mathbb{R}_{0}^{+}\) be defined by

$$ d_{b} ( x, y, z )=\max \bigl\{ \vert x-y \vert ^{2}, \vert x-z \vert ^{2}, \vert y-z \vert ^{2} \bigr\} $$

for all \(x, y, z\in X\), then (\(X, d_{b}, s =2\)) is a tripled b-metric space. Define \(f ( x )=4\), \(g ( x )=16 - 3 x\) if \(x\in [1,4]\) and \(f ( x )=8\) and \(g ( x )=9\) whenever \(x\in (4, \infty )\) and

$$ \alpha ( x, y, z )= \textstyle\begin{cases} 6, & x, y, z\in [1,4],\\ 0, & \text{otherwise}. \end{cases} $$

Let us consider \(\{ x_{n} \}\) to be a sequence such that \(\alpha ( x_{n}, x_{n+ 1}, x_{n+ 2} ) \geq s^{2}\), and let

$$ \lim_{n\rightarrow \infty } f ( x_{n} )= \lim_{n\rightarrow \infty } g ( x ), $$

then \(x_{n} =4\). It is clear that \(\lim_{n\rightarrow \infty } f ( x_{n} )= \lim_{n\rightarrow \infty } g ( x )=4\). We obtain that

$$\begin{aligned} \lim_{n\rightarrow \infty } d_{b} \bigl( fg ( x_{n} ), gf ( x_{n} ), f ( x_{n} )\bigr)&= \lim_{n\rightarrow \infty } d_{b} \bigl( fg ( x_{n} ), gf ( x_{n} ), g ( x_{n} )\bigr) \\ &= d_{b} (4,4,4)=0. \end{aligned}$$

Hence (\(f, g \)) is an \(\alpha _{s}\)-compatible pair. Now, if we consider \(x_{n} =4 - \frac{1}{n}\), then

$$ \lim_{n\rightarrow \infty } f ( x_{n} )= \lim_{n\rightarrow \infty } g ( x_{n} )=4. $$

But \(\lim_{n\rightarrow \infty } gf ( x_{n} )=4\),

$$ \lim_{n\rightarrow \infty } fg ( x_{n} )= \lim_{n\rightarrow \infty } f \biggl(16 - 3\biggl(4 - \frac{1}{n} \biggr)\biggr)= \lim _{n\rightarrow \infty } f \biggl(4 + \frac{3}{n} \biggr)=8, $$

and \(\lim_{n\rightarrow \infty } d_{b} ( fg ( x_{n} ), gf ( x_{n} ), f x_{n} ) \neq 0\). Consequently, (\(f, g \)) is not compatible.

Definition 2.14

Let \(f,g\), and T be self-mappings defined on a nonempty set X. If \(f(x)=g(x)=T(x)\) for some \(x\in X\), then x is called a coincidence point of \(f,g\), and T. Three self-mappings \(f,g\), and T defined on X are said to be weakly compatible if \(\{f,g\}\), \(\{g,T\}\), and \(\{f,T\}\) commute at their coincidence points.

Definition 2.15

Let (\(X, d_{b},s\)) be a tripled b-metric space. The space (\(X, d_{b},s\)) is said to be \(\alpha _{s}\)-regular if, for any sequence \(\{ x_{n} \}\) in X, the following condition holds: if \(x_{n} \rightarrow x\) as \(n\rightarrow \infty \) and \(\alpha _{s} ( x_{n}, x_{n+1}, x_{n+2} )\geq s^{2}\) for all \(n \in \mathbb{N}\), then \(\alpha _{s} ( x_{n},x,x)\geq s^{2}\) and \(\alpha _{s} ( x_{n}, s_{n},x)\geq s^{2}\) for all \(n \in \mathbb{N}\).

Now, we are ready to prove our results.

Lemma 2.16

Let (\(X, d_{b},s\)) be a tripled b-metric space. If there exist three sequence \(\{ x_{n} \}\), \(\{ y_{n} \}\), and \(\{ z_{n} \}\) such that \(\lim_{n\rightarrow \infty } d_{b} ( x_{n}, y_{n}, z_{n} )=0\) and \(\lim_{n\rightarrow \infty } x_{n} = \lim_{n\rightarrow \infty } y_{n} =t\) for some \(t\in X\), then \(\lim_{n\rightarrow \infty } z_{n} =t\).

Proof

By the triangle inequality, we have

$$ d_{b} ( z_{n}, t, t ) \leq s \bigl[ d_{b} ( z_{n}, x_{n}, y_{n} ) + d_{b} ( t, t, t ) + d_{b} ( t, y_{n}, t ) \bigr]. $$

By taking limit as \(n\rightarrow \infty \), the result follows. □

Definition 2.17

Let (\(X, d_{b},s\)) be a tripled b-metric space, \(f,g,h, S_{1}, S_{2}, S_{3}:X\rightarrow X\) be self-mappings, and \(\alpha _{s}\) be as defined in Definition 2.1. We define the set \(\lambda _{f,g,h, \alpha _{s}}\) by

$$ \begin{aligned}[b] \lambda _{f, g, h, \alpha _{s}} & = \bigl\{ ( \alpha , \beta , \gamma ) \in X^{3}: \alpha _{s} \bigl( S_{1} ( \alpha ), S_{2} ( \beta ), S_{3} ( \gamma )\bigr) \geq s^{2}, \\ & \quad \text{and } d_{b} \bigl( f ( \alpha ), g ( \beta ), h ( \gamma )\bigr)>0 \bigr\} . \end{aligned} $$
(2.1)

Let

$$ \begin{aligned}[b] & M ( \alpha , \beta , \gamma ) \\ &\quad =\max \biggl\{ d_{b} \bigl( S_{1} ( \alpha ), S_{2} ( \beta ), S_{3} ( \gamma ) \bigr), d_{b} \bigl( f ( \alpha ), S_{2} ( \alpha ), S_{3} ( \alpha ) \bigr), \\ &\qquad d_{b} \bigl( g ( \beta ), S_{1} ( \beta ), S_{3} ( \beta ) \bigr), d_{b} \bigl( h ( \gamma ), S_{1} ( \gamma ), S_{2} ( \gamma ) \bigr), \\ &\qquad \frac{d_{b} ( S_{1} ( \alpha ), g ( \beta ), h ( \gamma ) ) + d_{b} ( f ( \alpha ), S_{2} ( \beta ), h ( \gamma ) ) + d_{b} ( S_{3} ( \gamma ), g ( \beta ), f ( \alpha ) )}{3 s} \biggr\} . \end{aligned} $$
(2.2)

The following theorem is one of our main results.

Theorem 2.18

Let X be a nonempty set and \(\alpha _{s}\) be as defined in Definition 2.1. Let f, g, h, \(S_{1}\), \(S_{2}\), \(S_{3}\) be \(\alpha _{s} - b\)-continuous self-mappings defined an \(\alpha _{s}\)-complete tripled b-metric space (\(X, d_{b},s\)) such that \(f(X)\subseteq S_{1} (X)\), \(g(X)\subseteq S_{2} (X)\), and \(h(X)\subseteq S_{3} (X)\). Suppose that, for all \((x,y,z)\in \lambda _{f,g,h, \alpha _{s}}\), there exist \(F\in \mathcal{F}_{s}\) and \(r>0\) such that

$$ r+F \bigl( s d_{b} \bigl( f ( x ), g ( y ), h ( z )\bigr)\bigr) \leq F \bigl( M ( x, y, z )\bigr). $$
(2.3)

Assume that the pairs (\(f, S_{1} \)), (\(g, S_{2} \)), and (\(h, S_{3} \)) are \(\alpha _{s}\)-compatible and the triples (\(f, g, h \)), \(( g, f, h )\), and (\(h, g, f \)) are triangular partially weakly \(\alpha _{s}\)-admissible with respect to \(S_{1}\), \(S_{2}\), and \(S_{3}\), respectively. Then the pairs (\(f, S_{1} \)), \(( g, S_{2} )\), and (\(h, S_{3} \)) have the coincidence fixed point say v in X. Moreover, if \(\alpha _{s} ( S_{1} ( v ), S_{2} ( v ), S_{3} ( v )) \geq s^{2}\), then v is a common fixed point of f, g, h, \(S_{1}\), \(S_{2}\), \(S_{3}\).

Proof

Let \(x_{0} \in X\) be an arbitrary point. As \(f ( X ) \subseteq S_{1} ( X )\), there exists \(x_{1} \in X\) such that \(f ( x_{0} )= S_{1} ( x_{1} )\). Since \(g ( x_{1} ) \in S_{2} ( X )\), we can choose \(x_{2} \in X\) such that \(g ( x_{1} )= S_{2} ( x_{2} )\). Since \(h ( x_{2} ) \in S_{3} ( X )\), there exists \(x_{3} \in X\) such that \(h ( x_{2} )= S_{3} ( x_{3} )\). In general, \(x_{2 n}\), \(x_{2 n+ 1}\), and \(x_{2 n+ 2}\) are chosen in X such that \(f ( x_{2 n} )= S_{1} ( x_{2 n+ 1} )\), \(g ( x_{2 n+ 1} )= S_{2} ( x_{2 n+ 2} )\), and \(h ( x_{2 n+ 2} )= S_{3} ( x_{2 n+ 3} )\). Define a sequence \(\{ J_{n} \} \in X\) such that, for all \(n\in \mathbb{N}\), \(J_{2 n+ 1} = f ( x_{2 n} )= S_{1} ( x_{2 n+ 1} )\), \(J_{2 n+ 2} = g ( x_{2 n+ 1} )= S_{2} ( x_{2 n+ 2} )\), and \(J_{2 n+ 2} = h ( x_{2 n+ 2} )= S_{3} ( x_{2 n+ 3} )\). As \(x_{1} \in S_{1}^{- 1} ( f ( x_{0} ))\), \(x_{2} \in S_{2}^{- 1} ( g ( x_{1} ))\), \(x_{3} \in S_{3}^{- 1} ( h ( x_{2} ))\), and (\(f, g, h \)), (\(h, g, f \)), and (\(g, f, h \)) are triangular partially weakly \(\alpha _{s}\)-admissible triples of mappings with respect to \(S_{1}\), \(S_{2}\), and \(S_{3}\), respectively, we have

$$\begin{aligned}& \alpha _{s} \bigl( f ( x_{0} ), g ( x_{1} ), h ( x_{2} ) \bigr) = \alpha _{s} \bigl( S_{1} ( x_{1} ), S_{2} ( x_{2} ), S_{3} ( x_{3} ) \bigr) \geq s^{2}, \\& \alpha _{s} \bigl( h ( x_{2} ), g ( x_{1} ), f ( x_{0} ) \bigr) = \alpha _{s} \bigl( S_{3} ( x_{3} ), S_{2} ( x_{2} ), S_{1} ( x_{1} ) \bigr) \geq s^{2}, \end{aligned}$$

and

$$ \alpha _{s} \bigl( g ( x_{1} ), f ( x_{0} ), h ( x_{2} ) \bigr) = \alpha _{s} \bigl( S_{2} ( x_{2} ), S_{1} ( x_{1} ), S_{3} ( x_{3} ) \bigr) \geq s^{2}. $$

Continuing this way, we obtain

$$\begin{aligned}& \alpha _{s} \bigl( S_{1} ( x_{2 n+ 1} ), S_{2} ( x_{2 n+ 2} ), S_{3} ( x_{2 n+ 3} ) \bigr) \geq s^{2}, \\& \alpha _{s} \bigl( S_{3} ( x_{2 n+ 3} ), S_{2} ( x_{2 n+ 2} ), S_{1} ( x_{2 n+ 1} ) \bigr) \geq s^{2}, \end{aligned}$$

and \(\alpha _{s} ( S_{2} ( x_{2 n+ 2} ), S_{1} ( x_{2 n+ 1} ), S_{3} ( x_{2 n+ 3} ) ) \geq s^{2}\). Thus, we have

$$\begin{aligned}& \alpha _{s} ( J_{2 n+ 1}, J_{2 n+ 2}, J_{2 n+ 3} ) \geq s^{2}, \\& \alpha _{s} ( J_{2 n+ 3}, J_{2 n+ 2}, J_{2 n+ 1} ) \geq s^{2}, \end{aligned}$$

and \(\alpha _{s} ( J_{2 n+ 2}, J_{2 n+ 1}, J_{2 n+ 3} ) \geq s^{2}\) for all \(n\in \mathbb{N}\). At present, we prove that

$$ \lim_{l\rightarrow \infty } d_{b} ( J_{l}, J_{l+ 1}, J_{l+ 2} ) =0. $$

Set \(d_{l} = d_{b} ( J_{l}, J_{l+ 1}, J_{l+ 2} )\). Suppose that \(d_{l_{0}} =0\) for some \(l_{0}\). Then \(J_{l_{0}} = J_{l_{0} + 1}\). If \(l_{0} =2 n\), then \(J_{2 n} = J_{2 n+ 1}\) gives \(J_{2 n+ 1} = J_{2 n+ 2}\). Indeed, by contractive condition (2.3), we get

$$\begin{aligned} F \bigl( s d_{b} ( J_{2 n+ 1}, J_{2 n+ 2}, J_{2 n+ 3} ) \bigr) &= F \bigl( s d_{b} \bigl( f ( x_{2 n} ), g ( x_{2 n+ 1} ), h ( x_{2 n+ 2} ) \bigr) \bigr) \\ &\leq F \bigl( M ( x_{2 n}, x_{2 n+ 1}, x_{2 n+ 2} ) \bigr) -r \end{aligned}$$

for all \(n\in \mathbb{N} \cup \{0\}\), where

$$\begin{aligned} M ( x_{2 n}, x_{2 n+ 1}, x_{2 n+ 2} ) &=\max \biggl\{ d_{b} \bigl( S_{1} ( x_{2 n} ), S_{2} ( x_{2 n+ 1} ), S_{3} ( x_{2 n+ 2} ) \bigr), \\ &\quad d_{b} \bigl( f ( x_{2 n} ), S_{2} ( x_{2 n} ), S_{3} ( x_{2 n} ) \bigr), \\ &\quad d_{b} \bigl( g ( x_{2 n+ 1} ), S_{1} ( x_{2 n+ 1} ), S_{3} ( x_{2 n+ 1} ) \bigr), \\ &\quad d_{b} \bigl( h ( x_{2 n+ 2} ), S_{1} ( x_{2 n+ 2} ), S_{2} ( x_{2 n+ 2} ) \bigr), \\ &\quad \frac{1}{3 s} \bigl[ d_{b} \bigl( S_{1} ( x_{2 n} ), g ( x_{2 n+ 1} ), h ( x_{2 n+ 2} ) \bigr) \\ &\quad {}+ d_{b} \bigl( f ( x_{2 n} ), S_{2} ( x_{2 n+ 1} ), h ( x_{2 n+ 2} ) \bigr) \\ &\quad {}+ d_{b} \bigl( S_{3} ( x_{2 n+ 2} ), g ( x_{2 n+ 1} ), f ( x_{2 n} ) \bigr) \bigr]\biggr\} \\ &=\max \biggl\{ d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ), d_{b} ( J_{2 n+ 1}, J_{2 n}, J_{2 n- 1} ), \\ &\quad d_{b} ( J_{2 n+ 2}, J_{2 n+ 1}, J_{2 n} ), d_{b} ( J_{2 n+ 2}, J_{2 n+ 2}, J_{2 n+ 2} ), \\ &\quad \frac{1}{3 s} \bigl[ d_{b} ( J_{2 n}, J_{2 n+ 2}, J_{2 n+ 2} ) + d_{b} ( J_{2 n+ 1}, J_{2 n+ 1}, J_{2 n+ 2} ) \\ &\quad {}+ d_{b} ( J_{2 n+ 1}, J_{2 n+ 2}, J_{2 n+ 1} ) \bigr]\biggr\} . \end{aligned}$$

So

$$ \begin{aligned} M ( x_{2 n}, x_{2 n+ 1}, x_{2 n+ 2} ) & = \max \biggl\{ d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 1} ), d_{b} ( J_{2 n- 1}, J_{2 n}, J_{2 n+ 1} ), \\ &\quad d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ), \\ &\quad \frac{1}{3 s} \bigl[ d_{b} ( J_{2 n}, J_{2 n+ 2}, J_{2 n+ 2} ) + d_{b} ( J_{2 n+ 1}, J_{2 n+ 1}, J_{2 n+ 2} ) \\ &\quad{}+ d_{b} ( J_{2 n+ 1}, J_{2 n+ 1}, J_{2 n+ 2} ) \bigr]\biggr\} \\ & \leq \max \bigl\{ d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ), d_{b} ( J_{2 n- 1}, J_{2 n}, J_{2 n+ 1} ), \\ &\quad d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ), d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ) \bigr\} \\ & =\max \bigl\{ d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ), d_{b} ( J_{2 n- 1}, J_{2 n}, J_{2 n+ 1} ) \bigr\} . \end{aligned} $$

Since \(d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ) =0\), therefore \(M ( x_{2 n}, x_{2 n+ 1}, x_{2 n+ 2} ) = d_{b} ( J_{2 n- 1}, J_{2 n}, J_{2 n+ 1} )\). Then

$$ F \bigl( s d_{b} ( J_{2 n+ 1}, J_{2 n+ 2}, J_{2 n+ 3} ) \bigr) = F \bigl( d_{b} ( J_{2 n- 1}, J_{2 n}, J_{2 n+ 1} ) \bigr) -r. $$

By (\(F_{1} \)), we have

$$ s d_{b} ( J_{2 n+ 1}, J_{2 n+ 2}, J_{2 n+ 3} ) \leq d_{b} ( J_{2 n- 1}, J_{2 n}, J_{2 n+ 1} ) -r. $$

Let \(l =2 n\), then we have \(s d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ) \leq d_{b} ( J_{2 n- 2}, J_{2 n- 1}, J_{2 n} ) -r\). Thus, for all n,

$$ d_{b} ( J_{n}, J_{n+ 1}, J_{n+ 2} ) \leq \frac{1}{s} d_{b} ( J_{n- 1}, J_{n}, J_{n+ 1} ). $$

That is, a sequence \(\{ d_{b} ( J_{n}, J_{n+ 1}, J_{n+ 2} ) \} \) is nonincreasing and \(d_{b} ( J_{n}, J_{n+ 1}, J_{n+ 2} ) \rightarrow 0\) as \(n\rightarrow \infty \). Hence \(\lim_{l\rightarrow \infty } d_{b} ( J_{l}, J_{l+ 1}, J_{l+ 2} ) =0\) holds true. Now, suppose that \(d_{l} = d_{b} ( J_{l}, J_{l+ 1}, J_{l+ 2} ) >0\) for each \(l\in \mathbb{N}\). We claim that \(\lim_{n\rightarrow \infty } d_{b} ( J_{n}, J_{n+ 1}, J_{n+ 2} ) = -\infty \). Let \(l =2 n\). As

$$\alpha _{s} \bigl( S_{1} ( x_{2 n} ), S_{2} ( x_{2 n+ 1} ), S_{3} ( x_{2 n+ 2} ) \bigr) \geq s^{2}, $$

\(d_{b} ( f ( x_{2 n} ), g ( x_{2 n} ), h ( x_{2 n+ 1} ) ) >0\), so \(( x_{2 n- 1}, x_{2 n}, x_{2 n+ 1} ) \in \lambda _{f, g, h, \alpha _{s}}\), by (2.3), we obtain

$$ F \bigl( s d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ) \bigr) \leq F \bigl( d_{b} ( J_{2 n- 1}, J_{2 n}, J_{2 n+ 1} ) \bigr) -r $$
(2.4)

for all \(n\in \mathbb{N}\). Similarly, for \(\mathcal{l} =2 n- 1\),

$$ F \bigl( s d_{b} ( J_{2 n- 1}, J_{2 n}, J_{2 n+ 1} ) \bigr) \leq F \bigl( d_{b} ( J_{2 n- 2}, J_{2 n- 1}, J_{2 n} ) \bigr) -r $$
(2.5)

for all \(n\in \mathbb{N}\). Hence, by (2.4) and (2.5), we have

$$ F \bigl( s d_{b} ( J_{n}, J_{n+ 1}, J_{n+ 2} ) \bigr) \leq F \bigl( d_{b} ( J_{n- 1}, J_{n}, J_{n+ 1} ) \bigr) -r $$
(2.6)

for all \(n\in \mathbb{N}\). Let \(a_{n} = d_{b} ( J_{n}, J_{n+ 1}, J_{n+ 2} )\) for each \(n\in \mathbb{N}\). By (2.6) and property (\(F_{4}\)), we have \(r+F ( s^{n} a_{n} ) \leq F ( s^{n- 1} a_{n- 1} )\) for all \(n\in \mathbb{N}\). Continuing this process, we obtain

$$ F \bigl( s^{n} a_{n} \bigr) \leq F ( a_{n} ) -nr $$
(2.7)

for all \(n\in \mathbb{N}\). On taking limit \(n\rightarrow \infty \) in (2.7), we have \(\lim_{n\rightarrow \infty } F ( s^{n} a_{n} )= -\infty \). By property (\(F_{2}\)), we get \(\lim_{n\rightarrow \infty } s^{n} a_{n} =0\) and (\(F_{2}\)) implies that there exists \(k\in (0,1)\) such that \(\lim_{n\rightarrow \infty } ( s^{n} a_{n} )^{k} F ( s^{n} a_{n} )=0\). By (2.7), for all \(n\in \mathbb{N}\), we obtain

$$ \bigl( s^{n} a_{n} \bigr)^{k} F \bigl( s^{n} a_{n} \bigr) - \bigl( s^{n} a_{n} \bigr)^{k} F ( a_{0} ) \leq - \bigl( s^{n} a_{n} \bigr)^{k} nr\leq 0. $$
(2.8)

On taking limit \(n\rightarrow \infty \) in (2.8), we have \(\lim_{n\rightarrow \infty } n ( s^{n} a_{n} )^{k} =0\). This implies there exists \(n_{1} \in \mathbb{N}\) such that \(n ( s^{n} a_{n} )^{k} \leq 1\) for all \(n\geq n_{1}\), or \(s^{n} a_{n} \leq \frac{1}{n^{\frac{1}{k}}}\) for all \(n\geq n_{1}\). To prove \(\{ J_{n} \}\) is a Cauchy sequence, by the triangular inequality, we have

$$\begin{aligned} d_{b} ( x_{n}, x_{m}, x_{m} ) &\leq s \bigl[ d_{b} ( x_{n}, x_{n+ 1}, x_{n+ 2} ) + d_{b} ( x_{m}, x_{m}, x_{m} ), d_{b} ( x_{m}, x_{m+ 2}, x_{m} ) \bigr] \\ &= s d_{b} ( x_{n}, x_{n+ 1}, x_{n+ 2} ) +s d_{b} ( x_{n+ 2} ), x_{m}, x_{m} ) \\ &\leq s d_{b} ( x_{n}, x_{n+ 1}, x_{n+ 2} ) + s^{2} \bigl[ d_{b} ( x_{n+ 2}, x_{n+ 3}, x_{n+ 4} ) \\ &\quad {} + d_{b} ( x_{m}, x_{m}, x_{m} ) + d_{b} ( x_{m}, x_{n+ 3}, x_{n+ 1} ) \bigr] \\ &= s d_{b} ( x_{n}, x_{n+ 1}, x_{n+ 2} ) + s^{2} d_{b} ( x_{n+ 2}, x_{n+ 3}, x_{n+ 4} ) + s^{2} d_{b} ( x_{n+ 3}, x_{m}, x_{m} ) \\ &\leq s d_{b} ( x_{n}, x_{n+ 1}, x_{n+ 2} ) + s^{2} d_{b} ( x_{n+ 2}, x_{n+ 3}, x_{n+ 4} ) \\ &\quad{}+ s^{3} d_{b} ( x_{n+ 3}, x_{n+ 4}, x_{n+ 5} ) + s^{3} d_{b} ( x_{n+ 4}, x_{m}, x_{m} ). \end{aligned}$$

Take \(m = n+p\), (\(n, p\in \mathbb{N}\)), then we have

$$\begin{aligned} d_{b} ( x_{n}, x_{m}, x_{m} ) &\leq s d_{b} ( x_{n}, x_{n+ 1}, x_{n+ 2} ) + s^{2} d_{b} ( x_{n+ 2}, x_{n+ 3}, x_{n+ 4} ) \\ &\quad {}+ s^{3} d_{b} ( x_{n+ 3}, x_{n+ 4}, x_{n+ 5} ) +\cdots + s^{n- 1} d_{b} ( x_{n+p- 1}, x_{n+p}, x_{n+p} ) \\ &\leq \frac{s}{s^{n} n^{\frac{1}{k}}} + \frac{s^{2}}{s^{n+ 2} ( n+ 2 )^{\frac{1}{k}}} + \frac{s^{3}}{s^{n+ 3} ( n+ 3 )^{\frac{1}{k}}} \\ &\quad {}+\cdots + \frac{s^{p- 1}}{s^{n+P- 1} ( n+p- 1 )^{\frac{1}{k}}} \\ &= \frac{s^{1 -n}}{n^{\frac{1}{k}}} + \frac{s^{-n}}{( n+ 2 )^{\frac{1}{k}}} + \frac{s^{-n}}{( n+ 3 )^{\frac{1}{k}}} +\cdots + \frac{s^{-n}}{( n+p- 1 )^{\frac{1}{k}}} \\ &= \frac{s^{1 -n}}{n^{\frac{1}{k}}} + s^{-n} \sum_{i =2}^{p- 1} \frac{1}{( n+i )^{\frac{1}{k}}}. \end{aligned}$$

Since \(\sum_{i =2}^{p- 1} \frac{1}{( n+i )^{\frac{1}{k}}}\) is convergent and \(s^{-n} \rightarrow 0\) as \(n\rightarrow \infty \), thus we conclude that

$$ \lim_{n, m\rightarrow \infty } d_{b} ( x_{n}, x_{m}, x_{m} ) =0. $$

This implies that \(\{ J_{n} \}\) is a Cauchy sequence in the \(\alpha _{s}\)-complete tripled b-metric space X and

$$ \alpha _{s} ( J_{n}, J_{n+ 1}, J_{n+ 2} ) \geq s^{2}, $$

there exists \(v\in X\) such that

$$ \lim_{n\rightarrow \infty } d_{b} ( J_{2 n+ 1}, v, v ) = \lim _{n\rightarrow \infty } d_{b} ( f x_{2 n}, v, v ) = \lim _{n\rightarrow \infty } d_{b} \bigl( S_{1} ( x_{2 n+ 1} ), v, v \bigr) =0. $$

Consequently, \(f ( x_{2 n} ) \rightarrow v\) and \(S_{1} ( x_{2 n+ 1} ) \rightarrow v\) as \(n\rightarrow \infty \). So

$$ \lim_{n\rightarrow \infty } d_{b} ( J_{2 n+ 1}, v, v ) = \lim _{n\rightarrow \infty } d_{b} ( g x_{2 n}, v, v ) = \lim _{n\rightarrow \infty } d_{b} \bigl( S_{2} ( x_{2 n+ 1} ), v, v \bigr) =0. $$

Thus \(g ( x_{2 n} ) \rightarrow v\) and \(S_{2} ( x_{2 n+ 1} ) \rightarrow v\) as \(n\rightarrow \infty \). Again, we have

$$ \lim_{n\rightarrow \infty } d_{b} ( J_{2 n}, v, v ) = \lim _{n\rightarrow \infty } d_{b} ( h x_{2 n}, v, v ) = \lim _{n\rightarrow \infty } d_{b} \bigl( S_{3} ( x_{2 n+ 1} ), v, v \bigr) =0. $$

Hence \(h ( x_{2 n} ) \rightarrow v\) and \(S_{3} ( x_{2 n+ 1} ) \rightarrow v\) as \(n\rightarrow \infty \). Now, since (\(f, S_{1} \)) is an \(\alpha _{s}\)-compatible pair and

$$ \alpha _{s} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ) \geq s^{2}. $$

Therefore, we have \(\lim_{n\rightarrow \infty } d_{b} ( f S_{1} ( x_{2 n} ), S_{1} f ( x_{2 n} ), x_{2 n} ) =0\) and (\(g, S_{2} \)) is an \(\alpha _{s}\)-compatible pair and

$$ \alpha _{s} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ) \geq s^{2}. $$

We have \(\lim_{n\rightarrow \infty } d_{b} ( g S_{2} ( x_{2 n} ), S_{2} g ( x_{2 n} ), x_{2 n} ) =0\) and (\(h, S_{3} \)) is an \(\alpha _{s}\)-compatible pair, we get

$$ \lim_{n\rightarrow \infty } d_{b} \bigl( h S_{3} ( x_{2 n} ), S_{3} h ( x_{2 n} ), x_{2 n} \bigr) =0. $$

Since \(\lim_{n\rightarrow \infty } d_{b} ( f ( x_{2 n} ), v, v ) =0\), \(\lim_{n\rightarrow \infty } d_{b} ( S_{1} ( x_{2 n} ), v, v ) =0\), and f, \(S_{1}\) is \(\alpha _{s}\)-continuous. Thus \(\lim_{n\rightarrow \infty } d_{b} ( S_{1} f ( x_{2 n} ), S_{1} v, S_{1} v ) =0\), \(\lim_{n\rightarrow \infty } d_{b} ( f S_{1} ( x_{2 n} ), fv, fv ) =0\), and

$$ \lim_{n\rightarrow \infty } d_{b} \bigl( g ( x_{2 n} ), v, v \bigr) =0, $$

so g, \(S_{2}\) is \(\alpha _{s}\)-continuous, we have \(\lim_{n\rightarrow \infty } d_{b} ( S_{2} g ( x_{2 n} ), S_{2} v, S_{2} v ) =0\) and

$$ \lim_{n\rightarrow \infty } d_{b} \bigl( g S_{2} ( x_{2 n} ), gv, gv \bigr) =0. $$

Again in this way, \(\lim_{n\rightarrow \infty } d_{b} ( S_{3} h ( x_{2 n} ), S_{3} v, S_{3} v ) =0\) and \(\lim_{n\rightarrow \infty } d_{b} ( h S_{3} g ( x_{2 n} ), hv, hv ) =0\). By the triangle inequality, we have

$$ \begin{aligned}[b] d_{b} \bigl( fv, S_{1} v, S_{1} ( x_{2 n} ) \bigr) & \leq s \bigl[ d_{b} \bigl( fv, fv, f S_{1} ( x_{2 n} ) \bigr) + d_{b} \bigl( S_{1} v, S_{1} f ( x_{2 n} ), S_{1} v \bigr) \\ &\quad {} + d_{b} \bigl( S_{1} x_{2 n}, f S_{1} x_{2 n}, S_{1} f ( x_{2 n} ) \bigr) \bigr]. \end{aligned} $$
(2.9)

Applying limit as \(n\rightarrow \infty \), we obtain \(d_{b} ( fv, S_{1} v, v ) \leq 0\), which yields that \(fv = S_{1} v = v\). Thus v is a coincidence and common fixed point of f, \(S_{1}\). Arguing in a similar manner, we can prove that \(gv = S_{2} v = v\) and \(hv = S_{1} v = v\). Thus \(fv = gv = hv = S_{1} v = S_{2} v = S_{3} v = v\) and v is a common fixed point of f, g, h, \(S_{1}\), \(S_{2}\), and \(S_{3}\). □

Remark 2.19

If we suppose that \(\alpha _{s} ( v,w,w ) \geq s^{2}\) for each pair of common fixed points of f, g, h, \(S_{1}\), \(S_{2}\), and \(S_{3}\), then v is unique. Indeed, if w is another fixed point of f, g, h, \(S_{1}\), \(S_{2}\), and \(S_{3}\) and assuming on contrary \(d_{b} ( fv,gw,hw ) >0\), then from (2.3) we have

$$ F \bigl( d_{b} ( v, w, w ) \bigr) = F \bigl( s d_{b} \bigl( S_{1} ( v ), S_{2} ( w ), S_{3} ( w ) \bigr) \bigr) \leq F \bigl( M ( v, w, w ) \bigr) -r, $$
(2.10)

where

$$ \begin{aligned} M ( v, w, w ) & =\max \biggl\{ d_{b} \bigl( S_{1} ( v ), S_{2} ( w ), S_{3} ( w ) \bigr), d_{b} \bigl( f ( v ), S_{2} ( v ), S_{3} ( v ) \bigr), \\ &\quad d_{b} \bigl( g ( w ), S_{1} ( w ), S_{3} ( w ) \bigr), d_{b} \bigl( h ( w ), S_{1} ( w ), S_{2} ( w ) \bigr), \\ &\quad \frac{1}{3 s} \bigl[ d_{b} \bigl( S_{1} ( v ), g ( w ), h ( w ) \bigr) \\ &\quad {} + d_{b} \bigl( f ( v ), S_{2} ( w ), h ( w ) \bigr) + d_{b} \bigl( S_{3} ( w ), g ( w ), f ( v ) \bigr) \bigr]\biggr\} \\ & =\max \biggl\{ d_{b} ( v, w, w ), d_{b} ( v, v, v ), d_{b} ( w, w, w ), d_{b} ( w, w, w ), \\ &\quad \frac{1}{3 s} \bigl[ d_{b} ( v, w, w ), d_{b} ( v, w, w ) + d_{b} ( w, w, v ) \bigr]\biggr\} . \end{aligned} $$

Thus, by (2.10), we have \(F ( s d_{b} ( v, w, w ) ) \leq F ( d_{b} ( v, w, w ) ) -r < F ( d_{b} ( v, w, w ) )\), which is a contradiction. Hence \(v = w\) and v is a unique common fixed point of self-mappings f, g, h, \(S_{1}\), \(S_{2}\), and \(S_{3}\).

The following example elucidates Theorem 2.18.

Example 2.20

Let \(X= \mathbb{R}_{0}^{+}\) and \(d_{b}:X\times X\times X\rightarrow \mathbb{R}_{0}^{+}\) be defined by

$$ d_{b} ( x, y, z )=\max \bigl\{ \vert x-y \vert ^{2}, \vert x-z \vert ^{2}, \vert y-z \vert ^{2} \bigr\} $$

for all \(x, y, z\in X\). Define \(\alpha _{s}: X\times X\times X\rightarrow \mathbb{R}_{0}^{+}\) by

$$ \alpha _{s} ( x, y, z )= \textstyle\begin{cases} 4\max \{ e^{x-y}, e^{x-z}, e^{y-z} \} , & x\geq y\geq z,\\ 4\max \{ e^{y-x}, e^{z-x}, e^{z-y} \} , & x\leq y\leq z . \end{cases} $$

So (\(S, d_{b}, s \)) is an \(\alpha _{s}\)-complete tripled b-metric with \(s =2\). Define the mappings f, g, h, \(S_{1}\), \(S_{2}\), and \(S_{3}: X\rightarrow X\) for all \(x\in X\) by

$$ \begin{gathered} f ( x ) =\ln \biggl( 1 + \frac{x}{5} \biggr), \\ g ( x ) =\ln \biggl( 1 + \frac{x}{6} \biggr), \\ h ( x ) =\ln \biggl( 1 + \frac{x}{7} \biggr), \end{gathered} $$

\(S_{1} ( x )= e^{6 x} - 1\), \(S_{2} ( x )= e^{7 x} - 1\), and \(S_{3} ( x )= e^{8 x} - 1\). Clearly, f, g, h, \(S_{1}\), \(S_{2}\), and \(S_{3}\) are \(\alpha _{s}\)-continuous self-mappings complying with \(f ( X )= g ( X )= h ( X )= S_{1} ( X )= S_{2} ( X )= S_{3} ( X )\). We note that the pair (\(f, S_{1} \)) is \(\alpha _{s}\)-compatible. Indeed, let \(\{ x_{n} \}\) be a sequence in X satisfying \(\alpha _{s} ( x_{n}, x_{n+ 1}, x_{n+ 2} ) \geq s^{2}\) and

$$ \lim_{n\rightarrow \infty } f ( x_{n} )= \lim_{n\rightarrow \infty } \ln \biggl( 1 + \frac{x_{n}}{5} \biggr) = \lim_{n\rightarrow \infty } S_{1} ( x_{n} )= t $$

for some \(t\in X\). Then \(\lim_{n\rightarrow \infty } | f ( x_{n} ) -t | ^{2} = \lim_{n\rightarrow \infty } | S_{1} ( x_{n} ) -t | ^{2} =0\), equivalently

$$ \lim_{n\rightarrow \infty } \biggl\vert \ln \biggl( 1 + \frac{x_{n}}{5} \biggr) -t \biggr\vert ^{2} = \lim_{n\rightarrow \infty } \bigl\vert e^{6 x_{n}} - 1 -t \bigr\vert ^{2} =0 $$

implies

$$ \lim_{n\rightarrow \infty } \bigl\vert x_{n} - \bigl(5 e^{t} - 5\bigr) \bigr\vert ^{2} = \lim _{n\rightarrow \infty } \biggl\vert x_{n} - \frac{\ln ( t+ 1)}{6} \biggr\vert ^{2} =0. $$

Uniqueness of limit gives that \(5 e^{t} - 5= \frac{\ln ( t+ 1)}{6}\), thus \(t =0\) is only possible solution. Due to \(alph a_{s}\)-continuity of f and \(S_{1}\), for \(t =0 \in X\), we have

$$\begin{aligned}& \lim_{n\rightarrow \infty } d_{b} \bigl( f S_{1} ( x_{n} ), S_{1} f ( x_{n} ), f ( x_{n} ) \bigr)\\& \quad =\max \Bigl\{ \lim_{n\rightarrow \infty } \bigl\vert f S_{1} ( x_{n} ) - S_{1} f ( x_{n} ) \bigr\vert ^{2}, \\& \qquad \lim_{n\rightarrow \infty } \bigl\vert S_{1} f ( x_{n} ) -f ( x_{n} ) \bigr\vert ^{2}, \lim_{n\rightarrow \infty } \bigl\vert f S_{1} ( x_{n} ) -f ( x_{n} ) \bigr\vert ^{2} \Bigr\} \\& \quad =\max \bigl\{ \bigl\vert f ( t ) - S_{1} ( t ) \bigr\vert ^{2}, \bigl\vert S_{1} ( t ) -t \bigr\vert ^{2}, \bigl\vert f ( t ) -t \bigr\vert ^{2} \bigr\} \\& \quad =0. \end{aligned}$$

Similarly, the pair (\(g, S_{2} \)) and (\(h, S_{3} \)) is \(\alpha _{s}\)-compatible. To prove that (\(f, g, h \)) is a partially weakly \(\alpha _{s}\)-admissible triple of mappings with respect to \(S_{!}\), let \(x\in X\) and \(y\in S_{1}^{- 1} ( g ( f ( x )))\), that is, \(S_{1} ( y )= g ( f ( x ))\) and

$$ e^{6 y} - 1= g \biggl( \ln \biggl( 1 + \frac{x}{5} \biggr) \biggr) =\ln \biggl( 1 + \frac{\ln ( 1 + \frac{x}{5} )}{6} \biggr). $$

Thus \(y = \frac{1}{6} \ln ( 1 + \ln ( 1 + \frac{\ln ( 1 + \frac{x}{5} )}{6} ) )\). We have

$$ f ( x )=\ln \biggl( 1 + \frac{x}{5} \biggr) \geq g ( y )=\ln \biggl( 1 + \frac{y}{6} \biggr) =\ln \biggl( 1 + \frac{1}{36} \ln \biggl( 1 + \ln \biggl( 1 + \frac{\ln ( 1 + \frac{x}{5} )}{6} \biggr) \biggr) \biggr). $$

We have \(z\in S_{1}^{- 1} ( hg ( f ( x )) )\), that is, \(S_{1} ( z )= hg ( f ( x ))\), \(S_{1} ( z )= h ( S_{1} ( y ))\), \(e^{z} - 1=\ln ( 1 + \frac{S_{1} ( y )}{7} )\),

$$ e^{6 z} - 1=\ln \biggl( 1 + \frac{1}{7} \ln \biggl( 1 + \frac{\ln ( 1 + \frac{x}{5} )}{6} \biggr) \biggr), $$

and

$$ z = \frac{1}{6} \ln \biggl( 1 + \ln \biggl( 1 + \frac{1}{7} \ln \biggl( \frac{\ln ( 1 + \frac{x}{5} )}{5} \biggr) \biggr) \biggr). $$

We conclude that

$$\begin{aligned} g ( y )&=\ln \biggl( 1 + \frac{y}{6} \biggr) =\ln \biggl( 1 + \frac{1}{42} \ln \biggl( 1 + \ln \biggl( 1 + \frac{\ln ( 1 + \frac{x}{5} )}{6} \biggr) \biggr) \biggr) \\ &\geq h ( z )=\ln \biggl( 1 + \frac{z}{7} \biggr) \\ &=\ln \biggl( 1 + \frac{1}{42} \ln \biggl( 1 + \ln \biggl( 1 + \frac{1}{7} \ln \biggl( 1 + \frac{\ln ( 1 + \frac{x}{5} )}{6} \biggr) \biggr) \biggr) \biggr). \end{aligned}$$

Thus \(\alpha _{s} ( f ( x ), g ( y ), h ( z ) ) =4\max \{ e^{x-y}, e^{x-z}, e^{y-z} \} \geq s^{2}\). In this process, we can prove that (\(g, f, h \)) is a partially weakly \(\alpha _{s}\)-admissible triple of mappings with respect to \(S_{2}\) and (\(h, g, f \)) is a partially weakly \(\alpha _{s}\)-admissible triple of mappings with respect \(S_{1}\). Now, for each \(x, y, z\in X\), consider

$$\begin{aligned}& d_{b} \bigl( f ( x ), g ( y ), h ( z ) \bigr) =\max \bigl\{ \bigl\vert f ( x ) -g ( y ) \bigr\vert ^{2}, \bigl\vert g ( y ) -h ( z ) \bigr\vert ^{2}, \bigl\vert f ( x ) -h ( z ) \bigr\vert ^{2} \bigr\} , \\& \begin{aligned} \bigl\vert f ( x ) -g ( y ) \bigr\vert ^{2} &= \biggl\vert \ln \biggl( 1 + \frac{x}{5} \biggr) - \ln \biggl( 1 + \frac{y}{6} \biggr) \biggr\vert ^{2} \\ &\leq \biggl( \frac{x}{5} - \frac{y}{6} \biggr)^{2} \\ &= \frac{1}{900} ( 6 x- 5 y )^{2} \\ &\leq \frac{1}{900} \bigl( e^{6 x} - e^{5 y} \bigr)^{2}, \end{aligned} \\& \begin{aligned} \bigl\vert g ( y ) -h ( z ) \bigr\vert ^{2} &= \biggl\vert \ln \biggl( 1 + \frac{y}{6} \biggr) - \ln \biggl( 1 + \frac{z}{7} \biggr) \biggr\vert ^{2} \\ &\leq \biggl( \frac{y}{6} - \frac{z}{7} \biggr)^{2} \\ &= \frac{1}{1764} ( 7 y- 6 z )^{2} \\ &\leq \frac{1}{1764} \bigl( e^{7 y} - e^{6 z} \bigr)^{2}, \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \bigl\vert f ( x ) -h ( z ) \bigr\vert ^{2} &= \biggl\vert \ln \biggl( 1 + \frac{x}{5} \biggr) - \ln \biggl( 1 + \frac{z}{7} \biggr) \biggr\vert ^{2} \\ &\leq \biggl( \frac{x}{5} - \frac{z}{7} \biggr)^{2} \\ &= \frac{1}{1225} ( 7 x- 5 z )^{2} \\ &\leq \frac{1}{1225} \bigl( e^{7 x} - e^{5 z} \bigr)^{2}. \end{aligned}$$

Thus

$$\begin{aligned} d_{b} \bigl( f ( x ), g ( y ), h ( z ) \bigr) &\leq \frac{1}{900} \max \bigl\{ \bigl( e^{6 x} - e^{5 y} \bigr)^{2}, \bigl( e^{7 y} - e^{6 z} \bigr)^{2}, \bigl( e^{7 x} - e^{5 z} \bigr)^{2} \bigr\} \\ &= \frac{1}{900} d_{b} \bigl( S_{1} ( x ), S_{2} ( y ), S_{3} ( z ) \bigr). \end{aligned}$$

Define the function \(F: \mathbb{R}^{+} \rightarrow \mathbb{R}\) by \(F ( x )=\ln x\) for all \(x\in \mathbb{R}^{+}\). Hence, for all \(x, y, z\in X\) such that \(d_{b} ( f ( x ), g ( y ), h ( z ) ) >0\), \(r =\ln (900)\), we obtain

$$ r+F \bigl( d_{b} \bigl( f ( x ), g ( y ), h ( z ) \bigr) \bigr) \leq F \bigl( M ( x, y, z ) \bigr). $$

Thus the contractive condition (2.3) is satisfied for all \(x, y, z\in X\). Hence, all the hypotheses of Theorem 2.18 are satisfied. Note that f, g, h, \(S_{1}\), \(S_{2}\), and \(S_{3}\) have a unique common fixed point \(x =0\).

We have obtained some results from Theorem 2.18, which we express in order.

Corollary 2.21

Let X be a nonempty set and \(\alpha _{s}:X\times X\times X\rightarrow \mathbb{R}_{0}^{+}\) be a function. Let (\(X, d_{b},s\)) be an \(\alpha _{s}\)-complete tripled metric space and f, g, h, \(S_{1}\), \(S_{2}\), and \(S_{3}\) be \(\alpha _{s}\)-continuous self-mappings on (\(X, d_{b},s\)) such that for all \((x,y,z)\in \lambda _{f,g,h, \alpha _{s}}\) the inequality

$$ s d_{b} \bigl( f ( x ), g ( y ), h ( z ) \bigr) \leq kM ( x, y, z ) $$
(2.11)

holds. Assume that the pairs (\(f, S_{1} \)), \(( g, S_{2} )\), and (\(h, S_{3} \)) are \(\alpha _{s}\)-compatible and the triples of mappings (\(f, g, h \)), (\(g, f, h \)), and (\(h, g, f \)) are triangular partially weakly \(\alpha _{s}\)-admissible with respect to \(S_{1}\), \(S_{2}\), and \(S_{3}\), respectively. Then the pairs (\(f, S_{1} \)), (\(g, S_{2} \)), and (\(h, S_{3} \)) have the coincidence point v in X. Moreover, if \(\alpha _{s} ( S_{1} v, S_{2} v, S_{3} v ) \geq s^{2}\), then v is a common fixed point of f, g, h, \(S_{1}\), \(S_{2}\), and \(S_{3}\).

Proof

For all \(( x, y, z ) \in \lambda _{f, g, h, \alpha _{s}}\), we have \(s d_{b} ( f ( x ), g ( y ), h ( z ) ) \leq kM ( x, y, z )\). It follows that \(r+ \ln ( d_{b} ( f ( x ), g ( y ), h ( z ) ) ) \leq \ln ( M ( x, y, z ) )\), where \(r =\ln ( \frac{s}{k} )>0\). Then the contraction condition (2.11) reduces to (2.3) with \(F ( x )=\ln x\), and the application of Theorem 2.18 ensures the existence of a fixed point. □

If we set \(S = S_{1} = S_{2} = S_{3}\) in Theorem 2.18, we obtain the following corollaries.

Corollary 2.22

Let \(f,g,h\), and S be self-mappings defined on an \(\alpha _{s}\)-complete tripled metric space (\(X, d_{b},s\)) such that \(f(X)\cup g(X)\cup h(X)\subseteq S(X)\) with \(\alpha _{s}\)-continuous. Suppose that, for all \(x,y,z\in X\) with \(\alpha _{s} (Tx,Ty,Tz)\geq s^{2}\), there exist \(F\in \mathcal{F}_{s}\) and \(r>0\) such that \(d_{b} ( f(x),g(y),h(z) ) >0\), then

$$ r+F \bigl( s d_{b} \bigl( f ( x ), g ( y ), h ( z ) \bigr) \bigr) \leq F \bigl( M ( x, y, z ) \bigr), $$

where

$$\begin{aligned} M ( x, y, z ) &=\max \biggl\{ d_{b} \bigl( S ( x ), S ( y ), S ( z ) \bigr), d_{b} \bigl( f ( x ), S ( x ), S ( x ) \bigr), \\ &\quad d_{b} \bigl( g ( y ), S ( y ), S ( y ) \bigr), d_{b} \bigl( h ( z ), S ( z ), S ( z ) \bigr), \\ &\quad \frac{1}{3 s} \bigl[ d_{b} \bigl( S ( x ), g ( y ), h ( z ) \bigr) + d_{b} \bigl( f ( x ), S ( y ), h ( z ) \bigr) \\ &\quad {}+ d_{b} \bigl( S ( z ), g ( y ), f ( x ) \bigr) \bigr]\biggr\} . \end{aligned}$$

Assume that either the pair (\(f, S \)) is \(\alpha _{s}\)-compatible and f is \(\alpha _{s}\)-continuous or (\(g, S \)) is \(\alpha _{s}\)-compatible and g is \(\alpha _{s}\)-continuous, or (\(h, S \)) is \(\alpha _{s}\)-compatible and h is \(\alpha _{s}\)-continuous. Then the pairs (\(f, S \)), \(( g, S )\), and (\(h, S \)) have the coincidence point v in X provided that the triple of mappings (\(f, g, h \)) is triangular weakly \(\alpha _{s}\)-admissible with respect to S. Moreover, if \(\alpha _{s} ( Sv, Sv, Sv ) \geq s^{2}\), then v is a common fixed point of f, g, h, and S.

If we set \(S_{1} = S_{2} = S_{3}\) and \(f = g = h\) in Theorem 2.18, we obtain the following corollary.

Corollary 2.23

Let f and S be \(\alpha _{s}\)-continuous self-mappings defined on an \(\alpha _{s}\)-complete tripled metric space (\(X, d_{b},s\)) such that \(f(X)\subseteq S(X)\). Suppose that, for all \(x,y,z\in X\) with \(\alpha _{s} (Sx,Sy,Sz)\geq s^{2}\), there exist \(F\in \mathcal{F}_{s}\) and \(r>0\) such that \(d_{b} ( f(x),f(y),f(z) ) >0\), then

$$ r+F \bigl( s d_{b} \bigl( f ( x ), f ( y ), f ( z ) \bigr) \bigr) \leq F \bigl( M ( x, y, z ) \bigr), $$

where

$$\begin{aligned} M ( x, y, z ) &=\max \biggl\{ d_{b} \bigl( S ( x ), S ( y ), S ( z ) \bigr), d_{b} \bigl( f ( x ), S ( x ), S ( x ) \bigr), \\ &\quad d_{b} \bigl( f ( y ), S ( y ), S ( y ) \bigr), d_{b} \bigl( f ( z ), S ( z ), S ( z ) \bigr), \\ &\quad \frac{1}{3 s} \bigl[ d_{b} \bigl( S ( x ), f ( y ), f ( z ) \bigr) + d_{b} \bigl( f ( x ), S ( y ), f ( z ) \bigr) \\ &\quad {}+ d_{b} \bigl( S ( z ), f ( y ), f ( x ) \bigr) \bigr]\biggr\} . \end{aligned}$$

Assume that the pair (\(f, S \)) is \(\alpha _{s}\)-compatible. Then the mappings f and S have the coincidence fixed point in X provided that fg is a triangular weakly \(\alpha _{s}\)-admissible mapping with respect to S. Moreover, if \(\alpha _{s} ( Sv, Sv, Sv ) \geq s^{2}\), then f, S has a common point v.

Corollary 2.24

Let \(f,g,h\), and S be self-mappings defined on an \(\alpha _{s}\)-regular and \(\alpha _{s}\)-complete tripled metric space (\(X, d_{b},s\)) such that \(f(X),g(X),h(X)\subseteq S(X)\), and \(S(X)\) is a closed subset of X. Suppose that, for all \(x,y,z\in X\) with \(\alpha _{s} (Sx,Sy,Sz)\geq s^{2}\), there exist \(F\in \mathcal{F}_{s}\), and \(r>0\) such that \(d_{b} ( f(x),g(y),h(z) ) >0\), then \(r+F ( s d_{b} ( f(x),g(y),h(z) ) ) \leq F ( M(x,y,z) )\), where

$$\begin{aligned} M ( x, y, z ) &=\max \biggl\{ d_{b} \bigl( S ( x ), S ( y ), S ( z ) \bigr), d_{b} \bigl( f ( x ), S ( x ), S ( x ) \bigr), \\ &\quad d_{b} \bigl( g ( y ), S ( y ), S ( y ) \bigr), d_{b} \bigl( h ( z ), S ( z ), S ( z ) \bigr), \\ &\quad \frac{1}{3 s} \bigl[ d_{b} \bigl( S ( x ), g ( y ), h ( z ) \bigr) + d_{b} \bigl( f ( x ), S ( y ), h ( z ) \bigr) \\ &\quad {}+ d_{b} \bigl( S ( z ), g ( y ), f ( x ) \bigr) \bigr]\biggr\} . \end{aligned}$$

Assume that the pairs (\(f, S \)), (\(g, S \)), and (\(h, S \)) are weakly compatible and the triple of mappings (\(f, g, h \)) is triangular weakly \(\alpha _{s}\)-admissible with respect to S. Then the pairs (\(f, S \)), (\(g, S \)), and (\(h, S \)) have the coincidence point v in X. Moreover, if \(\alpha _{s} ( Sv, Sv, Sv ) \geq s^{2}\), then v is a coincidence point of f, g, h, and S.

Corollary 2.25

Let f and S be self-mappings defined on an \(\alpha _{s}\)-regular and \(\alpha _{s}\)-complete tripled metric space (\(X, d_{b},s\)) such that \(f(X)\subseteq S(X)\), and \(S(X)\) is a closed subset of X. Suppose that, for all \(x,y,z\in X\) with \(\alpha _{s} (Sx,Sy,Sz)\geq s^{2}\), there exist \(F\in \mathcal{F}_{s}\) and \(r>0\) such that \(d_{b} ( f(x),f(y),f(z) ) >0\), then \(r+F ( s d_{b} ( f(x),f(y),f(z) ) ) \leq F ( M(x,y,z) )\), where

$$\begin{aligned} M ( x, y, z ) &=\max \biggl\{ d_{b} \bigl( S ( x ), S ( y ), S ( z ) \bigr), d_{b} \bigl( f ( x ), S ( x ), S ( x ) \bigr), \\ &\quad d_{b} \bigl( f ( y ), S ( y ), S ( y ) \bigr), d_{b} \bigl( f ( z ), S ( z ), S ( z ) \bigr), \\ &\quad \frac{1}{3 s} \bigl[ d_{b} \bigl( S ( x ), f ( y ), f ( z ) \bigr) + d_{b} \bigl( f ( x ), S ( y ), f ( z ) \bigr) \\ &\quad {}+ d_{b} \bigl( S ( z ), f ( y ), f ( x ) \bigr) \bigr]\biggr\} . \end{aligned}$$

Assume that the pair (\(f, S \)) is weakly compatible and f is a triangular weakly \(\alpha _{s}\)-admissible mapping with respect to S. Then the pair (\(f, S \)) has the coincidence point v in X.

Corollary 2.26

Let \(f,g\), and h be self-mappings defined on a complete tripled metric space (\(X, d_{b},s\)). Suppose that, for all \(x,y,z\in X\) with \(\alpha _{s} (x,y,z)\geq s^{2}\), there exist \(F\in \mathcal{F}_{s}\) and \(r>0\) such that \(d_{b} ( f(x),g(y),h(z) ) >0\), then \(r+F ( s d_{b} ( f(x),g(y),h(z) ) ) \leq F ( M(x,y,z) )\), where

$$\begin{aligned} M ( x, y, z ) &=\max \biggl\{ d_{b} ( x, y, z ), d_{b} \bigl( f ( x ), x, x \bigr), \\ &\quad d_{b} \bigl( g ( y ), y, y \bigr), d_{b} \bigl( h ( z ), z, z \bigr), \\ &\quad \frac{1}{3 s} \bigl[ d_{b} \bigl( x, g ( y ), h ( z ) \bigr) + d_{b} \bigl( f ( x ), y, h ( z ) \bigr) \\ &\quad {}+ d_{b} \bigl( z, g ( y ), f ( x ) \bigr) \bigr]\biggr\} . \end{aligned}$$

Assume that the triple of mappings (\(f, g, h \)) is triangular weakly \(\alpha _{s}\)-admissible. Then f, g, and h have a common fixed point v in X provided that either f or g or h is \(\alpha _{s}\)-continuous, or X is \(\alpha _{s}\)-regular.

Theorem 2.27

Let f, g, h, \(S_{1}\), \(S_{2}\), and \(S_{3}\) be \(\alpha _{s}\)-continuous self-mappings defined on an \(\alpha _{s}\)-complete tripled b-metric space (\(X, d_{b},s\)) such that \(f(X)\subseteq S_{1} (X)\), \(g(X)\subseteq S_{2} (X)\), and \(h(X)\subseteq S_{3} (X)\). Suppose that, for all \(( x,y,z ) \in \lambda _{f,g,h, \alpha _{s}}\), there exist \(F\in \mathcal{F}_{s}\) and \(r>0\) such that

$$ r+F \bigl( s d_{b} \bigl( f ( x ), g ( y ), h ( z )\bigr)\bigr) \leq F \bigl( M_{i} ( x, y, z )\bigr) $$
(2.12)

holds for one of \(i =1,2,3,4,5\), where

$$ \begin{aligned} M_{1} ( x, y, z ) & = a_{1} d_{b} \bigl( S_{1} ( x ), S_{2} ( y ), S_{3} ( z ) \bigr) + a_{2} d_{b} \bigl( f ( x ), S_{2} ( x ), S_{3} ( x ) \bigr) \\ &\quad {} + a_{3} d_{b} \bigl( g ( y ), S_{1} ( y ), S_{3} ( y ) \bigr) + a_{4} d_{b} \bigl( h ( z ), S_{1} ( z ), S_{2} ( z ) \bigr) \\ &\quad {} + a_{5} \bigl[ d_{b} \bigl( S_{1} ( x ), g ( y ), h ( z ) \bigr) + d_{b} \bigl( f ( x ), S_{2} ( y ), h ( z ) \bigr) \\ &\quad {} + d_{b} \bigl( S_{3} ( z ), g ( y ), f ( x ) \bigr) \bigr] \end{aligned} $$

with \(a_{i} \geq 0\), \(i =1,2,3,4,5\), such that \(a_{1} + a_{2} + a_{3} + 3 a_{5} = s\),

$$ \begin{aligned} M_{2} ( x, y, z ) & = a_{1} d_{b} \bigl( S_{1} ( x ), S_{2} ( y ), S_{3} ( z ) \bigr) + a_{2} d_{b} \bigl( f ( x ), S_{2} ( x ), S_{3} ( x ) \bigr) \\ &\quad {} + a_{3} d_{b} \bigl( g ( y ), S_{1} ( y ), S_{3} ( y ) \bigr) + a_{4} d_{b} \bigl( h ( z ), S_{1} ( z ), S_{2} ( z ) \bigr) \end{aligned} $$

with \(a_{1} + a_{2} + a_{3} = s\),

$$ \begin{aligned} M_{3} ( x, y, z ) & = k \max \bigl\{ d_{b} \bigl( f ( x ), S_{2} ( x ), S_{3} ( x ) \bigr), d_{b} \bigl( g ( y ), S_{1} ( y ), S_{3} ( y ) \bigr), \\ &\quad d_{b} \bigl( h ( z ), S_{1} ( z ), S_{2} ( z ) \bigr) \bigr\} \end{aligned} $$

with \(k\in [0,1)\),

$$ \begin{aligned} M_{4} ( x, y, z ) & = a_{1} ( x, y, z ) d_{b} \bigl( S_{1} ( x ), S_{2} ( y ), S_{3} ( z ) \bigr) \\ &\quad {} + a_{2} ( x, y, z ) d_{b} \bigl( f ( x ), S_{2} ( x ), S_{3} ( x ) \bigr) \\ &\quad {} + a_{3} ( x, y, z ) d_{b} \bigl( g ( y ), S_{1} ( y ), S_{3} ( y ) \bigr) \\ &\quad {} + a_{4} d_{b} \bigl( h ( z ), S_{1} ( z ), S_{2} ( z ) \bigr) \\ &\quad{}+ a_{5} ( x, y, z )\bigl[ d_{b} \bigl( S_{1} ( x ), g ( y ), h ( z ) \bigr) \\ &\quad {} + d_{b} \bigl( f ( x ), S_{2} ( y ), h ( z ) \bigr) \\ &\quad {} + d_{b} \bigl( S_{3} ( z ), g ( y ), f ( x ) \bigr) \bigr] \end{aligned} $$

with \(a_{i} ( x, y, z )\), \(i =1,2,3,4,5\), are nonnegative functions such that

$$ \sup_{x, y, z\in X} \bigl[ a_{1} ( x, y, z ) + a_{2} ( x, y, z ) + a_{3} ( x, y, z ) + 3 a_{5} ( x, y, z ) \bigr] = s. $$

Suppose that the pairs (\(f, S_{1} \)), (\(g, S_{2} \)), and (\(h, S_{3} \)) are \(\alpha _{s}\)-compatible and the triples of mappings (\(f, g, h \)), \(( g, f, h )\), and (\(h, g, f \)) are triangular partially triple weakly \(\alpha _{s}\)-admissible with respect to \(S_{1}\), \(S_{2}\), and \(S_{3}\), respectively. Then the pairs (\(f, S_{1} \)), \(( g, S_{2} )\), and (\(h, S_{3} \)) have the coincidence point v in X. Moreover, if \(\alpha _{s} ( S_{1} ( v ), S_{2} ( v ), S_{3} ( v )) \geq s^{2}\), then v is a common fixed point of f, g, h, \(S_{1}\), \(S_{2}\), and \(S_{3}\).

Proof

In line with the beginning part of Theorem 2.18, for all \(( x, y, z ) \in \lambda _{f, g, h, \alpha _{s}}\) for some \(F\in \mathcal{F}_{s}\) and \(r >0\), from contractive condition (2.12) we get

$$ \begin{aligned}[b] F \bigl( s d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ) \bigr) & = F \bigl( s d_{b} \bigl( f ( x_{2 n} ), g ( x_{2 n+ 1} ), h ( x_{2 n+ 2} ) \bigr) \bigr) \\ & \leq F \bigl( M_{1} ( x_{2 n}, x_{2 n+ 1}, x_{2 n+ 2} ) \bigr) -r \end{aligned} $$
(2.13)

for all \(n\in \mathbb{N}\), where

$$ \begin{aligned} M_{1} ( x_{2 n}, x_{2 n+ 1}, x_{2 n+ 2} ) & = a_{1} d_{b} \bigl( S_{1} ( x_{2 n} ), S_{2} ( x_{2 n+ 1} ), S_{3} ( x_{2 n+ 2} ) \bigr) \\ &\quad {} + a_{2} d_{b} \bigl( f ( x_{2 n} ), S_{2} ( x_{2 n} ), S_{3} ( x_{2 n} ) \bigr) \\ &\quad {} + a_{3} d_{b} \bigl( g ( x_{2 n+ 1} ), S_{1} ( x_{2 n+ 1} ), S_{3} ( x_{2 n+ 1} ) \bigr) \\ &\quad {} + a_{4} d_{b} \bigl( h ( x_{2 n+ 2} ), S_{1} ( x_{2 n+ 2} ), S_{2} ( x_{2 n+ 2} ) \bigr) \\ &\quad {} + a_{5} \bigl[ d_{b} \bigl( S_{1} ( x_{2 n} ), g ( x_{2 n+ 1} ), h ( x_{2 n+ 2} ) \bigr) \\ &\quad {} + d_{b} \bigl( f ( x_{2 n} ), S_{2} ( x_{2 n+ 1} ), h ( x_{2 n+ 2} ) \bigr) \\ &\quad {} + d_{b} \bigl( S_{3} ( x_{2 n+ 2} ), g ( x_{2 n+ 1} ), f ( x_{2 n} ) \bigr) \bigr] \\ & = a_{1} d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 1} ) + a_{2} d_{b} ( J_{2 n+ 1}, J_{2 n}, J_{2 n- 1} ) \\ &\quad {} + a_{3} d_{b} ( J_{2 n+ 2}, J_{2 n+ 1}, J_{2 n} ) + a_{4} d_{b} ( J_{2 n+ 2}, J_{2 n+ 2}, J_{2 n+ 2} ) \\ &\quad {} + a_{5} \bigl[ d_{b} ( J_{2 n}, J_{2 n+ 2}, J_{2 n+ 2} ) + d_{b} ( J_{2 n+ 1}, J_{2 n+ 1}, J_{2 n+ 2} ) \\ &\quad {} + d_{b} ( J_{2 n+ 1}, J_{2 n+ 2}, J_{2 n+ 1} ) \bigr] \\ & \leq a_{1} d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 1} ) + a_{2} d_{b} ( J_{2 n+ 1}, J_{2 n}, J_{2 n- 1} ) \\ &\quad {} + a_{3} d_{b} ( J_{2 n+ 2}, J_{2 n+ 1}, J_{2 n} ) \\ &\quad {} + a_{5} \bigl[3 d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ) \bigr] \\ & =( a_{1} + a_{3} + 3 a_{5} ) d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ) \\ &\quad {} + a_{2} d_{b} ( J_{2 n+ 1}, J_{2 n}, J_{2 n- 1} ). \end{aligned} $$

Now from (2.13) we have

$$ \begin{aligned}[b] F \bigl( s d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ) \bigr) & = F \bigl( ( a_{1} + a_{3} + 3 a_{5} ) d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ) \\ &\quad {} + a_{2} d_{b} ( J_{2 n+ 1}, J_{2 n}, J_{2 n- 1} ) \bigr) -r. \end{aligned} $$
(2.14)

Since F is strictly increasing, (2.14) implies

$$ \begin{aligned} s d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ) & \leq ( a_{1} + a_{3} + 3 a_{5} ) d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ) \\ &\quad {} + a_{2} d_{b} ( J_{2 n+ 1}, J_{2 n}, J_{2 n- 1} ). \end{aligned} $$

So

$$ \begin{aligned} ( s- a_{1} - a_{3} - 3 a_{5} ) d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ) & \leq a_{2} d_{b} ( J_{2 n+ 1}, J_{2 n}, J_{2 n- 1} ). \end{aligned} $$

Hence

$$ \begin{aligned} d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ) & \leq \frac{a_{2}}{s- a_{1} - a_{3} - 3 a_{5}} d_{b} ( J_{2 n- 1}, J_{2 n}, J_{2 n+ 1} ). \end{aligned} $$

Since \(a_{1} + a_{2} + a_{3} + 3 a_{5} = s\), therefore \(d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ) \leq d_{b} ( J_{2 n- 1}, J_{2 n}, J_{2 n+ 1} )\). Thus from (2.14) we obtain

$$ \begin{aligned}[b] F \bigl( s d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ) \bigr) & \leq F \bigl( d_{b} ( J_{2 n- 1}, J_{2 n}, J_{2 n+ 1} ) \bigr) -r \end{aligned} $$
(2.15)

for all \(n\in \mathbb{N}\). Similarly,

$$ \begin{aligned}[b] F \bigl( s d_{b} ( J_{2 n- 1}, J_{2 n}, J_{2 n+ 1} ) \bigr) & \leq F \bigl( d_{b} ( J_{2 n- 2}, J_{2 n- 1}, J_{2 n} ) \bigr) -r \end{aligned} $$
(2.16)

for all \(n\in \mathbb{N}\). Hence, from (2.15) and (2.16), we have

$$ \begin{aligned}[b] F \bigl( s d_{b} ( J_{n}, J_{n+ 1}, J_{n+ 2} ) \bigr) & = F \bigl( d_{b} ( J_{n- 1}, J_{n}, J_{n+ 1} ) \bigr) -r. \end{aligned} $$
(2.17)

Inequality (2.17) leads to remark that \(\{ x_{n} \}\) is a Cauchy sequence, and the remaining part of the proof can easily be followed from the finishing part of the proof of Theorem 2.18. For \(M_{2} ( x, y, z )\), in line with the beginning part of the proof of Theorem 2.18, for all \(( x, y, z ) \in \lambda _{f, g, h, \alpha _{s}}\), for some \(F\in \mathcal{F}_{s}\), and \(r >0\), from contractive condition (2.11), we get

$$ \begin{aligned}[b] F \bigl( s d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ) \bigr) & = F \bigl( s d_{b} \bigl( f ( x_{2 n} ), g ( x_{2 n+ 1} ), h ( x_{2 n+ 2} ) \bigr) \bigr) \\ & \leq F \bigl( M_{2} ( x_{2 n}, x_{2 n+ 1}, x_{2 n+ 2} ) \bigr) -r \end{aligned} $$
(2.18)

for all \(n\in \mathbb{N} \cup \{0\}\), where

$$ \begin{aligned} M_{2} ( x_{2 n}, x_{2 n+ 1}, x_{2 n+ 2} ) & = a_{1} d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 1} ) + a_{2} d_{b} ( J_{2 n+ 1}, J_{2 n}, J_{2 n- 1} ) \\ &\quad{}+ a_{3} d_{b} ( J_{2 n+ 2}, J_{2 n+ 1}, J_{2 n} ) \\ &\quad{}+ a_{4} d_{b} ( J_{2 n+ 2}, J_{2 n+ 2}, J_{2 n+ 2} ) \\ & \leq a_{1} d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ) + a_{2} d_{b} ( J_{2 n+ 1}, J_{2 n}, J_{2 n- 1} ) \\ &\quad{}+ a_{3} d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ) \\ & =( a_{1} + a_{3} ) d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ) \\ &\quad{}+ a_{2} d_{b} ( J_{2 n+ 1}, J_{2 n}, J_{2 n- 1} ). \end{aligned} $$

From (2.18), we have

$$ \begin{aligned}[b] F \bigl( s d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ) \bigr) & \leq F \bigl( ( a_{1} + a_{3} ) d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ) \bigr) \\ &\quad{}+ a_{2} d_{b} ( J_{2 n+ 1}, J_{2 n}, J_{2 n- 1} ) -r. \end{aligned} $$
(2.19)

Since F is strictly increasing, (2.19) implies

$$ \begin{aligned} s d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ) & \leq ( a_{1} + a_{2} ) d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ) \\ &\quad{}+ a_{2} d_{b} ( J_{2 n+ 1}, J_{2 n}, J_{2 n- 1} ), \end{aligned} $$

so \(( s- a_{1} - a_{3} ) d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ) \leq a_{2} d_{b} ( J_{2 n- 1}, J_{2 n}, J_{2 n+ 1} )\). Hence

$$ \begin{aligned} d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ) & \leq \frac{a_{2}}{s- a_{1} - a_{3}} d_{b} ( J_{2 n- 1}, J_{2 n}, J_{2 n+ 1} ). \end{aligned} $$

Thus, from (2.19), we obtain

$$ \begin{aligned}[b] F \bigl( s d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ) \bigr) & \leq F \bigl( d_{b} ( J_{2 n- 1}, J_{2 n}, J_{2 n+ 1} ) \bigr) -r \end{aligned} $$
(2.20)

for all \(n\in \mathbb{N}\). Similarly,

$$ \begin{aligned}[b] F \bigl( s d_{b} ( J_{2 n- 1}, J_{2 n}, J_{2 n+ 1} ) \bigr) & \leq F \bigl( d_{b} ( J_{2 n- 2}, J_{2 n- 1}, J_{2 n} ) \bigr) -r \end{aligned} $$
(2.21)

for all \(n\in \mathbb{N}\). Hence, from (2.20) and (2.21), we have

$$ \begin{aligned}[b] F \bigl( s d_{b} ( J_{n}, J_{n+ 1}, J_{n+ 2} ) \bigr) & \leq F \bigl( d_{b} ( J_{n- 1}, J_{n}, J_{n+ 1} ) \bigr) -r. \end{aligned} $$
(2.22)

Inequality (2.22) leads to remark that \(\{ J_{n} \}\) is a Cauchy sequence, and the remaining part of the proof can easily be followed from the finishing part of the proof of Theorem 2.18. For \(M_{3} ( x, y, z )\), in line with the beginning part of the proof of Theorem 2.18, for all \(( x, y, z ) \in \lambda _{f, g, h, \alpha _{s}}\), for some \(F\in \mathcal{F}_{s}\), and \(r >0\), from contractive condition (2.12), we get

$$ \begin{aligned}[b] F \bigl( s d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ) \bigr) & = F \bigl( s d_{b} \bigl( f ( x_{2 n} ), g ( x_{2 n+ 1} ), h ( x_{2 n+ 2} ) \bigr) \bigr) \\ & \leq F \bigl( M_{3} ( x_{2 n}, x_{2 n+ 1}, x_{2 n+ 2} ) \bigr) -r \end{aligned} $$
(2.23)

for all \(n\in \mathbb{N} \cup \{0\}\), where

$$ \begin{aligned}[b] M_{3} ( x_{2 n}, x_{2 n+ 1}, x_{2 n+ 2} ) & = k \max \bigl\{ d_{b} ( J_{2 n- 1}, J_{2 n}, J_{2 n+ 1} ), d_{b} ( J_{2 n+ 2}, J_{2 n+ 1}, J_{2 n} ),0\bigr\} \\ & = k \max \bigl\{ d_{b} ( J_{2 n- 1}, J_{2 n}, J_{2 n+ 1} ), d_{b} ( J_{2 n+ 2}, J_{2 n+ 1}, J_{2 n} ) \bigr\} . \end{aligned} $$

If

$$ \max \bigl\{ d_{b} ( J_{2 n- 1}, J_{2 n}, J_{2 n+ 1} ), d_{b} ( J_{2 n+ 2}, J_{2 n+ 1}, J_{2 n} ) \bigr\} = d_{b} ( J_{2 n+ 2}, J_{2 n+ 1}, J_{2 n} ), $$

then from (2.23) we have \(F ( s d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ) ) \leq F ( k d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ) ) -r\). Since F is strictly increasing, we have \(s d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ) < k d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} )\). It is a contradiction. Thus we have

$$ \begin{aligned} F \bigl( s d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ) \bigr) & \leq F \bigl( k d_{b} ( J_{2 n- 1}, J_{2 n}, J_{2 n+ 1} ) \bigr) -r, \end{aligned} $$

and \(s d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ) \leq k d_{b} ( J_{2 n- 1}, J_{2 n}, J_{2 n+ 1} )\). So

$$ \begin{aligned} d_{b} ( J_{2 n}, J_{2 n+ 1}, J_{2 n+ 2} ) & \leq \frac{k}{s} d_{b} ( J_{2 n- 1}, J_{2 n}, J_{2 n+ 1} ). \end{aligned} $$

The emaining part of the proof can easily be followed from the proof of Theorem 2.18. Similar arguments hold from \(M_{4} ( x, y, z )\). □

Theorem 2.28

Let f, g, h, \(S_{1}\), \(S_{2}\), and \(S_{3}\) be self-mappings defined on a complete tripled b-metric space (\(X, d_{b},s\)) such that \(f(X)\subseteq S_{1} (X)\), \(g(X)\subseteq S_{2} (X)\), and \(h(X)\subseteq S_{3} (X)\). If there exist \(F\in \mathcal{F}_{s}\) and \(r>0\) such that \(d_{b} ( f(x),g(y),h(z) ) >0\), then

$$ r+F ( s d_{b} \bigl( f ( x ), g ( y ), h ( z ) \bigr) \leq F \bigl( M ( x, y, z )\bigr) $$

for all \(x, y, z\in X\). Then f, g, h, \(S_{1}\), \(S_{2}\), and \(S_{3}\) have a unique common fixed point in X provided that \(S_{1}\), \(S_{2}\), and \(S_{3}\) are continuous and pairs (\(f, S_{1} \)), (\(g, S_{2} \)), and (\(h, S_{3} \)) are compatible.

Proof

The arguments follow the same lines as in the proof of Theorem 2.18. □

Application to a system of integral equations

Let \(X = C ([0,1], \mathbb{R})\) be the space of all continuous real-valued functions defined on [\(0,1\)]. Let \(d_{b}: X\times X\times X\rightarrow \mathbb{R}_{0}^{+}\) be defined

$$ d_{b} ( u, v, w ) =\max \Bigl\{ \sup_{t\in [0,1]} \bigl\vert u ( t ) -v ( t ) \bigr\vert ^{2}, \sup_{t\in [0,1]} \bigl\vert u ( t ) -w ( t ) \bigr\vert ^{2}, \sup _{t\in [0,1]} \bigl\vert v ( t ) -w ( t ) \bigr\vert ^{2} \Bigr\} $$

for all \(u, v, w\in C ( [ 0,1 ], \mathbb{R} )\), and define \(\alpha _{s}: X\times X\times X\rightarrow \mathbb{R}_{0}^{+}\) by \(\alpha _{s} ( u, v, w )= s^{2}\) for all \(u, v, w\in X\). Obviously, (\(X, d_{b}, s \)) is an \(\alpha _{s}\)-complete tripled b-metric space. We will apply Theorem 2.18 to show the existence of a common solution of the system of Volterra-type integral equations given by

$$ \begin{gathered} u ( t ) = p ( t ) + \int _{0}^{t} K \bigl( t, r, S_{1} \bigl( u ( t )\bigr)\bigr) dr, \\ v ( t ) = p ( t ) + \int _{0}^{t} J \bigl( t, r, S_{2} \bigl( v ( t )\bigr)\bigr) dr, \\ w ( t ) = p ( t ) + \int _{0}^{t} I \bigl( t, r, S_{3} \bigl( w ( t )\bigr)\bigr) dr \end{gathered} $$
(3.1)

for all \(t\in [0,1]\), where \(p:[0,1] \rightarrow \mathbb{R}\) is a continuous function and \(K, J, I:[0,1] \times [0,1] \times X\rightarrow \mathbb{R}\) are lower semi-continuous operators. Now, we prove the following theorem to ensure the existence of solution for the system of integral equations.

Theorem 3.1

Let \(X=C([0,1], \mathbb{R)}\) and define the mappings \(f,g,h:X\rightarrow X\) by

$$ \begin{gathered} f \bigl( u ( t )\bigr) = p ( t ) + \int _{0}^{t} K \bigl( t, r, S_{1} \bigl( u ( t )\bigr)\bigr) dr, \\ g \bigl( v ( t )\bigr) = p ( t ) + \int _{0}^{t} J \bigl( t, r, S_{2} \bigl( v ( t )\bigr)\bigr) dr, \\ h \bigl( w ( t )\bigr) = p ( t ) + \int _{0}^{t} I \bigl( t, r, S_{3} \bigl( w ( t )\bigr)\bigr) dr \end{gathered} $$

for all \(t\in [0,1]\). Assume that the following conditions are satisfied.

  • There exists a continuous function \(\phi _{i}: X\rightarrow \mathbb{R}_{0}^{+}\), \(i =1,2,3\), such that

    $$ \begin{gathered} \bigl\vert K ( t, r, S_{1} ) -J ( t, r, S_{2} ) \bigr\vert \leq \phi _{1} ( r ) \bigl\vert S_{1} \bigl( u ( t )\bigr) - S_{2} \bigl( v ( t )\bigr) \bigr\vert , \\ \bigl\vert K ( t, r, S_{1} ) -I ( t, r, S_{3} ) \bigr\vert \leq \phi _{2} ( r ) \bigl\vert S_{1} \bigl( u ( t )\bigr) - S_{3} \bigl( w ( t )\bigr) \bigr\vert , \\ \bigl\vert J ( t, r, S_{2} ) -I ( t, r, S_{3} ) \bigr\vert \leq \phi _{3} ( r ) \bigl\vert S_{2} \bigl( v ( t ) \bigr) - S_{3} \bigl( w ( t )\bigr) \bigr\vert \end{gathered} $$

    for each \(t, r\in [0,1]\) and \(S_{1}\), \(S_{2}\), and \(S_{3} \in X\);

  • There exists \(\tau >0\) such that

    $$ \int _{0}^{t} \phi _{1} ( r ) dr, \int _{0}^{t} \phi _{2} ( r ) dr, \int _{0}^{t} \phi _{3} ( r ) dr\leq \sqrt{\frac{e^{-\tau }}{s}}. $$

Then the system of integral Eqs. (3.1) has a solution.

Proof

By assumptions (i) and (ii), we have

$$ \begin{aligned} d_{b} \bigl( f \bigl( u ( t )\bigr), g \bigl( v ( t ) \bigr), h \bigl( w ( t )\bigr) \bigr) & =\max \Bigl\{ \sup_{t\in [0,1]} \bigl\vert f \bigl( u ( t )\bigr) -g \bigl( v ( t )\bigr) \bigr\vert ^{2}, \\ &\quad \sup_{t\in [0,1]} \bigl\vert g \bigl( v ( t )\bigr) -h \bigl( w ( t )\bigr) \bigr\vert ^{2}, \\ &\quad \sup_{t\in [0,1]} \bigl\vert f \bigl( u ( t )\bigr) -h \bigl( w ( t )\bigr) \bigr\vert ^{2} \Bigr\} , \end{aligned} $$

where

$$\begin{aligned}& \begin{aligned} \sup_{t\in [0,1]} \bigl\vert f \bigl( u ( t ) \bigr) -g \bigl( v ( t )\bigr) \bigr\vert ^{2}& = \biggl( \sup _{t\in [0,1]} \int _{0}^{t} \bigl\vert K \bigl( t, r, S_{1} \bigl( u ( t ) \bigr) \bigr) - J \bigl( t, r, S_{2} \bigl( v ( t )\bigr)\bigr) \bigr\vert dr \biggr)^{2} \\ &\leq \biggl( \sup_{t\in [0,1]} \int _{0}^{t} \phi _{1} ( r ) \bigl\vert S_{1} \bigl( u ( t )\bigr) - S_{2} \bigl( v ( t ) \bigr) \bigr\vert dr \biggr)^{2} \\ &\leq \biggl( \sqrt{\sup_{t\in [0,1]} \bigl\vert S_{1} \bigl( u ( t )\bigr) - S_{2} \bigl( v ( t )\bigr) \bigr\vert ^{2}} \int _{0}^{t} \phi _{1} ( r ) dr \biggr)^{2} \\ &= \sup_{t\in [0,1]} \bigl\vert S_{1} \bigl( u ( t ) \bigr) - S_{2} \bigl( v ( t )\bigr) \bigr\vert ^{2} \biggl( \int _{0}^{t} \phi _{1} ( r ) dr \biggr)^{2}, \end{aligned} \\& \sup_{t\in [0,1]} \bigl\vert g \bigl( v ( t )\bigr) -h \bigl( w ( t )\bigr) \bigr\vert ^{2} \leq \sup_{t\in [0,1]} \bigl\vert S_{2} \bigl( v ( t )\bigr) - S_{3} \bigl( w ( t ) \bigr) \bigr\vert ^{2} \biggl( \int _{0}^{t} \phi _{2} ( r ) dr \biggr)^{2}, \\& \sup_{t\in [0,1]} \bigl\vert f \bigl( u ( t )\bigr) -h \bigl( w ( t )\bigr) \bigr\vert ^{2} \leq \sup_{t\in [0,1]} \bigl\vert S_{1} \bigl( u ( t )\bigr) - S_{3} \bigl( w ( t ) \bigr) \bigr\vert ^{2} \biggl( \int _{0}^{t} \phi _{3} ( r ) dr \biggr)^{2}. \end{aligned}$$

Consequently, we have

$$ \begin{aligned} d_{b} \bigl( f \bigl( u ( t )\bigr), g \bigl( v ( t ) \bigr), h \bigl( w ( t )\bigr) \bigr) &= \frac{e^{-\tau }}{s} \max \Bigl\{ \sup _{t\in [0,1]} \bigl\vert S_{1} \bigl( u ( t )\bigr) - S_{2} \bigl( v ( t )\bigr) \bigr\vert ^{2}, \\ &\quad \sup_{t\in [0,1]} \bigl\vert S_{2} \bigl( v ( t ) \bigr) - S_{3} \bigl( w ( t )\bigr) \bigr\vert ^{2}, \sup _{t\in [0,1]} \bigl\vert S_{1} \bigl( u ( t )\bigr) - S_{3} \bigl( w ( t )\bigr) \bigr\vert ^{2} \Bigr\} \\ & = \frac{e^{-\tau }}{s} d_{b} \bigl( S_{1} \bigl( u ( t ) \bigr), S_{2} \bigl( v ( t )\bigr), S_{3} \bigl( w ( t )\bigr) \bigr) \\ & \leq \frac{e^{-\tau }}{s} M \bigl( u ( t ), v ( t ), Sw ( t ) \bigr). \end{aligned} $$

Thus, we obtain

$$ s d_{b} \bigl( f \bigl( u ( t )\bigr), g \bigl( v ( t )\bigr), h \bigl( w ( t )\bigr) \bigr) \leq e^{-\tau } M \bigl( u ( t ), v ( t ), w ( t ) \bigr), $$

which implies that

$$ \tau + \ln \bigl( s d_{b} \bigl( f \bigl( u ( t )\bigr), g \bigl( v ( t )\bigr), h \bigl( w ( t )\bigr) \bigr) \bigr) \leq \ln \bigl( M \bigl( u ( t ), v ( t ), w ( t ) \bigr) \bigr). $$

For \(F ( r )=\ln r\), all the hypotheses of Theorem 2.28 are satisfied. Hence the system of integral equations has a unique common solution. □

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Ranjbar, G.K., Samei, M.E. Extraction new results of common fixed point theorems for \(({T}, {\alpha }_{{s}}, {F})\)-contraction of six mappings in a tripled b-metric space with an application of integral equations. J Inequal Appl 2020, 236 (2020). https://doi.org/10.1186/s13660-020-02503-9

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MSC

  • \(\alpha _{S}\)-complete tripled b-metric
  • (\(T, \alpha _{s}, F\))-contractions
  • Common fixed point

Keywords

  • 47H10
  • 54H25
  • 37C25