Novel results of $\alpha _{\ast }$-ψ-Λ-contraction multivalued mappings in F-metric spaces with an application

Abstract

The objective of this paper is to introduce a new motif of $\alpha _{\ast }$-ψ-Λ-contraction multivalued mappings, some novel fixed-point and coincidence-point results for this contraction will be investigated in the scope of F-metric spaces, and some examples are given to illustrate our main results and we derive the existence and uniqueness of a solution of a functional equation to support our main result.

1 Introduction and preliminaries

The conception of F-metric space (F-MS) was given by Jleli and Samet [21] in 2018 as a generalization of metric space (MS) [16], that has gained importance due to the development of the metric fixed-point theory; they proved that every metric space is an F-MS, but the converse is not true, confirming that F-MS is more general than the metric space with the help of concrete examples, and compared this concept with existing generalizations from the literature. They defined a natural topology $\tau _{F}$ on these spaces and studied their topological properties. Moreover, a new fixed-point theorem of the Banach Contraction Principle (BCP) was established in the scope of F-MS. This article is arranged into four sections. The first section contains a short history of the literature, providing motivation for this article and some basic definitions that will help readers understand our results. In Sect. 2, new fixed-point theorems for $\alpha _{\ast }$-ψ-Λ-contraction multivalued mappings in the scope of F-MS and the given example will be discussed. In Sect. 3, the coincidence-point results for said contraction mappings in F-MS are investigated as consequences. Section 4 is concerned with an application of the said results to the functional equations in dynamic programming with its example.

Definition 1.1

([16])

A mapping $d:\Upsilon \times \Upsilon \rightarrow {}[ 0,\infty )$ on a nonempty set ϒ, satisfying the following conditions for all $\gamma ,\delta ,\kappa \in \Upsilon$,

$(d_{1})$:

$d ( \gamma ,\delta ) =0\Longleftrightarrow \gamma =\delta$;

$(d_{2})$:

$d ( \gamma ,\delta ) =d ( \delta ,\gamma )$;

$(d_{3})$:

$d ( \gamma ,\delta ) \leq d ( \gamma ,\kappa ) +d ( \kappa ,\delta )$,

is called a metric on ϒ and the pair $( \Upsilon ,d )$ is said to be a MS.

We start with a brief recollection of basic ideas and the facts of F-MS. First, let Ξ be the set of functions $\mathcal{L} : ( 0,\infty ) \rightarrow \mathbb{R}$ satisfying the following stipulations:

$( \Xi 1 )$:

$\mathcal{L}$ is nondecreasing, i.e., $0<\vartheta <\varsigma \Longrightarrow \mathcal{L} ( \vartheta ) \leq \mathcal{L} ( \varsigma )$.

$( \Xi 2 )$:

For every sequence $\{ \varsigma _{\zeta } \} \subset ( 0,\infty )$, we have

$$ \lim_{\zeta \rightarrow \infty } \varsigma _{\zeta }=0\quad \Longleftrightarrow\quad \lim_{\zeta \rightarrow \infty } \mathcal{L} ( \varsigma _{\zeta } ) =-\infty . $$

Definition 1.2

([21])

Let ϒ be a nonempty set and $\mathbb{Q} :\Upsilon \times \Upsilon \rightarrow {}[ 0,\infty )$ be a given mapping. We postulate that there exists $( \mathcal{L} ,\alpha )\in \Xi \times {}[ 0,\infty )$ such that,

$(\mathbb{Q}1)$:

$( \gamma ,\delta ) \in \Upsilon \times \Upsilon$, $\mathbb{Q} ( \gamma ,\delta ) =0$ ⟺ $\gamma =\delta$;

$(\mathbb{Q}2)$:

$\mathbb{Q} ( \gamma ,\delta ) =\mathbb{Q} ( \delta ,\gamma )$;

$(\mathbb{Q}3)$:

for every $( \gamma ,\delta ) \in \Upsilon \times \Upsilon$, $\forall\nu \in \mathbb{N}$, $\nu \geq 2$, and ∀ $( \varsigma _{i} ) _{i=1}^{\nu }\subset \Upsilon$, $( \varsigma _{1},\varsigma _{\nu } ) = ( \gamma ,\delta )$, we have

$$ \mathbb{Q} ( \gamma ,\delta ) >0\quad \Longrightarrow \quad \mathcal{L} \bigl( \mathbb{Q} ( \gamma ,\delta ) \bigr) \leq \mathcal{L} \Biggl( \sum _{i=1}^{\nu -1}\mathbb{Q} ( \varsigma _{i},\varsigma _{i+1} ) \Biggr) +\alpha . $$

Then, $\mathbb{Q}$ is said to be an F-M on ϒ, and the pair $( \Upsilon ,\mathbb{Q} )$ is said to be an F-MS.

Example 1.3

([21])

Let $\Upsilon =\mathbb{N}$, and let $\mathbb{Q} :\Upsilon \times \Upsilon \rightarrow {}[ 0,\infty )$ be the mapping defined by

$$ \mathbb{Q} ( \gamma ,\delta ) = \textstyle\begin{cases} ( \gamma -\delta ) ^{2}, & \text{if } ( \gamma ,\delta ) \in [ 0,3 ] \times [ 0,3 ] , \\ \vert \gamma -\delta \vert & \text{if } ( \gamma ,\delta ) \notin [ 0,3 ] \times [ 0,3 ] ,\end{cases} $$

for all $( \gamma ,\delta ) \in \Upsilon \times \Upsilon$, with $\mathcal{L} ( \varsigma ) =\ln ( \varsigma )$ and $\alpha =\ln ( 3 )$. Then, $( \Upsilon ,\mathbb{Q} )$ is an F-MS.

Example 1.4

([21])

Let $\Upsilon =\mathbb{N}$, and let $\mathbb{Q} :\Upsilon \times \Upsilon \rightarrow {}[ 0,\infty )$ be the mapping defined by

$$ \mathbb{Q} ( \gamma ,\delta ) = \textstyle\begin{cases} e^{ \vert \gamma -\delta \vert }, & \text{if }\gamma =\delta , \\ 0 & \text{if }\gamma \neq \delta ,\end{cases} $$

for all $( \gamma ,\delta ) \in \Upsilon \times \Upsilon$, with $\mathcal{L} ( \varsigma ) =\frac{-1}{\varsigma }$ and $\alpha =1$. Then, $( \Upsilon ,\mathbb{Q} )$ is an F-MS.

Definition 1.5

([21])

Let $( \Upsilon ,\mathbb{Q} )$ be an F-MS, then:

• (i) Let $\{ \gamma _{\zeta } \}$ be a sequence in ϒ; we say that $\{ \gamma _{\zeta } \}$ is F-convergent to $\gamma \in \Upsilon$ if $\{ \gamma _{\zeta } \}$ is convergent to γ with respect to the F-MS $\mathbb{Q}$.

• (ii) A sequence $\{ \gamma _{\zeta } \}$ is F-Cauchy if $\lim_{\zeta ,\eta \rightarrow \infty } \mathbb{Q} ( \gamma _{\zeta },\gamma _{\eta } ) =0$.

• (iii) We say that $( \Upsilon ,\mathbb{Q} )$ is F-complete if every F-Cauchy sequence in ϒ is F-convergent to an assured element in ϒ.

Theorem 1.6

([21])

Let $(\Upsilon ,\mathbb{Q} )$ be an F-MS, and let $\Gamma :\Upsilon \rightarrow \Upsilon$ be a mapping. We postulate that the following affirmations hold:

• (i) $( \Upsilon ,\mathbb{Q} )$ is F-complete,

• (ii) there exists $k\in ( 0,1 )$ such that

  $$ \mathbb{Q} \bigl( \Gamma ( \gamma ) ,\Gamma ( \delta ) \bigr) \leq k\mathbb{Q} ( \gamma ,\delta ) ,\quad \forall ( \gamma ,\delta ) \in \Upsilon \times \Upsilon . $$

Then, Γ has a unique fixed point $\gamma ^{\ast }\in \Upsilon$. Moreover, for any $\gamma _{0}\in \Upsilon$, the sequence $\{ \gamma _{\zeta } \} \subset \Upsilon$ defined by $\gamma _{\zeta +1}=\Gamma ( \gamma _{\zeta } )$, $\zeta \in \mathbb{N}$ is F-convergent.

Many writers used the motif of F-MS to investigate powerful fixed-point results; for instance, Alnaser et al. [4] defined relation theoretic contractions and proved some generalized fixed-point theorems in F-metric spaces. Hussain and Kanwal [20] considered the notion of α-ψ-contraction and presented some fixed- and coupled fixed-point results in the setting of F-MSs. Lateef and Ahmad [24] defined Dass and Gupta’s contraction in the context of F-MSs and then proved some new fixed-point theorems to generalize and elaborate several known literature results. Mitrović et al. [26] proved certain common fixed-point theorems and some consequences to obtain the results of Banach, Jungck, Reich, and Berinde in F-MSs with an application for dynamic programming. Hussain [19] introduced the idea of fractional convex-type contraction and established some new fixed-point results for Reich-type α-η-contraction and Kannan-type α-η-contraction mappings in F-MS. He derived some consequences for Suzuki-type contractions, orbitally T-complete, and orbitally continuous mappings.

BCP [12] appeared in 1922 as the basis of functional analysis and plays a main role in several branches of mathematics and applied sciences, which asserts that every contraction mapping defined in complete MS has a fixed point. In many directions, this principle has been extended and generalized either by relaxing the contractive stipulations or imposing some more stipulations on space. Jungck [22] studied coincidence and common fixed points of commuting mappings and improved the BCP. In [35], coincidence-point and common fixed-point theorems for a class of Ćirić–Suzuki hybrid contractions involving a multivalued and two single-valued maps in an MS are obtained. Coincidence-point theorems for Geraghty contraction mappings have been introduced in different spaces [27–29, 33, 34, 37–39].

Theorem 1.7

([12])

Let $(\Upsilon ,\mathbb{Q} )$ be a complete MS and $\Gamma :\Upsilon \longrightarrow \Upsilon$ be a contraction mapping, that is $\forall \gamma ,\delta \in \Upsilon$, and $k\in ( 0,1 )$,

$$ \mathbb{Q} (\Gamma \gamma ,\Gamma \delta )\leq k\mathbb{Q} ( \gamma ,\delta ). $$

Then, Γ has a unique fixed point.

In 1973, Geraghty [17] generalized BCP and established its fixed-point results on complete MS.

Theorem 1.8

([17])

Let $(\Upsilon ,\mathbb{Q} )$ be a complete MS and $\Gamma :\Upsilon \longrightarrow \Upsilon$ be a mapping such that ∀ $\gamma ,\delta \in$ ϒ, and $\beta \in \mho$,

$$ \mathbb{Q} (\Gamma \gamma ,\Gamma \delta )\leq \beta \bigl(\mathbb{Q} (\gamma ,\delta )\bigr)\mathbb{Q} (\gamma ,\delta ), $$

where ℧ is a class of functions $\beta :[0,\infty )\rightarrow {}[ 0,1)$ satisfying $\beta (\varsigma _{\zeta })\rightarrow 1$ ⇒ $\varsigma _{\zeta }\rightarrow 0$ as $\zeta \rightarrow \infty$.

Then, Γ has a unique fixed point $\gamma ^{\ast }\in \Upsilon$.

In 2013, Cho et al. [14] presented the notion of α-Geraghty contraction-type mappings and deduced the unique fixed-point theorems for such mappings in a complete MS. In 2014, Popescu [31] opened a wide field in fixed-point theory by defining the concepts of α-orbital and triangular α-orbital admissible mappings and verified the unique fixed-point theorems for the said mappings, which are generalizations of α-Geraghty contraction-type mappings. In 2012, Wardowski [36] introduced the definition of F-contraction and proved fixed-point results as a generalization of the BCP in a complete MS, see also [1, 2, 5–7, 11, 18].

Definition 1.9

([31])

Let $\Gamma :\Upsilon \rightarrow \Upsilon$ be a map and $\alpha :\Upsilon \times \Upsilon \rightarrow \mathbb{R}$ be a function. Then, Γ is said to be α-orbital admissible if $\alpha (\gamma ,\Gamma \gamma )\geq 1$ implies $\alpha (\Gamma \gamma ,\Gamma ^{2}\gamma )\geq 1$.

Definition 1.10

([31])

Let $\Gamma :\Upsilon \rightarrow \Upsilon$ be a map and $\alpha :\Upsilon \times \Upsilon \rightarrow \mathbb{R}$ be a function. Then, Γ is said to be triangular α-orbital admissible if Γ is α-orbital admissible and $\alpha (\gamma ,\delta )\geq 1$ and $\alpha (\delta ,\Gamma \delta )\geq 1$ imply $\alpha (\gamma ,\Gamma \delta )\geq 1$.

Lemma 1.11

([31])

Let $\Gamma :\Upsilon \rightarrow \Upsilon$ be a triangular α-orbital admissible mapping. Assume that there exists $\gamma _{1}\in \Upsilon$ such that $\alpha (\gamma _{1},\Gamma \gamma _{1})\geq 1$. Define a sequence $\{ \gamma _{\zeta } \}$ by $\gamma _{\zeta +1}=\Gamma \gamma _{\zeta }$. Then, we have $\alpha ( \gamma _{\zeta },\gamma _{\eta } ) \geq 1$ for all $\zeta ,\eta \in \mathbb{N}$ with $\zeta <\eta$.

Definition 1.12

([23])

Let ϒ be a set. Assume that $\Im :\Upsilon \rightarrow \Upsilon$ and $\Gamma :\Upsilon \rightarrow 2^{\Upsilon }$. If $w=\Im \gamma \in \Gamma \gamma$ for some $\gamma \in \Upsilon$, then γ is called a coincidence point of ℑ and Γ, and w is called a point of coincidence of ℑ and Γ.

Mappings ℑ and Γ are called weakly compatible if $\Im \gamma \in \Gamma \gamma$ for some $\gamma \in \Upsilon$ implies $\Im \Gamma ( \gamma ) \subseteq \Gamma \Im ( \gamma )$.

Proposition 1.13

([23])

Let ϒ be a set. Assume that $\Im :\Upsilon \rightarrow \Upsilon$ and $\Gamma :\Upsilon \rightarrow 2^{\Upsilon }$ are weakly compatible mappings. If ℑ and Γ have a unique point of coincidence $w=\Im \gamma \in \Gamma \gamma$, then w is the unique common fixed point of ℑ and Γ.

Definition 1.14

([9])

Let $(\Upsilon ,d)$ be an MS. Let $CB(\Upsilon )$ be the family of all nonempty closed and bounded subsets of ϒ. Let $H:CB ( \Upsilon ) \times CB ( \Upsilon ) \rightarrow {}[ 0,\infty )$ be a function defined by

$$ H ( A,B ) =\max \Bigl\{ \sup_{\gamma \in A} \mathbb{Q} ( \gamma ,B ) ,\sup_{\delta \in B} \mathbb{Q} ( A,\delta ) \Bigr\} \quad \text{for all }A,B\in CB ( \Upsilon ) , $$

where $\mathbb{Q} ( \gamma ,B ) =\inf \{ d ( \gamma ,\delta ) ,\delta \in B \}$. Then, H defines a metric on $CB(\Upsilon )$ called the Hausdorff metric induced by d.

Asif et al. [10] obtain fixed points and common fixed-point results for Reich-type F-contractions for both single and set-valued mappings in F-MSs. Alansari et al. [3] studied a few fuzzy fixed-point theorems and discussed the corresponding fixed-point theorems of multivalued and single-valued mappings on F-complete F-MSs.

Lemma 1.15

([3])

Let A and B be nonempty closed and compact subsets of an F-metric space $( \Upsilon ,\mathbb{Q} )$. If $a\in A$, then $\mathbb{Q} ( a,B ) \leq H_{\mathbb{Q} } ( A,B )$.

Let Ψ be the family of nondecreasing functions $\psi :[0,\infty )\rightarrow {}[ 0,\infty )$ such that $\sum_{n=1}^{\infty }\psi ^{n} ( t ) <+\infty$, $\forall t>0$, where $\psi ^{n}$ is the $n{th}$ iterate of ψ.

Lemma 1.16

([8])

Let $\psi \in \Psi$. Then,

• 1. $\psi ( t ) < t$, $\forall t>0$;

• 2. $\psi ( 0 ) =0$.

Definition 1.17

([25])

Let $\Lambda : ( 0,\infty ) \longrightarrow ( 0,\infty )$ be a mapping verifying:

(Φ1):

Λ is nondecreasing;

(Φ2):

for each positive sequence $\{ t_{n} \}$,

$$ \lim_{n\rightarrow \infty }\Lambda (t_{n})=0\quad \text{if and only if}\quad \lim_{n\rightarrow \infty }t_{n}=0; $$

(Φ3):

Λ is continuous.

We denote by Φ the set of functions $\Lambda : ( 0,\infty ) \longrightarrow ( 0,\infty )$ satisfying the conditions $( \Phi 1 ) - ( \Phi 3 )$.

We modify the Definition 1.17 by adding a general condition $(\Phi 4)$ that is given in the following way:

(Φ4):

$\Lambda (\sum _{i=1}^{n}A_{i})\leq \sum _{i=1}^{n}\Lambda (A_{i})$, for all $A_{i}\in ( 0,\infty ), \ i=1,2,\cdots,n$,

where Φ is the set of functions $\Lambda : ( 0,\infty ) \longrightarrow ( 0,\infty )$ satisfying the conditions (Φ1), (Φ3), and (Φ4).

Example 1.18

Define the following functions for all $t\in ( 0,\infty )$,

• (1) $\Lambda ( t ) =at$, $a>0$;

• (2) $\Lambda ( t ) = \vert t \vert$.

Then $\Lambda \in \Phi$.

Now, we state and prove our main result.

2 Main results

In this section, we shall introduce a generalization of Geraghty contraction type mappings and establish some novel fixed-point theorems for $\alpha _{\ast }$-Λ-ψ-contraction multivalued mappings in the setting of F-MS.

Definition 2.1

Let $(\Upsilon ,\mathbb{Q} )$ be an F-MS, $\alpha :\Upsilon \times \Upsilon \rightarrow {}[ 0,\infty )$ be a function. A mapping $\Gamma :\Upsilon \rightarrow CB ( \Upsilon )$ is called a $\alpha _{\ast }$-Λ-ψ-contraction multivalued mapping if there exists $\beta \in \mho ,~\Lambda \in \Phi$ and $\psi \in \Psi$ such that

$$ \Lambda \bigl( \alpha _{\ast }(\Gamma \gamma ,\Gamma \delta )H_{\mathbb{Q} }(\Gamma \gamma ,\Gamma \delta ) \bigr) \leq \psi \bigl[ \Lambda \bigl( \beta \bigl( \aleph (\gamma ,\delta ) \bigr) \aleph (\gamma ,\delta ) \bigr) \bigr] , $$

(2.1)

where

$$ \aleph (\gamma ,\delta )=\max \biggl\{ \mathbb{Q} (\gamma ,\delta ),\mathbb{Q} (\gamma ,\Gamma \gamma ),\mathbb{Q} (\delta ,\Gamma \delta ),\frac{\mathbb{Q} (\gamma ,\Gamma \delta )+\mathbb{Q} (\delta ,\Gamma \gamma )}{2}\biggr\} , $$

for all $\gamma ,\delta \in \Upsilon$.

Theorem 2.2

Let $(\Upsilon , \mathbb{Q} )$ be an F-complete F-MS, $\alpha :\Upsilon \times \Upsilon \rightarrow {}[ 0,\infty )$ be a function, and $\Gamma :\Upsilon \rightarrow CB ( \Upsilon )$ a mapping. Postulating that the following affirmations hold:

• (1) Γ is $\alpha _{\ast }$-Λ-ψ-contraction;

• (2) Γ is triangular $\alpha _{\ast }$-orbital admissible;

• (3) there exists an $\gamma _{0}\in \Upsilon$ such that $\alpha _{\ast }(\gamma _{0},\Gamma \gamma _{0})\geq 1$;

• (4) Γ is continuous.

Then, Γ has a unique fixed point $\gamma ^{\ast }\in \Upsilon$.

Proof

Due to $( 3 )$, we define a sequence $\{\gamma _{n}\}_{n\in \mathbb{N} }$ by assuming that $\gamma _{1}\in \Gamma \gamma _{0}$ such that $\alpha ( \gamma _{0},\Gamma \gamma _{0} ) =\alpha ( \gamma _{0},\gamma _{1} ) \geq 1$ and $\gamma _{2}\in \Gamma \gamma _{1,}$ $\gamma _{3}\in \Gamma \gamma _{2},\ldots,\gamma _{\zeta +1}\in \Gamma \gamma _{\zeta }=\Gamma ^{\zeta }\gamma _{0}$, from $( 2 )$ and Lemma 1.11, we have $\alpha (\gamma _{\zeta },\gamma _{\zeta +1})\geq 1$ for all $\zeta \in \mathbb{N} \cup \{0\}$. Using Lemma 1.15, from $( 1 )$ and $( \Phi _{1} )$, we have

$$\begin{aligned} \Lambda \bigl( \mathbb{Q} ( \gamma _{\zeta },\gamma _{\zeta +1} ) \bigr) \leq & \Lambda \bigl( H_{\mathbb{Q} } ( \Gamma \gamma _{\zeta -1},\Gamma \gamma _{\zeta } ) \bigr) \\ \leq &\Lambda \bigl( \alpha _{\ast } ( \Gamma \gamma _{\zeta -1}, \Gamma \gamma _{\zeta } ) H_{\mathbb{Q} } ( \Gamma \gamma _{\zeta -1},\Gamma \gamma _{\zeta } ) \bigr) \\ \leq &\psi \bigl( \Lambda \bigl[ \beta \bigl( \aleph (\gamma _{\zeta -1}, \gamma _{\zeta }) \bigr) \aleph (\gamma _{\zeta -1},\gamma _{\zeta }) \bigr] \bigr) . \end{aligned}$$

(2.2)

We evaluate

$$\begin{aligned} \aleph ( \gamma _{\zeta -1},\gamma _{\zeta } ) =&\max \left \{ \textstyle\begin{array}{c} \mathbb{Q} ( \gamma _{\zeta -1},\gamma _{\zeta } ) ,\mathbb{Q} ( \gamma _{\zeta -1},\Gamma \gamma _{\zeta -1} ) , \\ \mathbb{Q} ( \gamma _{\zeta },\Gamma \gamma _{\zeta } ) ,\frac{\mathbb{Q} ( \gamma _{\zeta -1},\Gamma \gamma _{\zeta } ) +\mathbb{Q} ( \gamma _{\zeta },\Gamma \gamma _{\zeta -1} ) }{2}\end{array}\displaystyle \right \} \\ =&\max \left \{ \textstyle\begin{array}{c} \mathbb{Q} ( \gamma _{\zeta -1},\gamma _{\zeta } ) ,\mathbb{Q} ( \gamma _{\zeta -1},\gamma _{\zeta } ) ,\mathbb{Q} ( \gamma _{\zeta },\gamma _{\zeta +1} ) \\ \frac{\mathbb{Q} ( \gamma _{\zeta -1},\gamma _{\zeta +1} ) +\mathbb{Q} ( \gamma _{\zeta },\gamma _{\zeta } ) }{2}\end{array}\displaystyle \right \} , \end{aligned}$$

since

$$ \frac{\mathbb{Q} ( \gamma _{\zeta -1},\gamma _{\zeta +1} ) }{2}\leq \max \bigl\{ \mathbb{Q} ( \gamma _{\zeta -1},\gamma _{\zeta } ) ,\mathbb{Q} ( \gamma _{\zeta },\gamma _{\zeta +1} ) \bigr\} , $$

we conclude that

$$ \aleph ( \gamma _{\zeta -1},\gamma _{\zeta } ) =\max \bigl\{ \mathbb{Q} ( \gamma _{\zeta -1},\gamma _{\zeta } ) ,\mathbb{Q} ( \gamma _{\zeta },\gamma _{\zeta +1} ) \bigr\} . $$

Now, if $\max \{ \mathbb{Q} ( \gamma _{\zeta -1},\gamma _{\zeta } ) ,\mathbb{Q} ( \gamma _{\zeta },\gamma _{\zeta +1} ) \} =\mathbb{Q} ( \gamma _{\zeta },\gamma _{\zeta +1} )$ for $\zeta \geq 1$, then from (2.2), we obtain

$$ \Lambda \bigl( \mathbb{Q} ( \gamma _{\zeta },\gamma _{\zeta +1} ) \bigr) \leq \psi \bigl( \Lambda \bigl[ \beta \bigl( \mathbb{Q} ( \gamma _{\zeta },\gamma _{\zeta +1} ) \bigr) .\mathbb{Q} ( \gamma _{\zeta },\gamma _{\zeta +1} ) \bigr] \bigr) , $$

since $\beta \in \mho$ and from $( \Phi _{1} )$, we have

$$ \mathbb{Q} ( \gamma _{\zeta },\gamma _{\zeta +1} ) < \mathbb{Q} ( \gamma _{\zeta },\gamma _{\zeta +1} ), $$

which is a discrepancy as $\mathbb{Q} ( \gamma _{\zeta },\gamma _{\zeta +1} ) \geq 0$. Therefore,

$$ \max \bigl\{ \mathbb{Q} ( \gamma _{{\zeta }-1},\gamma _{{\zeta }} ) ,\mathbb{Q} ( \gamma _{{\zeta }},\gamma _{{\zeta }+1} ) \bigr\} =\mathbb{Q} ( \gamma _{{\zeta }-1},\gamma _{{\zeta }} ) , $$

by (2.2), we have

$$\begin{aligned} \Lambda \bigl( \mathbb{Q} ( \gamma _{{\zeta }},\gamma _{{\zeta }+1} ) \bigr) \leq &\Lambda \bigl( \alpha _{\ast } ( \Gamma \gamma _{{\zeta }-1},\Gamma \gamma _{{\zeta }} ) H_{\mathbb{Q} } ( \Gamma \gamma _{{\zeta }-1},\Gamma \gamma _{{\zeta }} ) \bigr) \\ \leq &\psi \bigl( \Lambda \bigl[ \beta \bigl( \mathbb{Q} ( \gamma _{{\zeta }-1},\gamma _{{\zeta }} ) \bigr) .\mathbb{Q} ( \gamma _{{\zeta }-1},\gamma _{{\zeta }} ) \bigr] \bigr) \\ \leq &\psi \bigl( \Lambda \bigl[ \beta \bigl( \mathbb{Q} ( \gamma _{{\zeta }-1},\gamma _{{\zeta }} ) \bigr) . \bigl( \alpha _{\ast } ( \Gamma \gamma _{{\zeta }-2},\Gamma \gamma _{{\zeta }-1} ) H_{\mathbb{Q} } ( T\gamma _{{\zeta }-2},T\gamma _{{\zeta }-1} ) \bigr) \bigr] \bigr) \\ \leq &\psi ^{2} \bigl( \Lambda \bigl[ \beta \bigl( \mathbb{Q} ( \gamma _{{\zeta }-1},\gamma _{{\zeta }} ) \bigr) \beta \bigl( \mathbb{Q} ( \gamma _{{\zeta }-2},\gamma _{{\zeta }-1} ) \bigr) \mathbb{Q} ( \gamma _{{\zeta }-2},\gamma _{{\zeta }-1} ) \bigr] \bigr) \\ &\vdots \\ \leq &\psi ^{\zeta } \bigl( \Lambda \bigl[ \beta \bigl( \mathbb{Q} ( \gamma _{{\zeta }-1},\gamma _{{\zeta }} ) \bigr) \beta \bigl( \mathbb{Q} ( \gamma _{{\zeta }-2},\gamma _{{\zeta }-1} ) \bigr) ...\beta \bigl( \mathbb{Q} ( \gamma _{0}, \gamma _{1} ) \bigr) \mathbb{Q} ( \gamma _{0},\gamma _{1} ) \bigr] \bigr) \\ =&\psi ^{\zeta } \Biggl( \Lambda \Biggl[ \Biggl( \prod _{i=1}^{{\zeta }}\beta \bigl( \mathbb{Q} ( \gamma _{i-1},\gamma _{i} ) \bigr) \Biggr) \mathbb{Q} ( \gamma _{0},\gamma _{1} ) \Biggr] \Biggr) \\ < &\psi ^{\zeta } \bigl( \Lambda \bigl[ \mathbb{Q} ( \gamma _{0},\gamma _{1} ) \bigr] \bigr) , \quad \text{for all }\zeta \in \mathbb{N} . \end{aligned}$$

Let $\epsilon >0$ be fixed and $( \mathcal{L} ,a ) \in \Xi \times {}[ 0,\infty )$ be such that $(\mathbb{Q} 3)$ is satisfied. By $( \Xi 2 )$, there exists $\eth >0$ such that

$$ 0< \varsigma < \eth \quad \text{implies}\quad \mathcal{L} ( \varsigma ) < \mathcal{L} ( \epsilon ) -a. $$

(2.3)

Let $\ell ( \epsilon ) \in \mathbb{N}$ such that $0<\sum_{\zeta \geq \ell ( \epsilon ) }\psi ^{\zeta } ( \Lambda [ \mathbb{Q} ( \gamma _{0},\gamma _{1} ) ] ) <\Lambda ( \eth )$.

Hence, by using properties of ψ, (2.3) and $( \Xi 1 )$, we have

$$\begin{aligned} \mathcal{L} \Biggl( \sum_{j=\zeta }^{\eta -1} \psi ^{j} \bigl( \Lambda \bigl[ \mathbb{Q} ( \gamma _{0},\gamma _{1} ) \bigr] \bigr) \Biggr) \leq & \mathcal{L} \biggl( \sum_{\zeta \geq \ell ( \epsilon ) }\psi ^{\zeta } \bigl( \Lambda \bigl[ \mathbb{Q} ( \gamma _{0},\gamma _{1} ) \bigr] \bigr) \biggr) \\ < & \mathcal{L} \bigl( \Lambda ( \epsilon ) \bigr) -a, \end{aligned}$$

(2.4)

where $\eta >\zeta >\ell ( \epsilon )$ with $\mathbb{Q} ( \gamma _{\zeta },\gamma _{\eta } ) >0$ using $( \mathbb{Q} 3 )$ and (2.4), we have

$$\begin{aligned} \mathcal{L} \bigl( \Lambda \bigl( \mathbb{Q} ( \gamma _{\zeta },\gamma _{\eta } ) \bigr) \bigr) \leq & \mathcal{L} \Biggl( \sum_{j=\zeta }^{\eta -1}\psi ^{j} \bigl( \Lambda \bigl[ \mathbb{Q} ( \gamma _{0},\gamma _{1} ) \bigr] \bigr) \Biggr) +a \\ \leq & \mathcal{L} \biggl( \sum_{\zeta \geq \ell ( \epsilon ) }\psi ^{\zeta } \bigl( \Lambda \bigl[ \mathbb{Q} ( \gamma _{0},\gamma _{1} ) \bigr] \bigr) \biggr) +a \\ < & \mathcal{L} \bigl( \Lambda ( \epsilon ) \bigr) -a+a \\ =& \mathcal{L} \bigl( \Lambda ( \epsilon ) \bigr) , \end{aligned}$$

which implies by $( \Xi 1 )$ and $( \Phi _{1} )$ that

$$ \mathbb{Q} ( \gamma _{\zeta },\gamma _{\eta } ) < \epsilon ,\quad \forall \eta >\zeta >\ell ( \epsilon ) . $$

Therefore, $\{ \gamma _{\zeta } \}$ is an F-Cauchy sequence in $( \Upsilon ,\mathbb{Q} )$. Since ϒ is F-complete, there exists $\gamma ^{\ast }\in \Upsilon$ such that $\gamma _{\zeta }\longrightarrow \gamma ^{\ast }$ as $\zeta \longrightarrow \infty$, implies

$$ \lim_{\zeta \rightarrow \infty } \mathbb{Q} \bigl( \gamma ^{\ast },\gamma _{\zeta } \bigr) =0. $$

(2.5)

Now, to show that $\gamma ^{\ast }\in \Gamma \gamma ^{\ast }$ is a fixed point of Γ, presume that $\mathbb{Q} ( \gamma ^{\ast },\Gamma \gamma ) >0$ such that $\gamma ^{\ast }\notin \Gamma \gamma ^{\ast }$ with $\alpha ( \gamma ^{\ast },\gamma _{\zeta } ) \geq 1$, $\zeta \in \mathbb{N}$. By $( \mathbb{Q} 3 )$ and $( \Phi 4 )$, we have

$$\begin{aligned} \mathcal{L} \bigl( \Lambda \bigl( \mathbb{Q} \bigl( \Gamma \gamma ^{\ast },\gamma ^{\ast } \bigr) \bigr) \bigr) \leq & \mathcal{L} \bigl( \Lambda \bigl( \mathbb{Q} \bigl( \Gamma \gamma ^{\ast },\Gamma \gamma _{\zeta } \bigr) +\mathbb{Q} \bigl( \Gamma \gamma _{\zeta },\gamma ^{\ast } \bigr) \bigr) \bigr) +a \\ \leq & \mathcal{L} \bigl( \Lambda \bigl( \mathbb{Q} \bigl( \Gamma \gamma ^{\ast },\Gamma \gamma _{\zeta } \bigr) \bigr) +\Lambda \bigl( \mathbb{Q} \bigl( \Gamma \gamma _{\zeta },\gamma ^{\ast } \bigr) \bigr) \bigr) +a \\ \leq & \mathcal{L} \bigl( \Lambda \bigl( \alpha _{\ast } \bigl( \Gamma \gamma ^{\ast },\Gamma \gamma _{\zeta } \bigr) H_{\mathbb{Q} } \bigl( \Gamma \gamma ^{\ast },\Gamma \gamma _{\zeta } \bigr) \bigr) +\Lambda \bigl( \mathbb{Q} \bigl( \Gamma \gamma _{\zeta },\gamma ^{\ast } \bigr) \bigr) \bigr) +a \\ \leq & \mathcal{L} \bigl( \psi \bigl( \Lambda \bigl[ \beta \bigl( \aleph \bigl( \gamma ^{\ast },\gamma _{\zeta } \bigr) \bigr) \aleph \bigl( \gamma ^{\ast },\gamma _{\zeta } \bigr) \bigr] \bigr) +\Lambda \bigl( \mathbb{Q} \bigl( \Gamma \gamma _{\zeta },\gamma ^{\ast } \bigr) \bigr) \bigr) +a \\ \leq & \mathcal{L} \bigl( \psi \bigl( \Lambda \bigl[ \aleph \bigl( \gamma ^{\ast },\gamma _{\zeta } \bigr) \bigr] \bigr) +\Lambda \bigl( \mathbb{Q} \bigl( \Gamma \gamma _{\zeta },\gamma ^{\ast } \bigr) \bigr) \bigr) +a \\ < & \mathcal{L} \bigl( \Lambda \bigl[ \aleph \bigl( \gamma ^{\ast }, \gamma _{\zeta } \bigr) \bigr] +\Lambda \bigl( \mathbb{Q} \bigl( \Gamma \gamma _{\zeta },\gamma ^{\ast } \bigr) \bigr) \bigr) +a, \end{aligned}$$

(2.6)

where

$$\begin{aligned} \aleph \bigl( \gamma ^{\ast },\gamma _{\zeta } \bigr) =&\max \left \{ \textstyle\begin{array}{c} \mathbb{Q} ( \gamma ^{\ast },\gamma _{\zeta } ) ,\mathbb{Q} ( \gamma ^{\ast },\Gamma \gamma ^{\ast } ) ,\mathbb{Q} ( \gamma _{\zeta },\Gamma \gamma _{\zeta } ) , \\ \frac{\mathbb{Q} ( \gamma ^{\ast },\Gamma \gamma _{\zeta } ) +\mathbb{Q} ( \Gamma \gamma ^{\ast },\gamma _{\zeta } ) }{2}\end{array}\displaystyle \right \} \\ =&\max \left \{ \textstyle\begin{array}{c} \mathbb{Q} ( \gamma ^{\ast },\gamma _{\zeta } ) ,\mathbb{Q} ( \gamma ^{\ast },\Gamma \gamma ^{\ast } ) ,\mathbb{Q} ( \gamma _{\zeta },\gamma _{\zeta +1} ) , \\ \frac{\mathbb{Q} ( \gamma ^{\ast },\gamma _{\zeta +1} ) +\mathbb{Q} ( \Gamma \gamma ^{\ast },\gamma _{\zeta } ) }{2}\end{array}\displaystyle \right \} , \end{aligned}$$

for all $\aleph ( \gamma ^{\ast },\gamma _{\zeta } )$ and using (2.5), $( \Phi 2 )$, and $( \Xi 2 )$, we obtain

$$ \lim_{\zeta \rightarrow \infty } \mathcal{L} \bigl( \Lambda \bigl( \aleph \bigl( \gamma ^{\ast },\gamma _{\zeta } \bigr) \bigr) + \Lambda \bigl( \mathbb{Q} \bigl( \Gamma \gamma _{\zeta },\gamma ^{\ast } \bigr) \bigr) \bigr) +a=-\infty , $$

which is a discrepancy. Hence, we have

$$ \mathbb{Q} \bigl( \gamma ^{\ast }, \Gamma \gamma \bigr) =0,\quad \text{that is }\gamma ^{\ast }\in \Gamma \gamma ^{\ast }. $$

(2.7)

For uniqueness, we postulate that $\gamma ^{\ast }$ and $\delta ^{\ast }$ are two fixed points of Γ in ϒ such that $\gamma ^{\ast }\neq$ $\delta ^{\ast }$. Then,

$$\begin{aligned} \Lambda \bigl( \mathbb{Q} \bigl( \gamma ^{\ast },\delta ^{\ast } \bigr) \bigr) =&\Lambda \bigl( \mathbb{Q} \bigl( \Gamma \gamma ^{\ast },\Gamma \delta ^{\ast } \bigr) \bigr) \\ \leq &\Lambda \bigl( \alpha _{\ast } \bigl( \Gamma \gamma ^{\ast },\Gamma \delta ^{\ast } \bigr) H_{\mathbb{Q} } \bigl( \Gamma \gamma ^{\ast },\Gamma \delta ^{\ast } \bigr) \bigr) \\ \leq &\psi \bigl( \Lambda \bigl[ \beta \bigl( \aleph \bigl( \gamma ^{\ast },\delta ^{\ast } \bigr) \bigr) \aleph \bigl( \gamma ^{\ast },\delta ^{\ast } \bigr) \bigr] \bigr) \\ < &\psi \bigl( \Lambda \bigl[ \aleph \bigl( \gamma ^{\ast },\delta ^{\ast } \bigr) \bigr] \bigr) \\ < &\Lambda \bigl[ \aleph \bigl( \gamma ^{\ast },\delta ^{\ast } \bigr) \bigr], \end{aligned}$$

where

$$\begin{aligned} \aleph \bigl( \gamma ^{\ast },\delta ^{\ast } \bigr) =&\max \left \{ \textstyle\begin{array}{c} \mathbb{Q} ( \gamma ^{\ast },\delta ^{\ast } ) ,\mathbb{Q} ( \gamma ^{\ast },\Gamma \gamma ^{\ast } ) ,\mathbb{Q} ( \delta ^{\ast },\Gamma \delta ^{\ast } ) , \\ \frac{\mathbb{Q} ( \gamma ^{\ast },\Gamma \delta ^{\ast } ) +\mathbb{Q} ( \delta ^{\ast },\Gamma \gamma ^{\ast } ) }{2}\end{array}\displaystyle \right \} = \mathbb{Q} \bigl( \gamma ^{\ast },\delta ^{\ast } \bigr) . \end{aligned}$$

From $( \Phi 1 )$, this yields that

$$ \mathbb{Q} \bigl( \gamma ^{\ast }, \delta ^{\ast } \bigr) < \mathbb{Q} \bigl( \gamma ^{\ast },\delta ^{\ast } \bigr), $$

a discrepancy. Therefore, $\gamma ^{\ast }=\delta ^{\ast }$ and Γ has a unique fixed point $\gamma ^{\ast }\in \Upsilon$. □

Example 2.3

Let $\Upsilon = \mathbb{R}$ be F-M and $\mathbb{Q}$ given by

$$ \mathbb{Q} ( \gamma ,\delta ) = \textstyle\begin{cases} ( \gamma -\delta ) ^{2} & \text{if } ( \gamma ,\delta ) \in {}[ 0,3]\times {}[ 0,3] ,\\ \vert \gamma -\delta \vert & \text{if } ( \gamma ,\delta ) \notin {}[ 0,3]\times {}[ 0,3], \end{cases} $$

with $\mathcal{L} ( \varsigma ) =\ln ( \varsigma )$ and $a=\ln ( 3 )$. Then, $( \Upsilon , \mathbb{Q} )$ is an F-complete F- MS. Define $\Gamma :\Upsilon \rightarrow CB ( \Upsilon )$ by

$$ \Gamma \gamma = \textstyle\begin{cases} \{ \frac{\gamma +1}{e^{10}} \} , & \text{if }\gamma \in {}[ 0,\infty ), \\ \{ 0 \} & \text{otherwise}, \end{cases} $$

and $\alpha :\Upsilon \times \Upsilon \rightarrow {}[ 0,\infty )$ by

$$ \alpha ( \gamma ,\delta ) = \textstyle\begin{cases} \frac{1}{\gamma }+1 & \text{if }\gamma ,\delta \in (0,\infty ) ,\\ 0 & \text{otherwise}, \end{cases} $$

let $\beta :\Upsilon \times \Upsilon \rightarrow {}[ 0,1)$ be as $\beta ( \gamma ,\delta ) =\frac{4 ( \gamma +1+e^{10} ) }{e^{20} ( \gamma +1 ) }$, $\Lambda ( t ) =t$ and $\psi ( t ) =\frac{3}{4}t$.

Now, for all $( \gamma ,\delta ) \in {}[ 0,3]\times {}[ 0,3]$, then

$$\begin{aligned}& \Lambda \bigl( \alpha _{\ast } ( \Gamma \gamma ,\Gamma \delta ) H_{\mathbb{Q} }(\Gamma \gamma ,\Gamma \delta ) \bigr) \\& \quad =\Lambda \biggl[ \frac{\gamma +1+e^{10}}{\gamma +1}\max \Bigl( \sup _{a\in \Gamma \delta } \mathbb{Q} ( a,\Gamma \delta ) ,\sup_{b\in \Gamma \delta } \mathbb{Q} ( \Gamma \gamma ,b ) \Bigr) \biggr] \\& \quad = \Lambda \biggl[ \frac{\gamma +1+e^{10}}{\gamma +1}\max \biggl( \sup _{a\in \Gamma \gamma } \mathbb{Q} \biggl( a, \biggl\{ \frac{\delta +1}{e^{10}} \biggr\} \biggr) ,\sup_{b\in \Gamma \delta } \mathbb{Q} \biggl( \biggl\{ \frac{\gamma +1}{e^{10}} \biggr\} ,b \biggr) \biggr) \biggr] \\& \quad =\Lambda \biggl[ \frac{\gamma +1+e^{10}}{\gamma +1}\max \biggl( \mathbb{Q} \biggl( \frac{\gamma +1}{e^{10}}, \biggl\{ \frac{\delta +1}{e^{10}} \biggr\} \biggr) ,\mathbb{Q} \biggl( \biggl\{ \frac{\gamma +1}{e^{10}} \biggr\} , \frac{\delta +1}{e^{10}} \biggr) \biggr) \biggr] \\& \quad =\Lambda \biggl[ \frac{\gamma +1+e^{10}}{\gamma +1}\max \biggl( \mathbb{Q} \biggl( \frac{\gamma +1}{e^{10}},\frac{\delta +1}{e^{10}} \biggr) ,\mathbb{Q} \biggl( \frac{\gamma +1}{e^{10}},\frac{\delta +1}{e^{10}} \biggr) \biggr) \biggr] \\& \quad \leq \frac{3}{4}\Lambda \biggl[ \frac{4 ( \gamma +1+e^{10} ) }{e^{20} ( \gamma +1 ) } ( \gamma - \delta ) ^{2} \biggr] \\& \quad \leq \psi \bigl[ \Lambda \bigl( \beta \bigl( \aleph ( \gamma ,\delta ) \bigr) \aleph ( \gamma ,\delta ) \bigr) \bigr] . \end{aligned}$$

Otherwise, we have

$$ \Lambda \bigl( \alpha _{\ast } ( \Gamma \gamma ,\Gamma \delta ) H_{\mathbb{Q} }(\Gamma \gamma ,\Gamma \delta ) \bigr) =0\leq \psi \bigl[ \Lambda \bigl( \beta \bigl( \aleph ( \gamma ,\delta ) \bigr) \aleph ( \gamma , \delta ) \bigr) \bigr] . $$

Now, for $( \gamma ,\delta ) \in (0,3]\times (0,3]$, $\alpha _{\ast }(\gamma ,\Gamma \gamma )\geq 1$ implies $\alpha _{\ast }(\Gamma \gamma ,\Gamma ^{2}\gamma )\geq 1$, then Γ is $\alpha _{\ast }$-orbital admissible and $\alpha (\gamma ,\delta )\geq 1$ and $\alpha _{\ast }(\delta ,\Gamma \delta )\geq 1$ imply $\alpha _{\ast }(\gamma ,\Gamma \delta )\geq 1$, therefore Γ is triangular $\alpha _{\ast }$-orbital admissible. Hence, all affirmations of Theorem 2.2 are satisfied and $\gamma ^{\ast }=\frac{1}{e^{10}-1}\in \Upsilon$ is the fixed point of Γ.

3 Consequences

In this part, some consequences are discussed in F-MS.

Theorem 3.1

Let $(\Upsilon , \mathbb{Q} )$ be an F-MS, set $\Im :\Upsilon \rightarrow \Upsilon$ and $\Gamma :\Upsilon \rightarrow CB ( \Upsilon )$. Presume that there exist functions $\beta \in \mho$, $\Lambda \in \Phi$, and $\psi \in \Psi$ such that $\forall \gamma ,\delta \in \Upsilon$,

$$ \Lambda \bigl( H_{\mathbb{Q} }(\Gamma \gamma ,\Gamma \delta ) \bigr) \leq \psi \bigl( \Lambda \bigl[ \beta \bigl( \aleph _{\Im }( \gamma ,\delta ) \bigr) \aleph _{\Im }(\gamma ,\delta ) \bigr] \bigr) , $$

(3.1)

where

$$ \aleph _{\Im }(\gamma ,\delta )=\max \biggl\{ \mathbb{Q} (\Im \gamma ,\Im \delta ),\mathbb{Q} (\Im \gamma ,\Gamma \gamma ),\mathbb{Q} (\Im \delta ,\Gamma \delta ),\frac{\mathbb{Q} (\Im \gamma ,\Gamma \delta )+\mathbb{Q} (\Im \delta ,\Gamma \gamma )}{2} \biggr\} . $$

If for any $\gamma \in \Upsilon$, $\Gamma \Upsilon \subseteq \Im \Upsilon$ and ℑϒ is an F-complete subspace of ϒ.

Then, Γ and ℑ have a unique point of coincidence. Indeed, if Γ and ℑ are weakly compatible, then Γ and ℑ have a unique common fixed point $\gamma ^{\ast }\in \Upsilon$.

Proof

Let $\gamma _{0}\in \Upsilon$, since $\Gamma \Upsilon \subseteq \Im \Upsilon$, we can construct a sequence $\{ \delta _{\zeta } \} _{\zeta \in \mathbb{N}}$ by

$$ \delta _{\zeta }\in \Gamma \gamma _{\zeta -1}=\Im \gamma _{\zeta },\quad \forall \zeta \in \mathbb{N}. $$

(3.2)

Now, if there exists some $\zeta _{0}\in \mathbb{N}$ such that $\mathbb{Q} ( \delta _{\zeta _{0}},\delta _{\zeta _{0}+1} ) =0$, then $\delta _{\zeta _{0}}=\delta _{\zeta _{0}+1}$, which implies that $\Im \gamma _{\zeta _{0}}=\Gamma \gamma _{\zeta _{0}}$, thus $\gamma _{\zeta _{0}}$ is a coincidence point of Γ and ℑ, so $w_{0}\in \Im \gamma _{\zeta _{0}}=\Gamma \gamma _{\zeta _{0}}$ is the point of coincidence of Γ and ℑ. We postulate that $\mathbb{Q} ( \delta _{\zeta },\delta _{\zeta +1} ) >0$ $\forall \zeta \in \mathbb{N}$. From (3.1) and (3.2), we have

$$\begin{aligned} \Lambda \bigl( \mathbb{Q} ( \delta _{\zeta },\delta _{\zeta +1} ) \bigr) \leq &\Lambda \bigl( H_{\mathbb{Q} } ( \Gamma \gamma _{\zeta -1},\Gamma \gamma _{\zeta } ) \bigr) \\ \leq &\psi \bigl( \Lambda \bigl( \beta \bigl( \aleph _{\Im }(\gamma _{\zeta -1},\gamma _{\zeta }) \bigr) \aleph _{\Im }( \gamma _{\zeta -1},\gamma _{\zeta }) \bigr) \bigr) , \end{aligned}$$

(3.3)

where

$$\begin{aligned} \aleph _{\Im }(\gamma _{\zeta -1},\gamma _{\zeta }) =& \max \left \{ \textstyle\begin{array}{c} \mathbb{Q} ( \Im \gamma _{\zeta -1},\Im \gamma _{\zeta } ) ,\mathbb{Q} ( \Im \gamma _{\zeta -1},\Gamma \gamma _{\zeta -1} ) , \\ \mathbb{Q} ( \Im \gamma _{\zeta },\Gamma \gamma _{\zeta } ) ,\frac{\mathbb{Q} ( \Im \gamma _{\zeta -1},\Gamma \gamma _{\zeta } ) +\mathbb{Q} ( \Im \gamma _{\zeta },\Gamma \gamma _{\zeta -1} ) }{2}\end{array}\displaystyle \right \} \\ =&\max \left \{ \textstyle\begin{array}{c} \mathbb{Q} ( \delta _{\zeta -1},\delta _{\zeta } ) ,\mathbb{Q} ( \delta _{\zeta -1},\delta _{\zeta } ) ,\mathbb{Q} ( \delta _{\zeta },\delta _{\zeta +1} ) , \\ \frac{\mathbb{Q} ( \delta _{\zeta -1},\delta _{\zeta +1} ) +\mathbb{Q} ( y_{\zeta },y_{\zeta } ) }{2}\end{array}\displaystyle \right \} \\ =&\max \bigl\{ \mathbb{Q} ( \delta _{\zeta -1},\delta _{\zeta } ) ,\mathbb{Q} ( \delta _{\zeta },\delta _{\zeta +1} ) \bigr\} . \end{aligned}$$

We conclude that

$$ \aleph _{\Im } ( \gamma _{\zeta -1},\gamma _{\zeta } ) = \max \bigl\{ \mathbb{Q} ( \delta _{\zeta -1},\delta _{\zeta } ) ,\mathbb{Q} ( \delta _{\zeta },\delta _{\zeta +1} ) \bigr\} . $$

Now, if $\max \{ \mathbb{Q} ( \delta _{\zeta -1},\delta _{\zeta } ) ,\mathbb{Q} ( \delta _{\zeta },\delta _{\zeta +1} ) \} =\mathbb{Q} ( \delta _{\zeta },\delta _{\zeta +1} )$ for $\zeta \geq 1$, then from (3.2), we obtain

$$ \Lambda \bigl( \mathbb{Q} ( \delta _{\zeta },\delta _{\zeta +1} ) \bigr) \leq \psi \bigl[ \Lambda \bigl( \beta \bigl( \mathbb{Q} ( \delta _{\zeta },\delta _{\zeta +1} ) \bigr) \mathbb{Q} ( \delta _{\zeta },\delta _{\zeta +1} ) \bigr) \bigr] . $$

Since $\beta \in \mho$ and from $( \Phi 1 )$, we have

$$ \mathbb{Q} ( \delta _{\zeta },\delta _{\zeta +1} ) < \mathbb{Q} ( \delta _{\zeta },\delta _{\zeta +1} ) , $$

which is a discrepancy as $\mathbb{Q} ( \delta _{\zeta },\delta _{\zeta +1} ) \geq 0$. Therefore,

$$ \max \bigl\{ \mathbb{Q} ( \delta _{\zeta -1},\delta _{\zeta } ) ,\mathbb{Q} ( \delta _{\zeta },\delta _{\zeta +1} ) \bigr\} =\mathbb{Q} ( \delta _{\zeta -1},\delta _{\zeta } ) , $$

(3.4)

by (3.3) and (3.4), we have

$$\begin{aligned}& \Lambda \bigl( \mathbb{Q} ( \delta _{\zeta },\delta _{\zeta +1} ) \bigr) \\& \quad =\Lambda \bigl( \mathbb{Q} ( \Gamma \gamma _{\zeta -1},\Gamma \gamma _{\zeta } ) \bigr) \leq \Lambda \bigl( H_{\mathbb{Q} } ( \Gamma \gamma _{\zeta -1},\Gamma \gamma _{\zeta } ) \bigr) \\& \quad \leq \psi \bigl[ \Lambda \bigl( \beta \bigl( \mathbb{Q} ( \Im \gamma _{\zeta -1},\Im \gamma _{\zeta } ) \bigr) .\mathbb{Q} ( \Im \gamma _{\zeta -1},\Im \gamma _{\zeta } ) \bigr) \bigr] \\& \quad =\psi \bigl[ \Lambda \bigl( \beta \bigl( \mathbb{Q} ( \delta _{\zeta -1},\delta _{\zeta } ) \bigr) .\mathbb{Q} ( \delta _{\zeta -1},\delta _{\zeta } ) \bigr) \bigr] \\& \quad =\psi \bigl[ \Lambda \bigl( \beta \bigl( \mathbb{Q} ( \delta _{\zeta -1},\delta _{\zeta } ) \bigr) .\mathbb{Q} ( \Gamma \gamma _{\zeta -2},\Gamma \gamma _{\zeta -1} ) \bigr) \bigr] \\& \quad \leq \psi \bigl[ \Lambda \bigl( \beta \bigl( \mathbb{Q} ( \delta _{\zeta -1},\delta _{\zeta } ) \bigr) .H_{\mathbb{Q} } ( \Gamma \gamma _{\zeta -2},\Gamma \gamma _{\zeta -1} ) \bigr) \bigr] \\& \quad \leq \psi ^{2} \bigl[ \Lambda \bigl( \beta \bigl( \mathbb{Q} ( \delta _{\zeta -1},\delta _{\zeta } ) \bigr) \beta \bigl( \mathbb{Q} ( \delta _{\zeta -2},\delta _{\zeta -1} ) \bigr) \mathbb{Q} ( \Gamma \gamma _{\zeta -3}, \Gamma \gamma _{\zeta -2} ) \bigr) \bigr] \\& \quad =\psi ^{2} \bigl[ \Lambda \bigl( \beta \bigl( \mathbb{Q} ( \delta _{\zeta -1},\delta _{\zeta } ) \bigr) \beta \bigl( \mathbb{Q} ( \delta _{\zeta -2},\delta _{\zeta -1} ) \bigr) \mathbb{Q} ( \delta _{\zeta -2},\delta _{\zeta -1} ) \bigr) \bigr] \\& \qquad \cdots \\& \quad \leq \psi ^{\zeta } \bigl[ \Lambda \bigl( \beta \bigl( \mathbb{Q} ( \Im \gamma _{\zeta -1}, \gamma _{\zeta } ) \bigr) \beta \bigl( \mathbb{Q} ( \gamma _{\zeta -2},\gamma _{\zeta -1} ) \bigr) ...\beta \bigl( \mathbb{Q} ( \gamma _{0}, \gamma _{1} ) \bigr) \mathbb{Q} ( \gamma _{0},\gamma _{1} ) \bigr) \bigr] \\& \quad =\psi ^{\zeta } \bigl[ \Lambda \bigl( \beta \bigl( \mathbb{Q} ( \delta _{\zeta -1},\delta _{\zeta } ) \bigr) \beta \bigl( \mathbb{Q} ( \delta _{\zeta -2},\delta _{\zeta -1} ) \bigr) ... \beta \bigl( \mathbb{Q} ( \delta _{0},\delta _{1} ) \bigr) \mathbb{Q} ( \delta _{0},\delta _{1} ) \bigr) \bigr] \\& \quad =\psi ^{\zeta } \Biggl[ \Lambda \Biggl( \Biggl[ \prod _{i=1}^{\zeta }\beta \bigl( \mathbb{Q} ( \delta _{i-1},\delta _{i} ) \bigr) \Biggr] \mathbb{Q} ( \delta _{0},\delta _{1} ) \Biggr) \Biggr] \\& \quad < \psi ^{\zeta } \bigl[ \Lambda \bigl( \mathbb{Q} ( \delta _{0},\delta _{1} ) \bigr) \bigr], \end{aligned}$$

for all $\zeta \in \mathbb{N}$. Let $\epsilon >0$ be fixed and $( \mathcal{L} ,a ) \in \Xi \times {}[ 0,\infty )$ be such that $(\mathbb{Q} 3)$ is satisfied. By $( \Xi 2 )$, there exists $\eth >0$ such that

$$ 0< \varsigma < \eth \quad \text{ implies } \mathcal{L} ( \varsigma ) < \mathcal{L} ( \epsilon ) -a. $$

(3.5)

Let $\ell ( \epsilon ) \in \mathbb{N}$ such that $0<\sum_{\zeta \geq \ell ( \epsilon ) }\psi ^{\zeta } [ \Lambda ( \mathbb{Q} ( \delta _{0},\delta _{1} ) ) ] <\Lambda ( \eth )$. Hence, by using the properties of ψ, (3.5), and $( \Xi 1 )$, we have

$$\begin{aligned} \mathcal{L} \Biggl( \sum_{j=\zeta }^{\eta -1} \psi ^{j} \bigl[ \Lambda \bigl( \mathbb{Q} ( \delta _{0},\delta _{1} ) \bigr) \bigr] \Biggr) \leq & \mathcal{L} \biggl( \sum_{\zeta \geq \ell }\psi ^{\zeta } \bigl[ \Lambda \bigl( \mathbb{Q} ( \delta _{0},\delta _{1} ) \bigr) \bigr] \biggr) \\ < & \mathcal{L} \bigl( \Lambda ( \epsilon ) \bigr) -a, \end{aligned}$$

(3.6)

where $\eta >\zeta >\ell$ with $\mathbb{Q} ( \delta _{\zeta },\delta _{\eta } ) >0$, using $( \mathbb{Q} 3 )$ and (3.6), we have

$$\begin{aligned} \mathcal{L} \bigl( \Lambda \bigl( \mathbb{Q} ( \delta _{\zeta },\delta _{\eta } ) \bigr) \bigr) \leq & \mathcal{L} \Biggl( \sum_{j=\zeta }^{\eta -1} \psi ^{j} \bigl[ \Lambda \bigl( \mathbb{Q} ( \delta _{0},\delta _{1} ) \bigr) \bigr] \Biggr) +a \\ \leq & \mathcal{L} \biggl( \sum_{\zeta \geq \ell }\psi ^{\zeta } \bigl[ \Lambda \bigl( \mathbb{Q} ( \delta _{0},\delta _{1} ) \bigr) \bigr] \biggr) +a \\ < & \mathcal{L} \bigl( \Lambda ( \epsilon ) \bigr) -a+a \\ =& \mathcal{L} \bigl( \Lambda ( \epsilon ) \bigr) , \end{aligned}$$

which yields by $( \Xi 1 )$ and $( \Phi 1 )$ that

$$ \mathbb{Q} ( \delta _{\zeta }, \delta _{\eta } ) < \epsilon ,\quad \forall \eta >\zeta >\ell. $$

Therefore, $\{ \delta _{\zeta } \} = \{ \Im \gamma _{\zeta } \}$ is an F-Cauchy sequence in ℑϒ. Since ℑϒ is F-complete, there exists $v^{\ast },u^{\ast }\in \Upsilon$ such that $v^{\ast }=\Im u^{\ast }$, which implies

$$ \lim_{\zeta \rightarrow \infty } \mathbb{Q} \bigl( v^{\ast },\delta _{\zeta } \bigr) =0=\lim_{\eta ,\zeta \rightarrow \infty } \mathbb{Q} ( \delta _{\eta },\delta _{\zeta } ) = \lim_{\zeta \rightarrow \infty } \mathbb{Q} \bigl( \Im u^{\ast },\delta _{\zeta } \bigr) =0. $$

(3.7)

Now, we show that $v^{\ast }\in \Gamma u^{\ast }$. Postulating that $\mathbb{Q} ( v^{\ast },\Gamma u^{\ast } ) >0$, by (3.1), we have

$$\begin{aligned} \Lambda \bigl( \mathbb{Q} \bigl( \delta _{\zeta },\Gamma u^{\ast } \bigr) \bigr) =&\Lambda \bigl( \mathbb{Q} \bigl(\Gamma \gamma _{\zeta -1},\Gamma u^{\ast }\bigr) \bigr) \leq \Lambda \bigl( H_{\mathbb{Q} }\bigl(\Gamma \gamma _{\zeta -1},\Gamma u^{\ast } \bigr) \bigr) \\ \leq &\psi \bigl( \Lambda \bigl[ \beta \bigl( \aleph _{\Im }\bigl( \gamma _{\zeta -1},u^{\ast }\bigr) \bigr) .\aleph _{\Im } \bigl(\gamma _{\zeta -1},u^{\ast }\bigr) \bigr] \bigr) , \end{aligned}$$

(3.8)

where

$$\begin{aligned} \aleph _{\Im }(\gamma _{\zeta -1},u) =&\max \left \{ \textstyle\begin{array}{c} \mathbb{Q} (\Im \gamma _{\zeta -1},\Im u^{\ast }),\mathbb{Q} (\Im \gamma _{\zeta -1},\Gamma \gamma _{\zeta -1}), \\ \mathbb{Q} (\Im u^{\ast },\Gamma u^{\ast }),\frac{\mathbb{Q} (\Im \gamma _{\zeta -1},\Gamma u^{\ast })+\mathbb{Q} (\Im u^{\ast },\Gamma \gamma _{\zeta -1})}{2}\end{array}\displaystyle \right \} \\ =&\max \left \{ \textstyle\begin{array}{c} \mathbb{Q} (\delta _{\zeta -1},v^{\ast }),\mathbb{Q} (\delta _{\zeta -1},\delta _{\zeta }),\mathbb{Q} (v^{\ast },\Gamma u^{\ast }), \\ \frac{\mathbb{Q} (\delta _{\zeta -1},\Gamma u^{\ast })+\mathbb{Q} (v^{\ast },\delta _{\zeta })}{2}\end{array}\displaystyle \right \} . \end{aligned}$$

(3.9)

Since $\beta \in \mho$, from $( \Phi 1 )$, letting $\zeta \rightarrow \infty$ in (3.8) and applying (3.9), we obtain

$$ \mathbb{Q} \bigl( v^{\ast },\Gamma u^{\ast } \bigr) < \mathbb{Q} \bigl(v^{\ast },\Gamma u^{\ast }\bigr), $$

which is a discrepancy. Therefore, $\mathbb{Q} ( v^{\ast },\Gamma u^{\ast } ) =0$, which implies that $v^{\ast }\in \Gamma u^{\ast }$. Thus, $v^{\ast }=\Im u^{\ast }\in \Gamma u^{\ast }$, and hence Γ and ℑ have a coincidence point $u^{\ast }$, and $v^{\ast }$ is a point of coincidence of Γ and ℑ. By $( \mathbb{Q} 1 )$, we have $\mathbb{Q} ( v^{\ast },v^{\ast } ) =0$. Postulating that $v_{1}^{\ast }$ is another point of coincidence of Γ and ℑ such that we can find $u_{1}^{\ast }\in \Upsilon$, such that $v_{1}^{\ast }=\Im u_{1}^{\ast }\in \Gamma u_{1}^{\ast }$ and by $( \mathbb{Q} 1 )$, $\mathbb{Q} ( v_{1}^{\ast },v_{1}^{\ast } ) =0$. Now, we prove that $\mathbb{Q} ( v^{\ast },v_{1}^{\ast } ) =0$ by contrast. Assume that $\mathbb{Q} ( v^{\ast },v_{1}^{\ast } ) >0$, from (3.1)

$$\begin{aligned}& \begin{aligned}[b] \Lambda \bigl( \mathbb{Q} \bigl( v^{\ast },v_{1}^{\ast } \bigr) \bigr) &\leq \Lambda \bigl( \mathbb{Q} \bigl( \Gamma u^{\ast },\Gamma u_{1}^{\ast } \bigr) \bigr) \leq \Lambda \bigl( H_{\mathbb{Q} } \bigl( \Gamma u^{\ast },\Gamma u_{1}^{\ast } \bigr) \bigr) \\ &\leq \psi \bigl( \Lambda \bigl[ \beta \bigl( \aleph _{\Im } \bigl( u^{\ast },u_{1}^{\ast } \bigr) \bigr) \aleph _{\Im } \bigl( u^{\ast },u_{1}^{\ast } \bigr) \bigr] \bigr) , \end{aligned} \end{aligned}$$

(3.10)

$$\begin{aligned}& \begin{aligned}[b] \aleph _{\Im } \bigl( u^{\ast },u_{1}^{\ast } \bigr) &=\max \left \{ \textstyle\begin{array}{c} \mathbb{Q} ( \Im u^{\ast },\Im u_{1}^{\ast } ) ,\mathbb{Q} ( \Im u^{\ast },\Gamma u^{\ast } ) , \\ \mathbb{Q} ( \Im u_{1}^{\ast },\Gamma u_{1}^{\ast } ) ,\frac{\mathbb{Q} ( \Im u^{\ast },\Gamma u_{1}^{\ast } ) +\mathbb{Q} ( \Gamma u^{\ast },\Im u_{1}^{\ast } ) }{2}\end{array}\displaystyle \right \} \\ &=\max \left \{ \textstyle\begin{array}{c} \mathbb{Q} ( v^{\ast },v_{1}^{\ast } ) ,\mathbb{Q} ( v^{\ast },v^{\ast } ) ,\mathbb{Q} ( v_{1}^{\ast },v_{1}^{\ast } ) , \\ \frac{\mathbb{Q} ( v^{\ast },v_{1}^{\ast } ) +\mathbb{Q} ( v^{\ast },v_{1}^{\ast } ) }{2}\end{array}\displaystyle \right \} \\ &=\mathbb{Q} \bigl( v^{\ast },v_{1}^{\ast } \bigr) . \end{aligned} \end{aligned}$$

(3.11)

Since $\beta \in \mho$, from $( \Phi 1 )$, (3.10), and (3.11), we obtain $\mathbb{Q} ( v^{\ast },v_{1}^{\ast } ) <\mathbb{Q} ( v^{\ast },v_{1}^{\ast } )$, which is a discrepancy. Therefore, $\mathbb{Q} ( v^{\ast },v_{1}^{\ast } ) =0$ implies that $v^{\ast }=v_{1}^{\ast }$. Thus, Γ and ℑ have a unique point of coincidence. Moreover, since Γ and ℑ are weakly compatible, we have $\Im v^{\ast }=\Gamma v^{\ast }$. Now, let $w=\Im v^{\ast }\in \Gamma v^{\ast }$. From the uniqueness of the point of coincidence, we have $w=v=\Im v^{\ast }\in \Gamma v^{\ast }$. Therefore, Γ and ℑ have a unique common fixed point. □

Corollary 3.2

Let $(\Upsilon ,\mathbb{Q} )$ be an F-complete F-MS, $\alpha :\Upsilon \times \Upsilon \rightarrow {}[ 0,\infty )$ be a function. A mapping $\Gamma :\Upsilon \rightarrow \Upsilon$ is called an improved α-Geraghty contraction mapping if there exist $\beta \in \mho$ such that for all $\gamma ,\delta \in \Upsilon$,

$$ \alpha (\gamma ,\delta )\mathbb{Q} (\Gamma \gamma ,\Gamma \delta )\leq \beta \bigl( \aleph (\gamma , \delta ) \bigr) .\aleph (\gamma ,\delta ), $$

where

$$ \aleph (\gamma ,\delta )=\max \biggl\{ \mathbb{Q} (\gamma ,\delta ),\mathbb{Q} (\gamma ,\Gamma \gamma ),\mathbb{Q} (\delta ,\Gamma \delta ),\frac{\mathbb{Q} (\gamma ,\Gamma \delta )+\mathbb{Q} (\delta ,\Gamma \gamma )}{2}\biggr\} , $$

for all $\gamma ,\delta \in \Upsilon$, satisfying the following stipulations:

• (1) Γ is an improved α-Geraghty contraction;

• (2) Γ is triangular α-orbital admissible;

• (3) there exists an $\gamma _{0}\in \Upsilon$ such that $\alpha (\gamma _{0},\Gamma \gamma _{0})\geq 1$;

• (4) Γ is continuous.

Then, Γ has a unique fixed point $\gamma ^{\ast }\in \Upsilon$.

Corollary 3.3

Let $(\Upsilon ,\mathbb{Q} )$ be an F-MS, and $\Gamma ,\Im :\Upsilon \rightarrow \Upsilon$ be two mappings with $\Gamma \Upsilon \subseteq \Im \Upsilon$ and ℑϒ is F-complete. The pair $( \Gamma ,\Im )$ is an improved Geraghty contraction if there exists $\beta \in \mho$ such that for all $\gamma ,\delta \in \Upsilon$,

$$ \mathbb{Q} (\Gamma \gamma ,\Gamma \delta ) \leq \beta \bigl( \aleph _{\Im }(\gamma ,\delta ) \bigr) .\aleph _{\Im }(\gamma ,\delta ), $$

where

$$ \aleph _{\Im }(\gamma ,\delta )=\max \biggl\{ \mathbb{Q} (\Im \gamma ,\Im \delta ),\mathbb{Q} (\Im \gamma ,\Gamma \gamma ),\mathbb{Q} (\Im \delta ,\Gamma \delta ),\frac{\mathbb{Q} (\Im \gamma ,\Gamma \delta )+\mathbb{Q} (\Im \delta ,\Gamma \gamma )}{2} \biggr\} . $$

Then, Γ and g have a unique point of coincidence. Indeed, if Γ and ℑ are weakly compatible, then Γ and ℑ have a unique common fixed point $\gamma ^{\ast }\in \Upsilon$.

Example 3.4

Let $\Upsilon =[0,\infty )$ and F-M $\mathbb{Q}$ given by

$$ \mathbb{Q} ( \gamma ,\delta ) = \textstyle\begin{cases} e^{ \vert \gamma -\delta \vert } & \text{if }\gamma \neq \delta ,\\ 0 & \text{if }\gamma =\delta, \end{cases} $$

with $\mathcal{L} ( \varsigma ) =\frac{-1}{\varsigma }$ and $a=1$. Then, $( \Upsilon ,\mathbb{Q} )$ is F-complete F-MS. Define $\Im :\Upsilon \rightarrow \Upsilon$ and $\Gamma :\Upsilon \rightarrow CB ( \Upsilon )$ by

$$ \Gamma \gamma = \textstyle\begin{cases} \{ \frac{\gamma }{8} \} , & \text{if }\gamma \in \mathbb{N} \cup \{ 0 \} ,\\ \{ 0 \} & \text{otherwise}, \end{cases}\displaystyle \quad \text{and}\quad \Im \gamma = \textstyle\begin{cases} \frac{3\gamma }{2} & \text{if }\gamma \in \mathbb{N} \cup \{ 0 \} ,\\ 0 & \text{otherwise}. \end{cases} $$

Clearly, for all $\gamma \in \mathbb{N} \cup \{ 0 \}$, $\Gamma ( \Upsilon ) \subseteq \Im ( \Upsilon )$ and $\Im ( \Upsilon )$ is an F-complete subset of ϒ; let $\beta :\Upsilon \times \Upsilon \rightarrow {}[ 0,1)$ be as $\beta ( \gamma ,\delta ) =\frac{1}{2},~\Lambda ( t ) =t$ and $\psi ( t ) =\frac{2}{3}t$. Now, for all $( \gamma ,\delta ) \in \mathbb{N} \cup \{ 0 \}$ with $\gamma \neq \delta$, then

$$\begin{aligned} \Lambda \bigl( H_{\mathbb{Q} } ( \Gamma \gamma ,\Gamma \delta ) \bigr) =& \Lambda \bigl( \max \bigl( \sup_{a\in \Gamma \gamma } \mathbb{Q} ( a,\Gamma \delta ) ,\sup_{b\in \Gamma \delta } \mathbb{Q} ( \Gamma \gamma ,b ) \bigr) \bigr) \\ =&\Lambda \biggl( \max \biggl( \sup_{a\in \Gamma \gamma } \mathbb{Q} \biggl( a, \biggl\{ \frac{\delta }{8} \biggr\} \biggr) ,\sup_{b\in \Gamma \delta } \mathbb{Q} \biggl( \biggl\{ \frac{\gamma }{8} \biggr\} ,b \biggr) \biggr) \biggr) \\ =&\Lambda \biggl( \max \biggl( \mathbb{Q} \biggl( \frac{\gamma }{8}, \biggl\{ \frac{\delta }{8} \biggr\} \biggr) ,\mathbb{Q} \biggl( \biggl\{ \frac{\gamma }{8} \biggr\} ,\frac{\delta }{8} \biggr) \biggr) \biggr) \\ =&\Lambda \biggl( \max \biggl( \mathbb{Q} \biggl( \frac{\gamma }{8},\frac{\delta }{8} \biggr) ,\mathbb{Q} \biggl( \frac{\gamma }{8},\frac{\delta }{8} \biggr) \biggr) \biggr) \\ =&\Lambda \biggl( \mathbb{Q} \biggl( \frac{\gamma }{8},\frac{\delta }{8} \biggr) \biggr) \\ =&\Lambda \bigl( e^{ \vert \frac{\gamma }{8}-\frac{\delta }{8} \vert } \bigr) =\Lambda \bigl( e^{\frac{1}{4} \vert \frac{\gamma }{2}-\frac{\delta }{2} \vert } \bigr) \\ \leq &\frac{2}{3}\Lambda \biggl( \frac{1}{2}e^{ \vert \frac{3\gamma }{2}-\frac{3\delta }{2} \vert } \biggr) \\ \leq &\psi \bigl( \Lambda \bigl[ \beta \bigl( \aleph _{\Im }(\gamma , \delta ) \bigr) .\aleph _{\Im }(\gamma ,\delta ) \bigr] \bigr) . \end{aligned}$$

If $\gamma =\delta$, then we have

$$ \Lambda \bigl( H_{\mathbb{Q} } ( \Gamma \gamma ,\Gamma \delta ) \bigr) =0\leq \psi \bigl( \Lambda \bigl[ \beta \bigl( \aleph _{\Im }(\gamma ,\delta ) \bigr) .\aleph _{\Im }(\gamma ,\delta ) \bigr] \bigr) . $$

Otherwise, we have that (3.1) trivially holds. Therefore, all stipulations of Theorem 3.1 are satisfied. Since $\Upsilon 0=\Im 0=0$, thus $\gamma =0$ is a common fixed point of Γ and ℑ.

4 Application for the existence of a solution to a functional equation

In this section, we use our main results to verify the existence and uniqueness of a solution to the functional equation:

$$ \varrho ( \gamma ) =\sup_{\delta \in \Upsilon } \bigl\{ \digamma (\gamma , \delta )+\Pi \bigl( \gamma ,\delta ,\varrho \bigl( \mu (\gamma ,\delta ) \bigr) \bigr) \bigr\} ,\quad \gamma \in \Game , $$

(4.1)

where $\digamma :\Game \times \Upsilon \rightarrow \mathbb{R}$ and $\Pi :\Game \times \Upsilon \times \mathbb{R} \rightarrow \mathbb{R}$ are bounded, $\mu :\Game \times \Upsilon \rightarrow \Game$, ⅁ and ϒ are BSs. Equations of the type $( 4.1 )$ have applications in mathematical optimization, computer programming, and in dynamic programming, giving tools for solutions to boundary value problems arising in engineering and physical sciences. Bhakta and Mitra [13] introduced the existence theorems that proved the existence and uniqueness of the solution of a functional equation under certain conditions in Banach spaces. Deepmala [15] utilized the fixed-point theorems to establish the existence, uniqueness, and iterative approximation of the solution for a functional equation in Banach spaces and complete metric spaces. In [30, 32], common solutions of certain functional equations arising in dynamic programming and common fixed-point theorems for a quadruple of self-mappings satisfying weak compatibility and JH-operator pairs on a complete metric space were discussed.

Let denote the set of all bounded real-valued functions on ⅁. The pair , where , is a BS along with the metric $\mathbb{Q}$ given by

$$ \mathbb{Q} ( h,k ) =\sup _{\varsigma \in \Game } \bigl\vert h ( \varsigma ) -k ( \varsigma ) \bigr\vert = \Vert h-k \Vert . $$

To show the existence of a solution to (4.1), we put in place the following stipulations:

$( S1 )$ Ϝ and Π are bounded,

$( S2 )$ for all and $\gamma \in \Game$, we define the operator as

$$ ( \Gamma h ) ( \gamma ) =\sup_{\delta \in \Upsilon } \bigl\{ \digamma (\gamma ,\delta )+\Pi \bigl( \gamma ,\delta ,h \bigl( \mu (\gamma ,\delta ) \bigr) \bigr) \bigr\} . $$

(4.2)

Undoubtedly, Γ is well defined since Ϝ and Π are bounded,

$( S3 )$ for $a>1$, and $\varsigma \in \Game$, we have

$$ \bigl\vert \Pi \bigl( \gamma ,\delta ,h ( \varsigma ) \bigr) -\Pi \bigl( \gamma ,\delta ,k ( \varsigma ) \bigr) \bigr\vert \leq e^{-a}\aleph ( h,k ) , $$

(4.3)

where

$$ \aleph ( h,k ) =\max \biggl\{ \mathbb{Q} ( h,k ) ,\mathbb{Q} ( h,\Gamma h ) ,\mathbb{Q} ( k,\Gamma k ) , \frac{\mathbb{Q} ( k,\Gamma h ) +\mathbb{Q} ( h,\Gamma k ) }{2} \biggr\} . $$

We shall verify the following theorem.

Theorem 4.1

Postulate that the stipulations $( S1 ) - ( S3 )$ hold, then the functional Eq. (4.1) has a bounded solution.

Proof

Let $\lambda >0$ be arbitrary, $\gamma \in \Game$ and . The space is an F-complete F-MS. There exist $\delta _{1},\delta _{2}\in \Upsilon$ such that

$$\begin{aligned}& ( \Gamma h ) ( \gamma ) < \digamma ( \gamma ,\delta _{1} ) +\Pi \bigl( \gamma ,\delta _{1},h \bigl( \mu ( \gamma ,\delta _{1} ) \bigr) \bigr) +\lambda , \end{aligned}$$

(4.4)

$$\begin{aligned}& ( \Gamma k ) ( \gamma ) < \digamma ( \gamma ,\delta _{2} ) +\Pi \bigl( \gamma ,\delta _{2},k \bigl( \mu ( \gamma ,\delta _{2} ) \bigr) \bigr) +\lambda , \end{aligned}$$

(4.5)

$$\begin{aligned}& ( \Gamma h ) ( \gamma ) \geq \digamma ( \gamma ,\delta _{2} ) +\Pi \bigl( \gamma ,\delta _{2},h \bigl( \mu ( \gamma ,\delta _{2} ) \bigr) \bigr) , \end{aligned}$$

(4.6)

$$\begin{aligned}& ( \Gamma k ) ( \gamma ) \geq \digamma ( \gamma ,\delta _{1} ) +\Pi \bigl( \gamma ,\delta _{1},k \bigl( \mu ( \gamma ,\delta _{1} ) \bigr) \bigr) . \end{aligned}$$

(4.7)

Then from (4.4) and (4.7), we obtain

$$\begin{aligned} ( \Gamma h ) ( \gamma ) - ( \Gamma k ) ( \gamma ) < &\Pi \bigl( \gamma ,\delta _{1},h \bigl( \mu ( \gamma ,\delta _{1} ) \bigr) \bigr) -\Pi \bigl( \gamma ,\delta _{1},k \bigl( \mu ( \gamma ,\delta _{1} ) \bigr) \bigr) +\lambda \\ \leq & \bigl\vert \Pi \bigl( \gamma ,\delta _{1},h \bigl( \mu ( \gamma ,\delta _{1} ) \bigr) \bigr) -\Pi \bigl( \gamma ,\delta _{1},k \bigl( \mu ( \gamma ,\delta _{1} ) \bigr) \bigr) \bigr\vert +\lambda \\ \leq &e^{-a}\aleph ( h,k ) +\lambda . \end{aligned}$$

(4.8)

Similarly from (4.5) and (4.6), we obtain

$$\begin{aligned} ( \Gamma k ) ( \gamma ) - ( \Gamma h ) ( \gamma ) < &\Pi \bigl( \gamma ,\delta _{2},k \bigl( \mu ( \gamma ,\delta _{2} ) \bigr) \bigr) -\Pi \bigl( \gamma ,\delta _{2},h \bigl( \mu ( \gamma , \delta _{2} ) \bigr) \bigr) +\lambda \\ \leq & \bigl\vert \Pi \bigl( \gamma ,\delta _{2},k \bigl( \mu ( \gamma ,\delta _{2} ) \bigr) \bigr) -\Pi \bigl( \gamma ,\delta _{2},h \bigl( \mu ( \gamma ,\delta _{2} ) \bigr) \bigr) \bigr\vert +\lambda \\ \leq &e^{-a}\aleph ( h,k ) +\lambda . \end{aligned}$$

(4.9)

Combining (4.8) and (4.9), we obtain

$$ \bigl\vert ( \Gamma h ) ( \gamma ) - ( \Gamma k ) ( \gamma ) \bigr\vert \leq e^{-a}\aleph ( h,k ) +\lambda , $$

which implies for $\lambda >0$ and $\gamma \in \Game$ such that

$$ e\times \mathbb{Q} ( \Gamma h,\Gamma k ) \leq \frac{1}{e^{a-1}}\aleph ( h,k ). $$

Taking $\alpha ( h,k ) =e\geq 1$ and $\beta ( h,k ) =\frac{1}{e^{a-1}}\in {}[ 0,1)$, we have

$$ \alpha ( h,k ) \mathbb{Q} ( \Gamma h,\Gamma k ) \leq \beta \bigl( \aleph ( h,k ) \bigr) \aleph ( h,k ). $$

All the stipulations of Corollary 3.2 are fulfilled, and Γ has a unique fixed point, so Eq. (4.1) has a bounded solution. □

Example 4.2

Let $\Game =\Upsilon = \mathbb{R}$ be a BS with the standard norm $\Vert \gamma \Vert = \vert \gamma \vert$, for all $\gamma \in \Game$. Postulating that $S=[0,1]\subseteq \Game$ is the state space and $D=[0,\infty )\subseteq \Upsilon$ the decision space. Define $\mu :S\times D\rightarrow S$ and $\digamma :S\times D\rightarrow \mathbb{R}$ by

$$ \mu ( \gamma ,\delta ) =\frac{\gamma \delta ^{2}}{1+\delta ^{2}}\quad \text{and}\quad \digamma ( \gamma ,\delta ) =0,\quad \forall \gamma \in S, \text{ and } \delta \in D. $$

Define $\varrho :S\rightarrow \mathbb{R}$ by

Now, for all , $\gamma \in S$, we define a map as,

$$\begin{aligned} \Gamma h ( \gamma ) =&\sup_{\delta \in D} \bigl\{ \digamma ( \gamma ,\delta ) +\Pi \bigl( \gamma ,\delta ,h \bigl( \mu ( \gamma ,\delta ) \bigr) \bigr) \bigr\} , \\ \Gamma k ( \gamma ) =&\sup_{\delta \in D} \bigl\{ \digamma ( \gamma ,\delta ) +\Pi \bigl( \gamma ,\delta ,k \bigl( \mu ( \gamma ,\delta ) \bigr) \bigr) \bigr\} , \end{aligned}$$

where $\Pi :S\times D\times \mathbb{R} \rightarrow \mathbb{R}$ is defined by

$$ \Pi ( \gamma ,\delta ,\varsigma ) =\frac{1}{32}\varsigma \sin \biggl( \frac{\delta }{\delta +2} \biggr) . $$

Hence,

Similarly,

Note that Π and Ϝ are bounded; this implies that stipulations $( S_{1} )$ and $( S_{2} )$ of Theorem 4.1 are satisfied. Now,

$$\begin{aligned} \bigl\vert \Pi \bigl( \gamma ,\delta ,h ( \varsigma ) \bigr) -\Pi \bigl( \gamma ,\delta ,k ( \varsigma ) \bigr) \bigr\vert =& \biggl\vert \frac{1}{32}h ( \varsigma ) \sin \biggl( \frac{\delta }{\delta +2} \biggr) - \frac{1}{32}h ( \varsigma ) \sin \biggl( \frac{\delta }{\delta +2} \biggr) \biggr\vert \\ =&\frac{1}{32} \biggl\vert \sin \biggl( \frac{\delta }{\delta +2} \biggr) \biggr\vert \bigl\vert h ( \varsigma ) -k ( \varsigma ) \bigr\vert \\ \leq &\frac{1}{32} \bigl\vert h ( \varsigma ) -k ( \varsigma ) \bigr\vert \\ \leq &\frac{1}{e^{2}} \Vert h-k \Vert . \end{aligned}$$

Thus, all the assertions of Theorem 4.1 are satisfied and the functional Eq. (4.1) has a bounded solution in .

5 Conclusion

In this paper, we introduced a new notion of $\alpha _{\ast }$-ψ-Λ-contraction multivalued mappings and proved some novel fixed-point theorems for such contraction in F-MSs. Some consequences are studied to investigate coincidence-point results for this contraction in F-MSs. Also, we gave some examples to clarify our obtained results; we utilized the main results to discuss the existence and uniqueness of a solution to a functional equation. The new concepts lead to further investigations and applications.