Creating new contractive mappings to obtain fixed points and data-dependence results under auxiliary functions

Abstract

This manuscript is concerned with obtaining results for fixed points that arise from new contractive mappings on controlled metric spaces. These mappings are a mixture of Wardowski’s contractions with both multivalued, nonlinear mappings and auxiliary functions. It is also proved that the obtained fixed-point outcomes are well-posed. Additionally, a data-dependence result for fixed points is given. To aid with understanding, several illustrative examples are also provided. Numerous findings that are currently in the literature are specific instances of the findings that were made.

1 Introduction

Fixed-point (FP) theory is a mathematical discipline that studies the existence, uniqueness, and properties of solutions to equations of the form $(\Upsilon \upsilon =\upsilon )$, where $(\Upsilon )$ is a given function. Although this equation appears simple, it has profound implications and finds applications across various domains, from pure mathematics to real-world problem solving in economics, physics, computer science, and beyond. On metric spaces (MSs), Stephan Banach established the renowned Banach contraction principle (BCP) [1] in 1922.

There are two ways to build new FP results: either with the contraction inequality or utilize a more generalized space. BCP is expanded upon and altered in a variety of ways. For instance, the authors of [2, 3] and [4, 5] altered the underlined space and examined Kannan-type contractions to support specific fixed-point conclusions.

In another direction, many authors have extended numerous previous findings using F-contractions. In 2015, Klim and Wardowski [6] extended an F-contraction in terms of nonlinear F-contractions. By using the dynamic processes, the same authors demonstrated a FP theorem and expanded the concept of F-contractive mappings to the situation of nonlinear F-contractions. Subsequently, Wardowski [7] eliminated one of the criteria on the F-mappings to create nonlinear F-contractions. Some theorems on the presence of fixed points of nonlinear F-contractions and the sum of mappings of this kind with a compact operator can be found in another paper by Wardowski [8].

FP theory has found significant applications in the realm of fractional calculus, particularly in the analysis of fractional differential equations. By establishing the existence and uniqueness of solutions to these equations, fixed-point theorems provide a powerful framework for investigating various phenomena in fields such as physics, engineering, and biology. The nonlocal nature of fractional derivatives, combined with the ability of fixed-point theorems to handle nonlinear operators, makes them well-suited for addressing complex problems involving fractional dynamics, see [9–14]

The concept of b-metric space (bMS) was first introduced by Bakhtin in 1989 [4]. Czerwick [15] added to it in order to provide certain FP outcomes made possible by this environment. In order to reduce the triangle inequality of a bMS, Kamran et al. [5] established a new route using a function $z:\eta \times \eta \rightarrow \lbrack 1,\infty )$. By defining an extended bMS as a controlled metric space (CMS) and expanding upon its concept, Mlaiki et al. [16] achieved another breakthrough in this regard.

Estimating the separation between the sets of FPs of two mappings is a data-dependence problem. This notion becomes important only if we are certain that these two operators have nonempty FP sets. Since multivalued mappings (MVMs) frequently have larger FP sets than single-valued mappings, the data-dependence problem primarily affects set-valued mappings. Iqbal et al. [17] addressed data dependence, strict FPs, and the well-posedness of certain multivalued generalized contractions in the context of complete MSs in 2021. They also covered the existence of FPs. In the setting of CMSs, we generalize and unify the findings of Iqbal et al. [17] in this study under new contractive mappings.

2 Basic facts

This section is devoted to recalling some basic facts, which are needed to understand the manuscript. We shall consider $\left ( \eta ,\varpi \right )$, $Q\left ( \eta \right )$, $\eta ^{c}$, $\eta ^{cb}$, and $\eta ^{cp}$ to denote, respectively, a MS, containing all subsets of η, the sets of nonempty, closed subsets of η, nonempty, closed, and bounded subsets of η, and nonempty, compact subsets of η.

Assume that $\mho :\eta \rightarrow Q\left ( \eta \right )$ is a MVM, the point $\vartheta \in \eta$ is called a FP of ℧ if $\vartheta \in \mho \vartheta$. The point $\widehat{\vartheta }\in \eta$ is said to be a strict FP if $\{\widehat{\vartheta }\}=\mho \vartheta$. The set of all (s.o.a.) FPs, and the s.o.a. strict FPs of ℧ are denoted by $F_{ix}\left ( \mho \right )$ and $SF_{ix}\left ( \mho \right )$, respectively.

Definition 2.1

[16] Assume that $\eta \neq \emptyset$ and $\gamma :\eta \times \eta \rightarrow \lbrack 1,\infty )$ is a given function. The distance mapping $\varpi :\eta \times \eta \rightarrow \lbrack 0,\infty )$ is called a CMS if the assertions below hold, for all $\vartheta _{1},\vartheta _{2},\vartheta _{3}\in \eta$,

$(\varpi _{1})$:

$\varpi \left ( \vartheta _{1},\vartheta _{2}\right ) =0$ if and only if $\vartheta _{1}=\vartheta _{2}$;

$(\varpi _{2})$:

$\varpi \left ( \vartheta _{1},\vartheta _{2}\right ) =\varpi \left ( \vartheta _{2},\vartheta _{1}\right )$;

$(\varpi _{3})$:

$\varpi \left ( \vartheta _{1},\vartheta _{2}\right ) \leq \gamma \left ( \vartheta _{1},\vartheta _{3}\right ) \varpi \left ( \vartheta _{1},\vartheta _{3}\right ) +\gamma \left ( \vartheta _{3}, \vartheta _{2}\right ) \varpi \left ( \vartheta _{3},\vartheta _{2} \right )$.

Then, the trio $\left ( \eta ,\varpi ,\gamma \right )$ is called a CMS.

The definition of a Pompei–Hausdorff (PH) MS is defined by the authors in [18], where they considered that $\Re ,\Theta \in NCB(\eta )$ and defined the mapping $\Upsilon :\eta ^{cb}\times \eta ^{cb}\rightarrow \lbrack 0,\infty )$ by

$$ \Upsilon \left ( \Re ,\Theta \right ) =\max \left \{ \sup _{ \vartheta \in \Re }B\left ( \vartheta ,\Theta \right ) ,\sup _{ \widetilde{\vartheta }\in \Theta }B\left ( \widetilde{\vartheta }, \Re \right ) \right \} , $$

where $B\left ( \vartheta ,\Theta \right ) =\left \{ \inf \varpi \left ( \vartheta ,\widetilde{\vartheta }\right ) :\widetilde{\vartheta }\in \Theta \right \}$ and $\Upsilon \left ( \Re ,\Theta \right )$ is called a Hausdorff distance.

In [7], Wardowski presented a wonderful definition (F-contraction) by selecting a strictly increasing function to the Banach contraction mapping: the function $F:(0,\infty )\rightarrow \mathbb{R}$ fulfills the axioms below:

($\heartsuit _{1}$):

for each $\vartheta ,\widetilde{\vartheta }\in (0,\infty )$, if $\vartheta <\widetilde{\vartheta }$, then $F(\vartheta )< F(\widetilde{\vartheta })$, that is, F is strictly increasing;

($\heartsuit _{2}$):

$\lim _{u\rightarrow \infty }\Phi _{u}=0\Leftrightarrow \lim _{u \rightarrow \infty }F(\Phi _{u})=-\infty$, for all sequences $\Phi _{u}\subseteq (0,\infty )$;

($\heartsuit _{3}$):

there is $l\in (0,1)$ such that $\lim _{u\rightarrow 0^{+}}\Phi ^{l}F\left ( \Phi \right ) =0$.

Assume that $\triangledown (\Xi )$ is the s.o.a. functions F, which satisfy $(\heartsuit _{1})$, $(\heartsuit _{2})$, and $(\heartsuit _{3})$. Further, let

$$ \triangledown (Z)=\left \{ F\in \triangledown (\Xi ):(\heartsuit _{3}) \text{ is true for }F\right \} , $$

where

($\heartsuit _{4}$):

for all $\Re \in (0,\infty )$ with $\inf \left ( \Re \right ) >0$, $F\left ( \inf \Re \right ) =\inf F\left ( \Re \right )$.

The results of Turinici [19] can be obtained if we change the axiom $(\heartsuit _{2})$ to

($\heartsuit _{2}^{\prime }$):

$\lim _{u\rightarrow \infty }F(s)=-\infty$.

Let us consider $\triangledown (\widetilde{Z})$ to denote the s.o.a. functions F that fulfill $(\heartsuit _{1})$, $(\heartsuit _{2}^{\prime })$, $(\heartsuit _{3})$, and $(\heartsuit _{4})$.

Now, for all $\vartheta ,\widetilde{\vartheta }\in \eta$, if there are $\nu >0$ and $F\in \triangledown (Z)$ such that

$$ \Upsilon \left ( \mho \vartheta ,\mho \widetilde{\vartheta }\right ) >0 \Rightarrow \nu +F\left ( \Upsilon \left ( \mho \vartheta ,\mho \widetilde{\vartheta }\right ) \right ) \leq F\left ( \varpi \left ( \vartheta , \widetilde{\vartheta }\right ) \right ) , $$

then the mapping $\mho :\eta \rightarrow \eta ^{cb}$ is said to be a multivalued F-contraction [20].

Definition 2.2

[21] Assume that there exist $F\in \triangledown (Z)$ and a function $\sigma :(0,\infty )\rightarrow (0,\infty )$ such that the assumptions below hold:

(A₁):

for all $\Phi >0$, $\lim \inf _{\kappa \rightarrow \Phi ^{+}}\sigma \left ( \kappa \right ) >0$;

(A₂):

for all $\vartheta ,\widetilde{\vartheta }\in \eta$ with $\mho \vartheta \neq \mho \widetilde{\vartheta }$,

$$ \sigma \left ( \varpi \left ( \vartheta ,\widetilde{\vartheta } \right ) \right ) +F\left ( \Upsilon \left ( \mho \vartheta ,\mho \widetilde{\vartheta }\right ) \right ) \leq F\left ( \varpi \left ( \vartheta ,\widetilde{\vartheta }\right ) \right ) . $$

Then, the mapping $\mho :\eta \rightarrow \eta$ is called a $\left ( \sigma ,F\right )$-contraction.

Definition 2.3

[17] Assume that Ψ represents the s.o.a functions $\psi :(0,\infty )\rightarrow (0,\infty )$ such that, for each $\Phi \geq 0$, we have

$$ \lim \inf _{\kappa \rightarrow \Phi ^{+}}\psi \left ( \kappa \right ) >0. $$

3 Auxiliary functions

The following new definitions are very important in the following.

Definition 3.1

Suppose that ℵ refers to the s.o.a. continuous mappings $\ell :[0,\infty )^{7}\rightarrow \lbrack 0,\infty )$ such that the hypotheses below hold:

($\ell _{1}$):

for $\lambda ,\upsilon \geq 1$, $\ell \left ( 1,1,1,\lambda +\upsilon ,0,1, \frac{\lambda +\upsilon }{2}\right ) \in \lbrack 0,1)$;

($\ell _{2}$):

for all $\left ( \vartheta _{1},\vartheta _{2},\vartheta _{3},\vartheta _{4}, \vartheta _{5},\vartheta _{6},\vartheta _{7}\right ) \in \lbrack 0, \infty )^{7}$ and $m\geq 0$, we obtain

$$ \ell \left ( m\vartheta _{1},m\vartheta _{2},m\vartheta _{3},m \vartheta _{4},m\vartheta _{5},m\vartheta _{6},m\vartheta _{7}\right ) \leq m\ell \left ( \vartheta _{1},\vartheta _{2},\vartheta _{3}, \vartheta _{4},\vartheta _{5},\vartheta _{6},\vartheta _{7}\right ) , $$

that is, ℓ is subhomogeneous;

($\ell _{3}$):

for $\vartheta _{i},\widetilde{\vartheta }_{i}\in \lbrack 0,\infty )$ with $\vartheta _{i}\leq \widetilde{\vartheta }_{i}$ $\left ( i=1,2,\ldots ,7\right )$, we obtain

$$ \ell \left ( \vartheta _{1},\vartheta _{2},\vartheta _{3},\vartheta _{4}, \vartheta _{5},\vartheta _{6},\vartheta _{7}\right ) \leq \ell \left ( \widetilde{\vartheta }_{1},\widetilde{\vartheta }_{2}, \widetilde{\vartheta }_{3},\widetilde{\vartheta }_{4},\widetilde{\vartheta }_{5}, \widetilde{\vartheta }_{6},\widetilde{\vartheta }_{7}\right ) , $$

that is, ℓ is nondecreasing. Moreover, if $\vartheta _{i}<\widetilde{\vartheta }_{i}$ $\left ( i=1,2,3,4,6,7\right )$, we have

$$ \ell \left ( \vartheta _{1},\vartheta _{2},\vartheta _{3},\vartheta _{4},0, \vartheta _{6},\vartheta _{7}\right ) < \ell \left ( \widetilde{\vartheta }_{1},\widetilde{\vartheta }_{2},\widetilde{\vartheta }_{3},\widetilde{\vartheta }_{4},0,\widetilde{\vartheta }_{6}, \widetilde{\vartheta }_{7}\right ) $$

and

$$ \ell \left ( \vartheta _{1},\vartheta _{2},\vartheta _{3},0, \vartheta _{4},\vartheta _{6},\vartheta _{7}\right ) < \ell \left ( \widetilde{\vartheta }_{1},\widetilde{\vartheta }_{2},\widetilde{\vartheta }_{3},0, \widetilde{\vartheta }_{4},\widetilde{\vartheta }_{6},\widetilde{\vartheta }_{7} \right ) . $$

Further, let $\widetilde{\aleph }=\left \{ \ell \in \aleph :\ell \left ( 1,0,0, \lambda ,\upsilon ,0,\frac{\lambda }{2}\right ) \in (0,1]\right \}$, where $\widetilde{\aleph }\subseteq \aleph$.

Example 3.2

The following functions support the above definition:

• (1) Describe $\ell _{1}:[0,\infty )^{7}\rightarrow \lbrack 0,\infty )$ as

  $$ \ell _{1}\left ( \vartheta _{1},\vartheta _{2},\vartheta _{3}, \vartheta _{4},\vartheta _{5},\vartheta _{6},\vartheta _{7}\right ) = \hbar \min \left \{ \vartheta _{1}, \frac{\vartheta _{2}+\vartheta _{3}}{2}, \frac{\vartheta _{4}+\vartheta _{5}}{2}, \frac{\vartheta _{6}+\vartheta _{7}}{2}\right \} , $$

  where $\hbar \in (0,1)$, then, $\ell _{1}\in \aleph$. Since $\ell _{1}\left ( 1,0,0,\lambda ,\upsilon ,0,\frac{\lambda }{2} \right ) =0\notin (0,1]$. Hence, $\ell _{1}\notin \widetilde{\aleph }$. This proves that $\widetilde{\aleph }\subseteq \aleph$, but the converse is not true.

• (2) Describe $\ell _{2}:[0,\infty )^{7}\rightarrow \lbrack 0,\infty )$ as

  $$ \ell _{2}\left ( \vartheta _{1},\vartheta _{2},\vartheta _{3}, \vartheta _{4},\vartheta _{5},\vartheta _{6},\vartheta _{7}\right ) = \frac{\vartheta _{1}}{2}+\frac{\vartheta _{2}+\vartheta _{3}}{4}+ \frac{\vartheta _{6}}{8}, $$

  then $\ell _{2}\left ( 1,0,0,\lambda ,\upsilon ,0,\frac{\lambda }{2} \right ) =\frac{1}{2}\in (0,1]$. Thus, $\ell _{2}\in \widetilde{\aleph }$.

• (3) Describe $\ell _{3}:[0,\infty )^{7}\rightarrow \lbrack 0,\infty )$ as

  $$ \ell _{3}\left ( \vartheta _{1},\vartheta _{2},\vartheta _{3}, \vartheta _{4},\vartheta _{5},\vartheta _{6},\vartheta _{7}\right ) = \hbar \min \left \{ \frac{\vartheta _{1}+\vartheta _{3}}{2}, \frac{\vartheta _{4}+\vartheta _{5}}{2},\frac{\vartheta _{6}+\vartheta _{7}}{2}\right \} , $$

  where $\hbar \in (0,1)$, then $\ell _{3}\left ( 1,0,0,\lambda ,\upsilon ,0,\frac{\lambda }{2}\right ) =\frac{1}{2}\in (0,1]$. Thus, $\ell _{3}\in \widetilde{\aleph }$.

The Lemma below is very important in the following:

Lemma 3.3

Assume that $\ell \in \aleph$, $\beta ,\theta \in \lbrack 0,\infty )$, $\lambda ,\upsilon \geq 1$, and

$$\begin{aligned} \beta \leq &\max \left \{ \ell \left ( \theta ,\theta ,\beta , \lambda \theta +\upsilon \beta ,0,\frac{\theta +\beta }{2}, \frac{\lambda \theta +\upsilon \beta }{2}\right ) ,\right . \\ &\ell \left ( \theta ,\theta ,\beta ,0,\lambda \theta +\upsilon \beta ,\frac{\theta +\beta }{2},\frac{\lambda \theta +\upsilon \beta }{2}\right ) , \\ &\ell \left ( \theta ,\beta ,\theta ,\lambda \theta +\upsilon \beta ,0, \frac{\theta +\beta }{2},\frac{\lambda \theta +\upsilon \beta }{2}\right ) , \\ &\left . \ell \left ( \theta ,\beta ,\theta ,0,\lambda \theta + \upsilon \beta ,\frac{\theta +\beta }{2}, \frac{\lambda \theta +\upsilon \beta }{2}\right ) \right \} . \end{aligned}$$

Then, $\beta \leq \theta$.

Proof

Keeping the generalization intact, we can suppose that

$$ \beta \leq \ell \left ( \theta ,\theta ,\beta ,\lambda \theta + \upsilon \beta ,0,\frac{\theta +\beta }{2}, \frac{\lambda \theta +\upsilon \beta }{2}\right ) . $$

(3.1)

Conversely, let us assume that $\theta <\beta$. Now, we examine

$$\begin{aligned} \ell \left ( \theta ,\theta ,\beta ,\lambda \theta +\upsilon \beta ,0, \frac{\theta +\beta }{2},\frac{\lambda \theta +\upsilon \beta }{2}\right ) < & \ell \left ( \beta ,\beta ,\beta ,\lambda \beta +\upsilon \beta ,0, \beta ,\frac{\lambda \beta +\upsilon \beta }{2}\right ) \\ \leq &\beta \ell \left ( 1,1,1,\lambda +\upsilon ,0,1, \frac{\lambda +\upsilon }{2}\right ) \\ \leq &\beta \left ( 1\right ) . \end{aligned}$$

Hence,

$$ \ell \left ( \theta ,\theta ,\beta ,\lambda \theta +\upsilon \beta ,0, \frac{\theta +\beta }{2},\frac{\lambda \theta +\upsilon \beta }{2}\right ) < \beta , $$

which contradicts (3.1). Therefore, $\beta \leq \theta$. □

4 Existence of fixed points

According to a new definition 3.1, we present our contraction mapping here as follows:

Definition 4.1

($\widetilde{{\psi }}{F} \text{{-contraction}}$)

We say that the mapping $\mho :\eta \rightarrow \eta ^{cb}$ is an ψ̃F-contraction if

$\left ( \widetilde{\psi }F\right ) _{i}$:

for all $q>0$, $F_{1}(q)\leq F_{2}(q)$;

$\left ( \widetilde{\psi }F\right ) _{ii}$:

$\Upsilon \left ( \mho \vartheta ,\mho \widetilde{\vartheta }\right ) >0$, implies

$$\begin{aligned} &\widetilde{\psi }\left ( \varpi \left ( \vartheta , \widetilde{\vartheta }\right ) \right ) +F_{2}\left ( \Upsilon \left ( \mho \vartheta , \mho \widetilde{\vartheta }\right ) \right ) \\ \leq &F_{1}\left \{ \ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta ,\widetilde{\vartheta }\right ) ,B\left ( \vartheta ,\mho \vartheta \right ) ,B\left ( \widetilde{\vartheta }, \mho \widetilde{\vartheta }\right ) ,B\left ( \vartheta ,\mho \widetilde{\vartheta } \right ) , \\ B\left ( \widetilde{\vartheta },\mho \vartheta \right ) , \frac{B\left ( \vartheta ,\mho \vartheta \right ) +B\left ( \widetilde{\vartheta },\mho \widetilde{\vartheta }\right ) }{2}, \frac{B\left ( \vartheta ,\mho \widetilde{\vartheta }\right ) +B\left ( \widetilde{\vartheta },\mho \vartheta \right ) }{2}\end{array}\displaystyle \right ) \right \} , \end{aligned}$$

for all $\vartheta ,\widetilde{\vartheta }\in \eta$, where ϖ is described in Definition 2.1, $F_{1},F_{2}$ are real-valued functions on $(0,\infty )$, $\ell \in \aleph$, and $\widetilde{\boldsymbol{\psi }}\in \Psi$.

Theorem 4.2

Let $\mho :\eta \rightarrow \eta ^{cp}$ be an ψ̃F-contraction defined on a complete CMS $\left ( \eta ,\varpi ,\gamma \right )$. If the conditions below hold:

$(C_{1})$:

$F_{1}$ is a nondecreasing function;

$(C_{2})$:

$F_{2}$ fulfills axioms $(\heartsuit _{2}^{\prime })$, and $(\heartsuit _{3})$;

$(C_{3})$:

for $\vartheta _{0}\in \eta$, define the Picard sequence $\left \{ \vartheta _{u}=\mho ^{u}\vartheta _{0}\right \}$ such that

$$ \sup _{n\geq 1}\lim _{j\rightarrow \infty } \frac{\gamma \left ( \vartheta _{j+1},\vartheta _{j+2}\right ) \gamma \left ( \vartheta _{j+1},\vartheta _{n}\right ) }{\gamma \left ( \vartheta _{j},\vartheta _{j+1}\right ) }< 1; $$

$(C_{4})$:

for $\vartheta \in \eta$, $\lim _{u\rightarrow \infty }\gamma \left ( \vartheta _{u},\vartheta \right ) \leq 1$.

Then, ℧ has at least one FP, that is, $F_{ix}\left ( \mho \right ) \neq \emptyset$.

Proof

Assume that $\vartheta _{0}\in \eta$ and $\vartheta _{1}\in \mho \vartheta _{0}$. Clearly, if $\vartheta _{1}\in \mho \vartheta _{1}$, $\vartheta _{1}\in F_{ix}\left ( \mho \right )$ and the proof is completed. Hence, let $\vartheta _{1}\notin \mho \vartheta _{1}$, which means $B\left ( \vartheta _{1},\mho \vartheta _{1}\right ) >0$. Thus, $\Upsilon \left ( \mho \vartheta _{0},\mho \vartheta _{1}\right ) >0$. As $\mho \vartheta _{1}$ is compact, there is $\vartheta _{2}\in \mho \vartheta _{1}$ such that $\varpi \left ( \vartheta _{1},\vartheta _{2}\right ) =B\left ( \vartheta _{1},\mho \vartheta _{1}\right )$. Consider

$$\begin{aligned} F_{1}\left ( \varpi \left ( \vartheta _{1},\vartheta _{2}\right ) \right ) =&F_{1}\left ( B\left ( \vartheta _{1},\mho \vartheta _{1} \right ) \right ) \\ \leq &F_{1}\left ( \Upsilon \left ( \mho \vartheta _{0},\mho \vartheta _{1}\right ) \right ) \\ \leq &F_{2}\left ( \Upsilon \left ( \mho \vartheta _{0},\mho \vartheta _{1}\right ) \right ) \\ \leq &F_{1}\left \{ \ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta _{0},\vartheta _{1}\right ) ,B\left ( \vartheta _{0},\mho \vartheta _{0}\right ) ,B\left ( \vartheta _{1}, \mho \vartheta _{1}\right ) ,B\left ( \vartheta _{0},\mho \vartheta _{1} \right ) , \\ B\left ( \vartheta _{1},\mho \vartheta _{0}\right ) , \frac{B\left ( \vartheta _{0},\mho \vartheta _{0}\right ) +B\left ( \vartheta _{1},\mho \vartheta _{1}\right ) }{2}, \frac{B\left ( \vartheta _{0},\mho \vartheta _{1}\right ) +B\left ( \vartheta _{1},\mho \vartheta _{0}\right ) }{2}\end{array}\displaystyle \right ) \right \} \\ &-\widetilde{\psi }\left ( \varpi \left ( \vartheta _{0},\vartheta _{1} \right ) \right ) \\ < &F_{1}\left \{ \ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta _{0},\vartheta _{1}\right ) ,\varpi \left ( \vartheta _{0},\vartheta _{1}\right ) ,\varpi \left ( \vartheta _{1}, \vartheta _{2}\right ) ,\varpi \left ( \vartheta _{0},\vartheta _{2} \right ) , \\ \varpi \left ( \vartheta _{1},\vartheta _{1}\right ) , \frac{\varpi \left ( \vartheta _{0},\vartheta _{1}\right ) +\varpi \left ( \vartheta _{1},\vartheta _{2}\right ) }{2}, \frac{\varpi \left ( \vartheta _{0},\vartheta _{2}\right ) +\varpi \left ( \vartheta _{1},\vartheta _{1}\right ) }{2}\end{array}\displaystyle \right ) \right \} . \end{aligned}$$

Since $F_{1}$ is nondecreasing, one has

$$\begin{aligned} \varpi \left ( \vartheta _{1},\vartheta _{2}\right ) < &\ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta _{0},\vartheta _{1}\right ) ,\varpi \left ( \vartheta _{0},\vartheta _{1}\right ) ,\varpi \left ( \vartheta _{1}, \vartheta _{2}\right ) ,\varpi \left ( \vartheta _{0},\vartheta _{2} \right ) , \\ 0, \frac{\varpi \left ( \vartheta _{0},\vartheta _{1}\right ) +\varpi \left ( \vartheta _{1},\vartheta _{2}\right ) }{2}, \frac{\varpi \left ( \vartheta _{0},\vartheta _{2}\right ) }{2}\end{array}\displaystyle \right ) \\ \leq &\ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta _{0},\vartheta _{1}\right ) ,\varpi \left ( \vartheta _{0},\vartheta _{1}\right ) ,\varpi \left ( \vartheta _{1}, \vartheta _{2}\right ) , \\ \gamma \left ( \vartheta _{0},\vartheta _{1}\right ) \varpi \left ( \vartheta _{0},\vartheta _{1}\right ) +\gamma \left ( \vartheta _{1}, \vartheta _{2}\right ) \varpi \left ( \vartheta _{1},\vartheta _{2} \right ) , \\ 0, \frac{\varpi \left ( \vartheta _{0},\vartheta _{1}\right ) +\varpi \left ( \vartheta _{1},\vartheta _{2}\right ) }{2}, \frac{\gamma \left ( \vartheta _{0},\vartheta _{1}\right ) \varpi \left ( \vartheta _{0},\vartheta _{1}\right ) +\gamma \left ( \vartheta _{1},\vartheta _{2}\right ) \varpi \left ( \vartheta _{1},\vartheta _{2}\right ) }{2}\end{array}\displaystyle \right ) . \end{aligned}$$

Based on Lemma 3.3, we conclude that

$$ \varpi \left ( \vartheta _{1},\vartheta _{2}\right ) < \varpi \left ( \vartheta _{0},\vartheta _{1}\right ) . $$

In the same way, we have $\vartheta _{3}\in \mho \vartheta _{2}$ such that $\varpi \left ( \vartheta _{2},\vartheta _{3}\right ) =B\left ( \vartheta _{2},\mho \vartheta _{2}\right )$ with $B\left ( \vartheta _{2},\mho \vartheta _{2}\right ) >0$ and

$$ \varpi \left ( \vartheta _{2},\vartheta _{3}\right ) < \varpi \left ( \vartheta _{1},\vartheta _{2}\right ) . $$

Repeating this technique, we have a sequence $\{\vartheta _{u}\}\subset \eta$ in order that $\vartheta _{u+1}\in \mho \vartheta _{u}$ fulfills $\varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) =B\left ( \vartheta _{u},\mho \vartheta _{u}\right )$ with $B\left ( \vartheta _{u},\mho \vartheta _{u}\right ) >0$ and

$$ \varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) < \varpi \left ( \vartheta _{u-1},\vartheta _{u}\right ) ,\text{ }\forall u\in \mathbb{N} . $$

It follows that $\{\varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) \}_{s\in \mathbb{N} }$ is a decreasing sequence. Next, we can write

$$\begin{aligned} &\widetilde{\psi }\left ( \varpi \left ( \vartheta _{u},\vartheta _{u+1} \right ) \right ) +F_{2}\left ( \Upsilon \left ( \mho \vartheta _{u}, \mho \vartheta _{u+1}\right ) \right ) \\ \leq &F_{1}\left \{ \ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) ,B\left ( \vartheta _{u},\mho \vartheta _{u}\right ) ,B\left ( \vartheta _{u+1}, \mho \vartheta _{u+1}\right ) ,B\left ( \vartheta _{u},\mho \vartheta _{u+1}\right ) , \\ B\left ( \vartheta _{u+1},\mho \vartheta _{u}\right ) , \frac{B\left ( \vartheta _{u},\mho \vartheta _{u}\right ) +B\left ( \vartheta _{u+1},\mho \vartheta _{u+1}\right ) }{2}, \frac{B\left ( \vartheta _{u},\mho \vartheta _{u+1}\right ) +B\left ( \vartheta _{u+1},\mho \vartheta _{u}\right ) }{2}\end{array}\displaystyle \right ) \right \} \\ \leq &F_{1}\left \{ \ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) ,\varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) ,\varpi \left ( \vartheta _{u+1}, \vartheta _{u+2}\right ) ,\varpi \left ( \vartheta _{u},\vartheta _{u+2} \right ) , \\ \varpi \left ( \vartheta _{u+1},\vartheta _{u+1}\right ) , \frac{\varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) +\varpi \left ( \vartheta _{u+1},\vartheta _{u+2}\right ) }{2}, \frac{\varpi \left ( \vartheta _{u},\vartheta _{u+2}\right ) +\varpi \left ( \vartheta _{u+1},\vartheta _{u+1}\right ) }{2}\end{array}\displaystyle \right ) \right \} \\ \leq &F_{1}\left \{ \ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) ,\varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) ,\varpi \left ( \vartheta _{u+1}, \vartheta _{u+2}\right ) , \\ \gamma \left ( \vartheta _{u},\vartheta _{u+1}\right ) \varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) +\gamma \left ( \vartheta _{u+1}, \vartheta _{u+2}\right ) \varpi \left ( \vartheta _{u+1},\vartheta _{u+2} \right ) , \\ 0, \frac{\varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) +\varpi \left ( \vartheta _{u+1},\vartheta _{u+2}\right ) }{2}, \frac{\gamma \left ( \vartheta _{u},\vartheta _{u+1}\right ) \varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) +\gamma \left ( \vartheta _{u+1},\vartheta _{u+2}\right ) \varpi \left ( \vartheta _{u+1},\vartheta _{u+2}\right ) }{2}\end{array}\displaystyle \right ) \right \} \\ < &F_{1}\left \{ \ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) ,\varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) ,\varpi \left ( \vartheta _{u}, \vartheta _{u+1}\right ) , \\ \gamma \left ( \vartheta _{u},\vartheta _{u+1}\right ) \varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) +\gamma \left ( \vartheta _{u+1}, \vartheta _{u+2}\right ) \varpi \left ( \vartheta _{u},\vartheta _{u+1} \right ) , \\ 0, \frac{\varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) +\varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) }{2}, \frac{\gamma \left ( \vartheta _{u},\vartheta _{u+1}\right ) \varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) +\gamma \left ( \vartheta _{u+1},\vartheta _{u+2}\right ) \varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) }{2}\end{array}\displaystyle \right ) \right \} \\ \leq &F_{1}\left \{ \varpi \left ( \vartheta _{u},\vartheta _{u+1} \right ) \ell \left ( \textstyle\begin{array}{c} 1,1,1,\gamma \left ( \vartheta _{u},\vartheta _{u+1}\right ) +\gamma \left ( \vartheta _{u+1},\vartheta _{u+2}\right ) , \\ 0,1, \frac{\gamma \left ( \vartheta _{u},\vartheta _{u+1}\right ) +\gamma \left ( \vartheta _{u+1},\vartheta _{u+2}\right ) }{2}\end{array}\displaystyle \right ) \right \} \\ \leq &F_{1}\left ( \varpi \left ( \vartheta _{u},\vartheta _{u+1} \right ) \right ) \\ =&F_{1}\left ( B\left ( \vartheta _{u},\mho \vartheta _{u}\right ) \right ) \\ \leq &F_{1}\left ( \Upsilon \left ( \mho \vartheta _{u-1},\mho \vartheta _{u}\right ) \right ) \\ \leq &F_{2}\left ( \Upsilon \left ( \mho \vartheta _{u-1},\mho \vartheta _{u}\right ) \right ) . \end{aligned}$$

Hence, for each $u\in \mathbb{N}$, we conclude that

$$ F_{2}\left ( \Upsilon \left ( \mho \vartheta _{u},\mho \vartheta _{u+1} \right ) \right ) \leq F_{2}\left ( \Upsilon \left ( \mho \vartheta _{u-1}, \mho \vartheta _{u}\right ) \right ) -\widetilde{\psi }\left ( \varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) \right ) . $$

(4.1)

Since $\widetilde{\boldsymbol{\psi }}\in \Psi$, there is $\zeta >0$ so that $\widetilde{\psi }\left ( \varpi \left ( \vartheta _{u},\vartheta _{u+1} \right ) \right ) >\zeta$, for all $u\geq u_{0}$. From (4.1), we obtain

$$\begin{aligned} F_{2}\left ( \Upsilon \left ( \mho \vartheta _{u},\mho \vartheta _{u+1} \right ) \right ) \leq &F_{2}\left ( \Upsilon \left ( \mho \vartheta _{u-1},\mho \vartheta _{u}\right ) \right ) - \widetilde{\psi }\left ( \varpi \left ( \vartheta _{u},\vartheta _{u+1} \right ) \right ) \\ < &F_{2}\left ( \Upsilon \left ( \mho \vartheta _{u-2},\mho \vartheta _{u-1}\right ) \right ) -\widetilde{\psi }\left ( \varpi \left ( \vartheta _{u-1},\vartheta _{u}\right ) \right ) - \widetilde{\psi }\left ( \varpi \left ( \vartheta _{u},\vartheta _{u+1} \right ) \right ) \\ &\vdots \\ < &F_{2}\left ( \Upsilon \left ( \mho \vartheta _{0},\mho \vartheta _{1} \right ) \right ) -\sum \limits _{j=1}^{u}\widetilde{\psi }\left ( \varpi \left ( \vartheta _{j},\vartheta _{j+1}\right ) \right ) \\ =&F_{2}\left ( \Upsilon \left ( \mho \vartheta _{0},\mho \vartheta _{1} \right ) \right ) -\sum \limits _{j=1}^{u_{0}-1}\widetilde{\psi } \left ( \varpi \left ( \vartheta _{j},\vartheta _{j+1}\right ) \right ) -\sum \limits _{j=u_{0}}^{u}\widetilde{\psi }\left ( \varpi \left ( \vartheta _{j},\vartheta _{j+1}\right ) \right ) \\ < &F_{2}\left ( \Upsilon \left ( \mho \vartheta _{0},\mho \vartheta _{1} \right ) \right ) -(u-u_{0})\zeta ,\text{ }u\geq u_{0}. \end{aligned}$$

Hence,

$$ F_{2}\left ( \Upsilon \left ( \mho \vartheta _{u},\mho \vartheta _{u+1} \right ) \right ) < F_{2}\left ( \Upsilon \left ( \mho \vartheta _{0}, \mho \vartheta _{1}\right ) \right ) -(u-u_{0})\zeta ,\text{ }u\geq u_{0}. $$

(4.2)

Letting $u\rightarrow \infty$ in (4.2), we have

$$ \lim _{u\rightarrow \infty }F_{2}\left ( \Upsilon \left ( \mho \vartheta _{u},\mho \vartheta _{u+1}\right ) \right ) =-\infty . $$

Applying $(\heartsuit _{2}^{\prime })$, we obtain

$$ \lim _{u\rightarrow \infty }\Upsilon \left ( \mho \vartheta _{u}, \mho \vartheta _{u+1}\right ) =0, $$

which yields,

$$ \lim _{u\rightarrow \infty }\varpi \left ( \vartheta _{u},\vartheta _{u+1} \right ) =\lim _{u\rightarrow \infty }D\left ( \vartheta _{u},\mho \vartheta _{u}\right ) \leq \lim _{u\rightarrow \infty }\Upsilon \left ( \mho \vartheta _{u-1},\mho \vartheta _{u}\right ) =0. $$

(4.3)

According to $(\heartsuit _{3})$, there is $l\in (0,1)$ so that

$$ \lim _{u\rightarrow \infty }\left ( \Upsilon \left ( \mho \vartheta _{u}, \mho \vartheta _{u+1}\right ) \right ) ^{l}F_{2}\left ( \Upsilon \left ( \mho \vartheta _{u},\mho \vartheta _{u+1}\right ) \right ) =0. $$

(4.4)

For all $u\geq u_{0}$, from (4.2), one has

$$\begin{aligned} &\left ( \Upsilon \left ( \mho \vartheta _{u},\mho \vartheta _{u+1} \right ) \right ) ^{l}F_{2}\left ( \Upsilon \left ( \mho \vartheta _{u}, \mho \vartheta _{u+1}\right ) \right ) -\left ( \Upsilon \left ( \mho \vartheta _{u},\mho \vartheta _{u+1}\right ) \right ) ^{l}F_{2} \left ( \Upsilon \left ( \mho \vartheta _{0},\mho \vartheta _{1} \right ) \right ) \\ \leq &\left ( \Upsilon \left ( \mho \vartheta _{u},\mho \vartheta _{u+1} \right ) \right ) ^{l}\left [ F_{2}\left ( \Upsilon \left ( \mho \vartheta _{0},\mho \vartheta _{1}\right ) \right ) -(u-u_{0})\zeta \right ] \\ &-\left ( \Upsilon \left ( \mho \vartheta _{u},\mho \vartheta _{u+1} \right ) \right ) ^{l}F_{2}\left ( \Upsilon \left ( \mho \vartheta _{0}, \mho \vartheta _{1}\right ) \right ) \\ =&-\left ( \Upsilon \left ( \mho \vartheta _{u},\mho \vartheta _{u+1} \right ) \right ) ^{l}(u-u_{0})\zeta \\ \leq &0. \end{aligned}$$

As $u\rightarrow \infty$ in (4.3) and (4.4), we can write

$$ 0\leq \lim _{u\rightarrow \infty }u\left ( \Upsilon \left ( \mho \vartheta _{u},\mho \vartheta _{u+1}\right ) \right ) ^{l}\leq 0 $$

and it follows that

$$ \lim _{u\rightarrow \infty }u\left ( \Upsilon \left ( \mho \vartheta _{u}, \mho \vartheta _{u+1}\right ) \right ) ^{l}=0. $$

(4.5)

Based on (4.5), there is $u_{1}\in \mathbb{N}$ so that $u\left ( \Upsilon \left ( \mho \vartheta _{u},\mho \vartheta _{u+1} \right ) \right ) ^{l}\leq 1$, for all $u\geq u_{1}$. Thus, we have

$$ \Upsilon \left ( \mho \vartheta _{u},\mho \vartheta _{u+1}\right ) \leq \frac{1}{u^{\frac{1}{l}}}\text{ for all }u\geq u_{1}. $$

Therefore,

$$ \varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) =B\left ( \vartheta _{u},\mho \vartheta _{u}\right ) \leq \Upsilon \left ( \mho \vartheta _{u-1},\mho \vartheta _{u}\right ) \leq \frac{1}{u^{\frac{1}{l}}}\text{ for all }u\geq u_{1}. $$

Now, we prove that $\{\vartheta _{u}\}$ is a Cauchy sequence (CS). In this regard, let $v,u\in \mathbb{N}$ in order that $v>u>u_{1}$. Then,

$$\begin{aligned} \varpi \left ( \vartheta _{u},\vartheta _{v}\right ) =&\gamma \left ( \vartheta _{u},\vartheta _{u+1}\right ) \varpi \left ( \vartheta _{u}, \vartheta _{u+1}\right ) +\gamma \left ( \vartheta _{u+1},\vartheta _{v} \right ) \varpi \left ( \vartheta _{u+1},\vartheta _{v}\right ) \\ \leq &\gamma \left ( \vartheta _{u},\vartheta _{u+1}\right ) \varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) +\gamma \left ( \vartheta _{u+1},\vartheta _{v}\right ) \gamma \left ( \vartheta _{u+1}, \vartheta _{u+2}\right ) \varpi \left ( \vartheta _{u+1},\vartheta _{u+2} \right ) \\ &+\gamma \left ( \vartheta _{u+1},\vartheta _{v}\right ) \gamma \left ( \vartheta _{u+2},\vartheta _{v}\right ) \varpi \left ( \vartheta _{u+2},\vartheta _{v}\right ) \\ \leq &\gamma \left ( \vartheta _{u},\vartheta _{u+1}\right ) \varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) +\gamma \left ( \vartheta _{u+1},\vartheta _{v}\right ) \gamma \left ( \vartheta _{u+1}, \vartheta _{u+2}\right ) \varpi \left ( \vartheta _{u+1},\vartheta _{u+2} \right ) \\ &+\gamma \left ( \vartheta _{u+1},\vartheta _{v}\right ) \gamma \left ( \vartheta _{u+2},\vartheta _{v}\right ) \gamma \left ( \vartheta _{u+2},\vartheta _{u+3}\right ) \varpi \left ( \vartheta _{u+2}, \vartheta _{u+3}\right ) \\ &+\gamma \left ( \vartheta _{u+1},\vartheta _{v}\right ) \gamma \left ( \vartheta _{u+2},\vartheta _{v}\right ) \gamma \left ( \vartheta _{u+3},\vartheta _{v}\right ) \varpi \left ( \vartheta _{u+3}, \vartheta _{v}\right ) \\ \leq & \\ &\vdots \\ \leq &\gamma \left ( \vartheta _{u},\vartheta _{u+1}\right ) \varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) +\sum _{j=u+1}^{v-2} \left ( \prod \limits _{b=u+1}^{j}\gamma \left ( \vartheta _{b}, \vartheta _{v}\right ) \right ) \gamma \left ( \vartheta _{j}, \vartheta _{j+1}\right ) \varpi \left ( \vartheta _{j},\vartheta _{j+1} \right ) \\ &+\left ( \prod \limits _{b=u+1}^{v-1}\gamma \left ( \vartheta _{j}, \vartheta _{v}\right ) \right ) \varpi \left ( \vartheta _{v-1}, \vartheta _{v}\right ) \\ \leq &\gamma \left ( \vartheta _{u},\vartheta _{u+1}\right ) \varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) +\sum _{j=u+1}^{v-2} \left ( \prod \limits _{b=u+1}^{j}\gamma \left ( \vartheta _{b}, \vartheta _{v}\right ) \right ) \gamma \left ( \vartheta _{j}, \vartheta _{j+1}\right ) \varpi \left ( \vartheta _{j},\vartheta _{j+1} \right ) \\ &+\left ( \prod \limits _{b=u+1}^{v-1}\gamma \left ( \vartheta _{j}, \vartheta _{v}\right ) \right ) \gamma \left ( \vartheta _{v-1}, \vartheta _{v}\right ) \varpi \left ( \vartheta _{v-1},\vartheta _{v} \right ) \\ =&\gamma \left ( \vartheta _{u},\vartheta _{u+1}\right ) \varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) +\sum _{j=u+1}^{v-2} \left ( \prod \limits _{b=u+1}^{j}\gamma \left ( \vartheta _{b}, \vartheta _{v}\right ) \right ) \gamma \left ( \vartheta _{j}, \vartheta _{j+1}\right ) \varpi \left ( \vartheta _{j},\vartheta _{j+1} \right ) \\ \leq &\gamma \left ( \vartheta _{u},\vartheta _{u+1}\right ) \varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) +\sum _{j=u+1}^{v-2} \left ( \prod \limits _{b=0}^{j}\gamma \left ( \vartheta _{b}, \vartheta _{v}\right ) \right ) \gamma \left ( \vartheta _{j}, \vartheta _{j+1}\right ) \varpi \left ( \vartheta _{j},\vartheta _{j+1} \right ) . \end{aligned}$$

Therefore,

$$ \varpi \left ( \vartheta _{u},\vartheta _{v}\right ) \leq \gamma \left ( \vartheta _{u},\vartheta _{u+1}\right ) \varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) +\sum _{j=u+1}^{v-2}\left ( \prod \limits _{b=0}^{j}\gamma \left ( \vartheta _{b},\vartheta _{v} \right ) \right ) \gamma \left ( \vartheta _{j},\vartheta _{j+1} \right ) \times \frac{1}{j^{\frac{1}{l}}}. $$

(4.6)

Consider

$$\begin{aligned} \sum _{j=u+1}^{v-2}\left ( \prod \limits _{b=0}^{j}\gamma \left ( \vartheta _{b},\vartheta _{v}\right ) \right ) \gamma \left ( \vartheta _{j},\vartheta _{j+1}\right ) \times \frac{1}{j^{\frac{1}{l}}} \leq &\sum _{j=u+1}^{\infty }\frac{1}{j^{\frac{1}{l}}}\left ( \prod \limits _{b=0}^{j}\gamma \left ( \vartheta _{b},\vartheta _{v}\right ) \right ) \gamma \left ( \vartheta _{j},\vartheta _{j+1}\right ) \\ =&\sum _{j=u+1}^{\infty }M_{j}N_{j}, \end{aligned}$$

where $M_{j}=\frac{1}{j^{\frac{1}{l}}}$ and $N_{j}=\left ( \prod _{b=0}^{j}\gamma \left ( \vartheta _{b}, \vartheta _{v}\right ) \right ) \gamma \left ( \vartheta _{j}, \vartheta _{j+1}\right )$. As $\frac{1}{l}>0$, the series $\sum _{j=u+1}^{\infty }\left ( \frac{1}{j^{\frac{1}{l}}}\right )$ converges. Since $\{N_{j}\}_{j}$ is bounded above and increasing, the nonzero $\lim _{j\rightarrow \infty }\{N_{j}\}$ exists. Hence, $\lim _{j\rightarrow \infty }\{M_{j}N_{j}\}$ converges.

Take the partial sums $\wp =\sum _{j=0}^{\infty }\left ( \prod _{b=0}^{j}\gamma \left ( \vartheta _{b},\vartheta _{v}\right ) \right ) \gamma \left ( \vartheta _{j},\vartheta _{j+1}\right ) \times \frac{1}{j^{\frac{1}{l}}}$. From (4.6), we can write

$$ \varpi \left ( \vartheta _{u},\vartheta _{v}\right ) \leq \gamma \left ( \vartheta _{u},\vartheta _{u+1}\right ) \varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) +\left ( \wp _{v-1}-\wp _{u} \right ) . $$

(4.7)

Utilizing the ratio test and Condition $(C_{3})$, we have that $\lim _{u\rightarrow \infty }\{\wp _{u}\}$ exists. Letting $u\rightarrow \infty$ in (4.7), we conclude that

$$ \lim _{u\rightarrow \infty }\varpi \left ( \vartheta _{u},\vartheta _{v} \right ) =0. $$

This proves that $\{\vartheta _{u}\}$ is a CS. Since η is complete, there is $\vartheta ^{\ast }\in \eta$ so that

$$ \lim _{u\rightarrow \infty }\vartheta _{u}=\vartheta ^{\ast }. $$

(4.8)

Consider

$$\begin{aligned} F_{1}\left ( \Upsilon \left ( \mho \vartheta ,\mho \widetilde{\vartheta }\right ) \right ) \leq &F_{2}\left ( \Upsilon \left ( \mho \vartheta ,\mho \widetilde{\vartheta }\right ) \right ) \leq \widetilde{\psi }\left ( \varpi \left ( \vartheta , \widetilde{\vartheta }\right ) \right ) +F_{2}\left ( \Upsilon \left ( \mho \vartheta ,\mho \widetilde{\vartheta }\right ) \right ) \\ \leq &F_{1}\left \{ \ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta ,\widetilde{\vartheta }\right ) ,B\left ( \vartheta ,\mho \vartheta \right ) ,B\left ( \widetilde{\vartheta }, \mho \widetilde{\vartheta }\right ) ,B\left ( \vartheta ,\mho \widetilde{\vartheta } \right ) , \\ B\left ( \widetilde{\vartheta },\mho \vartheta \right ) , \frac{B\left ( \vartheta ,\mho \vartheta \right ) +B\left ( \widetilde{\vartheta },\mho \widetilde{\vartheta }\right ) }{2}, \frac{B\left ( \vartheta ,\mho \widetilde{\vartheta }\right ) +B\left ( \widetilde{\vartheta },\mho \vartheta \right ) }{2}\end{array}\displaystyle \right ) \right \}. \end{aligned}$$

Since $F_{1}$ is a nondecreasing function, then for $\vartheta ,\widetilde{\vartheta }\in \xi$, one can write

$$ \Upsilon \left ( \mho \vartheta ,\mho \widetilde{\vartheta }\right ) \leq \ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta ,\widetilde{\vartheta }\right ) ,B\left ( \vartheta ,\mho \vartheta \right ) ,B\left ( \widetilde{\vartheta }, \mho \widetilde{\vartheta }\right ) ,B\left ( \vartheta ,\mho \widetilde{\vartheta } \right ) ,B\left ( \widetilde{\vartheta },\mho \vartheta \right ) , \\ \frac{B\left ( \vartheta ,\mho \vartheta \right ) +B\left ( \widetilde{\vartheta },\mho \widetilde{\vartheta }\right ) }{2}, \frac{B\left ( \vartheta ,\mho \widetilde{\vartheta }\right ) +B\left ( \widetilde{\vartheta },\mho \vartheta \right ) }{2}\end{array}\displaystyle \right ) . $$

(4.9)

Then, to illustrate the existence of the FP of ℧, assume the contrary, that is, $B\left ( \vartheta ^{\ast },\mho \vartheta ^{\ast }\right ) >0$. Using (4.9) and the compactness of $\mho \vartheta ^{\ast }$ implies that there is $\vartheta \in \mho \vartheta ^{\ast }$ such that

$$\begin{aligned} &B\left ( \vartheta ^{\ast },\mho \vartheta ^{\ast }\right ) \\ &\quad= \varpi \left ( \vartheta ^{\ast },\vartheta \right ) \\ &\quad\leq \gamma \left ( \vartheta ^{\ast },\vartheta _{u+1}\right ) \varpi \left ( \vartheta ^{\ast },\vartheta _{u+1}\right ) +\gamma \left ( \vartheta _{u+1},\vartheta \right ) \varpi \left ( \vartheta _{u+1}, \vartheta \right ) \\ &\quad=\gamma \left ( \vartheta ^{\ast },\vartheta _{u+1}\right ) \varpi \left ( \vartheta ^{\ast },\vartheta _{u+1}\right ) +\gamma \left ( \vartheta _{u+1},\vartheta \right ) B\left ( \vartheta _{u+1},\mho \vartheta ^{\ast }\right ) \\ &\quad\leq \gamma \left ( \vartheta ^{\ast },\vartheta _{u+1}\right ) \varpi \left ( \vartheta ^{\ast },\vartheta _{u+1}\right ) +\gamma \left ( \vartheta _{u+1},\vartheta \right ) \Upsilon \left ( \mho \vartheta _{u},\mho \vartheta ^{\ast }\right ) \\ &\quad\leq \gamma \left ( \vartheta ^{\ast },\vartheta _{u+1}\right ) \varpi \left ( \vartheta ^{\ast },\vartheta _{u+1}\right ) +\gamma \left ( \vartheta _{u+1},\vartheta \right ) \\ &\qquad{}\times \ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta _{u},\vartheta ^{\ast }\right ) ,B\left ( \vartheta _{u},\mho \vartheta _{u}\right ) ,B\left ( \vartheta ^{ \ast },\mho \vartheta ^{\ast }\right ) , \\ B\left ( \vartheta _{u},\mho \vartheta ^{\ast }\right ) ,B\left ( \vartheta ^{\ast },\mho \vartheta _{u}\right ) , \\ \frac{B\left ( \vartheta _{u},\mho \vartheta _{u}\right ) +B\left ( \vartheta ^{\ast },\mho \vartheta ^{\ast }\right ) }{2}, \frac{B\left ( \vartheta _{u},\mho \vartheta ^{\ast }\right ) +B\left ( \vartheta ^{\ast },\mho \vartheta _{u}\right ) }{2}\end{array}\displaystyle \right ) \\ &\quad\leq \gamma \left ( \vartheta ^{\ast },\vartheta _{u+1}\right ) \varpi \left ( \vartheta ^{\ast },\vartheta _{u+1}\right ) +\gamma \left ( \vartheta _{u+1},\vartheta \right )\\ &\qquad{}\times \ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta _{u},\vartheta ^{\ast }\right ) ,\varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) ,B\left ( \vartheta ^{ \ast },\mho \vartheta ^{\ast }\right ) , \\ \gamma \left ( \vartheta _{u},\vartheta ^{\ast }\right ) \varpi \left ( \vartheta _{u},\vartheta ^{\ast }\right ) +\gamma \left ( \vartheta ^{\ast },\mho \vartheta ^{\ast }\right ) \varpi \left ( \vartheta ^{\ast },\mho \vartheta ^{\ast }\right ) , \\ \varpi \left ( \vartheta ^{\ast },\vartheta _{u+1}\right ) , \frac{\varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) +B\left ( \vartheta ^{\ast },\mho \vartheta ^{\ast }\right ) }{2}, \\ \frac{\gamma \left ( \vartheta _{u},\vartheta ^{\ast }\right ) \varpi \left ( \vartheta _{u},\vartheta ^{\ast }\right ) +\gamma \left ( \vartheta ^{\ast },\mho \vartheta ^{\ast }\right ) \varpi \left ( \vartheta ^{\ast },\mho \vartheta ^{\ast }\right ) +\varpi \left ( \vartheta ^{\ast },\vartheta _{u+1}\right ) }{2}\end{array}\displaystyle \right ) . \end{aligned}$$

In the above inequality, letting $u\rightarrow \infty$, using Condition $(C_{4})$, and (4.8), we have

$$\begin{aligned} B\left ( \vartheta ^{\ast },\mho \vartheta ^{\ast }\right ) \leq &(1) \ell \left ( \textstyle\begin{array}{c} 0,0,B\left ( \vartheta ^{\ast },\mho \vartheta ^{\ast }\right ) ,0+ \gamma \left ( \vartheta ^{\ast },\mho \vartheta ^{\ast }\right ) \varpi \left ( \vartheta ^{\ast },\mho \vartheta ^{\ast }\right ) , \\ 0, \frac{0+B\left ( \vartheta ^{\ast },\mho \vartheta ^{\ast }\right ) }{2},\frac{0+\gamma \left ( \vartheta ^{\ast },\mho \vartheta ^{\ast }\right ) \varpi \left ( \vartheta ^{\ast },\mho \vartheta ^{\ast }\right ) }{2}\end{array}\displaystyle \right ) \\ \leq &\ell \left ( 0,0,B\left ( \vartheta ^{\ast },\mho \vartheta ^{ \ast }\right ) ,\varpi \left ( \vartheta ^{\ast },\mho \vartheta ^{ \ast }\right ) ,0,\frac{B\left ( \vartheta ^{\ast },\mho \vartheta ^{\ast }\right ) }{2}, \frac{\varpi \left ( \vartheta ^{\ast },\mho \vartheta ^{\ast }\right ) }{2} \right ) . \end{aligned}$$

Applying Lemma 3.3, we obtain that $B\left ( \vartheta ^{\ast },\mho \vartheta ^{\ast }\right ) \leq 0$. Hence, $B\left ( \vartheta ^{\ast },\mho \vartheta ^{\ast }\right ) =0$. As $\mho \vartheta ^{\ast }$ is closed, we have $\vartheta ^{\ast }\in \mho \vartheta ^{\ast }$, and this completes the proof. □

Theorem 4.2 can be supported by the following example:

Example 4.3

Consider $\eta =\left \{ 0,\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5}\right \}$. Describe $\varpi :\eta \times \eta \rightarrow \lbrack 0,\infty )$ and $\gamma :\eta \times \eta \rightarrow \lbrack 1,\infty )$ as mapping $\varpi \left ( \vartheta ,\widetilde{\vartheta }\right ) =\left \vert \vartheta -\widetilde{\vartheta }\right \vert ^{2}$ and

$$ \gamma \left ( \vartheta ,\widetilde{\vartheta }\right ) =\left \{ \textstyle\begin{array}{c@{\quad}c} 1, \ \ \ \ \ \ \ & \text{if }\vartheta =\widetilde{\vartheta }=0, \ \ \ \ \ \\ \frac{1}{\left ( \vartheta +\widetilde{\vartheta }\right ) ^{2}}, & \text{if }\vartheta \neq 0\text{ or }\widetilde{\vartheta }\neq 0,\end{array}\displaystyle \right . $$

respectively. Clearly, $\left ( \eta ,\varpi ,\gamma \right )$ is a complete CMS. Further, define $F_{1},F_{2}:\mathbb{R} ^{+}\rightarrow \mathbb{R}$ by

$$ F_{1}\left ( h\right ) =\left \{ \textstyle\begin{array}{c@{\quad}c} \frac{-1}{h}, & \text{if }h\in (0,1), \\ h^{2}, & \text{if }h\in \lbrack 1,\infty )\end{array}\displaystyle \right . $$

and $F_{2}\left ( h\right ) =\ln \left ( h\right ) +h^{2}$, for $h\in \mathbb{R} ^{+}$. From the definition of $F_{1}$ and $F_{2}$, we find that $F_{1}$ is nondecreasing, $F_{2}$ fulfills the conditions $(\heartsuit _{2}^{\prime })$ and $(\heartsuit _{3})$, and for all $h\in \mathbb{R} ^{+}$, $F_{1}\left ( h\right ) \leq F_{2}\left ( h\right )$. Let us define $\mho :\eta \rightarrow \eta ^{cp}$, $\ell :[0,\infty )^{7}\rightarrow \lbrack 0,\infty )$, and $\widetilde{\psi }:\mathbb{R} ^{+}\rightarrow \mathbb{R} ^{+}$ by

$$ \mho \vartheta =\left \{ \textstyle\begin{array}{c@{\quad}c} \{0\} & \text{if }\vartheta =0, \\ \{0,\frac{1}{2},\frac{1}{3}\} & \text{if }\vartheta \neq 0,\end{array}\displaystyle \right . $$

$\ell \left ( \vartheta _{1},\vartheta _{2},\vartheta _{3},\vartheta _{4}, \vartheta _{5},\vartheta _{6},\vartheta _{7}\right ) = \frac{\vartheta _{1}}{2}+30\vartheta _{5}$, and $\widetilde{\psi }\left ( s\right ) =\frac{1}{s^{2}}$, $s\in \mathbb{R} ^{+}$, respectively. It is clear that $\ell \in \aleph$, $\widetilde{\psi }\in \Psi$. As $\Upsilon \left ( \mho \vartheta ,\mho \widetilde{\vartheta }\right ) >0$, it follows that

$$\begin{aligned} &\widetilde{\psi }\left ( \varpi \left ( \vartheta , \widetilde{\vartheta }\right ) \right ) +F_{2}\left ( \Upsilon \left ( \mho \vartheta , \mho \widetilde{\vartheta }\right ) \right )\\ &\quad \leq F_{1}\left \{ \ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta ,\widetilde{\vartheta }\right ) ,B\left ( \vartheta ,\mho \vartheta \right ) ,B\left ( \widetilde{\vartheta }, \mho \widetilde{\vartheta }\right ) ,B\left ( \vartheta ,\mho \widetilde{\vartheta } \right ) , \\ B\left ( \widetilde{\vartheta },\mho \vartheta \right ) , \frac{B\left ( \vartheta ,\mho \vartheta \right ) +B\left ( \widetilde{\vartheta },\mho \widetilde{\vartheta }\right ) }{2}, \frac{B\left ( \vartheta ,\mho \widetilde{\vartheta }\right ) +B\left ( \widetilde{\vartheta },\mho \vartheta \right ) }{2}\end{array}\displaystyle \right ) \right \} . \end{aligned}$$

Moreover, $\lim _{u\rightarrow \infty }\gamma \left ( \vartheta _{u},\vartheta \right ) \leq 1$. Therefore, all the requirements of Theorem 4.2 are fulfilled and $F_{ix}\left ( \mho \right ) =\{0,\frac{1}{2},\frac{1}{3}\}$.

We can relax the conditions of Theorem 4.2, by neglecting conditions $(\heartsuit _{3})$ and $(C_{3})$ as follows:

Theorem 4.4

Let $\mho :\eta \rightarrow \eta ^{cp}$ be an MVM described on a complete CMS $\left ( \eta ,\varpi ,\gamma \right )$. Assume that $F_{1}$ and $F_{2}$ are functions verifying ψ̃F-contraction. Also, suppose that the assertions below are true:

$(i)$:

$F_{1}$ is nondecreasing;

$(ii)$:

$F_{2}$ fulfills $(\heartsuit _{2}^{\prime })$;

$(iii)$:

for $\vartheta \in \eta$, $\lim _{l\rightarrow \infty }\gamma \left ( \vartheta _{v_{l}}, \vartheta _{u_{l}}\right ) \leq 1$.

Then, $F_{ix}\left ( \mho \right ) \neq \emptyset$.

Proof

Assume that $\vartheta _{0}\in \eta$ and $\vartheta _{1}\in \mho \vartheta _{0}$. Similar to the proof of Theorem 4.2, consider that $\{\vartheta _{u}\}\subset \eta$ is a sequence such that $\vartheta _{u+1}\in \mho \vartheta _{u}$. It fulfills $\varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) =B\left ( \vartheta _{u},\mho \vartheta _{u}\right )$ with $B\left ( \vartheta _{u},\mho \vartheta _{u}\right ) >0$ and

$$ \varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) < \varpi \left ( \vartheta _{u-1},\vartheta _{u}\right ) ,\text{ }\forall u\in \mathbb{N} . $$

Also, we have

$$ F_{2}\left ( \Upsilon \left ( \mho \vartheta _{u},\mho \vartheta _{u+1} \right ) \right ) < F_{2}\left ( \Upsilon \left ( \mho \vartheta _{0}, \mho \vartheta _{1}\right ) \right ) -(u-u_{0})\zeta ,\text{ }u\geq u_{0}. $$

(4.10)

Passing $u\rightarrow \infty$ in (4.10), we obtain

$$ \lim _{u\rightarrow \infty }F_{2}\left ( \Upsilon \left ( \mho \vartheta _{u},\mho \vartheta _{u+1}\right ) \right ) =-\infty . $$

From $(\heartsuit _{2}^{\prime })$, we have

$$ \lim _{u\rightarrow \infty }\Upsilon \left ( \mho \vartheta _{u}, \mho \vartheta _{u+1}\right ) =0, $$

(4.11)

which implies that

$$ \lim _{u\rightarrow \infty }\varpi \left ( \vartheta _{u},\vartheta _{u+1} \right ) =\lim _{u\rightarrow \infty }D\left ( \vartheta _{u},\mho \vartheta _{u}\right ) \leq \lim _{u\rightarrow \infty }\Upsilon \left ( \mho \vartheta _{u-1},\mho \vartheta _{u}\right ) =0. $$

Now, we claim that

$$ \lim _{u,v\rightarrow \infty }\varpi \left ( \vartheta _{u}, \vartheta _{v}\right ) =0. $$

(4.12)

Assume the converse, i.e., there is $\theta >0$ so that for each $\widehat{r}\geq 0$, there exists $v_{l}>u_{l}>\widehat{r}$ such that

$$ \varpi \left ( \vartheta _{u_{l}},\vartheta _{v_{l}}\right ) >\theta . $$

Further, there is $\widehat{r}_{0}\in \mathbb{N}$ in order that

$$ m_{\widehat{r}_{0}}=\varpi \left ( \vartheta _{u-1},\vartheta _{u} \right ) < \theta ,\text{ }\forall u\geq \widehat{r}_{0}. $$

Also, there are two subsequences $\{\vartheta _{u_{l}}\}$ and $\{\vartheta _{v_{l}}\}$ of $\{\vartheta _{u}\}$ in order that

$$ \widehat{r}_{0}\leq u_{l}\leq v_{l}+1\text{ and }\varpi \left ( \vartheta _{u_{l}},\vartheta _{v_{l}}\right ) >\theta ,\ \forall l\geq 0. $$

(4.13)

It should be noted that

$$ \varpi \left ( \vartheta _{v_{l}-1},\vartheta _{u_{l}}\right ) < \theta ,\text{ }\forall l $$

(4.14)

and $v_{l}$ is the minimal index in order that (4.14) is satisfied. From (4.13) and (4.14), it is impossible to verify that $\vartheta _{u}+1\leq \vartheta _{u}$, then, $\vartheta _{u}+2\leq v_{l}$. This proves that

$$ \vartheta _{u}+1< v_{l}< v_{l}+1,\text{ }\forall l. $$

Again, using (4.13), (4.14), and $(\varpi _{3})$, one can write

$$\begin{aligned} \theta < &\varpi \left ( \vartheta _{v_{l}},\vartheta _{u_{l}}\right ) \\ \leq &\gamma \left ( \vartheta _{v_{l}},\vartheta _{v_{l}-1}\right ) \varpi \left ( \vartheta _{v_{l}},\vartheta _{v_{l}-1}\right ) + \gamma \left ( \vartheta _{v_{l}-1},\vartheta _{u_{l}}\right ) \varpi \left ( \vartheta _{v_{l}-1},\vartheta _{u_{l}}\right ) \\ \leq &\gamma \left ( \vartheta _{v_{l}},\vartheta _{v_{l}-1}\right ) \varpi \left ( \vartheta _{v_{l}},\vartheta _{v_{l}-1}\right ) + \theta \gamma \left ( \vartheta _{v_{l}-1},\vartheta _{u_{l}}\right ) . \end{aligned}$$

Passing $l\rightarrow \infty$ in the above inequality and using the condition (iii) of Theorem 4.4, one has

$$\begin{aligned} \theta < &\lim _{l\rightarrow \infty }\varpi \left ( \vartheta _{v_{l}}, \vartheta _{u_{l}}\right ) \\ \leq &0+\theta \lim _{l\rightarrow \infty }\gamma \left ( \vartheta _{v_{l}-1}, \vartheta _{u_{l}}\right ) \\ =&\theta \lim _{l\rightarrow \infty }\gamma \left ( \vartheta _{v_{l}-1}, \vartheta _{u_{l}}\right ) \\ \leq &\theta . \end{aligned}$$

This proves that

$$ \lim _{l\rightarrow \infty }\varpi \left ( \vartheta _{v_{l}}, \vartheta _{u_{l}}\right ) =\theta . $$

(4.15)

From (4.11) and (4.15), we deduce that

$$ \lim _{l\rightarrow \infty }\varpi \left ( \vartheta _{v_{l}+1}, \vartheta _{u_{l}+1}\right ) =\theta . $$

(4.16)

Let

$$\begin{aligned} &\widetilde{\psi }\left ( \varpi \left ( \vartheta _{v_{l}}, \vartheta _{u_{l}}\right ) \right ) +F_{1}\left ( \varpi \left ( \vartheta _{v_{l}+1},\vartheta _{u_{l}+1}\right ) \right ) \\ =&\widetilde{\psi }\left ( \varpi \left ( \vartheta _{v_{l}}, \vartheta _{u_{l}}\right ) \right ) +F_{1}\left ( B\left ( \vartheta _{v_{l}+1}, \mho \vartheta _{u_{l}}\right ) \right ) \\ \leq &\widetilde{\psi }\left ( \varpi \left ( \vartheta _{v_{l}}, \vartheta _{u_{l}}\right ) \right ) +F_{1}\left ( \Upsilon \left ( \mho \vartheta _{v_{l}},\mho \vartheta _{u_{l}}\right ) \right ) \\ \leq &\widetilde{\psi }\left ( \varpi \left ( \vartheta _{v_{l}}, \vartheta _{u_{l}}\right ) \right ) +F_{2}\left ( \Upsilon \left ( \mho \vartheta _{v_{l}},\mho \vartheta _{u_{l}}\right ) \right ) \\ \leq &F_{1}\left \{ \ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta _{v_{l}},\vartheta _{u_{l}}\right ) ,B\left ( \vartheta _{v_{l}},\mho \vartheta _{v_{l}}\right ) ,B\left ( \vartheta _{u_{l}},\mho \vartheta _{u_{l}}\right ) ,B\left ( \vartheta _{v_{l}},\mho \vartheta _{u_{l}}\right ) , \\ B\left ( \vartheta _{u_{l}},\mho \vartheta _{v_{l}}\right ) , \frac{B\left ( \vartheta _{v_{l}},\mho \vartheta _{v_{l}}\right ) +B\left ( \vartheta _{u_{l}},\mho \vartheta _{u_{l}}\right ) }{2}, \frac{B\left ( \vartheta _{v_{l}},\mho \vartheta _{u_{l}}\right ) +B\left ( \vartheta _{u_{l}},\mho \vartheta _{v_{l}}\right ) }{2}\end{array}\displaystyle \right ) \right \} \\ =&F_{1}\left \{ \ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta _{v_{l}},\vartheta _{u_{l}}\right ) ,\varpi \left ( \vartheta _{v_{l}},\vartheta _{v_{l}+1}\right ) ,\varpi \left ( \vartheta _{u_{l}},\vartheta _{u_{l}+1}\right ) ,\varpi \left ( \vartheta _{v_{l}},\vartheta _{u_{l}+1}\right ) , \\ \varpi \left ( \vartheta _{u_{l}},\vartheta _{v_{l}+1}\right ) , \frac{\varpi \left ( \vartheta _{v_{l}},\vartheta _{v_{l}+1}\right ) +\varpi \left ( \vartheta _{u_{l}},\vartheta _{u_{l}+1}\right ) }{2}, \frac{\varpi \left ( \vartheta _{v_{l}},\vartheta _{u_{l}+1}\right ) +\varpi \left ( \vartheta _{u_{l}},\vartheta _{v_{l}+1}\right ) }{2}\end{array}\displaystyle \right ) \right \} \\ \leq &F_{1}\left \{ \ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta _{v_{l}},\vartheta _{u_{l}}\right ) ,\varpi \left ( \vartheta _{v_{l}},\vartheta _{v_{l}+1}\right ) ,\varpi \left ( \vartheta _{u_{l}},\vartheta _{u_{l}+1}\right ) , \\ \gamma \left ( \vartheta _{u_{l}},\vartheta _{u_{l}+1}\right ) \varpi \left ( \vartheta _{u_{l}},\vartheta _{u_{l}+1}\right ) + \gamma \left ( \vartheta _{u_{l}},\vartheta _{v_{l}}\right ) \varpi \left ( \vartheta _{u_{l}},\vartheta _{v_{l}}\right ) , \\ \gamma \left ( \vartheta _{u_{l}},\vartheta _{u_{l}+1}\right ) \varpi \left ( \vartheta _{u_{l}},\vartheta _{u_{l}+1}\right ) + \gamma \left ( \vartheta _{u_{l}+1},\vartheta _{v_{l}+1}\right ) \varpi \left ( \vartheta _{u_{l}+1},\vartheta _{v_{l}+1}\right ) , \\ \frac{\varpi \left ( \vartheta _{v_{l}},\vartheta _{v_{l}+1}\right ) +\varpi \left ( \vartheta _{u_{l}},\vartheta _{u_{l}+1}\right ) }{2}, \frac{\gamma \left ( \vartheta _{u_{l}},\vartheta _{u_{l}+1}\right ) \varpi \left ( \vartheta _{u_{l}},\vartheta _{u_{l}+1}\right ) +\gamma \left ( \vartheta _{u_{l}},\vartheta _{v_{l}}\right ) \varpi \left ( \vartheta _{u_{l}},\vartheta _{v_{l}}\right ) }{2} \\ + \frac{\gamma \left ( \vartheta _{u_{l}},\vartheta _{u_{l}+1}\right ) \varpi \left ( \vartheta _{u_{l}},\vartheta _{u_{l}+1}\right ) +\gamma \left ( \vartheta _{u_{l}+1},\vartheta _{v_{l}+1}\right ) \varpi \left ( \vartheta _{u_{l}+1},\vartheta _{v_{l}+1}\right ) }{2}\end{array}\displaystyle \right ) \right \} . \end{aligned}$$

Since $F_{1}$ is continuous, letting $l\rightarrow \infty$, and using (4.15) and (4.16), we have

$$\begin{aligned} &\lim _{l\rightarrow \infty }\widetilde{\psi }\left ( \varpi \left ( \vartheta _{v_{l}},\vartheta _{u_{l}}\right ) \right ) +F_{1}\left ( \theta \right ) \\ \leq &F_{1}\left \{ \ell \left ( \textstyle\begin{array}{c} \theta ,0,0,0+\theta \gamma \left ( \vartheta _{u_{l}},\vartheta _{v_{l}} \right ) ,0+\theta \gamma \left ( \vartheta _{u_{l}+1},\vartheta _{v_{l}+1} \right ) , \\ 0, \frac{0+\theta \gamma \left ( \vartheta _{u_{l}},\vartheta _{v_{l}}\right ) +\theta \gamma \left ( \vartheta _{u_{l}+1},\vartheta _{v_{l}+1}\right ) }{2}\end{array}\displaystyle \right ) \right \} \\ =&F_{1}\left \{ \ell \left ( \theta ,0,0,\theta \gamma \left ( \vartheta _{u_{l}},\vartheta _{v_{l}}\right ) ,\theta \gamma \left ( \vartheta _{u_{l}+1},\vartheta _{v_{l}+1}\right ) ,0, \frac{\theta \gamma \left ( \vartheta _{u_{l}},\vartheta _{v_{l}}\right ) +\theta \gamma \left ( \vartheta _{u_{l}+1},\vartheta _{v_{l}+1}\right ) }{2} \right ) \right \} \\ \leq &F_{1}\left \{ \theta \ell \left ( 1,0,0,\gamma \left ( \vartheta _{u_{l}},\vartheta _{v_{l}}\right ) ,\gamma \left ( \vartheta _{u_{l}+1},\vartheta _{v_{l}+1}\right ) ,0, \frac{\gamma \left ( \vartheta _{u_{l}},\vartheta _{v_{l}}\right ) +\gamma \left ( \vartheta _{u_{l}+1},\vartheta _{v_{l}+1}\right ) }{2} \right ) \right \} \\ \leq &F_{1}(\theta ), \end{aligned}$$

since $\ell \in \widetilde{\aleph }$, thus, $\ell \left ( 1,0,0,\gamma \left ( \vartheta _{u_{l}},\vartheta _{v_{l}} \right ) ,\gamma \left ( \vartheta _{u_{l}+1},\vartheta _{v_{l}+1} \right ) ,0, \frac{\gamma \left ( \vartheta _{u_{l}},\vartheta _{v_{l}}\right ) +\gamma \left ( \vartheta _{u_{l}+1},\vartheta _{v_{l}+1}\right ) }{2} \right ) \in (0,1]$. Hence,

$$ \lim _{l\rightarrow \infty }\widetilde{\psi }\left ( \varpi \left ( \vartheta _{v_{l}},\vartheta _{u_{l}}\right ) \right ) +F_{1}\left ( \theta \right ) \leq F_{1}(\theta ), $$

which implies that

$$ \lim _{l\rightarrow \infty }\widetilde{\psi }\left ( \varpi \left ( \vartheta _{v_{l}},\vartheta _{u_{l}}\right ) \right ) \leq 0. $$

Therefore,

$$ \lim _{\wp \rightarrow \theta ^{+}}\inf \widetilde{\psi }(\wp )\leq 0, $$

which is a contradiction. Hence, (4.13) is true. Thus, $\{\vartheta _{u}\}$ is a CS and the completeness of η implies that there is $\vartheta ^{\ast }\in \eta$ in order that $\vartheta _{u} \rightarrow \vartheta ^{\ast }$ as $u\rightarrow \infty$. Theorem 4.2 provides the remainder of the proof, which leads to $\vartheta ^{\ast }\in \mho \vartheta ^{\ast }$. □

If we take $F\in \triangledown (\widetilde{Z})$, we can present the following theorem:

Theorem 4.5

Let $\mho :\eta \rightarrow \eta ^{c}$ be an MVM defined on a complete CMS $\left ( \eta ,\varpi ,\gamma \right )$. Assume that the following conditions are true:

(c_i):

$\widetilde{\psi }\in \Psi$ and $F\in \triangledown (\widetilde{Z})$;

(c_ii):

for all $\vartheta >0$, $F\left ( \vartheta \right ) \leq \Game \left ( \vartheta \right )$, where ⅁ is a real-valued function on $\mathbb{R} ^{+}$;

(c_iii):

$\Upsilon \left ( \mho \vartheta ,\mho \widetilde{\vartheta }\right ) >0$, implies

$$\begin{aligned} &\widetilde{\psi }\left ( \varpi \left ( \vartheta , \widetilde{\vartheta }\right ) \right ) +\Game \left ( \Upsilon \left ( \mho \vartheta , \mho \widetilde{\vartheta }\right ) \right ) \\ \leq &F\left \{ \ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta ,\widetilde{\vartheta }\right ) ,B\left ( \vartheta ,\mho \vartheta \right ) ,B\left ( \widetilde{\vartheta }, \mho \widetilde{\vartheta }\right ) ,B\left ( \vartheta ,\mho \widetilde{\vartheta } \right ) , \\ B\left ( \widetilde{\vartheta },\mho \vartheta \right ) , \frac{B\left ( \vartheta ,\mho \vartheta \right ) +B\left ( \widetilde{\vartheta },\mho \widetilde{\vartheta }\right ) }{2}, \frac{B\left ( \vartheta ,\mho \widetilde{\vartheta }\right ) +B\left ( \widetilde{\vartheta },\mho \vartheta \right ) }{2}\end{array}\displaystyle \right ) \right \} , \end{aligned}$$

for all $\vartheta ,\widetilde{\vartheta }\in \eta$ and $\ell \in \aleph$;

(c_iv):

for $\vartheta _{0}\in \eta$, define the Picard sequence $\left \{ \vartheta _{u}=\mho ^{u}\vartheta _{0}\right \}$ such that

$$ \sup _{n\geq 1}\lim _{j\rightarrow \infty } \frac{\gamma \left ( \vartheta _{j+1},\vartheta _{j+2}\right ) \gamma \left ( \vartheta _{j+1},\vartheta _{n}\right ) }{\gamma \left ( \vartheta _{j},\vartheta _{j+1}\right ) }< 1; $$

(c_v):

for all $\vartheta \in \eta$, $\lim _{u\rightarrow \infty }\gamma \left ( \vartheta _{u},\vartheta \right ) \leq 1$.

Then, $F_{ix}\left ( \mho \right ) \neq \emptyset$.

Proof

Suppose that $\vartheta _{0}\in \eta$ and $\vartheta _{1}\in \mho \vartheta _{0}$. If $\vartheta _{1}\in \mho \vartheta _{1}$, $\vartheta _{1}\in F_{ix}\left ( \mho \right )$ and this completes the proof. Hence, consider that $\vartheta _{1}\notin \mho \vartheta _{1}$, that is, $B\left ( \vartheta _{1},\mho \vartheta _{1}\right ) >0$. Thus, $\Upsilon \left ( \mho \vartheta _{0},\mho \vartheta _{1}\right ) >0$. From $(\heartsuit _{4})$, one has

$$ F\left ( B\left ( \vartheta _{1},\mho \vartheta _{1}\right ) \right ) = \inf _{a\in \mho \vartheta _{1}}F\left ( \varpi \left ( \vartheta _{1},a \right ) \right ) . $$

(4.17)

It follows from (4.17), (c_ii), and (c_iii) that

$$\begin{aligned} \inf _{a\in \mho \vartheta _{1}}F\left ( \varpi \left ( \vartheta _{1},a \right ) \right ) =&F\left ( B\left ( \vartheta _{1},\mho \vartheta _{1} \right ) \right ) \\ \leq &F\left ( \Upsilon \left ( \mho \vartheta _{0},\mho \vartheta _{1} \right ) \right ) \\ \leq &\Game \left ( \Upsilon \left ( \mho \vartheta _{0},\mho \vartheta _{1}\right ) \right ) \\ \leq &F\left \{ \ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta _{0},\vartheta _{1}\right ) ,B\left ( \vartheta _{0},\mho \vartheta _{0}\right ) ,B\left ( \vartheta _{1}, \mho \vartheta _{1}\right ) ,B\left ( \vartheta _{0},\mho \vartheta _{1} \right ) , \\ B\left ( \vartheta _{1},\mho \vartheta _{0}\right ) , \frac{B\left ( \vartheta _{0},\mho \vartheta _{0}\right ) +B\left ( \vartheta _{1},\mho \vartheta _{1}\right ) }{2}, \frac{B\left ( \vartheta _{0},\mho \vartheta _{1}\right ) +B\left ( \vartheta _{1},\mho \vartheta _{0}\right ) }{2}\end{array}\displaystyle \right ) \right \} \\ &-\widetilde{\psi }\left ( \varpi \left ( \vartheta _{0},\vartheta _{1} \right ) \right ) \\ < &F\left \{ \ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta _{0},\vartheta _{1}\right ) ,\varpi \left ( \vartheta _{0},\vartheta _{1}\right ) ,\varpi \left ( \vartheta _{1}, \vartheta _{2}\right ) ,\varpi \left ( \vartheta _{0},\vartheta _{2} \right ) ,0, \\ \frac{\varpi \left ( \vartheta _{0},\vartheta _{1}\right ) +\varpi \left ( \vartheta _{1},\vartheta _{2}\right ) }{2}, \frac{\varpi \left ( \vartheta _{0},\vartheta _{2}\right ) +\varpi \left ( \vartheta _{1},\vartheta _{1}\right ) }{2}\end{array}\displaystyle \right ) \right \} . \end{aligned}$$

Hence, there is $\vartheta _{2}\in \mho \vartheta _{1}$ in order that

$$ F\left ( \varpi \left ( \vartheta _{1},\vartheta _{2}\right ) \right ) < F\left \{ \ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta _{0},\vartheta _{1}\right ) ,\varpi \left ( \vartheta _{0},\vartheta _{1}\right ) ,\varpi \left ( \vartheta _{1}, \vartheta _{2}\right ) ,\varpi \left ( \vartheta _{0},\vartheta _{2} \right ) ,0, \\ \frac{\varpi \left ( \vartheta _{0},\vartheta _{1}\right ) +\varpi \left ( \vartheta _{1},\vartheta _{2}\right ) }{2}, \frac{\varpi \left ( \vartheta _{0},\vartheta _{2}\right ) +\varpi \left ( \vartheta _{1},\vartheta _{1}\right ) }{2}\end{array}\displaystyle \right ) \right \}. $$

(4.18)

Since F is nondecreasing, it follows from (4.18) and ($\varpi _{3}$) that

$$\begin{aligned} \varpi \left ( \vartheta _{1},\vartheta _{2}\right ) < &\ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta _{0},\vartheta _{1}\right ) ,\varpi \left ( \vartheta _{0},\vartheta _{1}\right ) ,\varpi \left ( \vartheta _{1}, \vartheta _{2}\right ) ,\varpi \left ( \vartheta _{0},\vartheta _{2} \right ) , \\ 0, \frac{\varpi \left ( \vartheta _{0},\vartheta _{1}\right ) +\varpi \left ( \vartheta _{1},\vartheta _{2}\right ) }{2}, \frac{\varpi \left ( \vartheta _{0},\vartheta _{2}\right ) }{2}\end{array}\displaystyle \right ) \\ \leq &\ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta _{0},\vartheta _{1}\right ) ,\varpi \left ( \vartheta _{0},\vartheta _{1}\right ) ,\varpi \left ( \vartheta _{1}, \vartheta _{2}\right ) , \\ \gamma \left ( \vartheta _{0},\vartheta _{1}\right ) \varpi \left ( \vartheta _{0},\vartheta _{1}\right ) +\gamma \left ( \vartheta _{1}, \vartheta _{2}\right ) \varpi \left ( \vartheta _{1},\vartheta _{2} \right ) ,0, \\ \frac{\varpi \left ( \vartheta _{0},\vartheta _{1}\right ) +\varpi \left ( \vartheta _{1},\vartheta _{2}\right ) }{2}, \frac{\gamma \left ( \vartheta _{0},\vartheta _{1}\right ) \varpi \left ( \vartheta _{0},\vartheta _{1}\right ) +\gamma \left ( \vartheta _{1},\vartheta _{2}\right ) \varpi \left ( \vartheta _{1},\vartheta _{2}\right ) }{2}\end{array}\displaystyle \right ) . \end{aligned}$$

From Lemma 3.3, we obtain

$$ \varpi \left ( \vartheta _{1},\vartheta _{2}\right ) < \varpi \left ( \vartheta _{0},\vartheta _{1}\right ) . $$

Similarly, we have $\vartheta _{3}\in \mho \vartheta _{2}$ with $B\left ( \vartheta _{2},\mho \vartheta _{2}\right ) >0$. Using Lemma 3.3, (c_ii), and (c_iii), we have

$$ \varpi \left ( \vartheta _{2},\vartheta _{3}\right ) < \varpi \left ( \vartheta _{1},\vartheta _{2}\right ) . $$

As we stated before, we have a sequence $\{\vartheta _{u}\}\subset \eta$ in order that $\vartheta _{u+1}\in \mho \vartheta _{u}$ with $B\left ( \vartheta _{u},\mho \vartheta _{u}\right ) >0$ and

$$ \varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) < \varpi \left ( \vartheta _{u-1},\vartheta _{u}\right ) ,\text{ }\forall u\in \mathbb{N} . $$

(4.19)

Inequality ((4.19) proves that $\{\varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) \}_{s\in \mathbb{N} }$ is a decreasing sequence. From $(\heartsuit _{4})$, one can write

$$\begin{aligned} &\inf _{a\in \mho \vartheta _{u}}F\left ( \varpi \left ( \vartheta _{u},a \right ) \right ) \\ &\quad=F\left ( B\left ( \vartheta _{u},\mho \vartheta _{u} \right ) \right ) \\ &\quad\leq F\left ( \Upsilon \left ( \mho \vartheta _{u-1},\mho \vartheta _{u}\right ) \right ) \\ &\quad\leq \Game \left ( \Upsilon \left ( \mho \vartheta _{u-1},\mho \vartheta _{u}\right ) \right ) \\ &\quad\leq F\left \{ \ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta _{u-1},\vartheta _{u}\right ) ,B\left ( \vartheta _{u-1},\mho \vartheta _{u-1}\right ) ,B\left ( \vartheta _{u}, \mho \vartheta _{u}\right ) ,B\left ( \vartheta _{u-1},\mho \vartheta _{u}\right ) , \\ B\left ( \vartheta _{u},\mho \vartheta _{u-1}\right ) , \frac{B\left ( \vartheta _{u-1},\mho \vartheta _{u-1}\right ) +B\left ( \vartheta _{u},\mho \vartheta _{u}\right ) }{2}, \frac{B\left ( \vartheta _{u-1},\mho \vartheta _{u}\right ) +B\left ( \vartheta _{u},\mho \vartheta _{u-1}\right ) }{2}\end{array}\displaystyle \right ) \right \} \\ &\qquad{}-\widetilde{\psi }\left ( \varpi \left ( \vartheta _{u-1}, \vartheta _{u}\right ) \right ) \\ &\quad\leq F\left \{ \ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta _{u-1},\vartheta _{u}\right ) ,\varpi \left ( \vartheta _{u-1},\vartheta _{u}\right ) ,\varpi \left ( \vartheta _{u}, \vartheta _{u+1}\right ) ,\varpi \left ( \vartheta _{u-1},\vartheta _{u+1} \right ) , \\ \varpi \left ( \vartheta _{u},\vartheta _{u}\right ) , \frac{\varpi \left ( \vartheta _{u-1},\vartheta _{u}\right ) +\varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) }{2}, \frac{\varpi \left ( \vartheta _{u-1},\vartheta _{u+1}\right ) +\varpi \left ( \vartheta _{u},\vartheta _{u}\right ) }{2}\end{array}\displaystyle \right ) \right \} \\ &\qquad{}-\widetilde{\psi }\left ( \varpi \left ( \vartheta _{u-1}, \vartheta _{u}\right ) \right ) \\ &\quad\leq F\left \{ \ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta _{u-1},\vartheta _{u}\right ) ,\varpi \left ( \vartheta _{u-1},\vartheta _{u}\right ) ,\varpi \left ( \vartheta _{u}, \vartheta _{u+1}\right ) , \\ \gamma \left ( \vartheta _{u-1},\vartheta _{u}\right ) \varpi \left ( \vartheta _{u-1},\vartheta _{u}\right ) +\gamma \left ( \vartheta _{u}, \vartheta _{u+1}\right ) \varpi \left ( \vartheta _{u},\vartheta _{u+1} \right ) ,0, \\ \frac{\varpi \left ( \vartheta _{u-1},\vartheta _{u}\right ) +\varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) }{2}, \frac{\gamma \left ( \vartheta _{u-1},\vartheta _{u}\right ) \varpi \left ( \vartheta _{u-1},\vartheta _{u}\right ) +\gamma \left ( \vartheta _{u},\vartheta _{u+1}\right ) \varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) +0}{2}\end{array}\displaystyle \right ) \right \} \\ &\qquad{}-\widetilde{\psi }\left ( \varpi \left ( \vartheta _{u-1}, \vartheta _{u}\right ) \right ) \\ &\quad< F\left \{ \ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta _{u-1},\vartheta _{u}\right ) ,\varpi \left ( \vartheta _{u-1},\vartheta _{u}\right ) ,\varpi \left ( \vartheta _{u-1}, \vartheta _{u}\right ) , \\ \gamma \left ( \vartheta _{u-1},\vartheta _{u}\right ) \varpi \left ( \vartheta _{u-1},\vartheta _{u}\right ) +\gamma \left ( \vartheta _{u}, \vartheta _{u+1}\right ) \varpi \left ( \vartheta _{u-1},\vartheta _{u} \right ) ,0, \\ \frac{\varpi \left ( \vartheta _{u-1},\vartheta _{u}\right ) +\varpi \left ( \vartheta _{u-1},\vartheta _{u}\right ) }{2}, \frac{\gamma \left ( \vartheta _{u-1},\vartheta _{u}\right ) \varpi \left ( \vartheta _{u-1},\vartheta _{u}\right ) +\gamma \left ( \vartheta _{u},\vartheta _{u+1}\right ) \varpi \left ( \vartheta _{u-1},\vartheta _{u}\right ) }{2}\end{array}\displaystyle \right ) \right \} \\ &\qquad{}-\widetilde{\psi }\left ( \varpi \left ( \vartheta _{u-1}, \vartheta _{u}\right ) \right ) \\ &\quad\leq F\left \{ \varpi \left ( \vartheta _{u-1},\vartheta _{u} \right ) \ell \left ( \textstyle\begin{array}{c} 1,1,1,\gamma \left ( \vartheta _{u-1},\vartheta _{u}\right ) +\gamma \left ( \vartheta _{u},\vartheta _{u+1}\right ) ,0, \\ 1, \frac{\gamma \left ( \vartheta _{u-1},\vartheta _{u}\right ) +\gamma \left ( \vartheta _{u},\vartheta _{u+1}\right ) }{2}\end{array}\displaystyle \right ) \right \} \\ &\qquad{}-\widetilde{\psi }\left ( \varpi \left ( \vartheta _{u-1}, \vartheta _{u}\right ) \right ) \\ &\quad\leq F\left ( \varpi \left ( \vartheta _{u-1},\vartheta _{u}\right ) \right ) -\widetilde{\psi }\left ( \varpi \left ( \vartheta _{u-1},\vartheta _{u} \right ) \right ) . \end{aligned}$$

Hence, for each $u\in \mathbb{N}$, we have

$$ \inf _{a\in \mho \vartheta _{u}}F\left ( \varpi \left ( \vartheta _{u},a \right ) \right ) \leq F\left ( \varpi \left ( \vartheta _{u-1}, \vartheta _{u}\right ) \right ) -\widetilde{\psi }\left ( \varpi \left ( \vartheta _{u-1},\vartheta _{u}\right ) \right ) ,\text{ } \forall u\in \mathbb{N} . $$

(4.20)

Since $\widetilde{\boldsymbol{\psi }}\in \Psi$, there is $\zeta >0$ and $u_{0}\in \mathbb{N}$ so that $\widetilde{\psi }\left ( \varpi \left ( \vartheta _{u},\vartheta _{u+1} \right ) \right ) >\zeta$, for all $u\geq u_{0}$. From (4.22), we obtain

$$\begin{aligned} F\left ( \varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) \right ) \leq &F\left ( \varpi \left ( \vartheta _{u-1},\vartheta _{u} \right ) \right ) -\widetilde{\psi }\left ( \varpi \left ( \vartheta _{u-1},\vartheta _{u} \right ) \right ) \\ \leq &F\left ( \varpi \left ( \vartheta _{u-2},\vartheta _{u-1} \right ) \right ) -\widetilde{\psi }\left ( \varpi \left ( \vartheta _{u-2}, \vartheta _{u-1}\right ) \right ) -\widetilde{\psi }\left ( \varpi \left ( \vartheta _{u-1},\vartheta _{u}\right ) \right ) \\ &\vdots \\ \leq &F\left ( \varpi \left ( \vartheta _{0},\vartheta _{1}\right ) \right ) -\sum \limits _{j=1}^{u-1}\widetilde{\psi }\left ( \varpi \left ( \vartheta _{j-1},\vartheta _{j}\right ) \right ) \\ =&F\left ( \varpi \left ( \vartheta _{0},\vartheta _{1}\right ) \right ) -\sum \limits _{j=1}^{u_{0}-1}\widetilde{\psi }\left ( \varpi \left ( \vartheta _{j-1},\vartheta _{j}\right ) \right ) - \sum \limits _{j=u_{0}}^{u}\widetilde{\psi }\left ( \varpi \left ( \vartheta _{j-1},\vartheta _{j}\right ) \right ) \\ =&F\left ( \varpi \left ( \vartheta _{0},\vartheta _{1}\right ) \right ) -(u-u_{0})\zeta ,\text{ }u\geq u_{0}. \end{aligned}$$

(4.21)

In (4.21), take $u\rightarrow \infty$, we have

$$ \lim _{u\rightarrow \infty }F\left ( \varpi \left ( \vartheta _{u-1}, \vartheta _{u}\right ) \right ) =-\infty . $$

From $(\heartsuit _{2}^{\prime })$, we obtain

$$ \lim _{u\rightarrow \infty }\varpi \left ( \vartheta _{u-1}, \vartheta _{u}\right ) =0. $$

(4.22)

Based on $(\heartsuit _{3})$, there is $l\in (0,1)$ such that that

$$ \lim _{u\rightarrow \infty }\left ( \varpi \left ( \vartheta _{u-1}, \vartheta _{u}\right ) \right ) ^{l}F_{2}\left ( \varpi \left ( \vartheta _{u-1},\vartheta _{u}\right ) \right ) =0. $$

(4.23)

For all $u\geq u_{0}$, by (4.22), one can write

$$\begin{aligned} &\left ( \varpi \left ( \vartheta _{u-1},\vartheta _{u}\right ) \right ) ^{l}F_{2}\left ( \varpi \left ( \vartheta _{u-1},\vartheta _{u} \right ) \right ) -\left ( \varpi \left ( \vartheta _{u-1},\vartheta _{u} \right ) \right ) ^{l}F\left ( \varpi \left ( \vartheta _{0}, \vartheta _{1}\right ) \right ) \\ \leq &\left ( \varpi \left ( \vartheta _{u-1},\vartheta _{u}\right ) \right ) ^{l}\left [ F\left ( \varpi \left ( \vartheta _{0}, \vartheta _{1}\right ) \right ) -(u-u_{0})\zeta \right ] \\ &-\left ( \varpi \left ( \vartheta _{u-1},\vartheta _{u}\right ) \right ) ^{l}F\left ( \varpi \left ( \vartheta _{0},\vartheta _{1} \right ) \right ) \\ =&-\left ( \varpi \left ( \vartheta _{0},\vartheta _{1}\right ) \right ) ^{l}(u-u_{0})\zeta \leq 0. \end{aligned}$$

(4.24)

Letting $u\rightarrow \infty$ in (4.24) and utilizing (4.22) and (4.23), we obtain that

$$ 0\leq -\lim _{u\rightarrow \infty }u\left ( \varpi \left ( \vartheta _{u-1}, \vartheta _{u}\right ) \right ) ^{l}\leq 0, $$

which yields

$$ \lim _{u\rightarrow \infty }u\left ( \varpi \left ( \vartheta _{u-1}, \vartheta _{u}\right ) \right ) ^{l}=0. $$

(4.25)

By (4.5), there is $u_{1}\in \mathbb{N}$ so that $u\left ( \varpi \left ( \vartheta _{u-1},\vartheta _{u}\right ) \right ) ^{l}\leq 1$, for all $u\geq u_{1}$. Thus, we have

$$ \varpi \left ( \vartheta _{u-1},\vartheta _{u}\right ) \leq \frac{1}{u^{\frac{1}{l}}}\text{ for all }u\geq u_{1}. $$

In order to demonstrate that $\{\vartheta _{u}\}_{u\in \mathbb{N} }$ is a CS, let us look at $v,u\in \mathbb{N}$ such that $v>u>u_{1}$. The remainder of the proof proceeds from Theorem 4.2. Using (c_iv) and the ratio test, we determine that $\{\vartheta _{u}\}_{u\in \mathbb{N} }$ is a CS and thus, there is $\vartheta ^{\ast }\in \eta$ so that

$$ \lim _{u\rightarrow \infty }\vartheta _{u}=\vartheta ^{\ast }. $$

Consider

$$\begin{aligned} F\left ( \Upsilon \left ( \mho \vartheta ,\mho \widetilde{\vartheta } \right ) \right ) \leq &L\left ( \Upsilon \left ( \mho \vartheta , \mho \widetilde{\vartheta }\right ) \right ) \leq \widetilde{\psi }\left ( \varpi \left ( \vartheta ,\widetilde{\vartheta }\right ) \right ) +L\left ( \Upsilon \left ( \mho \vartheta ,\mho \widetilde{\vartheta }\right ) \right ) \\ \leq &F\left \{ \ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta ,\widetilde{\vartheta }\right ) ,B\left ( \vartheta ,\mho \vartheta \right ) ,B\left ( \widetilde{\vartheta }, \mho \widetilde{\vartheta }\right ) ,B\left ( \vartheta ,\mho \widetilde{\vartheta } \right ) , \\ B\left ( \widetilde{\vartheta },\mho \vartheta \right ) , \frac{B\left ( \vartheta ,\mho \vartheta \right ) +B\left ( \widetilde{\vartheta },\mho \widetilde{\vartheta }\right ) }{2}, \frac{B\left ( \vartheta ,\mho \widetilde{\vartheta }\right ) +B\left ( \widetilde{\vartheta },\mho \vartheta \right ) }{2}\end{array}\displaystyle \right ) \right \}. \end{aligned}$$

Since $F_{1}$ is nondecreasing, one can write for all $\vartheta ,\widetilde{\vartheta }\in \eta$,

$$ \Upsilon \left ( \mho \vartheta ,\mho \widetilde{\vartheta }\right ) \leq \ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta ,\widetilde{\vartheta }\right ) ,B\left ( \vartheta ,\mho \vartheta \right ) ,B\left ( \widetilde{\vartheta }, \mho \widetilde{\vartheta }\right ) ,B\left ( \vartheta ,\mho \widetilde{\vartheta } \right ) ,B\left ( \widetilde{\vartheta },\mho \vartheta \right ) , \\ \frac{B\left ( \vartheta ,\mho \vartheta \right ) +B\left ( \widetilde{\vartheta },\mho \widetilde{\vartheta }\right ) }{2}, \frac{B\left ( \vartheta ,\mho \widetilde{\vartheta }\right ) +B\left ( \widetilde{\vartheta },\mho \vartheta \right ) }{2}\end{array}\displaystyle \right ) . $$

Finally, to find the FP of ℧, assume the contrary, that is, $B\left ( \vartheta ^{\ast },\mho \vartheta ^{\ast }\right ) >0$. Along the same lines as Theorem 4.2, we have $B\left ( \vartheta ^{\ast },\mho \vartheta ^{\ast }\right )$. Since $\mho \vartheta ^{\ast }$ is closed, $\vartheta ^{\ast }\in \mho \vartheta ^{\ast }$. This completes the proof. □

Now, if we take $\ell \in \widetilde{\aleph }$, we have the following theorem:

Theorem 4.6

Let $\mho :\eta \rightarrow \eta ^{c}$ be an MVM defined on a complete CMS $\left ( \eta ,\varpi ,\gamma \right )$. Assume that the following conditions are satisfied:

($\bigstar _{i}$):

$\widetilde{\psi }\in \Psi$, $\ell \in \widetilde{\aleph}$, and F satisfy condition $(\heartsuit _{2}^{\prime })$, where $F:\mathbb{R} ^{+}\rightarrow \mathbb{R}$ is a nondecreasing, continuous, and real-valued function;

($\bigstar _{ii}$):

for all $\vartheta >0$, $F\left ( \vartheta \right ) \leq \Game \left ( \vartheta \right )$, where ⅁ is a real-valued function on $\mathbb{R} ^{+}$;

($\bigstar _{iii}$):

$\Upsilon \left ( \mho \vartheta ,\mho \widetilde{\vartheta }\right ) >0$, implies

$$\begin{aligned} &\widetilde{\psi }\left ( \varpi \left ( \vartheta , \widetilde{\vartheta }\right ) \right ) +\Game \left ( \Upsilon \left ( \mho \vartheta , \mho \widetilde{\vartheta }\right ) \right ) \\ \leq &F\left \{ \ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta ,\widetilde{\vartheta }\right ) ,B\left ( \vartheta ,\mho \vartheta \right ) ,B\left ( \widetilde{\vartheta }, \mho \widetilde{\vartheta }\right ) ,B\left ( \vartheta ,\mho \widetilde{\vartheta } \right ) , \\ B\left ( \widetilde{\vartheta },\mho \vartheta \right ) , \frac{B\left ( \vartheta ,\mho \vartheta \right ) +B\left ( \widetilde{\vartheta },\mho \widetilde{\vartheta }\right ) }{2}, \frac{B\left ( \vartheta ,\mho \widetilde{\vartheta }\right ) +B\left ( \widetilde{\vartheta },\mho \vartheta \right ) }{2}\end{array}\displaystyle \right ) \right \} , \end{aligned}$$

for all $\vartheta ,\widetilde{\vartheta }\in \eta$;

($\bigstar _{iv}$):

for all $\vartheta \in \eta$, $\lim _{u\rightarrow \infty }\gamma \left ( \vartheta _{u},\vartheta \right ) \leq 1$.

Then, $F_{ix}\left ( \mho \right ) \neq \emptyset$.

Proof

Assume that $\vartheta _{0}\in \eta$ and $\vartheta _{1}\in \mho \vartheta _{0}$. Similar to the proof of Theorem 4.2, we have a sequence $\{\vartheta _{u}\}\subset \eta$ such that $\vartheta _{u+1}\in \mho \vartheta _{u}$ with $B\left ( \vartheta _{u},\mho \vartheta _{u+1}\right ) >0$, and

$$ \varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) < \varpi \left ( \vartheta _{u-1},\vartheta _{u}\right ) $$

and

$$ F\left ( \varpi \left ( \vartheta _{u-1},\vartheta _{u}\right ) \right ) < F\left ( \varpi \left ( \vartheta _{0},\vartheta _{1} \right ) \right ) -(u-u_{0})\zeta ,\text{ }\forall u\geq u_{0}. $$

(4.26)

In (4.26), letting $u\rightarrow \infty$, we have

$$ \lim _{u\rightarrow \infty }F\left ( \varpi \left ( \vartheta _{u-1}, \vartheta _{u}\right ) \right ) =-\infty . $$

By $(\heartsuit _{2}^{\prime })$, we obtain

$$ \lim _{u\rightarrow \infty }\varpi \left ( \vartheta _{u-1}, \vartheta _{u}\right ) =0. $$

Now, we show that

$$ \lim _{u,v\rightarrow \infty }\varpi \left ( \vartheta _{u}, \vartheta _{v}\right ) =0. $$

(4.27)

Assume that (4.27) is not true, there is $\theta >0$ so that for each $\widehat{r}\geq 0$, and we have $v_{l}>u_{l}>\widehat{r}$ and

$$ \varpi \left ( \vartheta _{u},\vartheta _{v}\right ) < \theta . $$

In addition, there is $\widehat{r}_{0}\in \mathbb{N}$ in order that

$$ m_{\widehat{r}_{0}}=\varpi \left ( \vartheta _{u-1},\vartheta _{u} \right ) < \theta ,\text{ }\forall u\geq \widehat{r}_{0}. $$

There are two subsequences $\{\vartheta _{u_{l}}\}$ and $\{\vartheta _{v_{l}}\}$ of $\{\vartheta _{u}\}$, and following the same steps as Theorem 4.4, we obtain that

$$ \lim _{l\rightarrow \infty }\varpi \left ( \vartheta _{v_{l}}, \vartheta _{u_{l}}\right ) =\theta $$

and

$$ \lim _{l\rightarrow \infty }\varpi \left ( \vartheta _{v_{l}+1}, \vartheta _{u_{l}+1}\right ) =\theta . $$

(4.28)

The monotonicity of F and the conditions ($\bigstar _{ii}$) and ($\bigstar _{iii}$) imply that

$$\begin{aligned} &\widetilde{\psi }\left ( \varpi \left ( \vartheta _{v_{l}}, \vartheta _{u_{l}}\right ) \right ) +F\left ( \varpi \left ( \vartheta _{v_{l}+1},\vartheta _{u_{l}+1}\right ) \right ) \\ =&\widetilde{\psi }\left ( \varpi \left ( \vartheta _{v_{l}}, \vartheta _{u_{l}}\right ) \right ) +F\left ( B\left ( \vartheta _{v_{l}+1}, \mho \vartheta _{u_{l}}\right ) \right ) \\ \leq &\widetilde{\psi }\left ( \varpi \left ( \vartheta _{v_{l}}, \vartheta _{u_{l}}\right ) \right ) +F\left ( \Upsilon \left ( \mho \vartheta _{v_{l}},\mho \vartheta _{u_{l}}\right ) \right ) \\ \leq &\widetilde{\psi }\left ( \varpi \left ( \vartheta _{v_{l}}, \vartheta _{u_{l}}\right ) \right ) +\Game \left ( \Upsilon \left ( \mho \vartheta _{v_{l}},\mho \vartheta _{u_{l}}\right ) \right ) \\ \leq &F\left \{ \ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta _{v_{l}},\vartheta _{u_{l}}\right ) ,\varpi \left ( \vartheta _{v_{l}},\vartheta _{v_{l}+1}\right ) ,\varpi \left ( \vartheta _{u_{l}},\vartheta _{u_{l}+1}\right ) , \\ \gamma \left ( \vartheta _{u_{l}},\vartheta _{u_{l}+1}\right ) \varpi \left ( \vartheta _{u_{l}},\vartheta _{u_{l}+1}\right ) + \gamma \left ( \vartheta _{u_{l}},\vartheta _{v_{l}}\right ) \varpi \left ( \vartheta _{u_{l}},\vartheta _{v_{l}}\right ) , \\ \gamma \left ( \vartheta _{u_{l}},\vartheta _{u_{l}+1}\right ) \varpi \left ( \vartheta _{u_{l}},\vartheta _{u_{l}+1}\right ) + \gamma \left ( \vartheta _{u_{l}+1},\vartheta _{v_{l}+1}\right ) \varpi \left ( \vartheta _{u_{l}+1},\vartheta _{v_{l}+1}\right ) , \\ \frac{\varpi \left ( \vartheta _{v_{l}},\vartheta _{v_{l}+1}\right ) +\varpi \left ( \vartheta _{u_{l}},\vartheta _{u_{l}+1}\right ) }{2}, \frac{\gamma \left ( \vartheta _{u_{l}},\vartheta _{u_{l}+1}\right ) \varpi \left ( \vartheta _{u_{l}},\vartheta _{u_{l}+1}\right ) +\gamma \left ( \vartheta _{u_{l}},\vartheta _{v_{l}}\right ) \varpi \left ( \vartheta _{u_{l}},\vartheta _{v_{l}}\right ) }{2} \\ + \frac{\gamma \left ( \vartheta _{u_{l}},\vartheta _{u_{l}+1}\right ) \varpi \left ( \vartheta _{u_{l}},\vartheta _{u_{l}+1}\right ) +\gamma \left ( \vartheta _{u_{l}+1},\vartheta _{v_{l}+1}\right ) \varpi \left ( \vartheta _{u_{l}+1},\vartheta _{v_{l}+1}\right ) }{2}\end{array}\displaystyle \right ) \right \} . \end{aligned}$$

In the above inequality, letting $l\rightarrow \infty$, and using the continuity of F and (4.28), we have

$$\begin{aligned} &\lim _{l\rightarrow \infty }\widetilde{\psi }\left ( \varpi \left ( \vartheta _{v_{l}},\vartheta _{u_{l}}\right ) \right ) +F\left ( \theta \right ) \\ \leq &F\left \{ \ell \left ( \textstyle\begin{array}{c} \theta ,0,0,0+\theta \gamma \left ( \vartheta _{u_{l}},\vartheta _{v_{l}} \right ) ,0+\theta \gamma \left ( \vartheta _{u_{l}+1},\vartheta _{v_{l}+1} \right ) , \\ 0, \frac{0+\theta \gamma \left ( \vartheta _{u_{l}},\vartheta _{v_{l}}\right ) +\theta \gamma \left ( \vartheta _{u_{l}+1},\vartheta _{v_{l}+1}\right ) }{2}\end{array}\displaystyle \right ) \right \} \\ =&F\left \{ \ell \left ( \theta ,0,0,\theta \gamma \left ( \vartheta _{u_{l}},\vartheta _{v_{l}}\right ) ,\theta \gamma \left ( \vartheta _{u_{l}+1},\vartheta _{v_{l}+1}\right ) ,0, \frac{\theta \gamma \left ( \vartheta _{u_{l}},\vartheta _{v_{l}}\right ) +\theta \gamma \left ( \vartheta _{u_{l}+1},\vartheta _{v_{l}+1}\right ) }{2} \right ) \right \} \\ \leq &F\left \{ \theta \ell \left ( 1,0,0,\gamma \left ( \vartheta _{u_{l}}, \vartheta _{v_{l}}\right ) ,\gamma \left ( \vartheta _{u_{l}+1}, \vartheta _{v_{l}+1}\right ) ,0, \frac{\gamma \left ( \vartheta _{u_{l}},\vartheta _{v_{l}}\right ) +\gamma \left ( \vartheta _{u_{l}+1},\vartheta _{v_{l}+1}\right ) }{2} \right ) \right \} \\ \leq &F(\theta ), \end{aligned}$$

since $\ell \in \widetilde{\aleph }$, and thus $\ell \left ( 1,0,0,\gamma \left ( \vartheta _{u_{l}},\vartheta _{v_{l}} \right ) ,\gamma \left ( \vartheta _{u_{l}+1},\vartheta _{v_{l}+1} \right ) ,0, \frac{\gamma \left ( \vartheta _{u_{l}},\vartheta _{v_{l}}\right ) +\gamma \left ( \vartheta _{u_{l}+1},\vartheta _{v_{l}+1}\right ) }{2} \right ) \in (0,1]$. Hence,

$$ \lim _{l\rightarrow \infty }\widetilde{\psi }\left ( \varpi \left ( \vartheta _{v_{l}},\vartheta _{u_{l}}\right ) \right ) +F\left ( \theta \right ) \leq F(\theta ), $$

which implies that

$$ \lim _{l\rightarrow \infty }\widetilde{\psi }\left ( \varpi \left ( \vartheta _{v_{l}},\vartheta _{u_{l}}\right ) \right ) \leq 0. $$

Therefore,

$$ \lim _{\wp \rightarrow \theta ^{+}}\inf \widetilde{\psi }(\wp )\leq 0, $$

which is a contradiction of the definition of Ψ. Hence, (4.26) is true. Thus, $\{\vartheta _{u}\}$ is a CS and the completeness of η implies that there is $\vartheta ^{\ast }\in \eta$ in order that $\vartheta _{u}\rightarrow \vartheta ^{\ast }$ as $u\rightarrow \infty$. Theorem 4.5 provides the remainder of the proof, which leads to $\vartheta ^{\ast }\in \mho \vartheta ^{\ast }$. □

5 Data-dependence result

The FP sets $F_{ix}\left ( \mho _{1}\right )$ and $F_{ix}\left ( \mho _{2}\right )$ are nonempty for a MS $(\eta ,\varpi )$ and mappings $\mho _{1},\mho _{2}:\eta \rightarrow Q(\eta )$. Numerous authors have tackled the topic of determining the PH distance ϒ between $F_{ix}\left ( \mho _{1}\right )$ and $F_{ix}\left ( \mho _{2}\right )$, provided that for $k>0$, $\Upsilon (\mho _{1}\vartheta ,\mho _{2}\vartheta )< k$ for all $\vartheta \in \eta$. For instance, see [22–24].

We provide a data-dependence result for the established result in this section.

Definition 5.1

Suppose that $(\eta ,\varpi )$ is a MS and $\mho :\eta \rightarrow \eta ^{c}$ is a MVM such that for all $\vartheta \in \eta$ and $\widetilde{\vartheta }\in \mho \vartheta$, there is a sequence $\{\vartheta _{u}\}_{u\in \mathbb{N} }$ satisfying

• (1) $\vartheta _{0}=\vartheta$ and $\vartheta _{1}=\widetilde{\vartheta }$;

• (2) $\vartheta _{u+1}\in \mho \vartheta _{u}$, for all $u\in \mathbb{N}$;

• (3) $\{\vartheta _{u}\}_{u\in \mathbb{N} }$ is convergent to a FP of ℧.

Then, ℧ is called a multivalued, weakly Picard operator (MWPO, for short). A sequence $\{\vartheta _{u}\}_{u\in \mathbb{N} }$ that satisfies conditions (2) and (3) of Definition 5.1 is described as a sequence of successive approximations (SAM).

Our main theorem in this section is as follows:

Theorem 5.2

Assume that $\mho _{1},\mho _{2}:\eta \rightarrow \eta ^{cp}$ are MVMs on a complete CMS $\left ( \eta ,\varpi ,\gamma \right )$ such that an ψ̃F-contraction is true for $\mho _{j}$, where $j=1,2$. Also, assume that the hypotheses below hold:

(D₁):

$F_{1}$ is a real-valued, nondecreasing function on $\mathbb{R} ^{+}$;

(D₂):

$F_{2}$ is a real-valued function on $\mathbb{R} ^{+}$ verifying $(\heartsuit _{2}^{\prime })$ and $(\heartsuit _{3})$;

(D₄):

for all $\vartheta \in \eta$, there exists $\zeta >0$ so that $\Upsilon \left ( \mho _{1}\vartheta ,\mho _{2}\vartheta \right ) \leq \zeta$;

(D₅):

for $\vartheta _{0}\in \eta$, define the Picard sequence $\left \{ \vartheta _{u}=\mho ^{u}\vartheta _{0}\right \}$ such that

$$ \sup _{n\geq 1}\lim _{j\rightarrow \infty } \frac{\gamma \left ( \vartheta _{j+1},\vartheta _{j+2}\right ) \gamma \left ( \vartheta _{j+1},\vartheta _{n}\right ) }{\gamma \left ( \vartheta _{j},\vartheta _{j+1}\right ) }< 1; $$

(D₆):

for all $\vartheta \in \eta$, $\lim _{u\rightarrow \infty }\gamma \left ( \vartheta _{u},\vartheta \right ) \leq 1$.

Then, the following results are obtained:

(R₁):

for $j\in \{1,2\}$, $F_{ix}\left ( \mho _{j}\right ) \in \eta ^{c}$;

(R₂):

$\mho _{1}$ are $\mho _{2}$ are MWPOs, and

$$\begin{aligned} &\Upsilon \left ( F_{ix}\left ( \mho _{1}\right ) ,F_{ix}\left ( \mho _{2} \right ) \right )\\ &\quad \leq \frac{\zeta }{1-\max \left \{ \ell _{1}\left ( 1,1,1,\lambda +\upsilon ,0,1,\frac{\lambda +\upsilon }{2}\right ) ,\ell _{2}\left ( 1,1,1,\lambda +\upsilon ,0,1,\frac{\lambda +\upsilon }{2}\right ) \right \} }, \end{aligned}$$

where $\lambda ,\upsilon \geq 1$.

Proof

(R₁) Thanks to Theorem 4.2, $F_{ix}\left ( \mho _{j}\right ) \neq \emptyset$ for $j\in \{1,2\}$. We claim that $F_{ix}\left ( \mho _{j}\right )$ is closed for $j\in \{1,2\}$. Assume that a there is a sequence $\{\vartheta _{u}\}$ in $F_{ix}\left ( \mho _{j}\right )$ such that $\lim _{u\rightarrow \infty }\vartheta _{u}=\vartheta$. Now,

$$\begin{aligned} &F_{1}\left ( \Upsilon \left ( \mho \vartheta ,\mho \widetilde{\vartheta }\right ) \right ) \\ \leq &F_{2}\left ( \Upsilon \left ( \mho \vartheta ,\mho \widetilde{\vartheta }\right ) \right ) \\ \leq &\widetilde{\psi }\left ( \varpi \left ( \vartheta , \widetilde{\vartheta }\right ) \right ) +F_{2}\left ( \Upsilon \left ( \mho \vartheta ,\mho \widetilde{\vartheta }\right ) \right ) \\ \leq &F_{1}\left \{ \ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta ,\widetilde{\vartheta }\right ) ,B\left ( \vartheta ,\mho \vartheta \right ) ,B\left ( \widetilde{\vartheta }, \mho \widetilde{\vartheta }\right ) ,B\left ( \vartheta ,\mho \widetilde{\vartheta } \right ) , \\ B\left ( \widetilde{\vartheta },\mho \vartheta \right ) , \frac{B\left ( \vartheta ,\mho \vartheta \right ) +B\left ( \widetilde{\vartheta },\mho \widetilde{\vartheta }\right ) }{2}, \frac{B\left ( \vartheta ,\mho \widetilde{\vartheta }\right ) +B\left ( \widetilde{\vartheta },\mho \vartheta \right ) }{2}\end{array}\displaystyle \right ) \right \} . \end{aligned}$$

The monotonicity of $F_{1}$, implies that

$$ \Upsilon \left ( \mho \vartheta ,\mho \widetilde{\vartheta }\right ) \leq \ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta ,\widetilde{\vartheta }\right ) ,B\left ( \vartheta ,\mho \vartheta \right ) ,B\left ( \widetilde{\vartheta }, \mho \widetilde{\vartheta }\right ) ,B\left ( \vartheta ,\mho \widetilde{\vartheta } \right ) , \\ B\left ( \widetilde{\vartheta },\mho \vartheta \right ) , \frac{B\left ( \vartheta ,\mho \vartheta \right ) +B\left ( \widetilde{\vartheta },\mho \widetilde{\vartheta }\right ) }{2}, \frac{B\left ( \vartheta ,\mho \widetilde{\vartheta }\right ) +B\left ( \widetilde{\vartheta },\mho \vartheta \right ) }{2}\end{array}\displaystyle \right ) , $$

(5.1)

for all $\vartheta ,\widetilde{\vartheta }\in \eta$. Let $B\left ( \widetilde{\vartheta },\mho \widetilde{\vartheta }\right ) >0$. Then, by (5.1), there is $\vartheta \in \mho \widetilde{\vartheta }$ such that

$$\begin{aligned} B\left ( \widetilde{\vartheta },\mho \widetilde{\vartheta }\right ) =& \varpi \left ( \widetilde{\vartheta },\vartheta \right ) \\ \leq &\gamma \left ( \widetilde{\vartheta },\vartheta _{u+1}\right ) \varpi \left ( \widetilde{\vartheta },\vartheta _{u+1}\right ) + \gamma \left ( \vartheta _{u+1},\vartheta \right ) \varpi \left ( \vartheta _{u+1},\vartheta \right ) \\ =&\gamma \left ( \widetilde{\vartheta },\vartheta _{u+1}\right ) \varpi \left ( \widetilde{\vartheta },\vartheta _{u+1}\right ) + \gamma \left ( \vartheta _{u+1},\vartheta \right ) B\left ( \vartheta _{u+1},\mho \widetilde{\vartheta }\right ) \\ \leq &\gamma \left ( \widetilde{\vartheta },\vartheta _{u+1}\right ) \varpi \left ( \widetilde{\vartheta },\vartheta _{u+1}\right ) + \gamma \left ( \vartheta _{u+1},\vartheta \right ) \Upsilon \left ( \mho \vartheta _{u},\mho \widetilde{\vartheta }\right ) \\ \leq &\gamma \left ( \widetilde{\vartheta },\vartheta _{u+1}\right ) \varpi \left ( \widetilde{\vartheta },\vartheta _{u+1}\right ) \\ &+\gamma \left ( \vartheta _{u+1},\vartheta \right ) \ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta _{u},\widetilde{\vartheta }\right ) ,B\left ( \vartheta _{u},\mho \vartheta _{u}\right ) ,B\left ( \widetilde{\vartheta },\mho \widetilde{\vartheta }\right ) ,B\left ( \vartheta _{u},\mho \widetilde{\vartheta }\right ) , \\ B\left ( \widetilde{\vartheta },\mho \vartheta _{u}\right ) , \frac{B\left ( \vartheta _{u},\mho \vartheta _{u}\right ) +B\left ( \widetilde{\vartheta },\mho \widetilde{\vartheta }\right ) }{2}, \frac{B\left ( \vartheta _{u},\mho \widetilde{\vartheta }\right ) +B\left ( \widetilde{\vartheta },\mho \vartheta _{u}\right ) }{2}\end{array}\displaystyle \right ) \\ \leq &\gamma \left ( \widetilde{\vartheta },\vartheta _{u+1}\right ) \varpi \left ( \widetilde{\vartheta },\vartheta _{u+1}\right ) \\ &+\gamma \left ( \vartheta _{u+1},\vartheta \right ) \ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta _{u},\widetilde{\vartheta }\right ) ,B\left ( \vartheta _{u},\mho \vartheta _{u}\right ) ,B\left ( \widetilde{\vartheta },\mho \widetilde{\vartheta }\right ) , \\ \gamma \left ( \vartheta _{u},\widetilde{\vartheta }\right ) B\left ( \vartheta _{u},\widetilde{\vartheta }\right ) +\gamma \left ( \widetilde{\vartheta },\mho \widetilde{\vartheta }\right ) B\left ( \widetilde{\vartheta }, \mho \widetilde{\vartheta }\right ) , \\ B\left ( \widetilde{\vartheta },\mho \vartheta _{u}\right ) , \frac{B\left ( \vartheta _{u},\mho \vartheta _{u}\right ) +B\left ( \widetilde{\vartheta },\mho \widetilde{\vartheta }\right ) }{2}, \\ \frac{\gamma \left ( \vartheta _{u},\widetilde{\vartheta }\right ) B\left ( \vartheta _{u},\widetilde{\vartheta }\right ) +\gamma \left ( \widetilde{\vartheta },\mho \widetilde{\vartheta }\right ) B\left ( \widetilde{\vartheta },\mho \widetilde{\vartheta }\right ) +B\left ( \vartheta _{u},\mho \widetilde{\vartheta }\right ) }{2}\end{array}\displaystyle \right ) \\ \leq &\gamma \left ( \widetilde{\vartheta },\vartheta _{u+1}\right ) \varpi \left ( \widetilde{\vartheta },\vartheta _{u+1}\right ) \\ &+\gamma \left ( \vartheta _{u+1},\vartheta \right ) \ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta _{u},\widetilde{\vartheta }\right ) ,\varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) ,B\left ( \widetilde{\vartheta },\mho \widetilde{\vartheta }\right ) , \\ \gamma \left ( \vartheta _{u},\widetilde{\vartheta }\right ) \varpi \left ( \vartheta _{u},\widetilde{\vartheta }\right ) +\gamma \left ( \widetilde{\vartheta },\vartheta \right ) B\left ( \widetilde{\vartheta },\mho \widetilde{\vartheta }\right ) , \\ \varpi \left ( \widetilde{\vartheta },\vartheta _{u+1}\right ) , \frac{\varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) +B\left ( \widetilde{\vartheta },\mho \widetilde{\vartheta }\right ) }{2}, \\ \frac{\gamma \left ( \vartheta _{u},\widetilde{\vartheta }\right ) \varpi \left ( \vartheta _{u},\widetilde{\vartheta }\right ) +\gamma \left ( \widetilde{\vartheta },\vartheta \right ) B\left ( \widetilde{\vartheta },\mho \widetilde{\vartheta }\right ) +\varpi \left ( \widetilde{\vartheta },\vartheta _{u+1}\right ) }{2}\end{array}\displaystyle \right ) . \end{aligned}$$

Letting $u\rightarrow \infty$ in the above inequality and using the definition of ℓ and $(D_{6})$, we have

$$\begin{aligned} &B\left ( \widetilde{\vartheta },\mho \widetilde{\vartheta }\right ) \\ \leq &(1)\ell \left ( 0,0,B\left ( \widetilde{\vartheta },\mho \widetilde{\vartheta }\right ) ,0+\gamma \left ( \widetilde{\vartheta }, \vartheta \right ) B\left ( \widetilde{\vartheta },\mho \widetilde{\vartheta }\right ) ,0, \frac{B\left ( \widetilde{\vartheta },\mho \widetilde{\vartheta }\right ) }{2}, \frac{0+\gamma \left ( \widetilde{\vartheta },\vartheta \right ) B\left ( \widetilde{\vartheta },\mho \widetilde{\vartheta }\right ) }{2}\right ) \\ =&\ell \left ( 0,0,B\left ( \widetilde{\vartheta },\mho \widetilde{\vartheta }\right ) ,\gamma \left ( \widetilde{\vartheta },\vartheta \right ) B \left ( \widetilde{\vartheta },\mho \widetilde{\vartheta }\right ) ,0, \frac{B\left ( \widetilde{\vartheta },\mho \widetilde{\vartheta }\right ) }{2}, \frac{\gamma \left ( \widetilde{\vartheta },\vartheta \right ) B\left ( \widetilde{\vartheta },\mho \widetilde{\vartheta }\right ) }{2} \right ). \end{aligned}$$

Using Lemma 3.3, we observe that $B\left ( \widetilde{\vartheta },\mho \widetilde{\vartheta }\right ) \leq 0$, hence $B\left ( \widetilde{\vartheta },\mho \widetilde{\vartheta }\right ) =0$. Since ℧ϑ̃ is closed, $\widetilde{\vartheta }\in \mho \widetilde{\vartheta }$. Therefore, $F_{ix}\left ( \mho _{j}\right )$ is closed for $j\in \{1,2\}$.

(R₂) According to Theorem 4.2, we conclude that $\mho _{1}$ are $\mho _{2}$ are MWPOs. It remains to prove that

$$\begin{aligned} &\Upsilon \left ( F_{ix}\left ( \mho _{1}\right ) ,F_{ix}\left ( \mho _{2} \right ) \right )\\ &\quad \leq \frac{\zeta }{1-\max \left \{ \ell _{1}\left ( 1,1,1,\lambda +\upsilon ,0,1,\frac{\lambda +\upsilon }{2}\right ) ,\ell _{2}\left ( 1,1,1,\lambda +\upsilon ,0,1,\frac{\lambda +\upsilon }{2}\right ) \right \} }. \end{aligned}$$

Let us consider $p>0$ and $\vartheta _{0}\in F_{ix}\left ( \mho _{2}\right )$. Then, there is $\vartheta _{1}\in \mho _{2}\left ( \vartheta _{0}\right )$ in order that

$$ \varpi \left ( \vartheta _{0},\vartheta _{1}\right ) =B\left ( \vartheta _{0},\mho _{2}\left ( \vartheta _{0}\right ) \right ) \text{ and }\varpi \left ( \vartheta _{1},\vartheta _{2}\right ) \leq p \Upsilon \left ( \mho _{1}\left ( \vartheta _{0}\right ) ,\mho _{2} \left ( \vartheta _{0}\right ) \right ) . $$

Now, there is $\vartheta _{2}\in \mho _{2}\left ( \vartheta _{1}\right )$ so that

$$ \varpi \left ( \vartheta _{0},\vartheta _{1}\right ) =B\left ( \vartheta _{0},\mho _{2}\left ( \vartheta _{0}\right ) \right ) \text{ and }\varpi \left ( \vartheta _{1},\vartheta _{2}\right ) \leq p \Upsilon \left ( \mho _{2}\left ( \vartheta _{0}\right ) ,\mho _{2} \left ( \vartheta _{1}\right ) \right ) . $$

Further, we obtain $\varpi \left ( \vartheta _{1},\vartheta _{2}\right ) <\varpi \left ( \vartheta _{0},\vartheta _{1}\right )$ and

$$\begin{aligned} &\varpi \left ( \vartheta _{1},\vartheta _{2}\right ) \\ &\quad\leq p \Upsilon \left ( \mho _{2}\left ( \vartheta _{0}\right ) ,\mho \left ( \vartheta _{1}\right ) \right ) \\ &\quad\leq p\ell _{1}\left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta _{0},\vartheta _{1}\right ) ,B\left ( \vartheta _{0},\mho _{2}\vartheta _{0}\right ) ,B\left ( \vartheta _{1}, \mho _{2}\vartheta _{1}\right ) ,B\left ( \vartheta _{0},\mho _{2} \vartheta _{1}\right ) , \\ B\left ( \vartheta _{1},\mho _{2}\vartheta _{0}\right ) , \frac{\varpi \left ( \vartheta _{0},\vartheta _{0}\right ) +B\left ( \vartheta _{1},\mho _{2}\vartheta _{1}\right ) }{2}, \frac{B\left ( \vartheta _{0},\mho _{2}\vartheta _{1}\right ) +B\left ( \vartheta _{1},\mho _{2}\vartheta _{0}\right ) }{2}\end{array}\displaystyle \right ) \\ &\quad\leq p\ell _{1}\left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta _{0},\vartheta _{1}\right ) ,\varpi \left ( \vartheta _{0},\vartheta _{1}\right ) ,\varpi \left ( \vartheta _{1}, \vartheta _{2}\right ) ,\varpi \left ( \vartheta _{0},\vartheta _{2} \right ) , \\ \varpi \left ( \vartheta _{1},\vartheta _{1}\right ) , \frac{\varpi \left ( \vartheta _{0},\vartheta _{1}\right ) +\varpi \left ( \vartheta _{1},\vartheta _{2}\right ) }{2}, \frac{\varpi \left ( \vartheta _{0},\vartheta _{2}\right ) +\varpi \left ( \vartheta _{1},\vartheta _{1}\right ) }{2}\end{array}\displaystyle \right ) \\ &\quad\leq p\ell _{1}\left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta _{0},\vartheta _{1}\right ) ,\varpi \left ( \vartheta _{0},\vartheta _{1}\right ) ,\varpi \left ( \vartheta _{1}, \vartheta _{2}\right ) , \\ \gamma \left ( \vartheta _{0},\vartheta _{1}\right ) \varpi \left ( \vartheta _{0},\vartheta _{1}\right ) +\gamma \left ( \vartheta _{1}, \vartheta _{2}\right ) \varpi \left ( \vartheta _{1},\vartheta _{2} \right ) , \\ 0, \frac{\varpi \left ( \vartheta _{0},\vartheta _{1}\right ) +\varpi \left ( \vartheta _{1},\vartheta _{2}\right ) }{2}, \frac{\gamma \left ( \vartheta _{0},\vartheta _{1}\right ) \varpi \left ( \vartheta _{0},\vartheta _{1}\right ) +\gamma \left ( \vartheta _{1},\vartheta _{2}\right ) \varpi \left ( \vartheta _{1},\vartheta _{2}\right ) +0}{2}\end{array}\displaystyle \right ) \\ &\quad< p\ell _{1}\left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta _{0},\vartheta _{1}\right ) ,\varpi \left ( \vartheta _{0},\vartheta _{1}\right ) ,\varpi \left ( \vartheta _{0}, \vartheta _{1}\right ) , \\ \gamma \left ( \vartheta _{0},\vartheta _{1}\right ) \varpi \left ( \vartheta _{0},\vartheta _{1}\right ) +\gamma \left ( \vartheta _{1}, \vartheta _{2}\right ) \varpi \left ( \vartheta _{0},\vartheta _{1} \right ) , \\ 0, \frac{\varpi \left ( \vartheta _{0},\vartheta _{1}\right ) +\varpi \left ( \vartheta _{0},\vartheta _{1}\right ) }{2}, \frac{\gamma \left ( \vartheta _{0},\vartheta _{1}\right ) \varpi \left ( \vartheta _{0},\vartheta _{1}\right ) +\gamma \left ( \vartheta _{1},\vartheta _{2}\right ) \varpi \left ( \vartheta _{0},\vartheta _{1}\right ) }{2}\end{array}\displaystyle \right ) \\ &\quad\leq p\varpi \left ( \vartheta _{0},\vartheta _{1}\right ) \ell _{1} \left ( 1,1,1,\gamma \left ( \vartheta _{0},\vartheta _{1}\right ) + \gamma \left ( \vartheta _{1},\vartheta _{2}\right ) ,0,1, \frac{\gamma \left ( \vartheta _{0},\vartheta _{1}\right ) +\gamma \left ( \vartheta _{1},\vartheta _{2}\right ) }{2} \right ) , \end{aligned}$$

where $\ell _{1}\in \ell \in \aleph$. Therefore, we obtain a sequence of SAM of ℧ at starting point $\vartheta _{0}$, which fulfills

$$\begin{aligned} &\varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) \leq \left ( p \ell _{1}\left ( 1,1,1,\lambda +\upsilon ,0,1, \frac{\lambda +\upsilon }{2}\right ) ^{u}\varpi \left ( \vartheta _{0}, \vartheta _{1}\right ) \right ) ,\\ &\quad\text{for all }\lambda ,\upsilon \geq 1\text{ and all }u\in \mathbb{N} . \end{aligned}$$

In another form, we can write

$$ \varpi \left ( \vartheta _{u},\vartheta _{u+n}\right ) \leq \frac{\left ( p\ell _{1}\left ( 1,1,1,\lambda +\upsilon ,0,1,\frac{\lambda +\upsilon }{2}\right ) \right ) ^{u}}{1-p\ell _{1}\left ( 1,1,1,\lambda +\upsilon ,0,1,\frac{\lambda +\upsilon }{2}\right ) } \varpi \left ( \vartheta _{0},\vartheta _{1}\right ) ,\text{ for all }u\in \mathbb{N} . $$

(5.2)

In (5.2), letting $u\rightarrow \infty$, we find that $\{\vartheta _{u}\}$ is a CS in η, and thus, converges to some $\sigma \in \eta$. From the proof of Theorem 4.2, we obtain that $\sigma \in F_{ix}\left ( \mho _{2}\right )$. Again, passing $n\rightarrow \infty$ in (5.2), one has

$$ \varpi \left ( \vartheta _{u},\sigma \right ) \leq \frac{\left ( p\ell _{1}\left ( 1,1,1,\lambda +\upsilon ,0,1,\frac{\lambda +\upsilon }{2}\right ) \right ) ^{u}}{1-p\ell _{1}\left ( 1,1,1,\lambda +\upsilon ,0,1,\frac{\lambda +\upsilon }{2}\right ) } \varpi \left ( \vartheta _{0},\vartheta _{1}\right ) ,\text{ for all }u\in \mathbb{N} . $$

Setting $u=0$, and using (D₄), we obtain

$$\begin{aligned} \varpi \left ( \vartheta _{0},\sigma \right )& \leq \frac{1}{1-p\ell _{1}\left ( 1,1,1,\lambda +\upsilon ,0,1,\frac{\lambda +\upsilon }{2}\right ) } \varpi \left ( \vartheta _{0},\vartheta _{1}\right ) \\ &\leq \frac{p\zeta }{1-p\ell _{1}\left ( 1,1,1,\lambda +\upsilon ,0,1,\frac{\lambda +\upsilon }{2}\right ) }. \end{aligned}$$

Switching the roles of $\mho _{1}$ and $\mho _{2}$, for every $\sigma _{0}\in F_{ix}\left ( \mho _{1}\right )$, one can write

$$\begin{aligned} \varpi \left ( \vartheta _{0},\sigma _{0}\right ) &\leq \frac{1}{1-p\ell _{2}\left ( 1,1,1,\lambda +\upsilon ,0,1,\frac{\lambda +\upsilon }{2}\right ) }\varpi \left ( \vartheta _{0},\vartheta _{1}\right )\\ & \leq \frac{p\zeta }{1-p\ell _{2}\left ( 1,1,1,\lambda +\upsilon ,0,1,\frac{\lambda +\upsilon }{2}\right ) }. \end{aligned}$$

Hence,

$$\begin{aligned} &\Upsilon \left ( F_{ix}\left ( \mho _{1}\right ) ,F_{ix}\left ( \mho _{2} \right ) \right )\\ &\quad \leq \frac{P\zeta }{1-\max \left \{ p\ell _{1}\left ( 1,1,1,\lambda +\upsilon ,0,1,\frac{\lambda +\upsilon }{2}\right ) ,p\ell _{2}\left ( 1,1,1,\lambda +\upsilon ,0,1,\frac{\lambda +\upsilon }{2}\right ) \right \} }, \end{aligned}$$

where $\lambda ,\upsilon \geq 1$. Letting $p\rightarrow 1$ in the above inequality, we have the result. □

6 Well-posednees and strict FPs

The definition of the well-posedness for the FP problem is presented in [25] as follows:

Definition 6.1

Assume that $\left ( \eta ,\varpi \right )$ is an MS, $\Lambda \in Q\left ( \eta \right )$, and $\mho :\Lambda \rightarrow \eta ^{c}$ is a MVM. The FP issue is called well-posed for ℧ with respect to (w.r.t.) B if

(w₁):

$F_{ix}\left ( \mho \right ) =\{\widehat{\vartheta }\}$;

(w₂):

if $\vartheta _{u}\in \Lambda$, for all $u\in \mathbb{N}$, $\lim _{u\rightarrow \infty }B\left ( \vartheta _{u},\mho \vartheta _{u} \right ) =0$.

Then, $\lim _{u\rightarrow \infty }\vartheta _{u}=\widehat{\vartheta }\in F_{ix} \left ( \mho \right )$.

Definition 6.2

Assume that $\left ( \eta ,\varpi \right )$ is an MS, $\Lambda \in Q\left ( \eta \right )$, and $\mho :\Lambda \rightarrow \eta ^{c}$ is a MVM. The FP issue is called well-posed for ℧ w.r.t. ϒ if

(w₁):

$SF_{ix}\left ( \mho \right ) =\{\widehat{\vartheta }\}$;

(w₂):

if $\vartheta _{u}\in \Lambda$, for all $u\in \mathbb{N}$, $\lim _{u\rightarrow \infty }\Upsilon \left ( \vartheta _{u},\mho \vartheta _{u}\right ) =0$.

Then, $\lim _{u\rightarrow \infty }\vartheta _{u}=\widehat{\vartheta }\in SF_{ix} \left ( \mho \right )$.

The main theorem in this part is as follows:

Theorem 6.3

Suppose that $\left ( \eta ,\varpi ,\gamma \right )$ is a complete CMS, $\mho :\eta \rightarrow \eta ^{cp}$ is an MVM and $F_{1},F_{2}$ are functions verifying an ψ̃F-contraction. Assume that the following presumptions hold:

(P₁):

$F_{1}$ is nondecreasing;

(P₂):

$F_{2}$ verifies $(\heartsuit _{2}^{\prime })$ with $\ell \left ( 1,0,0,1,1,0,1\right ) \in (0,1)$;

(P₃):

$SF_{ix}\left ( \mho \right )$ is nonempty;

(P₄):

for all $\vartheta \in \eta$, $\lim _{u\rightarrow \infty }\gamma \left ( \vartheta _{u},\vartheta \right ) \leq 1$.

Then,

• (I) $F_{ix}\left ( \mho \right ) =SF_{ix}\left ( \mho \right ) =\left \{ \widehat{\vartheta }\right \}$;

• (II) The FP problem is well-posed for the MVM ℧ w.r.t. ϒ.

Proof

(I) According to Theorem 4.4, we have $F_{ix}\left ( \mho \right ) \neq \emptyset$. Next, we will show that $F_{ix}\left ( \mho \right ) =\left \{ \widehat{\vartheta }\right \}$. Utilizing $\left ( \widetilde{\psi }F\right ) _{i}$ and $\left ( \widetilde{\psi }F\right ) _{ii}$, we can write

$$\begin{aligned} &F_{1}\left ( \Upsilon \left ( \mho \vartheta ,\mho \widetilde{\vartheta }\right ) \right ) \\ \leq &F_{2}\left ( \Upsilon \left ( \mho \vartheta ,\mho \widetilde{\vartheta }\right ) \right ) \\ \leq &\widetilde{\psi }\left ( \varpi \left ( \vartheta , \widetilde{\vartheta }\right ) \right ) +F_{2}\left ( \Upsilon \left ( \mho \vartheta ,\mho \widetilde{\vartheta }\right ) \right ) \\ \leq &F_{1}\left \{ \ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta ,\widetilde{\vartheta }\right ) ,B\left ( \vartheta ,\mho \vartheta \right ) ,B\left ( \widetilde{\vartheta }, \mho \widetilde{\vartheta }\right ) ,B\left ( \vartheta ,\mho \widetilde{\vartheta } \right ) , \\ B\left ( \widetilde{\vartheta },\mho \vartheta \right ) , \frac{B\left ( \vartheta ,\mho \vartheta \right ) +B\left ( \widetilde{\vartheta },\mho \widetilde{\vartheta }\right ) }{2}, \frac{B\left ( \vartheta ,\mho \widetilde{\vartheta }\right ) +B\left ( \widetilde{\vartheta },\mho \vartheta \right ) }{2}\end{array}\displaystyle \right ) \right \} . \end{aligned}$$

The monotonicity of $F_{1}$, implies that

$$ \Upsilon \left ( \mho \vartheta ,\mho \widetilde{\vartheta }\right ) \leq \ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta ,\widetilde{\vartheta }\right ) ,B\left ( \vartheta ,\mho \vartheta \right ) ,B\left ( \widetilde{\vartheta }, \mho \widetilde{\vartheta }\right ) ,B\left ( \vartheta ,\mho \widetilde{\vartheta } \right ) , \\ B\left ( \widetilde{\vartheta },\mho \vartheta \right ) , \frac{B\left ( \vartheta ,\mho \vartheta \right ) +B\left ( \widetilde{\vartheta },\mho \widetilde{\vartheta }\right ) }{2}, \frac{B\left ( \vartheta ,\mho \widetilde{\vartheta }\right ) +B\left ( \widetilde{\vartheta },\mho \vartheta \right ) }{2}\end{array}\displaystyle \right ) , $$

for all $\vartheta ,\widetilde{\vartheta }\in \eta$. Consider $\sigma \in F_{ix}\left ( \mho \right )$ with $\sigma \neq \widehat{\vartheta }$. Then, $B\left ( \widehat{\vartheta },\mho \sigma \right ) >0$. Now, we obtain

$$\begin{aligned} B\left ( \widehat{\vartheta },\mho \sigma \right ) =&\Upsilon \left ( \mho \widehat{\vartheta },\mho \sigma \right ) \\ \leq &\ell \left ( \textstyle\begin{array}{c} \varpi \left ( \widehat{\vartheta },\sigma \right ) ,B\left ( \widehat{\vartheta },\mho \widehat{\vartheta }\right ) ,B\left ( \sigma ,\mho \sigma \right ) ,B\left ( \widehat{\vartheta },\mho \sigma \right ) , \\ B\left ( \sigma ,\mho \widehat{\vartheta }\right ) , \frac{B\left ( \widehat{\vartheta },\mho \widehat{\vartheta }\right ) +B\left ( \sigma ,\mho \sigma \right ) }{2}, \frac{B\left ( \widehat{\vartheta },\mho \sigma \right ) +B\left ( \sigma ,\mho \widehat{\vartheta }\right ) }{2}\end{array}\displaystyle \right ) \\ \leq &\ell \left ( \varpi \left ( \widehat{\vartheta },\sigma \right ) ,0,0,B\left ( \widehat{\vartheta },\sigma \right ) ,B\left ( \sigma ,\widehat{\vartheta }\right ) ,0, \frac{B\left ( \widehat{\vartheta },\sigma \right ) +B\left ( \sigma ,\widehat{\vartheta }\right ) }{2}\right ) \\ \leq &\varpi \left ( \widehat{\vartheta },\sigma \right ) \ell \left ( 1,0,0,1,1,0,1\right ) . \end{aligned}$$

Applying the condition (P₂), we obtain

$$ \varpi \left ( \widehat{\vartheta },\sigma \right ) =B\left ( \widehat{\vartheta },\mho \sigma \right ) \leq \varpi \left ( \widehat{\vartheta },\sigma \right ) , $$

which is a contradiction. Hence, $\varpi \left ( \widehat{\vartheta },\sigma \right ) =0$, that is, $\widehat{\vartheta }=\sigma$.

(II) Assume that $\vartheta _{u}\in \Lambda$ and $u\in \mathbb{N}$ in order that

$$ \lim _{u\rightarrow \infty }B\left ( \vartheta _{u},\mho \vartheta _{u} \right ) =0. $$

(6.1)

We prove that

$$ \lim _{u\rightarrow \infty }\varpi \left ( \vartheta _{u}, \widehat{\vartheta }\right ) =0, $$

where $\widehat{\vartheta }\in F_{ix}\left ( \mho \right )$. Assume the contrary, then for each $u\in \mathbb{N}$, there is $\varepsilon >0$ so that

$$ \varpi \left ( \vartheta _{u},\widehat{\vartheta }\right ) > \varepsilon . $$

Equation (6.1) leads to the fact that there is $u_{\varepsilon }\in \mathbb{N} -\{0\}$ so that

$$ \lim _{u\rightarrow \infty }B\left ( \vartheta _{u},\mho \vartheta _{u} \right ) < \varepsilon ,\text{ for each }u>u_{\varepsilon }. $$

It follows that

$$ \varpi \left ( \vartheta _{u},\widehat{\vartheta }\right ) =B\left ( \vartheta _{u},\mho \widehat{\vartheta }\right ) ,\text{ for each }u>u_{ \varepsilon }. $$

Since ℧ϑ̂ is compact, there is $\vartheta \in \mho \widehat{\vartheta }$ so that

$$\begin{aligned} \varpi \left ( \vartheta _{u},\widehat{\vartheta }\right ) =&B\left ( \vartheta _{u},\mho \widehat{\vartheta }\right ) =\varpi \left ( \vartheta _{u},\vartheta \right ) \\ \leq &\gamma \left ( \vartheta _{u},\vartheta _{u+1}\right ) \varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) +\gamma \left ( \vartheta _{u+1},\vartheta \right ) \varpi \left ( \vartheta _{u+1}, \vartheta \right ) \\ =&\gamma \left ( \vartheta _{u},\vartheta _{u+1}\right ) B\left ( \vartheta _{u},\mho \vartheta _{u}\right ) +\gamma \left ( \vartheta _{u+1}, \vartheta \right ) B\left ( \vartheta _{u+1},\mho \widehat{\vartheta }\right ) \\ \leq &\gamma \left ( \vartheta _{u},\vartheta _{u+1}\right ) B\left ( \vartheta _{u},\mho \vartheta _{u}\right ) +\gamma \left ( \vartheta _{u+1}, \vartheta \right ) \Upsilon \left ( \mho \vartheta _{u},\mho \widehat{\vartheta }\right ) \\ \leq &\gamma \left ( \vartheta _{u},\vartheta _{u+1}\right ) B\left ( \vartheta _{u},\mho \vartheta _{u}\right ) \\ &+\gamma \left ( \vartheta _{u+1},\vartheta \right ) \ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta _{u},\widehat{\vartheta }\right ) ,B\left ( \vartheta _{u},\mho \vartheta _{u}\right ) ,B\left ( \widehat{\vartheta },\mho \widehat{\vartheta }\right ) ,B\left ( \vartheta _{u},\mho \widehat{\vartheta }\right ) , \\ B\left ( \widehat{\vartheta },\mho \vartheta _{u}\right ) , \frac{B\left ( \vartheta _{u},\mho \vartheta _{u}\right ) +B\left ( \widehat{\vartheta },\mho \widehat{\vartheta }\right ) }{2}, \frac{B\left ( \vartheta _{u},\mho \widehat{\vartheta }\right ) +B\left ( \widehat{\vartheta },\mho \vartheta _{u}\right ) }{2}\end{array}\displaystyle \right ) \\ \leq &\gamma \left ( \vartheta _{u},\vartheta _{u+1}\right ) B\left ( \vartheta _{u},\mho \vartheta _{u}\right ) \\ &+\gamma \left ( \vartheta _{u+1},\vartheta \right ) \ell \left ( \textstyle\begin{array}{c} \varpi \left ( \vartheta _{u},\widehat{\vartheta }\right ) ,B\left ( \vartheta _{u},\mho \vartheta _{u}\right ) ,\varpi \left ( \widehat{\vartheta },\widehat{\vartheta }\right ) ,B\left ( \vartheta _{u},\widehat{\vartheta } \right ) , \\ \gamma \left ( \widehat{\vartheta },\vartheta _{u}\right ) B\left ( \widehat{\vartheta },\vartheta _{u}\right ) +\gamma \left ( \vartheta _{u}, \mho \vartheta _{u}\right ) \varpi \left ( \vartheta _{u},\vartheta _{u+1} \right ) , \\ \frac{B\left ( \vartheta _{u},\mho \vartheta _{u}\right ) ,\varpi \left ( \widehat{\vartheta },\widehat{\vartheta }\right ) }{2}, \frac{B\left ( \vartheta _{u},\widehat{\vartheta }\right ) +\gamma \left ( \widehat{\vartheta },\vartheta _{u}\right ) B\left ( \widehat{\vartheta },\vartheta _{u}\right ) +\gamma \left ( \vartheta _{u},\mho \vartheta _{u}\right ) \varpi \left ( \vartheta _{u},\vartheta _{u+1}\right ) }{2}\end{array}\displaystyle \right ). \end{aligned}$$

From conditions (P₂) and (P₄), letting $u\rightarrow \infty$ in the above inequality and using (6.1), we have $\lim _{u\rightarrow \infty }\varpi \left ( \vartheta _{u}, \widehat{\vartheta }\right ) =0$, which is a contradiction. Therefore, the FP issue is well-posed for the MVM ℧ w.r.t. B. Additionally, $F_{ix}\left ( \mho \right ) =SF_{ix}\left ( \mho \right )$ and the FP issue is well-posed for the MVM ℧ w.r.t. ϒ. □

7 Conclusion

Several strict and FP results on CMSs have been established in this study. As we utilized the controlled metric setting platform and adhered to the plan of Iqbal et al. [17], the results presented in [17] are specific instances of those presented in this study. We have also given the theorems’ well-posedness. Additionally, the FP data-dependence issue of the considered mappings is established. For the sake of authenticity, numerous nontrivial examples are included.