The Existence of Fixed Points for Generalized ${\mathit{\omega }}_{\mathit{b}}^{\mathit{\phi }}$-Contractions and Applications

Abstract

This article introduces a new type of contractions via $\phi$-admissibility and ${\omega }_b$-distance called generalized ${\omega }_b^{\phi }$-contractions. We prove the existence of fixed points for this type of contractions under some conditions. Moreover, we give an example to demonstrate the applications of our results.

1. Introduction

The exploration of fixed point (FP) theory has emerged as a critical area for various research endeavors across nonlinear functional analysis, multidimensional calculus, and a wide range of other fields in science and technology, as highlighted in several studies. The recent advancements in the theory of FPs have significantly enhanced the mathematical modeling of intelligent systems, particularly in AI-based decision systems. Additionally, the stability of deep equilibrium models (DEMs) in AI, along with the validation of iterative protocols in cryptography, can be improved by leveraging these new findings. Given the contribution of the effective FP theory to solving many problems resulting from life applications and phenomena, attention has been paid to developing the famous Banach Contraction Principle (BCP) in several ways. Using the Hausdorff–Pompeiu metric (HPM), Nadler [1] introduced a notion of multivalued contractions and established a multivalued version of the well-known BCP. Since then a number of generalizations have appeared in the metric FP theory. In fact, existing results can be developed for many cases without utilizing the HPM, see [2,3,4] and others.

Metric FP theory has been broadened by editing the metric properties. In particular, b-metric space (bMS) or, as we call it, metric type space (MTS) is a useful extension of the metric space (MS). It was first introduced and studied by Bakhtin [5], and later Czerwik [6,7] modified the concept and studied the BCP within the structure of bMSs. Afterward, some FP discoveries were made for both single-valued and multivalued mappings in bMSs, for example [8,9,10,11,12,13,14,15].

In [16], Kada et al. introduced the idea of $\omega$-distance on MSs and ameliorated various findings by using the $\omega$-distance instead of the regular metric. Meanwhile, Suzuki and Takahashi defined the notions of single-valued and multivalued contractive mappings via $\omega$-distance in [17], and they extended some known results. In this direction, a great deal work has been undertaken; for instance, see [18,19,20]. In [21], the $\omega t$-distance was defined by Husain et al. on MTSs, and demonstrated some findings concerning FP for single-valued mappings involving $\omega t$-distance. For further findings, see [14,22,23,24,25] and others.

Although a number of interesting fixed point results have appeared in the literature (see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37]), many questions still remain open at the heart of the theory, and there are many unanswered questions regarding the existence of fixed points and common fixed points under generalized nonlinear mappings with respect to generalized distances. The main questions are as follows:

First, is it possible to establish some general fixed point results for multivalued mappings under somewhat weaker contractive inequalities in the setting of generalized metric spaces?

Second, which sufficient conditions, either for the mappings or the general spaces, could guarantee the existence of fixed points for multivalued mappings involving general inequalities in metric fixed point theory?

In this article, we introduce several FP findings for multivalued nonlinear generalized contractions along with supporting examples. In fact, our results either improve or generalize a number of known FP results.

Our research will contribute to the existing metric fixed point theory of the nonlinear functional analysis. Further, our positive results in these directions will be helpful for those working in this area of research.

2. Preliminaries

In this section, we reexamine some important principles. We consider $𝘍$ to be a MS with the metric $\Omega$, otherwise stated. We denote $2^{𝘍}=\{\Lambda \subset 𝘍:\Lambda \ne \varnothing \}$, $\mathcal{CL}\left(𝘍\right)=\{\varphi \ne \Lambda \subset 𝘍:\Lambda \mathrm{is}\mathrm{closed}\},\mathcal{CB}\left(𝘍\right)=\{\varphi \ne \Lambda \subset 𝘍:\Lambda \mathrm{is}\mathrm{closed}\mathrm{and}\mathrm{bounded}\}$. For any $\Delta ,\Gamma \in \mathcal{CB}\left(𝘍\right)$, define

$$H(\Delta ,\Gamma )=max\left\{\underset{\eta \in \Delta }{sup}\Omega (\eta ,\Gamma )\right.,\left.\underset{\vartheta \in \Gamma }{sup}\Omega (\vartheta ,\Delta )\right\},$$

where $\Omega (\eta ,\Gamma )=\underset{\vartheta \in \Gamma }{inf}\Omega (\eta ,\vartheta )$. It is recognized that H is a metric on $\mathcal{CB}\left(𝘍\right)$, termed as the Hausdorff–Pompeiu metric.

An element $\eta \in 𝘍$ is termed a FP of a mapping $G:𝘍\to 2^{𝘍}$ if $\eta$ belongs to the set $G\left(\eta \right)$. The set of all FPs of G will be referred to as $\mathcal{F}\left(G\right)$. Let g be a single-valued mapping from $𝘍$ into itself. We denote $C(G\cap g)=\{\eta \in 𝘍:g(\eta )\in G(\eta \left)\right\}$, the set of coincidence points of G and g. A point $\eta \in 𝘍$ is termed a common fixed point (CFP) of g and G if $\eta =g\left(\eta \right)$ belong to the set $G\left(\eta \right)$. The set of all CFPs of g and G will be referred to as $\mathcal{F}(g\cap G)$. A sequence $\left\{{\eta }_n\right\}$ in $𝘍$ is termed an orbit of $G$ at ${\eta }_0\in 𝘍$ if ${\eta }_n\in G\left({\eta }_{n-1}\right)$ for all $n\ge 1$. A map $\kappa :𝘍\to \mathbb{R}$ is described as lower semi-continuous (LSC) if for any sequence $\left\{{\eta }_n\right\}\subset 𝘍$ with ${\eta }_n\to \eta \in 𝘍,$ we have $\kappa \left(\eta \right)\le \underset{n\to \infty }{lim\; inf}\kappa \left({\eta }_n\right)$. We denote $[0,\infty )$ by ${\mathbb{R}}^{+}$.

The notion of $\omega$-distance on MSs was defined by Kada et al. as follows:

The function $\omega :𝘍\times 𝘍\to {\mathbb{R}}^{+}$ is referred to as $\omega$-distance if the following conditions are met for all $\eta ,\vartheta ,\mathfrak{z}\in 𝘍$

(i)

$\omega (\eta ,\mathfrak{z})\le \omega (\eta ,\vartheta )+\omega (\vartheta ,\mathfrak{z})$;

(ii)

the map $\omega (\eta ,·):𝘍\to {\mathbb{R}}^{+}$ is LSC;

(iii)

$\forall \iota >0\exists \sigma >0$, with $\omega (\mathfrak{z},\eta )\le \sigma$ and $\omega (\mathfrak{z},\vartheta )\le \sigma$ we have $\Omega (\eta ,\vartheta )\le \iota$.

Obviously, any metric $\Omega$ is an $\omega$-distance on $𝘍$. The $\omega$-distance $\omega$ on $𝘍$ is called an ${\omega }^0$-distance if $\omega (\rho ,\rho )=0$ for all $\rho \in 𝘍$ [28]. For details on the $\omega$-distance, we refer to [16,17].

In [6,7,11] Czerwik introduced the following notion of MTS or bMS:

Let $𝘍$ be a non-empty set, $b\ge 1$ and ${\Omega }_b:𝘍\times 𝘍\to {\mathbb{R}}^{+}$ be a map that meets the subsequent criteria for any $\eta ,\vartheta ,\mathfrak{z}\in 𝘍$:

(i)

${\Omega }_b(\eta ,\vartheta )=0⟺\eta =\vartheta$;

(ii)

${\Omega }_b(\eta ,\vartheta )={\Omega }_b(\vartheta ,\eta )$;

(iii)

${\Omega }_b(\eta ,\vartheta )\le b[{\Omega }_b(\eta ,\mathfrak{z})+{\Omega }_b(\mathfrak{z},\vartheta )]$.

Thus, ${\Omega }_b$ is referred to as a bM on $𝘍$, and the pair $(𝘍,{\Omega }_b)$ is termed a bMS (also known as a MTS [14]). Hereafter, we will also refer to it as a MTS.

Remark 1. 

Note that, any MS becomes a MTS, but not conversely, in general; see [6,21]. Thus, the class of MTSs contains the class of MSs.

Example 1 

([5]). If $𝘍=[0,1]$ and a function ${\Omega }_b:𝘍\times 𝘍\to {\mathbb{R}}^{+}$, defined by ${\Omega }_b(\eta ,\vartheta )={(\eta -\vartheta )}^2$ for any $\eta ,\vartheta \in 𝘍$, then $(𝘍, {\Omega }_b)$ is not a MS, while it is a MTS with $b=2$.

Commonly, ${\Omega }_b$ may fail to be continuous in each variable; for example, see [26]. However, it has been noted that a topology can be established based on convergence in these types of spaces [26]. In the framework of MTSs, the concepts of convergent, Cauchy sequences, etc., can be obtained under the model of MSs; for example, see [8,14,21].

A set $\Lambda$ in $(𝘍,{\Omega }_b)$ is termed open if and only if for any $\eta$ of $\Lambda$, there is a positive number $\sigma$ such that the open ball $B_o(\eta ,\sigma )$ is contained in $\Lambda$. We represent $\tau$ as the collection of all open subsets of $𝘍$, which forms a topology on the pair $(𝘍, {\Omega }_b)$. Additionally, a subset $\Lambda$ of $𝘍$ is considered closed if, for any sequence $\left\{{\vartheta }_n\right\}$ within $\Lambda$ that converges to a limit $\vartheta$, it follows that $\vartheta$ is also in $\Lambda$, as noted in [14]. Moreover, a map $g:𝘍\to \mathbb{R}$ is termed b-lower semi-continuous (b-LSC) if, for every sequence $\left\{{\eta }_n\right\}$ in $𝘍$ that converges to $\eta$ in $𝘍$, then $g\left(\eta \right)\le \underset{n\to \infty }{lim\; inf}\left(bg\left({\eta }_n\right)\right)$.

The subsequent fundamental findings related to MTSs are valuable.

Lemma 1 

([7]). For a closed set Λ in $(𝘍, {\Omega }_b)$ and an element η in 𝘍, the condition ${\Omega }_b(\eta ,\Lambda )=0$ is equivalent to η, being an element of $\overline{\Lambda }=\Lambda$, where ${\Omega }_b(\eta ,\Lambda )$ represents the infimum of distances ${\Omega }_b(\eta ,\vartheta )$ for all ϑ in Λ, and $\overline{\Lambda }$ denotes the closure of the set Λ.

Lemma 2 

([34,37]). Consider $(𝘍, {\Omega }_b)$ as a MTS and let $\left\{{\mathfrak{z}}_n\right\}$ be a sequence within 𝘍. Suppose there exists a constant c in the interval $[0,1)$ such that ${\Omega }_b({\mathfrak{z}}_{n+1},{\mathfrak{z}}_{n+2})\le c{\Omega }_b({\mathfrak{z}}_n,{\mathfrak{z}}_{n+1})$ holds for all n in the natural numbers. Under this condition, the sequence $\left\{{\mathfrak{z}}_n\right\}$ is Cauchy.

Applying Lemma 2, Suzuki [37] formulated a broad FP theorem for multivalued mappings, which consequently generalizes classical FP results due to Nadler [1] and Mizoguchi and Takahashi [35] for MTSs.

Inspired by the research of Kada et al. [16], Hussain et al. [21] proposed an $\omega$-distance in MTSs, referring to it as $wt$-distance (which will be referred to as ${\omega }_b$-distance in the following sections).

Let $(𝘍,{\Omega }_b)$ be a MTS. A function ${\omega }_b:𝘍\times 𝘍\to {\mathbb{R}}^{+}$ is termed ${\omega }_b$-distance on $𝘍$ if it meets the subsequent constraints for any $\eta ,\vartheta ,\mathfrak{z}\in 𝘍$:

(i)

${\omega }_b(\eta ,\mathfrak{z})\le b\left[{\omega }_b(\eta ,\vartheta )+{\omega }_b(\vartheta ,\mathfrak{z})\right]$;

(ii)

the map ${\omega }_b(\eta ,·):𝘍\to {\mathbb{R}}^{+}$ is b-LSC;

(iii)

$\forall \iota >0\exists \sigma >0$ with ${\omega }_b(\mathfrak{z},\eta )\le \sigma$ and ${\omega }_b(\mathfrak{z},\vartheta )\le \sigma$ implies ${\Omega }_b(\eta ,\vartheta )\le$ $\iota$.

It should be noted that each ${\omega }_b$-distance becomes equivalent to the $\omega$-distance when b is set to 1. The ${\omega }_b$-distance ${\omega }_b$ on $𝘍$ is termed a ${\omega }_b^0$-distance if ${\omega }_b(\rho ,\rho )=0$ for all $\rho \in 𝘍$.

Example 2 

([21]). Let $𝘍=\mathbb{R}$ and ${\Omega }_b(\eta ,\vartheta )={(\eta -\vartheta )}^2,\eta ,\vartheta \in 𝘍$. Then, the maps ${\omega }_{b_1},{\omega }_{b_2}:𝘍\times 𝘍\to {\mathbb{R}}^{+}$ defined by ${\omega }_{b_1}(\eta ,\vartheta )={\left|\eta \right|}^2+{\left|\vartheta \right|}^2$ and ${\omega }_{b_2}(\eta ,\vartheta )={\left|\vartheta \right|}^2$ for every $\eta ,\vartheta \in 𝘍$ are ${\omega }_b$-distances on 𝘍.

It has been observed that each MT ${\Omega }_b$ is a ${\omega }_b$-distance. However, the opposite may not hold, in general [29].

The subsequent findings are essential for proving our main outcomes of the next section.

Lemma 3 

([21]). Let $(𝘍, {\Omega }_b)$ be a MTS, and let ${\omega }_b$ be an ${\omega }_b^0$-distance on 𝘍. Let $\left\{{\eta }_n\right\}$ and $\left\{{\vartheta }_n\right\}$ be sequences in 𝘍. Let $\left\{{\sigma }_n\right\}$ and $\left\{{\beta }_n\right\}$ be sequences in $[0,\infty )$ such that ${\sigma }_n\to 0$ and ${\beta }_n\to 0$. Then, the subsequent assertions are fulfilled for every $\eta ,\vartheta ,\mathfrak{z}\in 𝘍$.

(a)

if ${\omega }_b\left({\eta }_n,\vartheta \right)\le {\sigma }_n$ and ${\omega }_b\left({\eta }_n,\mathfrak{z}\right)\le {\beta }_n$ for any $n\in \mathbb{N}$, then $\vartheta =\mathfrak{z}$. In particular, if ${\omega }_b(\eta ,\vartheta )=0$ and ${\omega }_b(\eta ,\mathfrak{z})=0$, then $\vartheta =\mathfrak{z}$;

(b)

if ${\omega }_b\left({\eta }_n,{\vartheta }_n\right)\le {\sigma }_n$ and ${\omega }_b\left({\eta }_n,\mathfrak{z}\right)\le {\beta }_n$ for any $n\in \mathbb{N}$, then ${\Omega }_b\left({\vartheta }_n,\mathfrak{z}\right)\to 0$;

(c)

if ${\omega }_b\left({\eta }_n,{\eta }_m\right)\le {\sigma }_n$ for any $n,m\in \mathbb{N}$ with $m>n$, then $\left\{{\eta }_n\right\}$ is a Cauchy sequence;

(d)

if ${\omega }_b\left(\vartheta ,{\eta }_n\right)\le {\sigma }_n$ for any $n\in \mathbb{N}$, then $\left\{{\eta }_n\right\}$ is a Cauchy sequence.

Lemma 4 

([24]). Let Λ be a closed subset of a MTS $(𝘍, {\Omega }_b)$, and let ${\omega }_b$ be a ${\omega }_b$-distance on 𝘍. Assume that there exists $\eta \in 𝘍$ such that ${\omega }_b(\eta ,\eta )=0$. Then ${\omega }_b(\eta ,\Lambda )=0\iff \eta \in$ Λ, where ${\omega }_b(\eta ,\Lambda )=inf\left\{{\omega }_b(\eta ,\vartheta ):\vartheta \in \Lambda \right\}$.

Lemma 5 

([30]). Let 𝘍 be a nonempty set and $\kappa :𝘍\to$ 𝘍 be a mapping. Then, there exists a subset $E\subseteq 𝘍$ such that $\kappa \left(E\right)=\kappa \left(𝘍\right)$ and $\kappa :E\to 𝘍$ is one-to-one.

Let us recall a few useful notions for multivalued mapping with respect to $\omega$-distance in the setting of MSs; see [27,32,36].

Let $G:𝘍\to 2^{𝘍}$ be a multivalued mapping, $\kappa :𝘍\to 𝘍$ be a single-valued mapping and $\phi :𝘍\times 𝘍\to$ $[0,\infty )$, then

(1)

The multivalued mapping G from $𝘍$ to $\mathcal{CL}\left(𝘍\right)$ is called a generalized $\kappa$-contraction if there exist an ${\omega }^0$-distance $\omega$ on $𝘍$ and $\mu \in (0,1)$ such that, for any $\rho ,\mathfrak{y}\in 𝘍$ and $\eta \in G\left(\rho \right)$, there is $\vartheta \in G\left(\mathfrak{y}\right)$ with

$$\begin{array}{cc}\hfill \omega (\eta ,\vartheta )& \le \mu max\left\{\omega \left(\kappa \right(\rho ),\kappa (\mathfrak{y}\left)\right)\omega \left(\kappa \right(\rho ),G(\rho \left)\right),\omega \left(\kappa \right(\mathfrak{y}),G(\mathfrak{y}\left)\right),\right.\hfill \\ \hfill & {\displaystyle \frac{1}{2}}[\omega (\kappa \left(\rho \right),G\left(\mathfrak{y}\right))+\omega (\kappa \left(\mathfrak{y}\right),G\left(\rho \right))]\}.\hfill \end{array}$$

(1)

(2)

G is called ${\phi }_{\ast }$-admissible if the following is satisfied for all $\rho ,\mathfrak{y}\in 𝘍$

$$\phi (\rho ,\mathfrak{y})\ge 1⟹{\phi }_{\ast }(G\left(\rho \right),G\left(\mathfrak{y}\right))\ge 1,$$

(2)

where ${\phi }_{\ast }(G\left(\rho \right),G\left(\mathfrak{y}\right))=inf\{\phi (\mathfrak{a},\mathfrak{b}):\mathfrak{a}\in G\left(\rho \right),\mathfrak{b}\in$ $G\left(\mathfrak{y}\right)\}.$

(3)

G is called $\phi$-admissible whenever, for each $\rho \in 𝘍$ and $\mathfrak{y}\in G\left(\rho \right)$ with $\phi (\rho ,\mathfrak{y})\ge 1$, one has $\phi (\mathfrak{y},\mathfrak{z})\ge 1$ for all $\mathfrak{z}\in G\left(\mathfrak{y}\right).$

Remark 2. 

The notion of φ and ${\phi }_{\ast }$-admissibility were first defined in [27,36] under the names α and ${\alpha }^{\ast }$-admissibility. Note that every ${\phi }_{\ast }$-admissible mapping is a φ-admissible mapping.

3. Main Results

In this section we introduce our new type of contraction mappings and present our new FP results and its corollaries.

Definition 1. 

Let $\left(𝘍, {\Omega }_b\right)$ be a MTS and $\phi :𝘍\times 𝘍⟶[0,\infty )$ be a mapping. The multivalued mapping $G:𝘍⟶\mathcal{CL}\left(𝘍\right)$ is said to be a generalized ${\omega }_b^{\phi }$-contraction if there exists a ${\omega }_b^0$-distance ${\omega }_b$ on 𝘍 and $\mu \in \left(0,{\displaystyle \frac{1}{b}}\right)$ such that for every $\rho ,\mathfrak{y}\in 𝘍$ and $\eta \in G\left(\rho \right)$, there is $\vartheta \in G\left(\mathfrak{y}\right)$ with

$$\begin{array}{cc}\hfill \phi \left(\eta ,\vartheta \right){\omega }_b\left(\eta ,\vartheta \right)& \le \mu max\{{\omega }_b\left(\rho ,\mathfrak{y}\right),{\omega }_b\left(\rho ,G\left(\rho \right)\right),{\omega }_b\left(\mathfrak{y},G\left(\mathfrak{y}\right)\right),\hfill \\ \hfill & {\displaystyle \frac{1}{2b}}\left[{\omega }_b\left(\rho ,G\left(\mathfrak{y}\right)\right)+{\omega }_b\left(\mathfrak{y},G\left(\rho \right)\right)\right]\}\hfill \end{array}$$

(3)

Remark 3. 

If $b=1$ in the previous definition, then the mapping G will reduced to a generalized ${\omega }^{\phi }$-contraction.

Now, we demonstrate a FP result for the class of multivalued generalized contraction type mappings, which extends the associated result in ([33], Theorem 16).

Theorem 1. 

Let $\left(𝘍, {\Omega }_b\right)$ be a complete MTS, $\phi :𝘍\times 𝘍⟶[0,\infty )$ be a mapping and $G:𝘍⟶\mathcal{CL}\left(𝘍\right)$ be a generalized ${\omega }_b^{\phi }$-contraction mapping. Assume that the subsequent conditions are met:

(a)

G is φ-admissible;

(b)

there exists ${\rho }_0\in 𝘍$ and ${\rho }_1\in G\left({\rho }_0\right)$ such that $\phi \left({\rho }_0,{\rho }_1\right)\ge 1;$

(c)

for every $\mathfrak{y}\in 𝘍$ with $\mathfrak{y}\notin G\left(\mathfrak{y}\right)$ we have

$$inf\left\{{\omega }_b\left(\rho ,\mathfrak{y}\right)+{\omega }_b\left(\rho ,G\left(\rho \right)\right):\rho \in 𝘍\right\}>0.$$

Then, $\mathcal{F}\left(G\right)\ne \varnothing .$

Proof. 

As G is a generalized ${\omega }_b^{\phi }$-contraction mapping and ${\rho }_1\in G\left({\rho }_0\right)$, so there exists ${\rho }_2\in G\left({\rho }_1\right)$ such that

$$\begin{array}{cc}\hfill \phi \left({\rho }_1,{\rho }_2\right){\omega }_b\left({\rho }_1,{\rho }_2\right)& \le \mu max\{{\omega }_b\left({\rho }_0,{\rho }_1\right),{\omega }_b\left({\rho }_0,G\left({\rho }_0\right)\right),{\omega }_b\left({\rho }_1,G\left({\rho }_1\right)\right),\hfill \\ \hfill & {\displaystyle \frac{1}{2b}}\left[{\omega }_b\left({\rho }_0,G\left({\rho }_1\right)\right)+{\omega }_b\left({\rho }_1,G\left({\rho }_0\right)\right)\right]\}.\hfill \end{array}$$

(4)

Now, as $\phi \left({\rho }_0,{\rho }_1\right)\ge 1$, so from (4) we have

$$\begin{array}{cc}\hfill {\omega }_b\left({\rho }_1,{\rho }_2\right)& \le \phi \left({\rho }_1,{\rho }_2\right){\omega }_b\left({\rho }_1,{\rho }_2\right)\hfill \\ \hfill & \le \mu max\{{\omega }_b\left({\rho }_0,{\rho }_1\right),{\omega }_b\left({\rho }_0,G\left({\rho }_0\right)\right),{\omega }_b\left({\rho }_1,G\left({\rho }_1\right)\right),\hfill \\ \hfill & {\displaystyle \frac{1}{2b}}\left[{\omega }_b\left({\rho }_0,G\left({\rho }_1\right)\right)+{\omega }_b\left({\rho }_1,G\left({\rho }_0\right)\right)\right]\}.\hfill \end{array}$$

In a comparable manner, by applying the definition of generalized ${\omega }_b^{\phi }$-contraction mapping, there exists ${\rho }_3\in G\left({\rho }_2\right)$ such that

$$\begin{array}{cc}\hfill \phi \left({\rho }_2,{\rho }_3\right){\omega }_b\left({\rho }_2,{\rho }_3\right)& \le \mu max\{{\omega }_b\left({\rho }_1,{\rho }_2\right),{\omega }_b\left({\rho }_1,G\left({\rho }_1\right)\right),{\omega }_b\left({\rho }_2,G\left({\rho }_2\right)\right),\hfill \\ \hfill & {\displaystyle \frac{1}{2b}}\left[{\omega }_b\left({\rho }_1,G\left({\rho }_2\right)\right)+{\omega }_b\left({\rho }_2,G\left({\rho }_1\right)\right)\right]\}.\hfill \end{array}$$

(5)

Using the $\phi$-admissibility of G, we have $\phi \left({\rho }_2,{\rho }_3\right)\ge 1$. Hence, from (5) we have

$$\begin{array}{cc}\hfill {\omega }_b\left({\rho }_2,{\rho }_3\right)& \le \phi \left({\rho }_2,{\rho }_3\right){\omega }_b\left({\rho }_2,{\rho }_3\right)\hfill \\ \hfill & \le \mu max\{{\omega }_b\left({\rho }_1,{\rho }_2\right),{\omega }_b\left({\rho }_1,G\left({\rho }_1\right)\right),{\omega }_b\left({\rho }_2,G\left({\rho }_2\right)\right),\hfill \\ \hfill & {\displaystyle \frac{1}{2b}}\left[{\omega }_b\left({\rho }_1,G\left({\rho }_2\right)\right)+{\omega }_b\left({\rho }_2,G\left({\rho }_1\right)\right)\right]\}.\hfill \end{array}$$

Continuing the same manner, we can construct a sequence $\left\{{\rho }_n\right\}$ in $𝘍$ such that ${\rho }_n\in G\left({\rho }_{n-1}\right)$, $\phi \left({\rho }_n,{\rho }_{n+1}\right)\ge 1$,

$$\begin{array}{cc}\hfill {\omega }_b\left({\rho }_n,{\rho }_{n+1}\right)& \le \phi \left({\rho }_n,{\rho }_{n+1}\right){\omega }_b\left({\rho }_n,{\rho }_{n+1}\right)\hfill \\ \hfill & \le \mu max\{{\omega }_b\left({\rho }_{n-1},{\rho }_n\right),{\omega }_b\left({\rho }_{n-1},G\left({\rho }_{n-1}\right)\right),{\omega }_b\left({\rho }_n,G\left({\rho }_n\right)\right),\hfill \\ \hfill & {\displaystyle \frac{1}{2b}}\left[{\omega }_b\left({\rho }_{n-1},G\left({\rho }_n\right)\right)+{\omega }_b\left({\rho }_n,G\left({\rho }_{n-1}\right)\right)\right]\}.\hfill \end{array}$$

Since ${\omega }_b$ is a ${\omega }_b^0$-distance, we deduce that

$${\omega }_b\left({\rho }_n,{\rho }_{n+1}\right)\le \mu max\left\{{\omega }_b\left({\rho }_{n-1},{\rho }_n\right),{\omega }_b\left({\rho }_n,{\rho }_{n+1}\right),{\displaystyle \frac{1}{2b}}{\omega }_b\left({\rho }_{n-1},{\rho }_{n+1}\right)\right\}$$

(6)

By using the b-lower semi-continuity of ${\omega }_b$, we obtain

$${\omega }_b\left({\rho }_n,{\rho }_{n+1}\right)\le \mu max\left\{{\omega }_b\left({\rho }_{n-1},{\rho }_n\right),{\omega }_b\left({\rho }_n,{\rho }_{n+1}\right),{\displaystyle \frac{1}{2}}\left[{\omega }_b\left({\rho }_{n-1},{\rho }_n\right)+{\omega }_b\left({\rho }_n,{\rho }_{n+1}\right)\right]\right\}$$

Now, if there exists $n_0$ such that ${\omega }_b\left({\rho }_{n_0-1},{\rho }_{n_0}\right)\le {\omega }_b\left({\rho }_{n_0},{\rho }_{n_0+1}\right)$, then

$$\begin{array}{cc}\hfill {\omega }_b\left({\rho }_{n_0},{\rho }_{n_0+1}\right)& \le \mu max\left\{{\omega }_b\left({\rho }_{n_0-1},{\rho }_{n_0}\right),{\omega }_b\left({\rho }_{n_0},{\rho }_{n_0+1}\right)\right\}\hfill \\ \hfill & =\mu {\omega }_b\left({\rho }_{n_0},{\rho }_{n_0+1}\right).\hfill \end{array}$$

As $\mu \in \left(0,{\displaystyle \frac{1}{b}}\right)$, then ${\omega }_b\left({\rho }_{n_0},{\rho }_{n_0+1}\right)=0.$ Therefore, ${\omega }_b\left({\rho }_{n_0-1},{\rho }_{n_0+1}\right)\le b[{\omega }_b\left({\rho }_{n_{0-1}},{\rho }_{n_0}\right)+{\omega }_b\left({\rho }_{n_0},{\rho }_{n_0+1}\right)]$. Hence, by Lemma 3, ${\rho }_{n_0}={\rho }_{n_0+1}$, and then ${\rho }_{n_0}\in G\left({\rho }_{n_0}\right)$. Therefore, ${\rho }_{n_0}$ is a FP of G. On the other hand, if ${\omega }_b\left({\rho }_{n_0},{\rho }_{n_0+1}\right)\le {\omega }_b\left({\rho }_{n_0-1},{\rho }_{n_0}\right)$, then

$$\begin{array}{cc}\hfill {\omega }_b\left({\rho }_n,{\rho }_{n+1}\right)& \le \mu max\left\{{\omega }_b\left({\rho }_{n-1},{\rho }_n\right),{\omega }_b\left({\rho }_n,{\rho }_{n+1}\right),{\displaystyle \frac{1}{2}}\left[{\omega }_b\left({\rho }_{n-1},{\rho }_n\right)+{\omega }_b\left({\rho }_n,{\rho }_{n+1}\right)\right]\right\}\hfill \\ \hfill & \le \mu {\omega }_b\left({\rho }_{n-1},{\rho }_n\right).\hfill \end{array}$$

Hence,

$${\omega }_b\left({\rho }_n,{\rho }_{n+1}\right)\le \mu {\omega }_b\left({\rho }_{n-1},{\rho }_n\right)\mathrm{for}\mathrm{all}n\in \mathbb{N}.$$

(7)

Repeating Equation (7) we obtain

$${\omega }_b\left({\rho }_n,{\rho }_{n+1}\right)\le {\mu }^n{\omega }_b\left({\rho }_0,{\rho }_1\right)\mathrm{for}\mathrm{all}n\in \mathbb{N}.$$

(8)

Now we prove that $\left\{{\rho }_n\right\}$ is a Cauchy sequence. For any $m,n\in \mathbb{N}$ with $m>n$ we get

$$\begin{array}{cc}\hfill {\omega }_b\left({\rho }_n,{\rho }_m\right)& \le b{\omega }_b\left({\rho }_n,{\rho }_{n+1}\right)+b^2{\omega }_b\left({\rho }_{n+1},{\rho }_{n+2}\right)+\dots b^{m-n-1}{\omega }_b\left({\rho }_{m-2},{\rho }_{m-1}\right)+b^{m-n}{\omega }_b\left({\rho }_{m-1},{\rho }_m\right)\hfill \end{array}$$

Using (8) we obtain

$${\omega }_b\left({\rho }_n,{\rho }_m\right)\le b{\mu }^n{\omega }_b\left({\rho }_0,{\rho }_1\right)+b^2{\mu }^{n+1}{\omega }_b\left({\rho }_0,{\rho }_1\right)+...b^{m-n}{\mu }^{m-1}{\omega }_b\left({\rho }_0,{\rho }_1\right).$$

As $b\mu <1$, the geometric series that appeared in the right hand side of the above inequality is convergent, so using its sum we acquire for all $m,n\in \mathbb{N}$ with $m>n$,

$${\omega }_b\left({\rho }_n,{\rho }_m\right)\le {\displaystyle \frac{b{\mu }^n}{1-b\mu }}{\omega }_b\left({\rho }_0,{\rho }_1\right)$$

(9)

Since ${\displaystyle \frac{b{\mu }^n}{1-b\mu }}$ goes zero as $n\to +\infty$, we derive from Lemma 3 that $\left\{{\rho }_n\right\}$ is a Cauchy sequence. The completeness of $𝘍$ guarantees the existence of $\mathfrak{z}\in 𝘍$ such that $\underset{n\to \infty }{lim}{\rho }_n=\mathfrak{z}.$ By LSC of ${\omega }_b\left({\rho }_n,·\right)$, we have

$${\omega }_b\left({\rho }_n,\mathfrak{z}\right)\le \underset{m\to \infty }{lim\; inf}\left({\rho }_n,{\rho }_m\right)\le {\displaystyle \frac{b{\mu }^n}{1-b\mu }}{\omega }_b\left({\rho }_0,{\rho }_1\right).$$

(10)

If we infer that $\mathfrak{z}\notin G\left(\mathfrak{z}\right)$, then by condition (c) and (10), we have

$$\begin{array}{cc}\hfill 0& <inf\left\{{\omega }_b\left(\rho ,\mathfrak{z}\right)+{\omega }_b\left(\rho ,G\left(\rho \right)\right):\rho \in 𝘍\right\}\hfill \\ \hfill & \le inf\left\{{\omega }_b\left({\rho }_n,\mathfrak{z}\right)+{\omega }_b\left({\rho }_n,G\left({\rho }_n\right)\right):n\in \mathbb{N}\right\}\hfill \\ \hfill & \le inf\left\{{\omega }_b\left({\rho }_n,\mathfrak{z}\right)+{\omega }_b\left({\rho }_n,{\rho }_{n+1}\right):n\in \mathbb{N}\right\}\hfill \\ \hfill & \le inf\left\{{\displaystyle \frac{b{\mu }^n}{1-b\mu }}{\omega }_b\left({\rho }_0,{\rho }_1\right)+b{\mu }^n{\omega }_b\left({\rho }_0,{\rho }_1\right):n\in \mathbb{N}\right\}\hfill \\ \hfill & =\left(\left\{{\displaystyle \frac{\left(2-\mu \right)b}{1-b\mu }}\right\}{\omega }_b\left({\rho }_0,{\rho }_1\right)\right)inf\left\{{\mu }^n:n\in \mathbb{N}\right\}=0\hfill \end{array}$$

which contradicts condition (c). Therefore, $\mathfrak{z}\in G\left(\mathfrak{z}\right)$, which implies that $\mathfrak{z}$ is a FP of G. □

From Theorem 1, we realize the next corollary.

Corollary 1. 

Let $\left(𝘍, {\Omega }_b\right)$ be a complete MTS, and ${\omega }_b$ be a ${\omega }_b^0$-distance on 𝘍. Let $G:𝘍⟶\mathcal{CL}\left(𝘍\right)$ be a multivalued mapping such that the following conditions are fulfilled:

(a)

for each $\rho ,\mathfrak{y}\in 𝘍$ and $\eta \in G\left(\rho \right)$, there exist $\vartheta \in G\left(\mathfrak{y}\right)$ and $\mu \in \left(0,{\displaystyle \frac{1}{b}}\right)$ such that

$$\begin{array}{cc}\hfill {\omega }_b\left(\eta ,\vartheta \right)& \le \mu max\{{\omega }_b\left(\rho ,\mathfrak{y}\right),{\omega }_b\left(\rho ,G\left(\rho \right)\right),{\omega }_b\left(\mathfrak{y},G\left(\mathfrak{y}\right)\right),\hfill \\ \hfill & {\displaystyle \frac{1}{2b}}\left[{\omega }_b\left(\rho ,G\left(\mathfrak{y}\right)\right)+{\omega }_b\left(\mathfrak{y},G\left(\rho \right)\right)\right]\}\hfill \end{array}$$

(11)

(b)

for every $\mathfrak{y}\in 𝘍$ with $\mathfrak{y}\notin G\left(\mathfrak{y}\right)$, we have

$$inf\left\{{\omega }_b\left(\rho ,\mathfrak{y}\right)+{\omega }_b\left(\rho ,G\left(\rho \right)\right):\rho \in 𝘍\right\}>0.$$

Then, $\mathcal{F}\left(G\right)\ne \varnothing .$

Proof. 

Taking $\phi \left(\rho ,\mathfrak{y}\right)=1$ for each $\rho ,\mathfrak{y}\in 𝘍$ in Theorem 1, the proof is complete. □

Remark 4. 

(1) Theorem 1 extends the FP findings of ([27], Theorem 2.3) and ([36], Theorem 3.4). Further, Corollary 1 contains ([32], Corollary 2.1) and ([33], Corollary 17) as a special case.

(2)

Note that using φ-admissibility in the previous results guarantees the existence of a monotonic sequence to handle the proof of the existence of FPs in the framework of nonlinear mappings.

Now, we define the ${\omega }_b^{\phi }$-contraction mapping and then prove the existence of FPs under some conditions for such mappings.

Definition 2. 

Let $\left(𝘍, {\Omega }_b\right)$ be a MTS and $\phi :𝘍\times 𝘍⟶[0,\infty )$ be a mapping. The multivalued mapping $G:𝘍⟶\mathcal{CL}\left(𝘍\right)$ is said to be a ${\omega }_b^{\phi }$-contraction if there exists a ${\omega }_b$-distance ${\omega }_b$ on 𝘍 and $\mu \in \left(0,{\displaystyle \frac{1}{b}}\right)$ such that for every $\rho ,\mathfrak{y}\in 𝘍$ and $\eta \in G\left(\rho \right)$, there is $\vartheta \in G\left(\mathfrak{y}\right)$ with

$$\phi \left(\eta ,\vartheta \right){\omega }_b\left(\eta ,\vartheta \right)\le \mu \left(\rho ,\mathfrak{y}\right).$$

(12)

Following the same manner in the proof of Theorem 1 we have the following FP result for ${\omega }_b^{\phi }$-contraction mappings.

Theorem 2. 

Let $\left(𝘍, {\Omega }_b\right)$ be a complete MTS, $\phi :𝘍\times 𝘍⟶[0,\infty )$ be a mapping and $G:𝘍⟶\mathcal{CL}\left(𝘍\right)$ be a ${\omega }_b^{\phi }$-contraction mapping. Assume that the following conditions hold:

(a)

G is φ-admissible;

(b)

there exists ${\rho }_0\in 𝘍$ and ${\rho }_1\in G\left({\rho }_0\right)$ such that $\phi \left({\rho }_0,{\rho }_1\right)\ge 1;$

(c)

for every $\mathfrak{y}\in 𝘍$ with $\mathfrak{y}\notin G\left(\mathfrak{y}\right)$ we have

$$inf\left\{{\omega }_b\left(\rho ,\mathfrak{y}\right)+{\omega }_b\left(\rho ,G\left(\rho \right)\right):\rho \in 𝘍\right\}>0.$$

Then, $\mathcal{F}\left(G\right)\ne \varnothing .$

Remark 5. 

Theorem $3.2$ extend the FP result in ([33], Theorem 19).

Let $𝘍$ be a non-empty set and $\kappa :𝘍\to 𝘍, G:$ $𝘍\to \mathcal{CL}\left(𝘍\right)$ such that $G\left(𝘍\right)\subseteq \kappa \left(𝘍\right)$. The multivalued mapping G is called $(\kappa ,\phi )$-admissible if, each $\kappa \left(\rho \right)\in \kappa \left(𝘍\right)$ and $\kappa \left(\mathfrak{y}\right)\in$ $G\left(\kappa \right(\rho \left)\right)$ with $\phi \left(\kappa \right(\rho ),\kappa (\mathfrak{y}\left)\right)\ge 1$ satisfies that $\phi \left(\kappa \right(\mathfrak{y}),\kappa (\mathfrak{z}\left)\right)\ge 1$ for all $\kappa \left(\mathfrak{z}\right)\in G\left(\kappa \right(\mathfrak{y}\left)\right)$, where $\phi$ is a functions from $\kappa \left(𝘍\right)\times \kappa \left(𝘍\right)$ to ${\mathbb{R}}^{+}$. Obviously, if G is $(\kappa ,\phi )$-admissible and $\kappa$ is the identity mapping, then G is $\phi$-admissible. The mappings $\kappa$ and G are said to be commute weakly if $\kappa \left(G\right(\rho \left)\right)\subseteq G\left(\kappa \right(\rho \left)\right)$ for all $\rho \in 𝘍$ [33]. We say a multivalued mapping $G:$ $𝘍\to \mathcal{CL}\left(𝘍\right)$ is a generalized $\left({\omega }_b^{\phi },\kappa \right)$-contraction if there exist a ${\omega }_b^0$-distance ${\omega }_b$ on $𝘍$ and $\mu \in (0,{\displaystyle \frac{1}{b}})$ such that, for any $\rho ,\mathfrak{y}\in 𝘍$ and $\eta \in G\left(\rho \right)$, there is $\vartheta \in G\left(\mathfrak{y}\right)$ with

$$\begin{array}{cc}\hfill \phi (\kappa \left(\rho \right),\kappa \left(\mathfrak{y}\right)){\omega }_b(\eta ,\vartheta )& \le \mu max\{{\omega }_b(\kappa \left(\rho \right),\kappa \left(\mathfrak{y}\right)),{\omega }_b(\kappa \left(\rho \right),G\left(\rho \right)),{\omega }_b(\kappa \left(\mathfrak{y}\right),G\left(\mathfrak{y}\right)),\hfill \\ \hfill & {\displaystyle \frac{1}{2b}}[{\omega }_b(\kappa \left(\rho \right),G\left(\mathfrak{y}\right))+{\omega }_b(\kappa \left(\mathfrak{y}\right),G\left(\rho \right))]\}.\hfill \end{array}$$

Note that if G is a generalized $\left({\omega }_b^{\phi },\kappa \right)$-contraction and $\kappa$ is the identity mapping, then G is generalized ${\omega }_b^{\phi }$-contraction.

Now, we present a coincidence point result for generalized $\left({\omega }_b^{\phi },\kappa \right)$-contraction mappings of MTSs.

Theorem 3. 

Let $(𝘍, {\Omega }_b)$ be a complete MTS, $\kappa :𝘍\to$ $𝘍, \phi :\kappa \left(𝘍\right)\times \kappa \left(𝘍\right)\to {\mathbb{R}}^{+}$, and $G:𝘍\to \mathcal{CL}\left(𝘍\right)$ be a generalized $\left({\omega }_b^{\phi },\kappa \right)$-contraction. Suppose that the following conditions hold:

(a)

$G\left(𝘍\right)\subseteq \kappa \left(𝘍\right)$;

(b)

G is $(\kappa ,\phi )$-admissible;

(c)

there exist ${\rho }_0,{\rho }_1\in 𝘍$ such that $\kappa \left({\rho }_1\right)\in G\left({\rho }_0\right)$ and $\phi \left(\kappa \left({\rho }_0\right),\kappa \left({\rho }_1\right)\right)\ge 1$;

(d)

for all $\mathfrak{y}\in 𝘍$ with $\kappa \left(\mathfrak{y}\right)\notin G\left(\mathfrak{y}\right)$, one has

$$\begin{array}{c}\hfill inf\{{\omega }_b(\kappa \left(\rho \right),\kappa \left(\mathfrak{y}\right))+{\omega }_b(\kappa \left(\rho \right),G\left(\rho \right)):\rho \in 𝘍\}>0.\end{array}$$

(13)

Then, $C(\kappa \cap G)\ne \varnothing$

Proof. 

Using Lemma 5, there exists $E\subseteq 𝘍$ such that $\kappa \left(E\right)=\kappa \left(𝘍\right)$ and ${\left.\kappa \right|}_E$ is one-to-one. Now, we can define a mapping $J:\kappa \left(E\right)\to \mathcal{CL}\left(𝘍\right)$ by

$$J\left(\kappa \right(\rho \left)\right)=G\left(\rho \right),$$

(14)

for all $\rho \in E$. Since ${\left.\kappa \right|}_E$ is one-to-one, then J is well defined. Since G is a generalized $\left({\omega }_b^{\phi },\kappa \right)$-contraction, there exist a ${\omega }_b^0$-distance ${\omega }_b$ on $𝘍$ and $\mu \in (0,{\displaystyle \frac{1}{b}})$ such that, for any $\rho ,\mathfrak{y}\in 𝘍$ and $\eta \in G\left(\rho \right)$, there is $\vartheta \in G\left(\mathfrak{y}\right)$ with

$$\begin{array}{cc}\hfill \phi (\kappa \left(\rho \right),\kappa \left(\mathfrak{y}\right)){\omega }_b(\eta ,\vartheta )& \le \mu max\{{\omega }_b(\kappa \left(\rho \right),\kappa \left(\mathfrak{y}\right)),{\omega }_b(\kappa \left(\rho \right),G\left(\rho \right)),{\omega }_b(\kappa \left(\mathfrak{y}\right),G\left(\mathfrak{y}\right)),\hfill \\ \hfill & {\displaystyle \frac{1}{2b}}[{\omega }_b(\kappa \left(\rho \right),G\left(\mathfrak{y}\right))+{\omega }_b(\kappa \left(\mathfrak{y}\right),G\left(\rho \right))]\}.\hfill \end{array}$$

(15)

By the construction of J, for any $\kappa \left(\rho \right),\kappa \left(\mathfrak{y}\right)\in \kappa \left(E\right)$, and $\eta \in J\left(\kappa \right(\rho \left)\right)$, there is $\vartheta \in J\left(\kappa \right(\mathfrak{y}\left)\right)$ such that

$$\begin{array}{cc}\hfill \phi (\kappa \left(\rho \right),\kappa \left(\mathfrak{y}\right)){\omega }_b(\eta ,\vartheta )& \le \mu max\{{\omega }_b(\kappa \left(\rho \right),\kappa \left(\mathfrak{y}\right)),{\omega }_b(\kappa \left(\rho \right),J\left(\kappa \left(\rho \right)\right),{\omega }_b(\kappa \left(\mathfrak{y}\right),J\left(\kappa \left(\mathfrak{y}\right)\right),\hfill \\ \hfill & {\displaystyle \frac{1}{2b}}[{\omega }_b(\kappa \left(\rho \right),J\left(\kappa \left(\mathfrak{y}\right)\right)+{\omega }_b(\kappa \left(\mathfrak{y}\right),J\left(\kappa \left(\rho \right)\right)]\}.\hfill \end{array}$$

(16)

This implies that J is a generalized ${\omega }_b^{\phi }$-contraction. Since G is $(\kappa ,\phi )$-admissible, we have J as $\phi$-admissible. It is obtained that condition (c) implies condition (b) in Theorem 1. From (d), for all $\kappa \left(\mathfrak{y}\right)\in \kappa \left(E\right)$ with $\kappa \left(\mathfrak{y}\right)\notin J\left(\kappa \right(\mathfrak{y}\left)\right)$, we have

$$inf\{{\omega }_b(\kappa \left(\rho \right),\kappa \left(\mathfrak{y}\right))+{\omega }_b(\kappa \left(\rho \right),J\left(\kappa \left(\rho \right)\right)):\rho \in 𝘍\}>0.$$

(17)

Using Theorem 1 with the mapping J, we can find a FP of the mapping J. Let $\mathfrak{z}$ be a FP of J; that is, $\mathfrak{z}\in J\left(\mathfrak{z}\right)$. Since $\mathfrak{z}\in$ $\kappa \left(E\right)$, we can find $\widehat{\mathfrak{z}}\in E$ such that $\mathfrak{z}=\kappa \left(\widehat{\mathfrak{z}}\right)$. Now, we have

$$\kappa \left(\widehat{\mathfrak{z}}\right)=\mathfrak{z}\in J\left(\mathfrak{z}\right)=J\left(\kappa \left(\widehat{\mathfrak{z}}\right)\right)=G\left(\widehat{\mathfrak{z}}\right).$$

(18)

Therefore, $\widehat{\mathfrak{z}}$ is a coincident point of $\kappa$ and G; that is, $C(\kappa \cap G)\ne \varnothing$. This completes the proof. □

Finally, we demonstrate a CFP result for generalized $\left({\omega }_b^{\phi },\kappa \right)$-contraction mappings.

Theorem 4. 

Suppose that all the hypotheses of Theorem 3 hold. Further, if κ and G commute weakly and satisfy the following condition for $\rho \in 𝘍$:

$$\kappa \left(\rho \right)\ne {\kappa }^2\left(\rho \right)⟹\kappa \left(\rho \right)\notin G\left(\rho \right).$$

(19)

Then, $\mathcal{F}(\kappa \cap G)\ne \varnothing$.

Proof. 

From Theorem 3, $\kappa$ and G have a coincidence point $\widehat{\mathfrak{z}}\in 𝘍$; that is, $\kappa \left(\widehat{\mathfrak{z}}\right)\in G\left(\widehat{\mathfrak{z}}\right)$. By the hypothesis, we get $\kappa \left(\widehat{\mathfrak{z}}\right)=$ ${\kappa }^2\left(\widehat{\mathfrak{z}}\right)$. It follows from $\kappa$ and G which commute weakly that

$$\kappa \left(\widehat{\mathfrak{z}}\right)=\kappa \left(\kappa \left(\widehat{\mathfrak{z}}\right)\right)\in \kappa \left(G\left(\widehat{\mathfrak{z}}\right)\right)\subseteq G\left(\kappa \left(\widehat{\mathfrak{z}}\right)\right)$$

This implies that $\kappa \left(\widehat{\mathfrak{z}}\right)$ is a CFP of $\kappa$ and G, and thus, $\mathcal{F}(\kappa \cap G)\ne \varnothing$. □

Remark 6. 

If we set $\phi (a,b)=1$ for all $a,b\in \kappa \left(\rho \right)$ in Theorems 3 and 4, then we get the FP results of Theorems 2.1 and 2.2 [32]. Further, Theorems 3 and 4 generalize the corresponding FP results in Theorems 25 and 27 [33].

Now we introduce some examples in defence of our main findings.

Example 3. 

Let $𝘍=[-1,\infty )$. For each $\rho ,\mathfrak{y}$ of 𝘍, we define ${\Omega }_b(\rho ,\mathfrak{y})={(\rho -\mathfrak{y})}^2$ and

$${\omega }_b(\rho ,\mathfrak{y})=\left\{\begin{array}{cc}{\mathfrak{y}}^2,\hfill & \mathrm{if}\rho \ne \mathfrak{y},\hfill \\ 0,\hfill & \mathrm{if}\rho =\mathfrak{y}.\hfill \end{array}\right.$$

Note that $(𝘍, {\Omega }_b)$ becomes a MTS with $b=2$ and ${\omega }_b$ is a ${\omega }_b^0$-distance on 𝘍. Define a multivalued mapping $G:𝘍\to \mathcal{CL}\left(𝘍\right)$ by

$$G\left(\rho \right)=\left\{\begin{array}{cc}\left\{{\displaystyle \frac{\rho }{6}}\right\},\hfill & \rho \in [0,1],\hfill \\ \{\rho ,5|\rho \left|\right\},\hfill & otherwise.\hfill \end{array}\right.$$

Define a function $\phi :𝘍\times 𝘍\to {\mathbb{R}}^{+}$ by

$$\phi (\rho ,\mathfrak{y})=\left\{\begin{array}{cc}{\rho }^2+{\mathfrak{y}}^2+1,\hfill & \rho ,\mathfrak{y}\in [0,1],\hfill \\ 0,\hfill & otherwise.\hfill \end{array}\right.$$

Now, we show that G is a generalized ${\omega }_b^{\phi }$-contraction multivalued mapping with $\mu ={\displaystyle \frac{1}{12}}$. For $\rho ,\mathfrak{y}\in [0,1]$ and $\rho \ne \mathfrak{y}$, let $\eta \in G\left(\rho \right)=\left\{{\displaystyle \frac{\rho }{6}}\right\}$, that is, $\eta ={\displaystyle \frac{\rho }{6}}$, we can find $\vartheta ={\displaystyle \frac{\mathfrak{y}}{6}}\in G\left(\mathfrak{y}\right)$ such that

$$\begin{array}{cc}\hfill \phi (\eta ,\vartheta ){\omega }_b(\eta ,\vartheta )& =\phi \left({\displaystyle \frac{\rho }{6}},{\displaystyle \frac{\mathfrak{y}}{6}}\right){\omega }_b\left({\displaystyle \frac{\rho }{6}},{\displaystyle \frac{\mathfrak{y}}{6}}\right)\hfill \\ & =\left({\displaystyle \frac{{\rho }^2}{36}}+{\displaystyle \frac{{\mathfrak{y}}^2}{36}}+1\right){\left({\displaystyle \frac{\mathfrak{y}}{6}}\right)}^2\hfill \\ & \le (1+1+1){\left({\displaystyle \frac{\mathfrak{y}}{6}}\right)}^2\hfill \\ & =\mu {\omega }_b(\rho ,\mathfrak{y})\hfill \\ & \le \mu max\{{\omega }_b(\rho ,\mathfrak{y}),{\omega }_b(\rho ,G\left(\rho \right)),{\omega }_b(\mathfrak{y},G\left(\mathfrak{y}\right)),\hfill \\ & {\displaystyle \frac{1}{2b}}[{\omega }_b(\rho ,G\left(\mathfrak{y}\right))+{\omega }_b(\mathfrak{y},G\left(\rho \right))]\}.\hfill \end{array}$$

Now if $\rho ,\mathfrak{y}\in [0,1]$ and $\rho =\mathfrak{y}$, then

$$\begin{array}{cc}\hfill \phi (\eta ,\vartheta ){\omega }_b(\eta ,\vartheta )& =\phi \left({\displaystyle \frac{\rho }{6}},{\displaystyle \frac{\mathfrak{y}}{6}}\right){\omega }_b\left({\displaystyle \frac{\rho }{6}},{\displaystyle \frac{\mathfrak{y}}{6}}\right)\hfill \\ & =0\hfill \\ & =\mu {\omega }_b(\rho ,\mathfrak{y})\hfill \\ & \le \mu max\{{\omega }_b(\rho ,\mathfrak{y}),{\omega }_b(\rho ,G\left(\rho \right)),{\omega }_b(\mathfrak{y},G\left(\mathfrak{y}\right)),\hfill \\ & {\displaystyle \frac{1}{2b}}[{\omega }_b(\rho ,G\left(\mathfrak{y}\right))+{\omega }_b(\mathfrak{y},G\left(\rho \right))]\}.\hfill \end{array}$$

Otherwise, obviously, the condition of generalized ${\omega }_b^{\phi }$-contractive holds. Therefore, G is a generalized ${\omega }_b^{\phi }$-contraction multivalued mapping.

Clearly, $(𝘍, {\Omega }_b)$ is a complete MTS also, it is apparent that G is φ-admissible; also, we can find ${\rho }_0=1$ such that ${\rho }_1=1/6\in G\left(1\right)$ and $\phi \left({\rho }_0,{\rho }_1\right)=\phi (1,1/6)\ge 1$. Ultimately, we observed that for $\mathfrak{y}\in 𝘍$ with $\mathfrak{y}\notin G\left(\mathfrak{y}\right)$, we acquire $\mathfrak{y}\in (0,1]$, and hence, $inf\{{\omega }_b(\rho ,\mathfrak{y})+{\omega }_b(\rho ,G\left(\rho \right)):$ $\rho \in 𝘍\}>0$.

Therefore, all the conditions of Theorem 1 are fulfilled, and so G has a FP.

Example 4. 

Let $𝘍=[0,\infty )$ with ${\Omega }_b$ as in Example 3. For any $\rho ,\mathfrak{y}$ of 𝘍, define ${\omega }_b(\rho ,\mathfrak{y})={\mathfrak{y}}^2$. Let $G:𝘍\to \mathcal{CL}\left(𝘍\right)$ be defined by

$$G\left(\rho \right)=\left\{\begin{array}{cc}\left\{{\displaystyle \frac{\rho }{3}}\right\},\hfill & \rho \in [0,1],\hfill \\ \{2x-{\displaystyle \frac{5}{3}}\},\hfill & \rho >1,\hfill \end{array}\right.$$

and define a function $\phi :𝘍\times 𝘍\to {\mathbb{R}}^{+}$ by

$$\phi (\rho ,\mathfrak{y})=\left\{\begin{array}{cc}\rho +\mathfrak{y}+1,\hfill & \rho ,\mathfrak{y}\in [0,1],\hfill \\ 0,\hfill & otherwise.\hfill \end{array}\right.$$

Now, we show that G is ${\omega }_b^{\phi }$-contraction multivalued mapping with $\mu ={\displaystyle \frac{1}{3}}$. For $\rho ,\mathfrak{y}\in [0,1]$, let $\eta \in G\left(\rho \right)=\left\{{\displaystyle \frac{\rho }{3}}\right\}$, that is, $\eta ={\displaystyle \frac{\rho }{3}}$, we can find $\vartheta ={\displaystyle \frac{\mathfrak{y}}{3}}\in G\left(\mathfrak{y}\right)$ such that

$$\begin{array}{cc}\hfill \phi (\eta ,\vartheta ){\omega }_b(\eta ,\vartheta )& =\phi \left({\displaystyle \frac{\rho }{3}},{\displaystyle \frac{\mathfrak{y}}{3}}\right){\omega }_b\left({\displaystyle \frac{\rho }{3}},{\displaystyle \frac{\mathfrak{y}}{3}}\right)\hfill \\ & =\left({\displaystyle \frac{\rho }{3}}+{\displaystyle \frac{\mathfrak{y}}{3}}+1\right){\left({\displaystyle \frac{\mathfrak{y}}{3}}\right)}^2\hfill \\ & \le (1+1+1){\left({\displaystyle \frac{\mathfrak{y}}{3}}\right)}^2\hfill \\ & ={\displaystyle \frac{1}{3}}{\mathfrak{y}}^2\hfill \\ & =\mu {\omega }_b(\rho ,\mathfrak{y}).\hfill \end{array}$$

Otherwise, obviously, the condition of ${\omega }_b^{\phi }$-contractive holds. Therefore, G is a ${\omega }_b^{\phi }$-contraction multivalued mapping.

Clearly, $(𝘍, {\Omega }_b)$ is a complete MTS. Also, it is apparent that G is φ-admissible, and we can find ${\rho }_0=1$ such that ${\rho }_1=1/3\in G\left(1\right)$ and $\phi \left({\rho }_0,{\rho }_1\right)=\phi (1,1/3)\ge 1$. Ultimately, we observed that for $\mathfrak{y}\in 𝘍$ with $\mathfrak{y}\notin G\left(\mathfrak{y}\right)$, we acquire $\mathfrak{y}\in (0,1]$, and hence, $inf\{{\omega }_b(\rho ,\mathfrak{y})+{\omega }_b(\rho ,G\left(\rho \right)):$ $\rho \in 𝘍\}>0$.

Therefore, all the conditions of Theorem 2 are fulfilled, so G has a FP.

4. Conclusions

Motivated by the work of Kutbi and Sintunavarat [33], we defined a new type of contraction, so-called generalized ${\omega }_b^{\phi }$-contraction, using $\phi$-admissibility in the framework of ${\omega }_b$-distance. After proving the existence of fixed points of such mappings, we introduced some examples to demonstrate the importance of our new contraction and the applicability of our main results. Thus, this work contributes to the development of FP theory and expands the range of possible applications. The results will unify and generalize many known useful results in fixed point theory. These results will also open a new research area in many fields of nonlinear analysis. Our new results will also help to promote mathematical research concerning the area of nonlinear analysis and operator theory.