New multivalued F-contraction mappings involving α-admissibility with an application

Abstract

In this article, we obtain some fixed-point results involving α-admissibility for multivalued F-contractions in the framework of partial $\mathfrak{b}$-metric spaces. Appropriate illustrations are provided to support the main results. Finally, an application is developed by demonstrating the existence of a solution to an integral equation.

1 Introduction and preliminaries

In 1922, Banach [6] proposed the well-known Banach contraction principle (BCP), which employed a contraction mapping in the domain of complete metric spaces. Later, it was regarded as an effective approach for locating unique fixed points. According to the BCP, in a complete metric space $(\mathcal{M}, d^{*})$, a mapping $f : \mathcal{M} \to \mathcal{M}$ satisfying the contraction condition on $\mathcal{M}$, i.e.,

$$ d^{*}(f\zeta , f\beta ) \leq c d^{*}(\zeta , \beta ), $$

for all $\zeta ,\beta \in \mathcal{M,}$ provided $c \in [0,1)$, has a unique fixed point.

The BCP was generalized using varieties of mappings on several extensions of metric spaces. In 1969, Nadler [7] generalized the BCP for multivalued mappings. In order to optimize a variety of approximation theory problems, it is much more advantageous to use proper fixed-point results for multivalued transformations. The notion of F-contractions was introduced by Wardowski [15]. Altun et al. [2] focused on the existence of the fixed point for multivalued F-contractions and proved certain fixed-point theorems on the setting of metric spaces. Many extensions and generalizations of BCP were produced and the existence and uniqueness of fixed-point were proved. Ali et al. [1] introduced the notion of α-F-admissible type mappings in the setting of uniform spaces. One can see many interesting results on α-F mappings in [3–5, 18].

In 2014, Shukla [12] gave a new direction for extending the metric space. He blended the principles of a partial metric space [9] and a $\mathfrak{b}$-metric space [10, 11] together and proposed a new notion of a partial $\mathfrak{b}$-metric space to present a fine interpretation of BCP in such a space. Kumar et al. [8] extended these results to partial metric spaces and proved fixed point results for multivalued F-contraction mappings. Kumar et al. [8] presented an article in April 2021, using multivalued F-mappings in partial metric spaces. A sound generalization of BCP under this new direction was given. One can see more work in the papers [16, 17, 19] and the references therein. Motivated by his work, an idea of extending the BCP in the globe of a partial $\mathfrak{b}$-metric space by integrating the notion of α-admissibility introduced by Samet et al. [13] under multivalued F-contractions, is presented.

Take $\mathbb{R}^{+}=[0,\infty )$ and denote by $\mathbb{N}$ the set of positive integers. Throughout the article, the compact subset of the underlying space $\mathcal{M}$ will be denoted by $K(\mathcal{M})$. Let us now look at some essential concepts and consequences that will set a foundation for our main result.

Definition 1.1

[12] Let $\mathcal{M} \neq \phi$ and $\mathfrak{b}\geq 1$ be any real number. A map $p_{\mathfrak{b}}: \mathcal{M} \times \mathcal{M} \to \mathbb{R}^{+}$ satisfying the following properties on $\mathcal{M}$ is called a partial $\mathfrak{b}$ metric on $\mathcal{M}$:

$\acute{p}_{b}(1)$::

$p_{\mathfrak{b}}(m_{1},m_{2})= p_{\mathfrak{b}}(m_{1},m_{1})=p_{ \mathfrak{b}}(m_{2},m_{2})$ if and only if $m_{1}=m_{2}$;

$\acute{p}_{b}(2)$::

$p_{\mathfrak{b}}(m_{1},m_{2})\geq p_{\mathfrak{b}}(m_{1},m_{1})$;

$\acute{p}_{b}(3)$::

$p_{\mathfrak{b}}(m_{1},m_{2})=p_{\mathfrak{b}}(m_{2},m_{1})$;

$\acute{p}_{b}(4)$::

$p_{\mathfrak{b}}(m_{1},m_{2})\leq \mathfrak{b}\{p_{\mathfrak{b}}(m_{1},m_{3})+p_{ \mathfrak{b}}(m_{3},m_{2})\}-p_{\mathfrak{b}}(m_{3},m_{3})$,for all $m_{1},m_{2},m_{3} \in \mathcal{M}$.

The pair $(\mathcal{M},p_{\mathfrak{b}})$ is said to be a partial $\mathfrak{b}$-metric space (P$\mathfrak{b}$MS).

Example 1.2

Let $\mathcal{M}=\mathbb{R}^{+}$. We define $p_{\mathfrak{b}}: \mathcal{M} \times \mathcal{M} \to \mathcal{M}$ by

$$ p_{\mathfrak{b}}(m_{1}, m_{2}) = \vert m_{1}-m_{2} \vert ^{q}+\bigl[\max \{m_{1}, m_{2}\}\bigr]^{q}, \quad \text{for all } m_{1}, m_{2} \in \mathcal{M}. $$

Let $q>1$ be any constant, then $(\mathcal{M},p_{\mathfrak{b}})$ is a P$\mathfrak{b}$MS with $\mathfrak{b}= 2^{q-1}$.

Definition 1.3

Let $(\mathcal{M},p_{\mathfrak{b}})$ be a P$\mathfrak{b}$MS with $\mathfrak{b}\geq 1$. Let $\{m_{\xi}\}$ be a sequence in $\mathcal{M}$ and $m_{0} \in \mathcal{M}$ be any arbitrary element.

• (1) The sequence $\{m_{\xi}\}$ is called a convergent sequence with limit $m_{0}$ if

  $$ \lim_{\xi \to \infty} p_{\mathfrak{b}}(m_{\xi},m_{0})=p_{ \mathfrak{b}}(m_{0},m_{0}). $$

  As an example, consider $\mathcal{M}=[0,1]$ and let $m_{\xi}= \{\frac{1}{\xi}: \xi \in \mathbb{N} \}$. Define a map $p_{\mathfrak{b}}: \mathcal{M} \times \mathcal{M} \to \mathbb{R}^{+}$ by $p_{\mathfrak{b}}(m_{1},m_{2})=|m_{1}-m_{2}|^{5}+v$, where $v >0$. It is easy to see that $(\mathcal{M},p_{\mathfrak{b}})$ is a P$\mathfrak{b}$MS with $\mathfrak{b}=2^{4}$. Now,

  $$ \lim_{\xi \to \infty}p_{\mathfrak{b}}(m_{\xi},0)=\lim _{\xi \to \infty}p_{\mathfrak{b}} \biggl(\frac{1}{\xi},0 \biggr)=\lim _{\xi \to \infty} \biggl[ \biggl\vert \frac{1}{\xi}-0 \biggr\vert +v \biggr]=p_{\mathfrak{b}}(0,0). $$

  That is, $\{m_{\xi}\}$ is a convergent sequence in $(\mathcal{M},p_{\mathfrak{b}})$.

• (2) A sequence $\{m_{k}\}$ in $\mathcal{M}$ becomes a Cauchy sequence if

  $$ \lim_{k,l \to \infty} p_{\mathfrak{b}}(m_{k},m_{l}) $$

  exists and is finite.

• (3) $(\mathcal{M},p_{\mathfrak{b}})$ is called a complete P$\mathfrak{b}$MS if every Cauchy sequence converges in $\mathcal{M}$.

Some useful ideas concerning Hausdorff distance under the structure of P$\mathfrak{b}$MSs have been suggested by Felhi [14] and recently revised by Anwar et al. [3].

Definition 1.4

Let $(\mathcal{M},p_{\mathfrak{b}})$ be a P$\mathfrak{b}$MS with $\mathfrak{b}\geq 1$, and $CB_{p_{\mathfrak{b}}}(\mathcal{M})$ be the collection of all nonempty bounded and closed subsets of $\mathcal{M}$. For $\mathcal{P}, \mathcal{Q} \in CB_{p_{\mathfrak{b}}}(\mathcal{M})$, the partial Hausdorff $\mathfrak{b}$-metric on $CB_{p_{\mathfrak{b}}}(\mathcal{M})$ induced by $p_{\mathfrak{b}}$ is given as follows:

$$ \mathcal{H}_{p_{\mathfrak{b}}}(\mathcal{P},\mathcal{Q})=\max \bigl\{ \delta _{p_{\mathfrak{b}}}(\mathcal{P},\mathcal{Q}),\delta _{p_{ \mathfrak{b}}}(\mathcal{Q}, \mathcal{P})\bigr\} , $$

where $\delta _{p_{\mathfrak{b}}}(\mathcal{P},\mathcal{Q})=\sup \{p_{ \mathfrak{b}}(p,\mathcal{Q}) :p \in \mathcal{P}\}$ and $\delta _{p_{\mathfrak{b}}}(\mathcal{Q},\mathcal{P})=\sup \{ p_{ \mathfrak{b}}(q,\mathcal{P}): q \in \mathcal{Q} \}$.

Lemma 1.5

Let $(\mathcal{M},p_{\mathfrak{b}})$ be a P$\mathfrak{b}$MS with $\mathfrak{b}\geq 1$. Consider two nonempty subsets $\mathcal{P},\mathcal{P^{*}} \in CB_{p_{\mathfrak{b}}}(\mathcal{M})$, and $k^{*}>1$. For some $p \in \mathcal{P}$, there exists $q \in \mathcal{P^{*}}$ so that

$$ {p_{\mathfrak{b}}}(p,q) \leq k^{*}\mathcal{H}_{p_{\mathfrak{b}}}\bigl( \mathcal{P},\mathcal{P^{*}}\bigr). $$

Lemma 1.6

Let $(\mathcal{M},p_{\mathfrak{b}})$ be a P$\mathfrak{b}$MS with $\mathfrak{b}\geq 1$, then for two nonempty subsets $\mathcal{P},\mathcal{P^{*}} \in CB_{p_{\mathfrak{b}}}(\mathcal{M})$, and for each $p \in \mathcal{P}$, we have

$$ p_{\mathfrak{b}}\bigl(p,\mathcal{P^{*}}\bigr) \leq \mathcal{H}_{p_{\mathfrak{b}}}\bigl( \mathcal{P},\mathcal{P^{*}}\bigr). $$

A new concept was given by Wardowski [15] in 2012 by introducing $\Delta _{f}$-family.

Definition 1.7

A mapping $\mathcal{F}$ from $(0,\infty )$ to $\mathbb{R}$ is a member of $\Delta _{f}$-family if $\mathcal{F}$ satisfies these properties:

$(F_{1})$: $\mathcal{F}$ is strictly increasing, i.e.,

$$ m_{1}< m_{2} \quad \implies\quad \mathcal{F}(m_{1})< \mathcal{F}(m_{2}), \quad \text{for all } m_{1},m_{2} \in \mathbb{R}. $$

$(F_{2})$: For every positive term sequence {$m_{\xi} :\xi \in \mathbb{N}$},

$$ \lim_{n \to \infty}m_{\xi}=0 \quad \iff \quad \lim_{n \to \infty} \mathcal{F}(m_{\xi})=-\infty . $$

$(F_{3})$: If we have $\gamma \in (0,1)$, then $\lim_{\xi \to 0^{+}}\xi ^{\gamma} \mathcal{F}(\xi )=0$.

Example 1.8

Let $\mathcal{F}: (0,\infty ) \to \mathbb{R}$ be defined as $\mathcal{F}(m)=\ln (m)$. $\mathcal{F}$ is a member of $\Delta _{f}$-family.

Let $(\mathcal{M},p_{\mathfrak{b}})$ be a P$\mathfrak{b}$MS with $\mathfrak{b}\geq 1$. This paper initiates the concept of new multivalued contraction mappings involving the $\Delta _{f}$-family and a given function $\alpha :\mathcal{M} \times \mathcal{M} \to \mathbb{R}^{+}$ in the context of a P$\mathfrak{b}$MS. We develop some fixed point results for such contractions. Furthermore, we illustrate our main result with concrete examples. An application is also presented for a deeper understanding of the obtained result.

2 Main results

We start with the following definition.

Definition 2.1

Consider a set $\mathcal{M}\neq \phi$ and let $S:\mathcal{M} \to 2^{\mathcal{M}}$ be a multivalued mapping. Given a function $\alpha :\mathcal{M} \times \mathcal{M} \to \mathbb{R}^{+}$. S is called a multivalued α-admissible mapping if for $m,n \in \mathcal{M}$, we have

$$ \alpha (m,n) \geq 1 \quad \implies\quad \alpha (m_{0},n_{0}) \geq 1, $$

where $m_{0} \in S(m)$ and $n_{0} \in S(n)$.

Definition 2.2

Let $(\mathcal{M},p_{\mathfrak{b}})$ be a P$\mathfrak{b}$MS with $\mathfrak{b}\geq 1$ and define a map $S: \mathcal{M} \to K(\mathcal{M})$. Then S is said to be a MV$\mathcal{F}$-contraction mapping if there are $\mathcal{F} \in \Delta _{f}-\text{family}$ and $\tau >0$ such that

$$ \mathcal{H}_{p{\mathfrak{b}}}(Sm_{1},Sm_{2})>0 \quad \implies \quad {}\tau + \mathcal{F}\bigl(\mathfrak{b}\mathcal{H}_{p{\mathfrak{b}}}(Sm_{1},Sm_{2}) \bigr) \leq \mathcal{F}\bigl(\mathbb{M}(m_{1},m_{2})\bigr), $$

(2.1)

where

$$\begin{aligned} \mathbb{M}(m_{1},m_{2}) =&\max \biggl\{ p_{\mathfrak{b}}(m_{1},m_{2}),p_{ \mathfrak{b}}(m_{1},Sm_{1}),p_{\mathfrak{b}}(m_{2},Sm_{2}), \\ & \frac{p_{\mathfrak{b}}(m_{1},Sm_{2})+p_{\mathfrak{b}}(m_{2},Sm_{1})}{2\mathfrak{b}} \biggr\} . \end{aligned}$$

Definition 2.3

Let $(\mathcal{M},p_{\mathfrak{b}})$ be a P$\mathfrak{b}$MS with $\mathfrak{b}\geq 1$. Given a function $\alpha :\mathcal{M} \times \mathcal{M} \to \mathbb{R}^{+}$. The mapping $S: \mathcal{M} \to K(\mathcal{M})$ is said to be a MV$\alpha \mathcal{F}$-contraction if there are $\mathcal{F} \in \Delta _{f}-\text{family}$ and $\tau >0$ such that

$$\begin{aligned}& \mathcal{H}_{p{\mathfrak{b}}}(Sm_{1},Sm_{2})>0 \\& \quad \implies\quad \tau + \mathcal{F}\bigl(\alpha (m_{1},m_{2}) \bigl(\mathfrak{b}\mathcal{H}_{p{ \mathfrak{b}}}(Sm_{1},Sm_{2}) \bigr)\bigr)\leq \mathcal{F}\bigl(\mathbb{M}(m_{1},m_{2}) \bigr), \end{aligned}$$

(2.2)

where

$$\begin{aligned} \mathbb{M}(m_{1},m_{2}) =&\max \biggl\{ p_{\mathfrak{b}}(m_{1},m_{2}),p_{ \mathfrak{b}}(m_{1},Sm_{1}),p_{\mathfrak{b}}(m_{2},Sm_{2}), \\ &{} \frac{p_{\mathfrak{b}}(m_{1},Sm_{2})+p_{\mathfrak{b}}(m_{2},Sm_{1})}{2\mathfrak{b}} \biggr\} . \end{aligned}$$

Lemma 2.4

Let $(\mathcal{M},p_{\mathfrak{b}})$ be a complete P$\mathfrak{b}$MS with $\mathfrak{b}\geq 1$ and $S: \mathcal{M} \to K(\mathcal{M})$ be a MV$\mathcal{F}$-contraction mapping, then

$$ \lim_{\xi \to \infty}\mathfrak{b}^{\xi} v_{\xi}=0, $$

where $v_{\xi}=p_{\mathfrak{b}}(m_{\xi +1}, m_{\xi +2})$ and $\xi =0,1,2,\ldots$ .

Proof

We take an arbitrary $m_{0} \in \mathcal{M}$. As $Sm_{0}$ is compact, it is nonempty, so we can choose $m_{1} \in Sm_{0}$. If $m_{1} \in Sm_{1}$, this means that $m_{1}$ is a fixed point of S trivially. Suppose $m_{1} \notin Sm_{1}$. As $Sm_{1}$ is closed, so we have $p_{\mathfrak{b}}(m_{1},Sm_{1}) >0$. Also, we know that

$$ p_{\mathfrak{b}}(m_{1},Sm_{1}) \leq \mathcal{H}_{p_{\mathfrak{b}}}(Sm_{0},Sm_{1}). $$

(2.3)

As $Sm_{1}$ is compact, so there exists $m_{2} \in Sm_{1}$ such that

$$ p_{\mathfrak{b}}(m_{1},m_{2})= p_{\mathfrak{b}}(m_{1},Sm_{1}). $$

Thus,

$$ p_{\mathfrak{b}}(m_{1},m_{2}) \leq \mathcal{H}_{p_{\mathfrak{b}}}(Sm_{0},Sm_{1}). $$

Similarly for $m_{3} \in Sm_{2}$, we get

$$ p_{\mathfrak{b}}(m_{2},m_{3}) \leq \mathcal{H}_{p_{\mathfrak{b}}}(Sm_{1},Sm_{2}), $$

which ultimately gives

$$ p_{\mathfrak{b}}(m_{\xi +1},m_{\xi +2}) \leq \mathcal{H}_{p_{ \mathfrak{b}}}(Sm_{\xi},Sm_{\xi +1}). $$

This leads to

$$ \mathfrak{b}\bigl(p_{\mathfrak{b}}(m_{\xi +1},m_{\xi +2})\bigr) \leq \mathfrak{b}\bigl(\mathcal{H}_{p_{\mathfrak{b}}}(Sm_{\xi},Sm_{\xi +1}) \bigr). $$

The condition $(F_{1})$ implies that

$$ \mathcal{F}\bigl(\mathfrak{b}\bigl(p_{\mathfrak{b}}(m_{\xi +1},m_{\xi +2}) \bigr)\bigr) \leq \mathcal{F}\bigl(\mathfrak{b}\bigl(\mathcal{H}_{p_{\mathfrak{b}}}(Sm_{\xi},Sm_{ \xi +1}) \bigr)\bigr). $$

(2.4)

By (2.1), we have

$$ \mathcal{F}\bigl(\mathfrak{b}\bigl(p_{\mathfrak{b}}(m_{\xi +1},m_{\xi +2}) \bigr)\bigr) \leq \mathcal{F}\bigl(\mathbb{M}(m_{\xi +1},m_{\xi}) \bigr) - \tau , $$

(2.5)

where

$$\begin{aligned} \mathbb{M}(m_{\xi},m_{\xi +1}) =&\max \biggl\{ p_{\mathfrak{b}}(m_{\xi},m_{ \xi +1}), p_{\mathfrak{b}}(m_{\xi},Sm_{\xi}), p_{\mathfrak{b}}(m_{ \xi +1},Sm_{\xi +1}), \\ & {}\frac{p_{\mathfrak{b}}(m_{\xi},Sm_{\xi +1})+p_{\mathfrak{b}}(m_{\xi +1},Sm_{\xi})}{2\mathfrak{b}} \biggr\} \\ =&\max \biggl\{ p_{\mathfrak{b}}(m_{\xi},m_{\xi +1}), p_{\mathfrak{b}}(m_{ \xi},m_{\xi +1}), p_{\mathfrak{b}}(m_{\xi +1},m_{\xi +2}), \\ &{} \frac{p_{\mathfrak{b}}(m_{\xi},m_{\xi +1})+p_{\mathfrak{b}}(m_{\xi +1},m_{\xi +2})}{2\mathfrak{b}} \biggr\} \\ \leq & \max \biggl\{ p_{\mathfrak{b}}(m_{\xi},m_{\xi +1}), p_{\mathfrak{b}}(m_{ \xi},m_{\xi +1}), p_{\mathfrak{b}}(m_{\xi +1},m_{\xi +2}), \\ &{} \mathfrak{b}\biggl[ \frac{p_{\mathfrak{b}}(m_{\xi},m_{\xi +1})+p_{\mathfrak{b}}(m_{\xi +1},m_{\xi +2})}{2\mathfrak{b}}\biggr] \biggr\} . \\ =&\max \bigl\{ p_{\mathfrak{b}}(m_{\xi},m_{\xi +1}), p_{\mathfrak{b}}(m_{ \xi +1},m_{\xi +2})\bigr\} . \end{aligned}$$

Assume that

$$ \max \bigl\{ p_{\mathfrak{b}}(m_{\xi},m_{\xi +1}), p_{\mathfrak{b}}(m_{ \xi +1},m_{\xi +2})\bigr\} =p_{\mathfrak{b}}(m_{\xi +1},m_{\xi +2}). $$

The inequality (2.5) yields

$$ \tau +\mathcal{F}\bigl(\mathfrak{b}\bigl(p_{\mathfrak{b}}(m_{\xi +1},m_{\xi +2}) \bigr)\bigr) \leq \mathcal{F}\bigl(p_{\mathfrak{b}}(m_{\xi +1},m_{\xi +2}) \bigr), $$

which is a contradiction. Therefore,

$$ \max \bigl\{ p_{\mathfrak{b}}(m_{\xi},m_{\xi +1}), p_{\mathfrak{b}}(m_{ \xi +1},m_{\xi +2})\bigr\} =p_{\mathfrak{b}}(m_{\xi},m_{\xi +1}). $$

It implies that

$$ \mathcal{F}\bigl(\mathfrak{b}\bigl(p_{\mathfrak{b}}(m_{\xi +1},m_{\xi +2}) \bigr)\bigr) \leq \mathcal{F}\bigl(p_{\mathfrak{b}}(m_{\xi},m_{\xi +1}) \bigr). $$

For convenience, we are setting $v_{\xi}=p_{\mathfrak{b}}(m_{\xi +1}, m_{\xi +2})$, where $\xi =0,1,\ldots$ . Clearly, $v_{\xi} >0$ for all $\xi \in \mathbb{N}$. Now, substituting this into the above equation, we have

$$ \tau +\mathcal{F}\bigl(\mathfrak{b} (v_{\xi})\bigr) \leq \mathcal{F}(v_{\xi -1}). $$

Iteratively,

$$ \tau +\mathcal{F}\bigl(\mathfrak{b}^{\xi} (v_{\xi})\bigr)\leq \mathcal{F}\bigl( \mathfrak{b}^{\xi -1} (v_{\xi -1})\bigr). $$

We will get

$$ \mathcal{F}(\mathfrak{b}^{\xi}(v_{\xi}) \leq \mathcal{F}\bigl(\mathfrak{b}^{ \xi -1}(v_{\xi -1})\bigr)-\tau \leq \mathcal{F}\bigl(\mathfrak{b}^{\xi -2}(v_{ \xi -2})\bigr)-2\tau \leq \cdots \leq \mathcal{F}(v_{0})-\xi \tau . $$

(2.6)

Hence,

$$ \lim_{\xi \to \infty}\mathcal{F}\mathfrak{b}^{\xi} (v_{\xi}) = - \infty , $$

we have

$$ \lim_{\xi \to \infty}\mathfrak{b}^{\xi} v_{\xi}=0, \quad \text{by } (F_{2}). $$

 □

Theorem 2.5

Let $(\mathcal{M},p_{\mathfrak{b}})$ be a complete P$\mathfrak{b}$MS with $\mathfrak{b}\geq 1$, such that $p_{\mathfrak{b}}$ is a continuous mapping and $S: \mathcal{M} \to K(\mathcal{M})$ is a multivalued $\alpha \mathcal{F}$-contraction mapping. Suppose that

• (1) S is continuous;

• (2) S is an α-admissible mapping;

• (3) there exist $m_{0} \in \mathcal{M}$ and $m_{1} \in Sm_{0}$ such that $\alpha (m_{0},m_{1})\geq 1$.

Then S has a fixed point.

Proof

For $m_{0} \in \mathcal{M}$, we have by assumption $\alpha (m_{0},m_{1}) \geq 1$ for some $m_{1}\in Sm_{0}$. Similarly, for $m_{2}\in Sm_{1}$, we have $\alpha (m_{1},m_{2}) \geq 1$ and for any sequence $m_{\xi +1} \in Sm_{\xi}$, we get

$$ \alpha (m_{\xi},m_{\xi +1}) \geq 1 \quad \text{for all } \xi \in \mathbb{N}\cup \{0\}. $$

(2.7)

Now, by the contraction condition (2.2), we have

$$ \tau + \mathcal{F}\bigl(\alpha (m_{\xi},m_{\xi +1})\mathfrak{b} \bigl( \mathcal{H}_{p{\mathfrak{b}}}(m_{\xi +1},m_{\xi +2})\bigr)\bigr) \leq \mathcal{F}\bigl(\mathbb{M}(m_{\xi +1},m_{\xi})\bigr). $$

The inequality (2.7) implies that

$$ \tau + \mathcal{F}\bigl(\mathfrak{b}\bigl(\mathcal{H}_{p{\mathfrak{b}}}(m_{\xi +1},m_{ \xi +2}) \bigr)\bigr)\leq \mathcal{F}\bigl(\mathbb{M}(m_{\xi +1},m_{\xi}) \bigr), $$

where $\mathfrak{b} \geq 1$. We have

$$ \mathcal{F}\bigl(\mathfrak{b}\bigl(p_{\mathfrak{b}}(m_{\xi +1},m_{\xi +2}) \bigr)\bigr) \leq \mathcal{F}\bigl(\mathbb{M}(m_{\xi +1},m_{\xi}) \bigr) - \tau . $$

(2.8)

By lemma 2.4, one writes

$$ \lim_{\xi \to \infty}\mathfrak{b}^{\xi} v_{\xi}=0. $$

By $(F_{3})$, for any $\gamma \in (0,1)$

$$ \lim_{\xi \to \infty}\bigl(\mathfrak{b}^{\xi} v_{\xi} \bigr)^{\gamma} \mathcal{F}\mathfrak{b}^{\xi}(v_{\xi})=0,\quad \forall \xi \in \mathbb{N}. $$

Using (2.6), one writes

$$ \bigl(\mathfrak{b}^{\xi} v_{\xi}\bigr)^{\gamma}\bigl( \mathcal{F}\mathfrak{b}^{\xi} (v_{ \xi})-\mathcal{F}(v_{0}) \bigr) \leq -\bigl(\mathfrak{b}^{\xi} {v_{\xi}} \bigr)^{ \gamma}\xi \tau \leq 0. $$

(2.9)

Now, as $\tau >0$, we have

$$ \lim_{\xi \to \infty }\bigl({\mathfrak{b}^{\xi}v_{\xi}} \bigr)^{\gamma}\xi = 0. $$

So, there exists $\xi _{1} \in \mathbb{N}$, such that

$$ \bigl(\mathfrak{b}^{\xi}v_{\xi}\bigr)^{\gamma}\xi \leq 1,\quad \forall \xi \geq \xi _{1}. $$

It implies that

$$ \mathfrak{b}^{\xi} v_{\xi} \leq \frac{1}{\xi ^{\frac{1}{\gamma}}}. $$

(2.10)

Now, we will prove that $\{m_{\xi}\}$ is a Cauchy sequence in $\mathcal{M}$. For this, let $\xi ,l \in \mathbb{N}$ provided that $\xi >l\geq \xi _{1}$. Using the triangular inequality of a P$\mathfrak{b}$MS, we have

$$\begin{aligned} p_{\mathfrak{b}}(m_{\xi},m_{\eta}) \leq & \mathfrak{b}\bigl\{ p_{ \mathfrak{b}}(m_{\xi},m_{\xi +1})+p_{\mathfrak{b}}(m_{\xi +1},m_{\eta}) \bigr\} -p_{\mathfrak{b}}(m_{\xi +1},m_{\xi +1}) \\ \leq & \mathfrak{b}\bigl\{ p_{\mathfrak{b}}(m_{\xi},m_{\xi +1})+ p_{ \mathfrak{b}}(m_{\xi +1},m_{\eta})\bigr\} \\ \leq & \mathfrak{b}p_{\mathfrak{b}}(m_{\xi},m_{\xi +1})+ \mathfrak{b}^{2} \bigl\{ p_{\mathfrak{b}}(m_{\xi +1},m_{\xi +2})+ p_{\mathfrak{b}}(m_{\xi +2},m_{ \eta})\bigr\} \\ &{} -p_{\mathfrak{b}}(m_{\xi +2},m_{\xi +2}) \\ \leq & \mathfrak{b}p_{\mathfrak{b}}(m_{\xi},m_{\xi +1})+ \mathfrak{b}^{2} \bigl\{ p_{\mathfrak{b}}(m_{\xi +1},m_{\xi +2})+ p_{\mathfrak{b}}(m_{\xi +2},m_{ \eta})\bigr\} \\ &{}\vdots \\ =&\mathfrak{b}p_{\mathfrak{b}}(m_{\xi},m_{\xi +1})+ \mathfrak{b}^{2} \{p_{\mathfrak{b}}(m_{\xi +1},m_{\xi +2})+ \cdots +\mathfrak{b}^{l- \xi}p_{\mathfrak{b}}(m_{\eta -1},m_{\eta}) \\ =& \sum_{\beta =\xi}^{\eta -1}\mathfrak{b}^{\beta -\xi +1}p_{ \mathfrak{b}}(m_{\beta},m_{\beta +1}) \\ \leq & \sum_{\beta =\xi}^{\infty} \mathfrak{b}^{\beta}p_{ \mathfrak{b}}(m_{\beta +1},m_{\beta +2}) \\ =&\sum_{\beta =\xi}^{\infty}\mathfrak{b}^{\beta}v_{\beta} \\ \leq & \sum_{\beta =\xi}^{\infty} \frac{1}{\beta ^{\frac{1}{\gamma}}}. \end{aligned}$$

The convergence of the series $\sum_{\beta =1}^{\infty} \frac{1}{\beta ^{\frac{1}{\gamma}}}$ implies that $\lim_{\xi \to \infty} p_{\mathfrak{b}}(m_{\xi},m_{\eta}) =0$, which shows $\{m_{\xi}\}$ is a Cauchy sequence in $\mathcal{M}$. Since $\mathcal{M}$ is complete, there exists $m^{*} \in \mathcal{M}$ such that

$$ \lim_{\xi \to \infty} p_{\mathfrak{b}} \bigl(m_{\xi},m^{*}\bigr) = p_{ \mathfrak{b}} \bigl(m^{*},m^{*}\bigr)=0 . $$

(2.11)

We claim that $m^{*}$ is a fixed point of S, that is,

$$ p_{\mathfrak{b}}\bigl(m^{*},Sm^{*}\bigr) =p_{\mathfrak{b}} \bigl(m^{*},m^{*}\bigr). $$

Suppose $p_{\mathfrak{b}}(m^{*},Sm^{*}) > 0$. So, there exists $k_{0} \in \mathbb{N}$ such that $p_{\mathfrak{b}}(m_{\xi},Sm^{*}) > 0$ for all $\xi > k_{0}$. We have

$$ p_{\mathfrak{b}}\bigl(m_{\xi},Sm^{*}\bigr) \leq \mathcal{H}_{p{\mathfrak{b}}}\bigl(Sm_{ \xi +1},Sm^{*}\bigr). $$

By using our contraction condition and taking limit $\xi \to \infty$, we have

$$ \begin{aligned} \tau +{\mathcal{F}}\bigl(p_{\mathfrak{b}} \bigl(m^{*},Sm^{*}\bigr)\bigr)& \leq \tau +{\mathcal{F}}\bigl( \alpha \bigl(m^{*},m^{*}\bigr)\mathcal{H}_{p{ \mathfrak{b}}} \bigl(Sm^{*},Sm^{*}\bigr)\bigr) \\ & \leq {\mathcal{F}}\bigl(\mathbb{M}\bigl(m^{*},m^{*}\bigr) \bigr) \\ &\leq {\mathcal{F}}\bigl(p_{\mathfrak{b}}\bigl(m^{*},Sm^{*} \bigr)\bigr), \end{aligned} $$

where,

$$\begin{aligned} \mathbb{M}\bigl(m^{*},m^{*}\bigr) =&\max \biggl\{ p_{\mathfrak{b}}\bigl(m^{*},m^{*}\bigr), p_{ \mathfrak{b}} \bigl(m^{*},Sm^{*}\bigr), p_{\mathfrak{b}} \bigl(m^{*},Sm^{*}\bigr), \\ &{} \frac{p_{\mathfrak{b}}(m^{*},Sm^{*})+p_{\mathfrak{b}}(Sm^{*},m^{*})}{2\mathfrak{b}} \biggr\} \\ \leq & p_{\mathfrak{b}}\bigl(m^{*},Sm^{*}\bigr). \end{aligned}$$

It yields that

$$ \tau +{\mathcal{F}}\bigl(p_{\mathfrak{b}}\bigl(m^{*},Sm^{*} \bigr)\bigr) \leq {\mathcal{F}}\bigl(p_{ \mathfrak{b}}\bigl(m^{*},Sm^{*} \bigr)\bigr). $$

Since $\tau > 0$, the above relation yields a contradiction, therefore $p_{\mathfrak{b}}(m^{*},Sm^{*}) =0$. Also,

$$ p_{\mathfrak{b}}\bigl(m^{*},m^{*}\bigr)=0. $$

This gives $m^{*} \in \bar{S}m^{*} =Sm^{*}$. Proving that $m^{*}$ is a fixed point of S. □

Example 2.6

Let $\mathcal{M}=\{0,1,2,3, \ldots\}$ and $p_{\mathfrak{b}}:\mathcal{M} \times \mathcal{M} \to \mathbb{R}^{+}$ be defined as

$$ p_{\mathfrak{b}}(\zeta , \nu ) = \vert \zeta -\nu \vert ^{q}+ \bigl[\max \{ \zeta , \nu \}\bigr]^{q}\quad \text{for all } \zeta , \nu \in \mathcal{M}. $$

It is easy to check that $(\mathcal{M},p_{\mathfrak{b}})$ is a complete P$\mathfrak{b}$MS with $\mathfrak{b}=2^{q-1}$, where $q >1$. We also define a multivalued map $S:\mathcal{M} \to 2^{\mathcal{M}}$ by

$$ S\zeta = \textstyle\begin{cases} \{0,1\} , & \text{if } \zeta =0,1, \\ \{\zeta -1,\zeta \} & \text{otherwise}. \end{cases} $$

Consider $\alpha :\mathcal{M} \times \mathcal{M} \to [0,\infty )$ as

$$ \alpha (\zeta ,\nu )= \textstyle\begin{cases} 2, & \text{if } \zeta ,\nu \in \{0,1\}, \\ \frac{1}{2}, & \text{otherwise}. \end{cases} $$

Let $\zeta _{0}=0$, $\zeta _{1}=1$, then $S\zeta _{0}= \{0,1\}$ and $\zeta _{1}= \{0,1\}$. Giving $\alpha (\zeta _{0},\zeta _{1})=\alpha (0,1)=2 > 1$, for some $\zeta _{2}=0 \in S\zeta _{1}$, we get $\alpha (\zeta _{1},\zeta _{2})=\alpha (1,0)=2 > 1$. That is, S is an α-admissible map.

Define $\mathcal{F}: (0,\infty ) \to \mathbb{R}$ as $\mathcal{F}(\zeta )=\ln (\zeta )+\zeta$. It can be observed easily that $\mathcal{F}$ is a member of $\Delta _{f}$-family. Now, applying $\mathcal{F}$ on our contraction condition, one gets

$$ \tau + \mathcal{F}\bigl(\alpha (\zeta ,\nu )\mathcal{H}_{p_{\mathfrak{b}}}(S \zeta ,S\nu )\bigr) \leq \mathcal{F}\bigl(\mathbb{M}(\zeta ,\nu )\bigr). $$

That is,

$$ \begin{aligned} &\tau + \ln \bigl\{ \alpha (\zeta ,\nu ) \mathcal{H}_{p_{ \mathfrak{b}}}(S\zeta ,S\nu )\bigr\} +\alpha (\zeta ,\nu ) \mathcal{H}_{p_{ \mathfrak{b}}}(S\zeta ,S\nu ) \\ &\quad \leq \ln \bigl(\mathbb{M}(\zeta ,\nu )\bigr)+\mathbb{M}(\zeta ,\nu ). \end{aligned} $$

Hence,

$$ \tau +\alpha (\zeta ,\nu )\mathcal{H}_{p_{\mathfrak{b}}}(S\zeta ,S \nu )- \mathbb{M}(\zeta ,\nu ) \leq \ln \bigl(\mathbb{M}(\zeta ,\nu )\bigr)- \ln \bigl\{ \alpha (\zeta ,\nu )\mathcal{H}_{p_{\mathfrak{b}}}(S\zeta ,S \nu )\bigr\} . $$

Therefore,

$$ e^{ \tau +\alpha (\zeta ,\nu )\mathcal{H}_{p_{\mathfrak{b}}}(S\zeta ,S \nu )- \mathbb{M}(\zeta ,\nu )} \leq \frac{\mathbb{M}(\zeta ,\nu )}{\alpha (\zeta ,\nu )\mathcal{H}_{p_{\mathfrak{b}}}(S\zeta ,S\nu )} $$

That is,

$$ \frac{\alpha (\zeta ,\nu )\mathcal{H}_{p_{\mathfrak{b}}}(S\zeta ,S\nu )}{\mathbb{M}(\zeta ,\nu )}e^{ \alpha (\zeta ,\nu )\mathcal{H}_{p_{\mathfrak{b}}}(S\zeta ,S\nu )- \mathbb{M}(\zeta ,\nu )} \leq e^{ -\tau}. $$

(2.12)

Now,

$$ \begin{aligned} \delta _{p_{\mathfrak{b}}}\bigl(\mathcal{P}, \mathcal{P^{*}}\bigr)&= \delta _{p_{\mathfrak{b}}}(S\zeta ,S\nu ) \\ &= \max \bigl\{ p_{\mathfrak{b}}(\zeta ,S\nu ),p_{\mathfrak{b}}(\zeta -1,S \nu ) \bigr\} \\ &= \max \bigl\{ \inf \bigl\{ p_{\mathfrak{b}}(\zeta ,\nu ),{p_{\mathfrak{b}}( \zeta ,\nu -1)}\bigr\} ,\inf \bigl\{ p_{\mathfrak{b}}(\zeta -1,\nu ),{p_{ \mathfrak{b}}(\zeta -1,\nu -1)}\bigr\} \bigr\} \\ &= \max \bigl\{ \vert \zeta -\nu \vert ^{q}+\zeta ^{q}, \vert \zeta -\nu -2 \vert ^{q}+\zeta ^{q} \bigr\} \\ & = \vert \zeta -\nu \vert ^{q}+\zeta ^{q}. \end{aligned} $$

Similarly, we can calculate

$$ \delta _{p_{\mathfrak{b}}}\bigl(\mathcal{P^{*}},\mathcal{P}\bigr)= \vert \zeta -\nu \vert ^{q}+ \zeta ^{q}. $$

Hence,

$$ \begin{aligned} \mathcal{H}_{p_{\mathfrak{b}}}\bigl( \mathcal{P}, \mathcal{P^{*}}\bigr)&=\max \bigl\{ \vert \zeta -\nu \vert ^{q}+\zeta ^{q}, \vert \zeta -\nu \vert ^{q}+ \zeta ^{q}\bigr\} \\ &= \vert \zeta -\nu \vert ^{q}+\zeta ^{q}. \end{aligned} $$

(2.13)

Also,

$$ \mathbb{M}(\zeta ,\nu )\geq p_{\mathfrak{b}}(\zeta ,\nu )= \vert \zeta - \nu \vert ^{q}+\zeta ^{q}. $$

(2.14)

Setting these both in the contraction condition, we get

$$\begin{aligned}& \frac{\alpha (\zeta ,\nu )\mathcal{H}_{p_{\mathfrak{b}}}(S\zeta ,S\nu )}{\mathbb{M}(\zeta ,\nu )}e^{( \alpha (\zeta ,\nu )\mathcal{H}_{p_{\mathfrak{b}}}(S\zeta ,S\nu ))- \mathbb{M}(\zeta ,\nu )} \\& \quad = \frac{ \vert \zeta -\nu \vert ^{q}+\zeta ^{q}}{2\mathbb{M}(\zeta ,\nu )} e^{ \frac{1}{2}( \vert \zeta -\nu \vert ^{q}+\zeta ^{q})- \mathbb{M}(\zeta ,\nu )} \quad \text{using (2.25)} \\& \quad \leq \frac{ \vert \zeta -\nu \vert ^{q}+\zeta ^{q}}{2 \vert \zeta -\nu \vert ^{q}+\zeta ^{q}} e^{ \frac{1}{2}( \vert \zeta -\nu \vert ^{q}+\zeta ^{q})- \vert \zeta -\nu \vert ^{q}+\zeta ^{q}} \quad \text{using (2.26)} \\& \quad = \frac{1}{2} e^{\frac{-1}{2}( \vert \zeta -\nu \vert ^{q}+\zeta ^{q})} \\& \quad = \frac{1}{2} e^{-\tau} \\& \quad < e^{-\tau}. \end{aligned}$$

This implies that (2.12) is satisfied with $\tau = {\frac{1}{2}(|\zeta -\nu |^{q}+\zeta ^{q})}$, which is a positive number for $\zeta \neq \nu$. All conditions of Theorem 2.5 are true, and 0 and 1 are two fixed points of S.

Theorem 2.7

Let $(\mathcal{M},p_{\mathfrak{b}})$ be a complete P$\mathfrak{b}$MS with $\mathfrak{b}\geq 1$ such that $p_{\mathfrak{b}}$ is a continuous mapping. Let $S :\mathcal{M} \to CB_{p_{\mathfrak{b}}}(\mathcal{M})$ be a MV$\alpha \mathcal{F}$-contraction mapping and $B \subset (0, \infty )$ with $\inf B > 0$. Suppose that

• (1) S is continuous;

• (2) S is an α-admissible mapping;

• (3) there exist $m_{0} \in \mathcal{M}$ and $m_{1} \in Sm_{0}$ such that $\alpha (m_{0},m_{1})\geq 1$;

• (4) $\mathcal{F}(\inf B)= \inf \mathcal{F}(B)$, where $\mathcal{F} \in \Delta _{f}-\textit{family}$.

Then S has a fixed point.

Proof

We take an arbitrary $m_{0} \in \mathcal{M}$. As Sm, the set of all images of $m \in \mathcal{M}$, is nonempty for all values in $\mathcal{M}$, we can choose $m_{1} \in Sm_{0}$. If $m_{1} \in Sm_{1}$, this means that $m_{1}$ is a fixed point of S. So suppose $m_{1} \notin Sm_{1}$. As $Sm_{1}$ is closed, we have

$$ p_{\mathfrak{b}}(m_{1},Sm_{1}) >0. $$

Also, we know that

$$ p_{\mathfrak{b}}(m_{1},Sm_{1}) \leq \mathcal{H}_{p_{\mathfrak{b}}}(Sm_{0},Sm_{1}). $$

We have

$$ \mathcal{F}\bigl(p_{\mathfrak{b}}(m_{1},Sm_{1}) \bigr) \leq \mathcal{F}\bigl( \mathcal{H}_{p_{\mathfrak{b}}}(Sm_{0},Sm_{1}) \bigr), \text{by} F_{2}. $$

(2.15)

Using (4)

$$ \mathcal{F}\bigl(p_{\mathfrak{b}}(m_{1},Sm_{1})\bigr)=\inf _{g \in Sm_{1}} \mathcal{F}\bigl(p_{\mathfrak{b}}(m_{1},g) \bigr). $$

That is,

$$ \inf_{g \in Sm_{1}}\mathcal{F}\bigl(p_{\mathfrak{b}}(m_{1},g) \bigr) \leq \mathcal{F}\bigl(\mathcal{H}_{p_{\mathfrak{b}}}(Sm_{0},Sm_{1}) \bigr). $$

(2.16)

As $Sm_{1}$ is compact, so we can find a $m_{2} \in Sm_{1}$ such that

$$ \inf_{g \in Sm_{1}}\mathcal{F}\bigl(p_{\mathfrak{b}}(m_{1},g) \bigr)= \mathcal{F}\bigl(p_{\mathfrak{b}}(m_{1},m_{2}) \bigr). $$

From (2.15),

$$ \mathcal{F}\bigl(p_{\mathfrak{b}}(m_{1},m_{2})\bigr) \leq \mathcal{F}\bigl( \mathcal{H}_{p_{\mathfrak{b}}}(Sm_{0},Sm_{1}) \bigr). $$

(2.17)

Similarly, for $m_{3} \in Sm_{2}$, we get

$$ \mathcal{F}\bigl(p_{\mathfrak{b}}(m_{2},m_{3})\bigr) \leq \mathcal{F}\bigl( \mathcal{H}_{p_{\mathfrak{b}}}(Sm_{1},Sm_{2}) \bigr), $$

which ultimately gives

$$ \mathcal{F}\bigl(p_{\mathfrak{b}}(m_{\xi +1},m_{\xi +2})\bigr) \leq \mathcal{F}\bigl( \mathcal{H}_{p_{\mathfrak{b}}}(Sm_{\xi},Sm_{\xi +1}) \bigr). $$

As $\mathfrak{b}\geq 1$, so we can write

$$ \mathcal{F}\bigl(\mathfrak{b}\bigl(p_{\mathfrak{b}}(m_{\xi +1},m_{\xi +2}) \bigr)\bigr) \leq \mathcal{F}\bigl(\mathfrak{b}\bigl(\mathcal{H}_{p_{\mathfrak{b}}}(Sm_{\xi},Sm_{ \xi +1}) \bigr)\bigr). $$

(2.18)

For $m_{0} \in \mathcal{M}$ by assumption, $\alpha (m_{0},m_{1}) \geq 1$ for some $m_{1}\in Sm_{0}$. Similarly, for some $m_{2}\in Sm_{1}$, we have $\alpha (m_{1},m_{2}) \geq 1$ and for any sequence $m_{\xi +1} \in Sm_{\xi}$, we may write

$$ \alpha (m_{\xi},m_{\xi +1}) \geq 1 \quad \text{for all } \xi \in \mathbb{N}\cup \{0\}. $$

(2.19)

Using (2.2), we have

$$ \tau + \mathcal{F}\bigl(\alpha (m_{\xi},m_{\xi +1}) \bigl( \mathcal{H}_{p{ \mathfrak{b}}}(m_{\xi +1},m_{\xi +2})\bigr)\bigr)\leq \mathcal{F}\bigl(\mathbb{M}(m_{ \xi +1},m_{\xi})\bigr), $$

The inequality (2.19) implies that

$$ \tau + \mathcal{F}\bigl(\mathfrak{b}\bigl(\mathcal{H}_{p{\mathfrak{b}}}(m_{\xi +1},m_{ \xi +2}) \bigr)\bigr)\leq \mathcal{F}\bigl(\mathbb{M}(m_{\xi +1},m_{\xi}) \bigr). $$

Using (2.18), we have

$$ \mathcal{F}\bigl(\mathfrak{b}\bigl(p_{\mathfrak{b}}(m_{\xi +1},m_{\xi +2}) \bigr)\bigr) \leq \mathcal{F}\bigl(\mathbb{M}(m_{\xi +1},m_{\xi}) \bigr) - \tau . $$

(2.20)

Now, using Lemma 2.4, one writes

$$ \lim_{\xi \to \infty}\mathfrak{b}^{\xi} v_{\xi}=0, $$

Now, by $(F_{3})$, for any $\gamma \in (0,1)$ and for all $\xi \in \mathbb{N}$,

$$ \lim_{\xi \to \infty}\bigl(\mathfrak{b}^{\xi} v_{\xi} \bigr)^{\gamma} \mathcal{F}\mathfrak{b}^{\xi}(v_{\xi})=0. $$

It implies that

$$ \bigl(\mathfrak{b}^{\xi} v_{\xi}\bigr)^{\gamma}\bigl( \mathcal{F}\mathfrak{b}^{\xi} (v_{ \xi})-\mathcal{F}(v_{0}) \bigr) \leq -\bigl(\mathfrak{b}^{\xi} {v_{\xi}} \bigr)^{ \gamma}\xi \tau \leq 0. $$

(2.21)

As $\tau > 0$, we have

$$ \lim_{\xi \rightarrow \infty}\bigl({\mathfrak{b}^{\xi}v_{\xi}} \bigr)^{\gamma} \xi = 0. $$

So there exists $\xi _{1} \in \mathbb{N}$ such that $(\mathfrak{b}^{\xi}v_{\xi})^{\gamma}\xi \leq 1$ for all $\xi \geq \xi _{1}$. Then

$$ \mathfrak{b}^{\xi} v_{\xi} \leq \frac{1}{\xi ^{\frac{1}{\gamma}}}. $$

(2.22)

Next, we prove that $\{m_{\xi}\}$ is a Cauchy sequence in $\mathcal{M}$. For this, following the same steps as done in Theorem 2.5, one can easily have

$$ \lim_{\xi \to \infty} p_{\mathfrak{b}} \bigl(m_{\xi},m^{*}\bigr) = p_{ \mathfrak{b}} \bigl(m^{*},m^{*}\bigr)=0 . $$

(2.23)

We claim that $m^{*}$ is a fixed point of S. Suppose that $p_{\mathfrak{b}}(m^{*},Sm^{*}) > 0$, this means there exists $k_{0} \in \mathbb{N}$ such that we have $p_{\mathfrak{b}}(m_{\xi},Sm^{*}) > 0$ for all $\xi > k_{0}$. One writes

$$ p_{\mathfrak{b}}\bigl(m_{\xi},Sm^{*}\bigr) \leq \mathcal{H}_{p{\mathfrak{b}}}\bigl(Sm_{ \xi +1},Sm^{*}\bigr) . $$

Using (2.2) and taking limit $\xi \to \infty$, we have

$$ \begin{aligned} \tau +{\mathcal{F}}\bigl(p_{\mathfrak{b}} \bigl(m^{*},Sm^{*}\bigr)\bigr)& \leq \tau +{\mathcal{F}}\bigl( \alpha \bigl(m^{*},m^{*}\bigr)\mathcal{H}_{p{ \mathfrak{b}}} \bigl(Sm^{*},Sm^{*}\bigr)\bigr) \\ & \leq {\mathcal{F}}\bigl(\mathbb{M}\bigl(m^{*},m^{*}\bigr) \bigr) \\ &\leq {\mathcal{F}}\bigl(p_{\mathfrak{b}}\bigl(m^{*},Sm^{*} \bigr)\bigr), \end{aligned} $$

where

$$\begin{aligned} \mathbb{M}\bigl(m^{*},m^{*}\bigr) =&\max \biggl\{ p_{\mathfrak{b}}\bigl(m^{*},m^{*}\bigr), p_{ \mathfrak{b}} \bigl(m^{*},Sm^{*}\bigr), p_{\mathfrak{b}} \bigl(m^{*},Sm^{*}\bigr), \\ &{} \frac{p_{\mathfrak{b}}(m^{*},Sm^{*})+p_{\mathfrak{b}}(Sm^{*},m^{*})}{2\mathfrak{b}} \biggr\} \\ \leq & p_{\mathfrak{b}}\bigl(m^{*},Sm^{*}\bigr). \end{aligned}$$

It implies that

$$ \tau +{\mathcal{F}}\bigl(p_{\mathfrak{b}}\bigl(m^{*},Sm^{*} \bigr)\bigr) \leq {\mathcal{F}}\bigl(p_{ \mathfrak{b}}\bigl(m^{*},Sm^{*} \bigr)\bigr). $$

Since $\tau > 0$, the above relation yields a contradiction. Thus,

$$ p_{\mathfrak{b}}\bigl(m^{*},Sm^{*}\bigr) =0. $$

Also, $p_{\mathfrak{b}}(m^{*},m^{*})=0$. This gives $m^{*} \in \bar{S}m^{*} =Sm^{*}$. Hence, $m^{*}$ is a fixed point of S. □

Example 2.8

Let $\mathcal{M}= \{m_{\zeta} = 1- (\frac{1}{2} )^{\zeta} : \zeta \in \mathbb{N} \}$ and $p_{\mathfrak{b}}:\mathcal{M} \times \mathcal{M} \to [0,\infty )$ be defined by

$$ p_{\mathfrak{b}}(\zeta , \nu ) = \vert \zeta -\nu \vert ^{2}+ \bigl[\max \{ \zeta , \nu \}\bigr]^{2} \text{for all} \zeta , \nu \in \mathcal{M}. $$

One can easily verify that $(\mathcal{M},p_{\mathfrak{b}})$ is a complete P$\mathfrak{b}$MS with $\mathfrak{b}=2$. We also define a multivalued map $S:\mathcal{M} \to 2^{\mathcal{M}}$ by

$$ Sm= \textstyle\begin{cases} \{m_{1}\} , & m=m_{1}, \\ \{m_{\zeta},m_{\zeta +1}\}, & m=m_{\zeta}, \zeta =2,3,\ldots . \end{cases} $$

Consider $\alpha (m_{\zeta},m_{\nu})=1$ and $\mathbb{M}(m_{\zeta},m_{\nu})= p_{\mathfrak{b}}(m_{\zeta},m_{\nu})$. Take $\mathcal{F}: (0,\mathrm{infty})\to \mathbb{R}$ as $\mathcal{F}(\zeta )=\ln (\zeta )+\zeta$. Hence, the contraction condition will take the following form:

$$ \frac{ \mathcal{H}_{p_{\mathfrak{b}}}(Sm_{\zeta},Sm_{\nu})}{\mathbb{M}(m_{\zeta},m_{\nu})}e^{ \mathcal{H}_{p_{\mathfrak{b}}}(Sm_{\zeta},Sm_{\nu})- \mathbb{M}(m_{ \zeta},m_{\nu})} \leq e^{ -\tau}. $$

(2.24)

Now, we verify this condition for the following two possible cases:

Case I

If $\mathcal{H}_{p_{\mathfrak{b}}}(Sm_{\zeta},Sm_{1}) >0$ and $\nu =1$, we have

$$ \begin{aligned} \delta _{p_{\mathfrak{b}}}(Sm_{\zeta},Sm_{1}) &= \max \bigl\{ p_{ \mathfrak{b}}(m_{\zeta},Sm_{1}),p_{\mathfrak{b}}(m_{\zeta +1},Sm_{1}) \bigr\} \\ &= \max \bigl\{ \vert m_{\zeta}-m_{1} \vert ^{2}+(m_{\zeta})^{2}, \vert m_{\zeta +1}-m_{1} \vert ^{2}+(m_{ \zeta +1})^{2}\bigr\} \\ & = \vert m_{\zeta +1}-m_{1} \vert ^{2}+(m_{\zeta +1})^{2}. \end{aligned} $$

In the same manner,

$$ \delta _{p_{\mathfrak{b}}}(Sm_{1},Sm_{\zeta})= \vert m_{\zeta}-m_{1} \vert ^{2}+(m_{ \zeta})^{2}. $$

It implies that

$$ \mathcal{H}_{p_{\mathfrak{b}}}(Sm_{\zeta},Sm_{1})= \vert m_{\zeta +1}-m_{1} \vert ^{2}+({m_{ \zeta +1}})^{2}. $$

(2.25)

Also,

$$ \mathbb{M}(m_{\zeta},m_{1})= \vert m_{\zeta}-m_{1} \vert ^{2}+(m_{\zeta})^{2} \leq \vert m_{\zeta}-m_{1} \vert ^{2}+(m_{\zeta +2})^{2}. $$

(2.26)

One writes

$$ \begin{aligned} & \frac{ \mathcal{H}_{p_{\mathfrak{b}}}(Sm_{\zeta},Sm_{1})}{\mathbb{M}(m_{\zeta},m_{1})}e^{( \mathcal{H}_{p_{\mathfrak{b}}}(Sm_{\zeta},Sm_{1})- \mathbb{M}(m_{ \zeta},m_{1}))} \\ &\quad \leq \frac{ \vert m_{\zeta +1}-m_{1} \vert ^{2}+(m_{\zeta +1})^{2}}{ \vert m_{\zeta}-m_{1} \vert ^{2}+(m_{\zeta +2})^{2}} e^{( \vert m_{\zeta +1}-m_{1} \vert ^{2}+(m_{\zeta +1})^{2})-( \vert m_{\zeta}-m_{1} \vert ^{2}+(m_{ \zeta +2})^{2})} \\ &\quad = \frac{ \vert (\frac{1}{2} )- (\frac{1}{2} )^{\zeta +1} \vert ^{2} + (1- (\frac{1}{2} )^{\zeta +1})^{2} }{ \vert (\frac{1}{2} )- (\frac{1}{2} )^{\zeta} \vert ^{2}+ (1- (\frac{1}{2} )^{\zeta +2} )^{2}} e^{ \vert (\frac{1}{2} )- (\frac{1}{2} )^{\zeta +1} \vert ^{2} + (1- (\frac{1}{2} )^{\zeta +1} )^{2}- \vert (\frac{1}{2} )- (\frac{1}{2} )^{\zeta} \vert ^{2} - (1- (\frac{1}{2} )^{\zeta +2} ) ^{2}} \\ &\quad \leq e^{ ( 2 (\frac{1}{2} )^{2\zeta +2} + (\frac{1}{2} )^{\zeta} - [ (\frac{1}{2} )^{2\zeta}+ (\frac{1}{2} )^{2\zeta +4}+ 3 (\frac{1}{2} )^{\zeta +1}+2 ( \frac{1}{2} )^{\zeta +2} ] ) } \\ &\quad < e^{-\tau}, \end{aligned} $$

for some $\tau >0$.

Case II

If $\mathcal{H}_{p_{\mathfrak{b}}}(Sm_{\zeta},Sm_{\nu}) >0$ with $\zeta \geq \nu >1$, we have

$$ \mathcal{H}_{p_{\mathfrak{b}}}(Sm_{\zeta},Sm_{\nu})= \vert m_{\zeta +1}-m_{ \nu +1} \vert ^{2}+({m_{\zeta +1}})^{2}, $$

and

$$ \mathbb{M}(m_{\zeta},m_{\nu})= \vert m_{\zeta}-m_{\nu} \vert ^{2}+(m_{\zeta})^{2} \leq \vert m_{\zeta}-m_{\nu} \vert ^{2}+(m_{\zeta +2})^{2}. $$

From (2.24), we have

$$ \begin{aligned} & \frac{ \mathcal{H}_{p_{\mathfrak{b}}}(Sm_{\zeta},Sm_{\nu})}{\mathbb{M}(m_{\zeta},m_{\nu})}e^{( \mathcal{H}_{p_{\mathfrak{b}}}(Sm_{\zeta},Sm_{\nu})- \mathbb{M}(m_{ \zeta},m_{\nu}))} \\ &\quad \leq \frac{ \vert m_{\zeta +1}-m_{\nu +1} \vert ^{2}+(m_{\zeta +1})^{2}}{ \vert m_{\zeta}-m_{\nu} \vert ^{2}+(m_{\zeta +2})^{2}} e^{( \vert m_{\zeta +1}-m_{\nu +1} \vert ^{2}+(m_{\zeta +1})^{2})-( \vert m_{\zeta}-m_{ \nu} \vert ^{2}+(m_{\zeta +2})^{2})} \\ &\quad = \frac{ \vert (\frac{1}{2} )^{\nu +1}- (\frac{1}{2} )^{\zeta +1} \vert ^{2} + ( 1- (\frac{1}{2} )^{\zeta +1} ) ^{2} }{ \vert (\frac{1}{2} )^{\nu}- (\frac{1}{2} )^{\zeta} \vert ^{2}+ (1- (\frac{1}{2} )^{\zeta +2} ) ^{2}} e^{ \vert (\frac{1}{2} )^{\nu +1}- (\frac{1}{2} )^{ \zeta +1} \vert ^{2} + ( 1- (\frac{1}{2} )^{\zeta +1} ) ^{2}- \vert (\frac{1}{2} )^{\nu}- (\frac{1}{2} )^{\zeta} \vert ^{2} - (1- (\frac{1}{2} )^{\zeta +2} ) ^{2}} \\ &\quad \leq e^{ ( \vert (\frac{1}{2} )^{\nu +1}- (\frac{1}{2} )^{\zeta +1} \vert ^{2} + ( 1- (\frac{1}{2} )^{\zeta +1} ) ^{2} )- ( \vert (\frac{1}{2} )^{\nu}- ( \frac{1}{2} )^{\zeta} \vert ^{2} + (1- (\frac{1}{2} )^{ \zeta +2} ) ^{2} )} \\ &\quad < e^{-\tau}, \end{aligned} $$

which is true for all $\zeta , \nu \in \mathbb{N}$ provided that $\zeta \geq \nu >1$, where $\tau >0$. Thus, all the required conditions of Theorem 2.7 are satisfied. Here, the mapping S has a fixed point ($m_{1}$ and $m_{\zeta}$ are fixed points).

3 An application

Here, we apply our main result to find a solution to an integral equation of Fredholm type. Take $I=[0,1]$. Denote by $\mathcal{M} =\mathcal{C}(I,\mathbb{R}^{2})$ the space of all continuous functions defined from I to $\mathbb{R}^{2}$. We endow $\mathcal{M}$ with the usual sup-norm. We consider a partial $\mathfrak{b}$ metric on $\mathcal{M}$ defined by

$$ p_{\mathfrak{b}}(\phi , \psi )= { \Vert \phi -\psi \Vert }_{ \infty}= \sup_{m \in I} \bigl\{ {e^{-mp}} { \bigl\vert \phi (m) - \psi (m) \bigr\vert }^{q}\bigr\} \quad p,q>1, $$

for all $\phi , \psi \in \mathcal{M}$. It is easy to verify that $(\mathcal{M},p_{\mathfrak{b}})$ is a complete P$\mathfrak{b}$MS. Consider the Fredholm integral inclusion

$$ \phi (\zeta )\in f(\zeta )+ \int _{0}^{1}k_{\phi}\bigl(\zeta ,x^{*},\phi \bigl(x^{*}\bigr)\bigr)\,dx^{*}, $$

(3.1)

such that for every $\mathcal{K_{\phi}}: I \times I \times \mathbb{R}^{2} \to K( \mathcal{M})$ there exists

$$ k_{\phi} \bigl(\zeta ,x^{*}, \phi ^{*}\bigr) \in \mathcal{K_{\phi}}\bigl(\zeta ,x^{*}, \phi ^{*}\bigr). $$

Define a multivalued mapping $S :\mathcal{M} \to K(\mathcal{M})$ as

$$ S\bigl(\phi (\zeta )\bigr)= \biggl\{ \phi ^{*}(\zeta ) : \phi ^{*}(\zeta ) \in \omega (\zeta )+ \int _{0}^{1}\mathcal{K_{\phi}}\bigl(\zeta ,x^{*},\phi \bigl(x^{*}\bigr)\bigr)\,dx^{*} \biggr\} . $$

(3.2)

Theorem 3.1

Suppose that the following conditions hold:

• (1) $\mathcal{K_{\phi}}:I\times I \times \mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ and $f:I\to \mathbb{R}^{2}$ are continuous;

• (2) there exists $\phi _{0} \in \mathcal{M}$ such that $\phi _{k} \in S\phi _{k-1}$;

• (3) there exists a continuous function $\mathfrak{f}: I \times I \to I$ such that

  $$ \bigl\vert k_{\phi}\bigl(\zeta ,x^{*},\phi \bigl(x^{*}\bigr)\bigr) -k_{\psi}\bigl(\zeta ,x^{*}, \psi \bigl(x^{*}\bigr)\bigr) \bigr\vert ^{q} \leq \sup _{x^{*} \in I} \mathfrak{f}\bigl(\phi \bigl(x^{*}\bigr),\psi \bigl(x^{*}\bigr)\bigr) \bigl\vert \phi \bigl(x^{*}\bigr)- \psi \bigl(x^{*}\bigr) \bigr\vert ^{q}, $$

  for each $\zeta ,x^{*} \in I$ and $\mathfrak{f}(\phi (x^{*}),\psi (x^{*}))\leq \gamma$.

Then the integral inclusion (3.1) has a solution.

Proof

Let $(\mathcal{M},p_{\mathfrak{b}})$ be a complete P$\mathfrak{b}$MS. We choose

$$ \mathcal{F}(\zeta )=\ln (\zeta ), $$

for all $\zeta \in (0,\infty )$. So after going through a natural logarithm, our condition will be

$$ \mathcal{H}_{p_{\mathfrak{b}}}\bigl(S\bigl(\phi (\zeta ),S\psi (\zeta )\bigr)\bigr) \leq e^{- \tau}M(\phi , \psi ), $$

with $\alpha (\phi ,\psi )=1$. Next, to show that S satisfies this condition, let $p >1$ such that

$$ \frac{1}{p}+ \frac{1}{q}=1, $$

then for $\phi ^{*} \in S(\phi )$, we have

$$\begin{aligned}& p_{\mathfrak{b}}\bigl(\bigl(\phi ^{*}(\zeta ),S\bigl(\psi (\zeta )\bigr)\bigr)\bigr) \leq p_{\mathfrak{b}}\bigl(\phi ^{*}(\zeta ),\bigl(\psi ^{*}(\zeta )\bigr)\bigr) \\& \quad = \sup_{\zeta \in I}{e^{-\zeta \gamma}} { \bigl\vert \phi ^{*}(\zeta ) -\psi ^{*}( \zeta ) \bigr\vert }^{q} \\& \quad =\sup_{ \zeta \in I}e^{-\zeta \gamma} \biggl\vert \int _{0}^{1}k_{\phi}\bigl( \zeta ,x^{*},\phi \bigl(x^{*}\bigr)\bigr)-k_{\psi}\bigl( \zeta ,x^{*},\psi \bigl(x^{*}\bigr)\bigr) \biggr\vert ^{q}\,dx^{*} \\& \quad \leq \sup_{ \zeta \in I}e^{-\zeta \gamma} \biggl[\biggl( \int _{0}^{1} \vert 1 \vert ^{p}\,dx^{*}\biggr)^{ \frac{1}{p}} \int _{0}^{1} \bigl( \bigl\vert k_{\phi} \bigl(\zeta ,x^{*},\phi \bigl(x^{*}\bigr)\bigr)-k_{ \psi} \bigl(\zeta ,x^{*},\psi \bigl(x^{*}\bigr)\bigr) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \biggr]^{q}\,dx^{*} \\& \quad =\sup_{ \zeta \in I}e^{-\zeta \gamma} \int _{0}^{1} \bigl\vert k_{\phi}\bigl( \zeta ,x^{*}, \phi \bigl(x^{*}\bigr)\bigr)-k_{\psi} \bigl(\zeta ,x^{*},\psi \bigl(x^{*}\bigr)\bigr) \bigr\vert ^{q}\,dx^{*} \\& \quad =\sup_{ \zeta \in I}e^{-\zeta \gamma} \int _{0}^{1} \bigl\vert e^{-x^{*}\gamma +x^{*} \gamma} k_{\phi}\bigl(\zeta ,x^{*},\phi \bigl(x^{*}\bigr) \bigr)-k_{\psi}\bigl(\zeta ,x^{*}, \psi \bigl(x^{*} \bigr)\bigr) \bigr\vert ^{q}\,dx^{*} \\& \quad \leq \sup_{ \zeta \in I}e^{-\zeta \gamma} \int _{0}^{1} e^{x^{*} \gamma}\mathfrak{f}\bigl( \phi \bigl(x^{*}\bigr),\psi \bigl(x^{*}\bigr)\bigr) \sup _{ x^{*} \in I}e^{-x^{*}\gamma} \bigl\vert \phi \bigl(x^{*} \bigr)-\psi \bigl(x^{*}\bigr) \bigr\vert ^{q}\,dx^{*} \\& \quad = \gamma \bigl\Vert \phi \bigl(x^{*}\bigr) - \psi \bigl(x^{*}\bigr) \bigr\Vert _{\infty} \sup_{ \zeta \in I} e^{-\zeta \gamma} \int _{0}^{1}e^{x^{*}\gamma}\,dx^{*} \\& \quad = p_{\mathfrak{b}}\bigl(\phi \bigl(x^{*}\bigr),\psi \bigl(x^{*}\bigr)\bigr) (1) \bigl(e^{\gamma} -1\bigr) \\& \quad \leq p_{\mathfrak{b}}\bigl(\phi \bigl(x^{*}\bigr),\psi \bigl(x^{*}\bigr)\bigr)e^{\gamma} \\& \quad \leq e^{\gamma} \mathbb{M}\bigl(\phi \bigl(x^{*}\bigr),\psi \bigl(x^{*}\bigr)\bigr), \end{aligned}$$

where

$$\begin{aligned}& \mathbb{M}\bigl(\phi \bigl(x^{*}\bigr),\psi \bigl(x^{*} \bigr)\bigr) \\& \quad = \max \biggl\{ p_{\mathfrak{b}}\bigl( \phi \bigl(x^{*} \bigr),\psi \bigl(x^{*}\bigr)\bigr),p_{\mathfrak{b}}\bigl(\phi \bigl(x^{*}\bigr),S\bigl(\phi \bigl(x^{*}\bigr)\bigr) \bigr),p_{ \mathfrak{b}}\bigl(\psi \bigl(x^{*}\bigr),S\bigl(\psi \bigl(x^{*}\bigr)\bigr)\bigr), \\& \qquad {} \frac{p_{\mathfrak{b}}(\phi (x^{*}),S(\psi (x^{*})))+p_{\mathfrak{b}}(\psi (x^{*}),S(\phi (x^{*})))}{2\mathfrak{b}} \biggr\} . \end{aligned}$$

Also, as $\phi ^{*}$ is arbitrary, we have

$$ \delta _{p_{\mathfrak{b}}}\bigl(S(\phi ),S(\psi )\bigr) \leq e^{\gamma} \mathbb{M}(\phi ,\psi ). $$

Similarly, one finds

$$ \delta _{p_{\mathfrak{b}}}\bigl(S(\psi ),S(\phi )\bigr)\leq e^{\gamma} \mathbb{M}(\psi ,\phi ). $$

Then

$$ \mathcal{H}_{p_{\mathfrak{b}}}\bigl(S(\phi ),S(\psi )\bigr) \leq e^{\gamma} \mathbb{M}(\phi ,\psi ). $$

That is, $\mathcal{H}_{p_{\mathfrak{b}}}(S(\phi ),S(\psi ))\leq e^{-\tau}M( \phi ,\psi )$.

Our desired contraction condition is then satisfied by choosing $-\tau =\gamma$. Thus, all conditions of Theorem 2.5 are satisfied, and so the integral inclusion (3.1) has a solution, and 0 is a fixed point of S. □