On multivalued maps for $\varphi$-contractions involving orbits with application

1.   Introduction and preliminaries

Let $(\hat{U}, d)$ be a metric space. For $\widehat{\ell }\in \hat{U}$ and $\beta _{1}\subseteq \hat{U}$, let $d_{b}(\widehat{\ell }_{1}, \beta _{1}) = \inf \{d_{b}(\widehat{\ell }_{1}, \widehat{\ell }_{2}):\widehat{\ell } _{2}\in \beta _{1}\}.$ Denote $N(\hat{U}),$ $CL(\hat{U}),$ $CB(\hat{U})$ by the class all nonempty subsets of $\hat{U},$ the class of all nonempty closed subsets of $\hat{U}$ and the class of all nonempty closed and bounded subsets of $\hat{U}$ respectively. Define the Hausdorff-Pompeiu metric $\hat{ H}_{b}$ induced by $d_{b}$ on $CB(\hat{U})$ as follows:

for all $\beta _{1}, \beta _{2}\in CL(\hat{U})$. A point $\widehat{\ell }\in \hat{U}$ is said to be a fixed point of $\widetilde{T}:\hat{U}\rightarrow CL(\hat{U}),$ if $\widehat{\ell }\in \widetilde{T}\widehat{\ell }.$ If, for $\widehat{\ell }_{0}\in \hat{U},$ there exists a sequence $\{\widehat{\ell } _{i}\}$ in $\hat{U}$ such that $\widehat{\ell }_{i}\in \widetilde{T}\widehat{ \ell }_{i-1},$ then $O(\widetilde{T}, \widehat{\ell }_{0}) = \{\widehat{\ell } _{0}, \widehat{\ell }_{1}, \widehat{\ell }_{2}, ...\}$ is said to be an orbit of $\widetilde{T}:\hat{U}\rightarrow CL(\hat{U})$. A mapping $f:\hat{U} \rightarrow \mathbb{R}$ is said to be $\widetilde{T}$-orbitally lower semi-continuous (o.l.s.c) if $\{\widehat{\ell }_{i}\}$ is a sequence in $O(\widetilde{T}, \widehat{\ell }_{0})$ and $\widehat{\ell }_{i}\rightarrow \varrho$ implies $f(\varrho)\leq \liminf_{i}f(\widehat{\ell }_{i}).$

From now on, Nadler ^[13] realized the following multivalued version of BCP:

Theorem 1.1. ^[13] Let $(\hat{U}, d_{b})$ be a complete metric space and $T:\hat{U} \rightarrow CB(\hat{U})$ be a Nadler contraction, i.e., there is $\gamma \in \lbrack 0, 1)$ such that

Then $T$ possesses at least one fixed point.

We start the following results for main sequel.

Lemma 1.2. ^[13] Let $(\hat{U}, d_{b})$ be a metric space, $\beta _{2}\in CB(\hat{U})$ and $\widehat{\ell }\in \hat{U}$. Then, for each $\epsilon > 0,$ there exists $\nu \in \beta _{2}$ such that

Lemma 1.3. ^[19] Let $(\hat{U}, d_{b})$ be a metric space and $\beta _{1}$, $\beta _{2}\in CB(\hat{U})$ with $\hat{H}_{b}(\beta _{1}, \beta _{2}) > 0.$ Then for all $h > 1$ and $\widehat{\ell }\in \beta _{1}$, there exists $\nu = \nu (\widehat{\ell })\in \beta _{2}$ such that

There after, many researchers worked on existence of fixed point theorems of single valued mappings can improve in the module of multi-valued mappings that satisfying various classes of contractive mappings (see ^{[1,2,3,4,6,9,10,12,15,17,18,19,20]}).

Definition 1.4. ^[8] A $b$-metric space on a nonempty set $M$ is a function $b:\hat{U}\times \hat{U}\rightarrow {\mathbb{R}}^{+}$ such that for all $\widehat{\ell }_{1}, \widehat{\ell }_{2}, \widehat{\ell }_{3}\in \hat{U}$ and a given real number $s\geq 1$, the following conditions hold:

$\left(b_{i}\right)$ $d_{b}\left(\widehat{\ell }_{1}, \widehat{\ell } _{2}\right) = 0$ if and only if $\widehat{\ell }_{1} = \widehat{\ell }_{2};$

$\left(b_{ii}\right)$ $d_{b}\left(\widehat{\ell }_{1}, \widehat{\ell } _{2}\right) = d_{b}\left(\widehat{\ell }_{2}, \widehat{\ell }_{1}\right);$

$\left(b_{iii}\right)$ $d_{b}\left(\widehat{\ell }_{1}, \widehat{\ell } _{3}\right) \leq s[d_{b}\left(\widehat{\ell }_{1}, \widehat{\ell } _{2}\right) +d_{b}\left(\widehat{\ell }_{2}, \widehat{\ell }_{3}\right) ].$

The pair $\left(\hat{U}, d_{b}\right)$ is known as $b$-metric space.

The following examples present the context of $b$-metric spaces, which are essentially larger than the context of metric spaces ^[8].

Example 1.5. ^[8] Let $\hat{U} = l_{p}\left({\mathbb{R}}\right)$ with $p\in \left(0, 1\right)$ where $l_{p}\left({\mathbb{R}}\right) = \{\left\{ \widehat{\ell } _{i}\right\} \subset {\mathbb{R}} : \sum\limits_{i = 1}^{+\infty }\left\vert \widehat{\ell }_{i}\right\vert ^{p} < \infty \}.$ A function $b: \hat{U}\times \hat{U} \rightarrow {\mathbb{R}}^{+}$ is given by $b\left(\widehat{\ell }_{1}, \widehat{\ell } _{2}\right) = (\sum\limits_{i = 1}^{+\infty }\left\vert \widehat{\ell } _{i}\right\vert ^{p})^{\frac{1}{p}}$, where $\widehat{\ell }_{1} = \widehat{ \ell }_{i}$ and $\widehat{\ell }_{2} = \widehat{\ell }_{i^{\prime }}$. Then the pair $\left(\hat{U}, d_{b}\right)$ is known as $b$-metric space with $s = 2^{ \frac{1}{p}}.$

Example 1.6. ^[8] Let $\hat{U} = L_{p}\left[ 0, 1\right]$ be the space of all real valued functions $\widehat{\ell }\left(r\right)$, $0\leq r\leq 1$ in such a way that $\int\limits_{0}^{1}\left\vert \widehat{\ell }\left(r\right) \right\vert ^{\frac{1}{p}}dr < \infty$. A function $b:\hat{U}\times \hat{U}\rightarrow {\mathbb{R}}^{+}$ is given by $b\left(\widehat{\ell }_{1}, \widehat{ \ell }_{2}\right) = (\int\limits_{0}^{1}\left\vert \widehat{\ell }_{1}\left(r\right) -\widehat{\ell }_{2}\left(r\right) \right\vert ^{p})^{\frac{1}{p}}$. Then the pair $\left(\hat{U}, d_{b}\right)$ is known as $b$-metric space with $s = 2^{\frac{1}{p}}.$

Definition 1.7. ^[8] A sequence $\left\{ \widehat{\ell }_{i}\right\}$ in $b$-metric space $\hat{U}$ is said to be convergent if there is $\widehat{ \ell }\in \hat{U}$ such that $d_{b}\left(\widehat{\ell }_{i}, \widehat{\ell } \right) \rightarrow 0$ as $i\rightarrow +\infty$ and write $\lim_{i\rightarrow +\infty }(\widehat{\ell }_{i}) = \widehat{\ell }$. A sequence $\left\{ \widehat{\ell }_{i}\right\}$ in $\left(\hat{U}, d_{b}\right)$ is said to be Cauchy if $d_{b}\left(\widehat{\ell }_{i}, \widehat{\ell }_{i^{\prime }}\right) \rightarrow 0$ as $i, i^{\prime }\rightarrow +\infty$. A $b$-metric space $\left(\hat{U}, d_{b}\right)$ is said to be complete if every Cauchy sequence in $\hat{U}$ converges.

Note that, in general, the $b$-metric is not a continuous functional. Recently, Liu et al. ^[12] produced the following classical function:

Definition 1.8. Let $\varphi :(0, +\infty)\rightarrow (0, +\infty)$ satisfy the following conditions:

$(\varphi _{a})$ $\varphi$ is nondecreasing;

$(\varphi _{b})$ for all $\{\widehat{\ell }_{i}\}$ in $(0, +\infty)$, $\lim_{i\rightarrow +\infty }\varphi (\widehat{\ell }_{i}) = 0$ if and only if $\lim_{i\rightarrow +\infty }(\widehat{\ell }_{i}) = 0$;

$(\varphi _{c})$ $\varphi$ is continuous$.$

From now on, we denote by $\varphi ^{\ast }$ the set of all function that satisfying $(\varphi _{a})-(\varphi _{c}).$ The following well known two lammas of $\varphi$ functions will be needed in our forthcoming sequel:

Lemma 1.9. ^[12] Let $\left\{ \widehat{\ell }_{i}\right\} _{i}$ be a bounded sequence of real numbers and all its convergent subsequences have the same limit $\gamma$. Then $\left\{ \widehat{\ell }_{i}\right\} _{i}$ is convergent and $\lim_{i\rightarrow +\infty }(\widehat{\ell }_{i}) = \gamma.$

Lemma 1.10. Let $\varphi :(0, +\infty)\rightarrow (0, +\infty)$ be a nondecreasing and continuous function with $\inf_{\widehat{\ell }\in (0, +\infty)}\varphi (\widehat{\ell }) = 0$ and $\left\{ \widehat{\ell } _{i}\right\} _{i}\in (0, +\infty).$ Then

Proof. ($\Rightarrow$) Suppose $\lim_{i\rightarrow +\infty }\varphi (\widehat{ \ell }_{i}) = 0$. Then we claim that the sequence $\left\{ \widehat{\ell } _{i}\right\}$ is bounded. In fact, if the sequence is unbounded, then we may assume that $\widehat{\ell }_{i}\rightarrow +\infty$ and so for all $\delta > 0$, there is $i_{0}\in N$ such that $\widehat{\ell }_{i}$ $> \delta$ for all $i > i_{0}$. Hence $\varphi (\delta)\leq \varphi (\widehat{\ell }_{i})$ and so $\varphi (\delta)\leq \lim_{i\rightarrow +\infty }\varphi (\widehat{ \ell }_{i}) = 0$, which contradicts to $\varphi (\delta) > 0$. Thus $\left\{ \widehat{\ell }_{i}\right\}$ is bounded. Hence there exists a subsequence $\left\{ \widehat{\ell }_{i_{i}}\right\} \subset \left\{ \widehat{\ell }_{i}\right\}$ such that $\lim_{i\rightarrow +\infty }\left\{ \widehat{\ell }_{i_{i}}\right\} = k$ (where $k$ is nonnegative number). Clearly $k\geq 0$. If $k > 0$, then there is $i_{0}\in N$ such that $\left\{ \widehat{\ell }_{i_{i}}\right\} \in \left(\frac{k}{2}, \frac{3k}{2}\right)$ for all $i\geq i_{0}$. By $(\varphi _{a})$, we deduce that $\varphi (\frac{k }{2})\leq \lim_{i\rightarrow +\infty }\left\{ \widehat{\ell } _{i_{i}}\right\} = 0,$ which contradicts to $\varphi (\frac{k}{2}) > 0$. Consequently, setting $k = 0$ and by the above lemma, we have $\lim_{i\rightarrow +\infty }(\widehat{\ell }_{i}) = 0.$

($\Leftarrow$) Suppose that $inf_{\widehat{\ell }\in \left(0, +\infty \right) }\varphi (\widehat{\ell }) = 0$. If $\widehat{\ell }_{i}\rightarrow 0$, then for any given $\epsilon > 0$, there is $k > 0$ such that $\varphi (k)\in \left(0, \epsilon \right)$ and there exists $i_{1}\in N$ such that $\widehat{\ell } _{i} < k$ for all $i > i_{1}$. Therefore, $0 < \varphi (\widehat{\ell }_{i})\leq \varphi (k) < \epsilon$ for $i > i_{1}$. Hence $\varphi (\widehat{\ell } _{i})\rightarrow 0$ as $i \to +\infty$.

Throughout this paper $E$ denotes an interval on $\mathbb{R}^{+}$ containing $0$, that is, an interval of the form $[0, R]$, $[0, R)$, or $[0, +\infty).$ Proinov ^[14] introduced the following:

Lemma 1.11. ^[14] Let $\widehat{\ell }_{0}\in \Lambda$ ($\Lambda$ is a closed subset of $\hat{U}$) such that

and $\widehat{\ell }_{i}\in \Lambda$ for some $i\geq 0$. Then we have $d_{b}(\widehat{\ell }_{i}, \widetilde{T}\widehat{\ell }_{i})\in E.$

Definition 1.12. ^[14] Suppose $\widehat{\ell }_{0}\in \Lambda$ and $d_{b}(\widehat{\ell }_{0}, \widetilde{T}\widehat{\ell }_{0})\in E.$ Then for an iterate $\widehat{\ell }_{i}\ (i\geq 0)$ which belongs to $\Lambda,$ we define the closed ball $\overline{b}(\widehat{\ell }_{i}, \rho)$ with center $\widehat{\ell }_{i}$ and radius $\rho > 0.$

Lemma 1.13. ^[14] If an element $\widehat{\ell }_{0}\in \Lambda$ satisfies $d_{b}(\widehat{\ell }_{0}, \widetilde{T}\widehat{\ell }_{0})\in E$ and $\overline{b}(\widehat{\ell }_{i}, \rho)\subset \Lambda$ for some $i\geq 0,$ then $\widehat{\ell }_{i+1}\in \Lambda$ and $\overline{b}(\widehat{\ell } _{i+1}, \rho)\subset \overline{b}(\widehat{\ell }_{i}, \rho).$

Definition 1.14. ^[14] Let $i\geq 1.$ A function $\xi :E\rightarrow E$ is said to be a gauge function of order $i$ on $E$ if it satisfies the following conditions: (a) $\xi (\lambda \widehat{\ell }) < \lambda ^{i}\xi (\widehat{ \ell })$ for all $\lambda \in (0, 1)$ and $\widehat{\ell }\in E;$ (b) $\xi (\widehat{\ell }) < \widehat{\ell }$ for all $\widehat{\ell }\in E-\{0\}$.

It is easy to see that the first condition of Definition 1.14 is equivalent to the following: $\xi (0) = 0$ and $\xi (\widehat{\ell })/\widehat{ \ell }^{i}$ is nondecreasing on $E-\{0\}$.

Definition 1.15. ^[14] A gauge function $\xi :E\rightarrow E$ is said to be a B-GGF on $E$ if

Note that a B-GGF also satisfies the following functional equation:

Proinov ^[14] proved his main results by assuming B-GGF $\xi$ and the mapping $T:\Lambda \rightarrow X$ satisfying the contractive condition $d(T\left(x\right) T^{2}\left(x\right))\leq \xi (d(x; Tx))$ when the underlying space is endowed with a metric. But from now on, in the context of $b$-metric space for some technical dialectics, Samreen et al. ^[16] introduced the following class of GF.

Definition 1.16. ^[16] A nondecreasing function $\xi :E\rightarrow E$ is said to be a $b$-B-GGF on $E$ if

where $s$ is the coefficient of $b$-metric space. Moreover, note that a $b$ -B-GGF also satisfies the following functional equation:

Remark 1.17. Every $b$-B-GGF is also a B-GGF ^[7] but the converse may not hold. Furthermore, in ^[16], Samreen et al. introduced gauge functions in a $b$ -metric space of the form

where $s$ is the coefficient of $b$-metric space. For instance, we refer the following simple examples of gauge functions of order $i$ as:

(a) $\xi (\widehat{\ell }) = \frac{\lambda \widehat{\ell }}{s}$ for all $\lambda \in (0, 1)$ is a gauge function of order $1$ on $\widehat{\ell }\in E;$

(b) $\xi (\widehat{\ell }) = \frac{\lambda \widehat{\ell }^{k}}{s}$ $(\lambda > 0,$ $k > 0)$ is a gauge function of order $k$ on $E = [0, l)$ where $l = \left(\frac{1}{\lambda }\right) ^{\frac{1}{1-k}}$.

In 2015, Khojasteh et al. ^[11] introduced the concept of simulation function as follows:

Definition 1.18. ^[11] A function $\Gamma :\mathbb{R}^{+}\times \mathbb{R} ^{+}\rightarrow \mathbb{R}$ is called an $SF$ if

$(\Gamma 1)$ $\Gamma (0, 0) = 0;$

$(\Gamma 2)$ $\Gamma (\widehat{\ell }_{1}, \widehat{\ell }_{2}) < \widehat{\ell }_{2}-\widehat{\ell }_{1}$ for all $\widehat{\ell }_{1}^{'}, \widehat{\ell } _{2} > 0;$

$(\Gamma 3)$ if $\{\widehat{\ell }_{1i}\}$, $\{\widehat{\ell }_{2i}\}\in (0, +\infty)$ such that $\lim_{i\rightarrow +\infty }\widehat{\ell } _{1i} = \lim_{i\rightarrow +\infty }\widehat{\ell }_{2i} > 0$, then

Due to $(\Gamma 2),$ we have $\Gamma (\widehat{\ell }_{1}, \widehat{\ell } _{1}) < 0$ for all $\widehat{\ell }_{1} > 0.$ From now on, we denote by $\nabla$ the set of all functions satisfying ($\Gamma 1$)-($\Gamma 3$)$.$ Some well known examples of $\Gamma$ functions presented in the existing exposition are as follows:

Example 1.19. ^[11] For $i = 1, 2$, let $\vartheta _{i}:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$ be continuous functions with $\vartheta _{i}(\widehat{\ell } _{1}) = 0$ if and only if $\widehat{\ell }_{1} = 0.$ The following functions $\Gamma _{j}:\mathbb{R}^{+}\times \mathbb{R}^{+}\rightarrow \mathbb{R}$ ($j = 1, \cdots, 6$) are in $\nabla$:

(a) $\Gamma _{1}(\widehat{\ell }_{1}, \widehat{\ell }_{2}) =$ $\vartheta _{1}(\widehat{\ell }_{2})-\vartheta _{2}(\widehat{\ell }_{1})$ for all $\widehat{ \ell }_{1}, \widehat{\ell }_{2}\geq 0,$ where $\vartheta _{1}(\widehat{\ell } _{1})\leq \widehat{\ell }_{1}\leq \vartheta _{2}(\widehat{\ell }_{1})$ for all $\widehat{\ell }_{1} > 0;$

(b) $\Gamma _{6}(\widehat{\ell }_{1}, \widehat{\ell }_{2}) = \widehat{\ell } _{2}-\int_{0}^{\widehat{\ell }_{1}}\varsigma (u)du{\rm{ }}$for all${\rm{ }} \widehat{\ell }_{1}, \widehat{\ell }_{2}\geq 0,$ where $\varsigma :\mathbb{R} ^{+}\rightarrow \mathbb{R}^{+}$ is a function such that

Let $(\hat{U}, d_{b})$ be a metric space, $\widetilde{T}$ be a self mapping on $\hat{U}$ and $\Gamma \in \nabla$. $\widetilde{T}$ is said to be a $\nabla$-contraction with respect to $\Gamma,$ if

Due to $(\Gamma _{2}),$ we have $d_{b}(T\widehat{\ell }_{1}, T\widehat{\ell } _{2})\neq d_{b}(\widehat{\ell }_{1}, \widehat{\ell }_{2})$ for all distinct points $\widehat{\ell }_{1}, \widehat{\ell }_{2}\in \hat{U}.$ Thus $T$ is not an isometry, whenever $T$ is a $\nabla$-contraction with respect to $\Gamma.$ Conversely, if a $\nabla$-contraction mapping $T$ on a metric space possesses a fixed point, then it is necessarily unique.

In the recent year, Ali et al. ^[5] initiated the following definition which is a modification of the notion of $\alpha$-admissible.

Definition 1.20. ^[5] Let $(\hat{U}, d_{b})$ be a metric space and $\Lambda$ be a nonempty subset of $\hat{U}.$ A mapping $\widetilde{T}:\Lambda \rightarrow CB(\hat{U})$ is called $\alpha$-admissible if there exists a function $\alpha :\Lambda \times \Lambda \rightarrow \lbrack 0, +\infty)$ such that

for all $\widehat{\ell }\in \widetilde{T}a\cap \Lambda$ and $\nu \in \widetilde{T}b\cap \Lambda.$

In this manuscript, we prove the notion of multi-valued Suzuki (SU) type fixed point results via $\varphi _{\xi }$-contraction mapping and $\left(\nabla _{\alpha }-\xi \right)$-contraction mapping in the module of $b$ -metric spaces, where $\xi$ is a $b$-B-GGF on an interval $E$ with some tangible examples and certain important corollaries are adopted subsequently. Our newly proved results over recent ones chiefly due to Proinov ^[14] and Ali et al. ^[1]. As the end results of a succession, we promote our main results to prove the existence of solution for the system of integral inclusion.

2.   Multivalued SU-type $\varphi _{\xi}$-contraction

In this section, motivated by the notion of multivalued Suzuki type $\varphi$ -contraction, we define the notion of multivalued Suzuki type $\varphi _{\xi }$-contraction as follows:

Definition 2.1. Let $(\hat{U}, d_{b})$ be a $b$-metric space with $s\geq 1$, $\Lambda$ be a closed subset of $\hat{U}$ and $\xi$ be a $b$-B-GGF on an interval $E.$ A mapping $\widetilde{T}:\Lambda \rightarrow CB(\hat{U})$ is said to be a multivalued SU-type $\varphi$-contraction if there exists $\varphi \in \varphi ^{\ast }$ such that for $\widetilde{T}\widehat{\ell } \cap \Lambda \neq \emptyset$

implies that

where

for all $\widehat{\ell }\in \Lambda,$ $\nu \in \widetilde{T}\widehat{\ell } \cap \Lambda$ with $d_{b}(\widehat{\ell }, \nu)\in E,$ and $\hat{H}_{b}(\widetilde{T}\widehat{\ell }\cap \Lambda, \widetilde{T}\nu \cap \Lambda) > 0.$

Clearly in a class $b$-metric space, if an element $\widehat{\ell }_{0}\in \Lambda$ such that $O(\widehat{\ell }_{0})\subset \Lambda$ satisfies $d_{b}(\widehat{\ell }_{0}, \widetilde{T}\widehat{\ell }_{0})\in E$ and $\overline{b}(\widehat{\ell }_{i}, \rho _{i})\subset \Lambda$ for some $i\geq 0,$ then $\widehat{\ell }_{i+1}\in \Lambda$ and $\overline{b}(\widehat{\ell }_{i+1}, \rho _{i+1})\subset \overline{b}(\widehat{\ell }_{i}, \rho _{i}).$

Our first main result is as follows:

Theorem 2.2. Let $(\hat{U}, d_{b})$ be a complete $b$-metric space with $s\geq 1$, $\Lambda$ be a closed subset of $\hat{U}$ and $\widetilde{T} :\Lambda \rightarrow CB(\hat{U})$ be a multivalued SU-type $\varphi$ -contraction. Assume $\widehat{\ell }_{0}\in \Lambda$ such that $d_{b}(\widehat{\ell }_{0}, c^{\ast })\in E$ for some $c^{\ast }\in \widetilde{T} \widehat{\ell }_{0}\cap \Lambda.$ Then there exist an orbit $\{\widehat{ \ell }_{i}\}$ of $\widetilde{T}$ in $\Lambda$ and $\sigma ^{\ast }\in \Lambda$ such that $\lim_{i\rightarrow +\infty }\widehat{\ell }_{i} = \sigma ^{\ast }$. Moreover, $\sigma ^{\ast }$ is a fixed point of $\widetilde{T}$ if and only if the function $g(\widehat{\ell }): = d_{b}(\widehat{\ell }, \widetilde{T}\widehat{\ell }\cap \Lambda)$ is $\widetilde{T}$-o.l.s.c at $\sigma ^{\ast }$.

Proof. Choose $\widehat{\ell }_{1} = c^{\ast }\in \widetilde{T}\widehat{\ell } _{0}\cap \Lambda$. In the presence of this manner $d_{b}(\widehat{\ell } _{0}, \widehat{\ell }_{1}) = 0$, $\widehat{\ell }_{0}$ is a fixed point of $\widetilde{T}.$ Thus we assume that $d_{b}(\widehat{\ell }_{0}, \widehat{ \ell }_{1})\neq 0$. On the other hand, we have

Define $\rho = \sigma (d_{b}(\widehat{\ell }_{0}, \widehat{\ell }_{1})).$ From (1.16), we have $\sigma (r)\geq r.$ Hence $d_{b}(\widehat{\ell }_{0}, \widehat{ \ell }_{1})\leq \rho$ and so $\widehat{\ell }_{1}\in \overline{b}(\widehat{ \ell }_{0}, \rho).$ Since $d_{b}(\widehat{\ell }_{0}, \widehat{\ell }_{1})\in E$, from (2.1) and (2.2) it follows that

By the property of right continuity of $\varphi,$ there exists a real number $h_{1} > 1$ such that

From

by Lemma 1.3, there exists $\widehat{\ell }_{2}\in \widetilde{T} \widehat{\ell }_{1}\cap \Lambda$ such that $d_{b}(\widehat{\ell }_{1}, \widehat{\ell }_{2})\leq h_{1}H_{b}(\widetilde{T}\widehat{\ell }_{0}\cap \Lambda, \widetilde{T}\widehat{\ell }_{1}\cap \Lambda).$ Since $\varphi$ is nondecreasing, by (2.3), this inequality gives that

where

Now, we claim that

Let $\Delta = \max \left\{ d_{b}(\widehat{\ell }_{0}, \widehat{\ell } _{1}), d_{b}(\widehat{\ell }_{1}, \widetilde{T}\widehat{\ell }_{1})\right\}.$ Assume that $\Delta = d_{b}(\widehat{\ell }_{1}, \widetilde{T}\widehat{\ell }_{1})$. Since $\widehat{\ell }_{2}\in \widetilde{T}\widehat{\ell }_{1}\cap \Lambda$, we have

which is a contradiction. Hence (2.4) holds true. We assume that $d_{b}(\widehat{\ell }_{1}, \widehat{\ell }_{2})\neq 0$, otherwise, $\widehat{\ell } _{1}$ is a fixed point of $\widetilde{T}.$ From $(\varphi _{a})$, (2.4) implies that

and so $d_{b}(\widehat{\ell }_{1}, \widehat{\ell }_{2})\in E.$ Next, $\widehat{\ell }_{2}\in \overline{b}(\widehat{\ell }_{0}, \rho)$ since

Since

from (2.1), we have

Since $\varphi$ is right continuous, there exists a real number $h_{2} > 1$ such that

Next, from

by Lemma 1.3, there exists $\widehat{\ell }_{3}\in \widetilde{T} \widehat{\ell }_{2}\cap \Lambda$ such that $d_{b}(\widehat{\ell }_{2}, \widehat{\ell }_{3})\leq h_{2}H_{b}(\widetilde{T}\widehat{\ell }_{1}\cap \Lambda, \widetilde{T}\widehat{\ell }_{2}\cap \Lambda).$ By (2.5), this inequality gives that

where

This implies that

Let $\Delta = \max \left\{ d_{b}(\widehat{\ell }_{1}, \widehat{\ell } _{2}), d_{b}(\widehat{\ell }_{2}, \widetilde{T}\widehat{\ell }_{2})\right\}.$ Assume that $\Delta = d_{b}(\widehat{\ell }_{2}, \widetilde{T}\widehat{\ell }_{2})$. Since $\widehat{\ell }_{3}\in \widetilde{T}\widehat{\ell }_{2}\cap \Lambda$, we have

which is a contradiction. Hence (2.6) holds true. We assume that $d_{b}(\widehat{\ell }_{2}, \widehat{\ell }_{3})\neq 0$, otherwise, $\widehat{\ell } _{2}$ is a fixed point of $\widetilde{T}.$ From $(\varphi _{a})$, (2.6) implies that

and so $d_{b}(\widehat{\ell }_{2}, \widehat{\ell }_{3})\in E.$ Also, we have $\widehat{\ell }_{3}\in \overline{b}(\widehat{\ell }_{0}, \rho),$ since

Continuing this manner, we build two sequences $\left\{ \widehat{\ell } _{i}\right\} \subset \overline{b}(\widehat{\ell }_{0}, \rho)$ and $\left\{ h_{i}\right\} \subset (0, +\infty)$ such that $\widehat{\ell }_{i+1}\in \widetilde{T}\widehat{\ell }_{i}\cap \Lambda,$ $\widehat{\ell }_{i}\neq \widehat{\ell }_{i+1}$ with $d_{b}(\widehat{\ell }_{i}, \widehat{\ell } _{i+1})\in E$ and

for all $i\in \mathbb{N}.$ Then

Since $\varphi :\left(0, +\infty \right) \rightarrow \left(0, +\infty \right)$, it follows from (2.6) that

which implies that

By $(\varphi _{b})$ and Lemma 1.2, we have

Next, we prove that $\{\widehat{\ell }_{i}\}$ is a Cauchy sequence in $\hat{U }.$ Arguing by contradiction, we assume that there are $\epsilon > 0$ and sequences $\left\{ \delta _{i}\right\} _{i = 1}^{+\infty }$ and $\left\{ \kappa _{i}\right\} _{i = 1}^{+\infty }$ of natural numbers such that

Therefore,

Setting $i\rightarrow +\infty$ in (2.8),

From the trianguler inequality, we have

and

Letting the upper limit as $i\rightarrow +\infty$ in (2.10) and applying (2.7) and (2.9), we obtain

Again, setting the upper limit as $i\rightarrow +\infty$ in (2.11), we get

Therefore,

equivalently, we have

By the trianguler inequality,

Setting the limit as $i\rightarrow +\infty$ in (2.14), using (2.7) and (2.12), we have

Owing to above process, we find

From (2.15) and (2.16), we have

Owing to (2.7) and (2.9), we can choose a positive integer $j_{0}\geq 1$ such that

for all $i\geq j_{0}.$ From (2.1), we have

where

Setting the limit as $i\rightarrow +\infty$ and by (2.7), (2.9), (2.12) and (2.13), we have

By (2.15) and $\left(\varphi _{b}\right)$, we have

which is a contradiction. Therefore, we deduce that $\{\widehat{\ell }_{i}\}$ is a Cauchy sequence in the closed ball $\overline{b}(\widehat{\ell } _{0}, \rho).$ Since $\overline{b}(\widehat{\ell }_{0}, \rho)$ is closed in $\hat{U}$, there exists a $\sigma ^{\ast }\in \overline{b}(\widehat{\ell } _{0}, \rho)$ such that $\widehat{\ell }_{i}\rightarrow \sigma ^{\ast }$. Note that $\sigma ^{\ast }\in \Lambda$, since $\widehat{\ell }_{i+1}\in \widetilde{T}\widehat{\ell }_{i}\cap \Lambda.$ Next, we claim that

or

for all $i\in {\mathbb{N}}.$ Assume, on contrary, there exists $i^{\prime }\in {\mathbb{N}}$ such that

and

By (2.18), we have

which implies that

This together with (2.19) implies

So

From the contractive condition (2.1), we have

where

which yields

Let $\Delta = \max \left\{ d_{b}(\widehat{\ell }_{i^{\prime }}, \widehat{\ell } _{i^{\prime }+1}), d_{b}(\widehat{\ell }_{i^{\prime }+1}, \widehat{\ell } _{i^{\prime }+2})\right\}.$ Assume that $\Delta = d_{b}(\widehat{\ell }_{i^{\prime }+1}, \widehat{\ell }_{i^{\prime }+2}).$ Since $\widehat{\ell }_{i^{\prime }+2}\in \widetilde{T}\widehat{\ell }_{i^{\prime }+1}\cap \Lambda$, we have

which is a contradiction. Owing to $(\varphi _{a}),$ we have

From (2.19), (2.20) and (2.21), we obtain

which is a contradiction. Hence (2.17) holds true, that is,

Owing to (2.22), we have

Moreover, we know that $d_{b}(\widehat{\ell }_{i}, \widehat{\ell }_{i+1})\in E$ for all $i$. Thus, from (2.1), we have

where

which implies

Let $\Delta = \max \left\{ d_{b}(\widehat{\ell }_{i}, \widehat{\ell } _{i+1}), d_{b}(\widehat{\ell }_{i+1}, \widehat{\ell }_{i+2})\right\}.$ Assume that $\Delta = d_{b}(\widehat{\ell }_{i+1}, \widehat{\ell }_{i+2}).$ Since $\widehat{ \ell }_{i+2}\in \widetilde{T}\widehat{\ell }_{i+1}\cap \Lambda$, we have

which is a contradiction. Also, by $(\varphi _{a})$, we deduce that

Taking the limit $i\rightarrow +\infty$ in (2.23), we get

Since $g(\widehat{\ell }) = d_{b}(\widehat{\ell }, \widetilde{T}\widehat{\ell } \cap \Lambda)$ is $\widetilde{T}$-o.l.s.c at $\sigma ^{\ast }$,

Since $\widetilde{T}\sigma ^{\ast }$ is closed, we have $\sigma ^{\ast }\in \widetilde{T}\sigma ^{\ast }$. Conversely, if $\sigma ^{\ast }$ is a fixed point of $\widetilde{T}$ then $g(\sigma ^{\ast }) = 0\leq \liminf_{i}g(\widehat{\ell }_{i}),$ since $\sigma ^{\ast }\in \Lambda.$

Corollary 2.3. Let $(\hat{U}, d_{b})$ be a $b$-metric space with $s\geq 1$, $\Lambda$ be a closed subset of $\hat{U}$ and $\xi$ be a $b$-B-GGF on an interval $E.$ A mapping $\widetilde{T}:\Lambda \rightarrow CB(\hat{U})$ is said to be a multivalued SU-type $\varphi$-contraction if there exists $\varphi \in \varphi ^{\ast }$ such that for $\widetilde{T}\widehat{\ell }\cap \Lambda \neq \emptyset$

implies that

for all $\widehat{\ell }\in \Lambda,$ $\nu \in \widetilde{T}\widehat{\ell } \cap \Lambda$ with $d_{b}(\widehat{\ell }, \nu)\in E,$ where $H_{b}(\widetilde{T}\widehat{\ell }\cap \Lambda, \widetilde{T}\nu \cap \Lambda) > 0.$ Assume $\widehat{\ell }_{0}\in \Lambda$ such that $d_{b}(\widehat{\ell } _{0}, c^{\ast })\in E$ for some $c^{\ast }\in \widetilde{T}\widehat{\ell } _{0}\cap \Lambda.$ Then there exist an orbit $\{\widehat{\ell }_{i}\}$ of $\widetilde{T}$ in $\Lambda$ and $\sigma ^{\ast }\in \Lambda$ such that $\lim_{i\rightarrow +\infty }\widehat{\ell }_{i} = \sigma ^{\ast }$. Moreover, $\sigma ^{\ast }$ is a fixed point of $\widetilde{T}$ if and only if the function $g(\widehat{\ell }): = d_{b}(\widehat{\ell }, \widetilde{T}\widehat{ \ell }\cap \Lambda)$ is $\widetilde{T}$-o.l.s.c at $\sigma ^{\ast }$.

Corollary 2.4. Let $(\hat{U}, d_{b})$ be a $b$-metric space with $s\geq 1$, $\Lambda$ be a closed subset of $\hat{U}$ and $\xi$ be a $b$-B-GGF on an interval $E.$ A mapping $\widetilde{T}:\Lambda \rightarrow CB(\hat{U})$ is said to be a multivalued SU-type $\varphi$-contraction if there exists $\varphi \in \varphi ^{\ast }$ such that for $\widetilde{T}\widehat{\ell }\cap \Lambda \neq \emptyset$

implies that

for all $\widehat{\ell }\in \hat{U},$ $\nu \in \widetilde{T}\widehat{\ell }$ with $d_{b}(\widehat{\ell }, \nu)\in E.$ Suppose that $\widehat{\ell } _{0}\in \hat{U}$ such that $d_{b}(\widehat{\ell }_{0}, c^{\ast })\in E$ for some $c^{\ast }\in \widetilde{T}\widehat{\ell }_{0}.$ Then there exists an orbit $\{\widehat{\ell }_{i}\}$ of $\widetilde{T}$ in $\hat{U}$ which converges to the fixed point $\sigma ^{\ast }\in \mathcal{F} = \{\widehat{\ell }\in \hat{U}:d_{b}(\widehat{\ell }, \sigma ^{\ast })\in E\}$ of $\widetilde{T}.$

Example 2.5. Let $\hat{U} = [0, 1]$ be endowed with the metric $d_{b}$ with coefficient $s\geq \frac{\alpha ^{2}+7}{\alpha ^{2}-1} > 1$ [where $\alpha \geq 3$ is any positive integers] as defined by $d_{b}(\widehat{\ell }, \nu) = \left\vert \widehat{\ell }-\nu \right\vert ^{2}$ for all $\widehat{\ell }, \nu \in \hat{U}$ but not a metric $b_{d}$. For $\widehat{\ell }_{1} = 0,$ $\widehat{\ell }_{2} = \frac{1}{2}$ and $\widehat{\ell }_{3} = 1,$ we obtain

and let $E = [0, +\infty)$. Consider the mapping $\widetilde{T}:\hat{U} \rightarrow CB(\hat{U})$ defined by $\widetilde{T}(\widehat{\ell }) = [0, \widehat{\ell }^{2}]$. Clearly,

if and only if $\widehat{\ell }, \nu \in \lbrack 0, 1]$. Let $\widehat{\ell } _{0} = 1$. Then we have $c^{\ast } = \frac{1}{2}\in \widetilde{T}\widehat{\ell } _{0}$ such that $d_{b}(\widehat{\ell }_{0}, c^{\ast })\in E$ and

Set $\varphi (r) = re^{r}$ for all $r > 0$ and suppose that $\xi (r) = r^{2}$ is a $b$ -B-GGF of order $2$ on $E = [0, \frac{1}{\alpha -1}]$ with coefficient $\frac{ \alpha ^{2}+7}{\alpha ^{2}-1}$. For any $\widehat{\ell }\in \lbrack 0, 1]$ and $\nu \in \widetilde{T}\widehat{\ell }$, we get

Thus, all the conditions of Corollary 2.3 are fulfilled and $0$ is a fixed point of $\widetilde{T}.$

3.   Main results

In this section, motivated by the notion of multivalued Suzuki type $\nabla$ -contraction, we define the notion of multivalued Suzuki type $\left(\nabla _{\alpha }-\xi \right)$-contraction as follows:

Definition 3.1. Let $(\hat{U}, d_{b})$ be a $b$-metric space with $s\geq 1$, $\Lambda$ be a closed subset of $\hat{U}$ and $\xi$ be a $b$-B-GGF on an interval $E.$ A mapping $\widetilde{T}:\Lambda \rightarrow CB(\hat{U})$ is said to be a multivalued Suzuki type $(\nabla _{\alpha} -\xi)$-contraction if there exists $\Gamma \in \nabla$ such that for $\widetilde{T}\widehat{\ell }\cap \Lambda \neq \emptyset$

implies that

where

for all $\widehat{\ell }\in \Lambda,$ $\nu \in \widetilde{T}\widehat{\ell } \cap \Lambda$ with $d_{b}(\widehat{\ell }, \nu)\in E.$

The second one of our results is as follows.

Theorem 3.2. Let $(\hat{U}, d_{b})$ be a complete $b$-metric space with $s\geq 1$, $\Lambda$ be a closed subset of $\hat{U}$ and $\widetilde{T} :\Lambda \rightarrow CB(\hat{U})$ be a multivalued SU-type ($\alpha$-$\nabla$)-contraction. Suppose that the following conditions are satisfied:

$(i)$ $\widetilde{T}$ is $\alpha$-admissible;

$(ii)$ there exists $\widehat{\ell }_{0}\in \Lambda$ with $d_{b}(\widehat{\ell }_{0}, \widehat{\ell }_{1})\in E$ for some $\widehat{\ell } _{1}\in \widetilde{T}\widehat{\ell }_{0}\cap \Lambda$ such that $\alpha \left(\widehat{\ell }_{0}, \widehat{\ell }_{1}\right) \geq 1$.

Then there exist an orbit $\{\widehat{\ell }_{i}\}$ of $\widetilde{T}$ in $\Lambda$ and $\sigma ^{\ast }\in \Lambda$ such that $\lim_{i\rightarrow +\infty } \widehat{\ell }_{i} = \sigma ^{\ast }$. Moreover, $\sigma ^{\ast }$ is a fixed point of $\widetilde{T}$ if and only if the function $g(\widehat{\ell }): = d_{b}(\widehat{\ell }, \widetilde{T}\widehat{\ell }\cap \Lambda)$ is $\widetilde{T}$-o.l.s.c at $\sigma ^{\ast }$.

Proof. Owing to the hypothesis, there exists $\widehat{\ell }_{0}\in \Lambda$ with $d_{b}(\widehat{\ell }_{0}, \widehat{\ell }_{1})\in E$ for some $\widehat{\ell }_{1}\in \widetilde{T}\widehat{\ell }_{0}\cap \Lambda$ such that $\alpha \left(\widehat{\ell }_{0}, \widehat{\ell }_{1}\right) \geq 1.$ On the other hand, we have

If $d_{b}(\widehat{\ell }_{0}, \widehat{\ell }_{1}) = 0$, then $\widehat{\ell }_{0}$ is a fixed point of $\widetilde{T}.$ Thus, we assume that $d_{b}(\widehat{\ell }_{0}, \widehat{\ell }_{1})\neq 0.$ Define $\rho = \sigma (d_{b}(\widehat{\ell }_{0}, \widehat{\ell }_{1})).$ From (1.16), we have $\sigma (r)\geq r.$ Hence $d_{b}(\widehat{\ell }_{0}, \widehat{\ell } _{1})\leq \rho$ and so $\widehat{\ell }_{1}\in \overline{b}(\widehat{\ell } _{0}, \rho).$ Since $\alpha \left(\widehat{\ell }_{0}, \widehat{\ell } _{1}\right) \geq 1$ and $d_{b}(\widehat{\ell }_{0}, \widehat{\ell }_{1})\in E$, from (3.1) and (3.2), it follows that

which implies

We can choose an $\epsilon _{1} > 0$ such that

Thus

It follows from Lemma 1.2 that there exists $\widehat{\ell }_{2}\in \widetilde{T}\widehat{\ell }_{1}\cap \Lambda$ such that

From (3.3) and (3.4), we have

where

We claim that

Let $\Delta = \max \left\{ d_{b}(\widehat{\ell }_{0}, \widehat{\ell } _{1}), d_{b}(\widehat{\ell }_{1}, \widetilde{T}\widehat{\ell }_{1})\right\}.$ Assume that $\Delta = d_{b}(\widehat{\ell }_{1}, \widetilde{T}\widehat{\ell }_{1}).$ Since $\widehat{\ell }_{2}\in \widetilde{T}\widehat{\ell }_{1}\cap \Lambda$, we have

which is a contradiction. Hence (3.5) holds true. We assume that $d_{b}(\widehat{\ell }_{1}, \widehat{\ell }_{2})\neq 0$, otherwise, $\widehat{\ell } _{1}$ is a fixed point of $\widetilde{T}.$ Since $d_{b}(\widehat{\ell }_{1}, \widehat{\ell }_{2})\leq \xi (d_{b}(\widehat{\ell }_{0}, \widehat{\ell } _{1})) < d_{b}(\widehat{\ell }_{0}, \widehat{\ell }_{1}),$ we deduce that $d_{b}(\widehat{\ell }_{1}, \widehat{\ell }_{2})\in E.$ Next, $\widehat{\ell } _{2}\in \overline{b}(\widehat{\ell }_{0}, \rho)$ since

Since $\widetilde{T}$ is $\alpha$-admissible, $\alpha \left(\widehat{\ell }_{1}, \widehat{\ell }_{2}\right) \geq 1.$ Also, since

from the contractive condition (3.1), we get

This implies that

Now choose an $\epsilon _{2} > 0$ such that

Thus,

It follows from Lemma 1.2 that there exists $\widehat{\ell }_{3}\in \widetilde{T}\widehat{\ell }_{2}\cap \Lambda$ such that

From (3.6) and (3.7), we obtain

where

This implies that

Let $\Delta = \max \left\{ d_{b}(\widehat{\ell }_{1}, \widehat{\ell } _{2}), d_{b}(\widehat{\ell }_{2}, \widetilde{T}\widehat{\ell }_{2})\right\}.$ Assume that $\Delta = d_{b}(\widehat{\ell }_{2}, \widetilde{T}\widehat{\ell }_{2}).$ Since $\widehat{\ell }_{3}\in \widetilde{T}\widehat{\ell }_{2}\cap \Lambda$, we have

which is a contradiction. Hence (3.8) holds true. We assume that $d_{b}(\widehat{\ell }_{2}, \widehat{\ell }_{3})\neq 0$, otherwise, $\widehat{\ell } _{2}$ is a fixed point of $\widetilde{T}.$ From (3.8), we have $d_{b}(\widehat{\ell }_{2}, \widehat{\ell }_{3}) < d_{b}(\widehat{\ell }_{1}, \widehat{ \ell }_{2})$ and so $d_{b}(\widehat{\ell }_{2}, \widehat{\ell }_{3})\in E.$ Also, we have $\widehat{\ell }_{3}\in \overline{b}(\widehat{\ell }_{0}, \rho),$ since

Continuing this manner, we obtain a sequence $\left\{ \widehat{\ell } _{i}\right\} \subset \overline{b}(\widehat{\ell }_{0}, \rho)$ such that $\widehat{\ell }_{i+1}\in \widetilde{T}\widehat{\ell }_{i}\cap \Lambda,$ $\widehat{\ell }_{i}\neq \widehat{\ell }_{i+1}$ with $\alpha \left(\widehat{ \ell }_{i}, \widehat{\ell }_{i+1}\right) \geq 1$, $d_{b}(\widehat{\ell }_{i}, \widehat{\ell }_{i+1})\in E$ and by the above hypothesis, we have

For any $q\in \mathbb{N}$, by using the triangular inequality and (3.9), we get

Assume that

By (3.10) and (3.11), we get

Due to (3.11), (3.12) implies that $d_{b}(\widehat{\ell }_{i}, \widehat{\ell } _{i+q})\rightarrow 0$ as $i\rightarrow +\infty$. Hence $\{\widehat{\ell } _{i}\}$ is a Cauchy sequence in the closed ball $\overline{b}(\widehat{\ell } _{0}, \rho).$ Since $\overline{b}(\widehat{\ell }_{0}, \rho)$ is closed in $\hat{U}$, there exists an $\sigma ^{\ast }\in \overline{b}(\widehat{\ell } _{0}, \rho)$ such that $\widehat{\ell }_{i}\rightarrow \sigma ^{\ast }$. Note that $\sigma ^{\ast }\in \Lambda,$ since $\widehat{\ell }_{i+1}\in \widetilde{T}\widehat{\ell }_{i}\cap \Lambda.$ By the same argument as in Theorem 2.2, we have

Also, we know that $\alpha \left(\widehat{\ell }_{i}, \widehat{\ell } _{i+1}\right) \geq 1$ and $d_{b}(\widehat{\ell }_{i}, \widehat{\ell } _{i+1})\in E$ for all $n$. Thus, from (3.1), we have

which gives that

Since $\widehat{\ell }_{i+1}\in \widetilde{T}\widehat{\ell }_{i}\cap \Lambda,$ from (3.9), we get

Taking the limit $i\rightarrow +\infty$ in (3.13), we obtain

Since $g(\widehat{\ell }) = d_{b}(\widehat{\ell }, \widetilde{T}\widehat{\ell } \cap \Lambda)$ is $\widetilde{T}$-orbitally lower semi-continuous at $\sigma ^{\ast }$,

Since $\widetilde{T}\sigma ^{\ast }$ is closed, we have $\sigma ^{\ast }\in \widetilde{T}\sigma ^{\ast }$. Conversely, if $\sigma ^{\ast }$ is a fixed point of $\widetilde{T}$ then $g(\sigma ^{\ast }) = 0\leq \liminf_{i}g(\widehat{\ell }_{i}),$ since $\sigma ^{\ast }\in \Lambda.$

Setting $\Gamma (r, s) = s-\int_{0}^{r}\varsigma (t)dt\;{\rm{ for \;all }}\;r, s\geq 0$ in Theorem 3.2, we get the following result.

Corollary 3.3. Let $(\hat{U}, d_{b})$ be a complete $b$-metric space with $s\geq 1$, $\Lambda$ be a closed subset of $\hat{U},$ $\xi$ be a $b$-B-GGF on an interval $E$ and let $\widetilde{T}:\Lambda \rightarrow CB(\hat{U})$ be a given multivalued mapping. Suppose that for $\widetilde{T}\widehat{\ell }\cap \Lambda \neq \emptyset$ such that

implies that

for all $\widehat{\ell }\in \Lambda,$ $\nu \in \widetilde{T}\widehat{\ell } \cap \Lambda$ with $\widehat{d}(\widehat{\ell }, \nu)\in E,$ where $\varsigma :\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$ is a function such that $\int_{0}^{\epsilon }\varsigma (t)dt$ exists and $\int_{0}^{\epsilon }\varsigma (t)dt > \epsilon$ for all $\epsilon > 0$. Suppose that the following conditions are satisfied:

$(i)$ $\widetilde{T}$ is $\alpha$-admissible;

$(ii)$ there exists $\widehat{\ell }_{0}\in \Lambda$ with $d_{b}(\widehat{\ell }_{0}, \widehat{\ell }_{1})\in E$ for some $\widehat{\ell } _{1}\in \widetilde{T}\widehat{\ell }_{0}\cap \Lambda$ such that $\alpha \left(\widehat{\ell }_{0}, \widehat{\ell }_{1}\right) \geq 1$.

Then there exist an orbit $\{\widehat{\ell }_{i}\}$ of $\widetilde{T}$ in $\Lambda$ and $\sigma ^{\ast }\in \Lambda$ such that $\lim_{i\rightarrow +\infty } \widehat{\ell }_{i} = \sigma ^{\ast }$. Moreover, $\sigma ^{\ast }$ is a fixed point of $\widetilde{T}$ if and only if the function $g(\widehat{\ell }): = d_{b}(\widehat{\ell }, \widetilde{T}\widehat{\ell }\cap \Lambda)$ is $\widetilde{T}$-o.l.s.c at $\sigma ^{\ast }$.

Corollary 3.4. Let $(\hat{U}, d_{b})$ be a complete $b$-metric space with $s\geq 1$, $\xi$ be $b$ -B-GGF on an interval $E$ and let $\widetilde{T}: \hat{U}\rightarrow CB(\hat{U})$ be a given multivalued mapping. Suupose that there exist $\psi \in \Phi$ and $\Gamma \in \nabla$ such that

implies that

for all $\widehat{\ell }\in \hat{U},$ $\nu \in \widetilde{T}\widehat{\ell }$ with $d_{b}(\widehat{\ell }, \nu)\in E.$ Suppose that the following conditions are satisfied:

$(i)$ $\widetilde{T}$ is $\alpha$-admissible;

$(ii)$ there exists $\widehat{\ell }_{0}\in \hat{U}$ with $d_{b}(\widehat{\ell }_{0}, \widehat{\ell }_{1})\in E$ for some $\widehat{\ell } _{1}\in \widetilde{T}\widehat{\ell }_{0}$ such that$\alpha \left(\widehat{ \ell }_{0}, \widehat{\ell }_{1}\right) \geq 1$.

Then there exists an orbit $\{\widehat{\ell }_{i}\}$ of $\widetilde{T}$ in $\hat{U}$ which converges to the fixed point $\sigma ^{\ast }\in \mathcal{F} = \{\widehat{\ell }\in \hat{U} :d_{b}(\widehat{\ell }, \sigma ^{\ast })\in E\}$ of $\widetilde{T}.$

4.   An application

In the recent past, Banach's fixed point theorem has a broad family of important applications to an iteration methods for the system of linear algebraic equation and the most publicized application of Banach's fixed point theorem emarge in the module of function spaces. This yields the existence of solution for the system of differential and integral equations (see ^[3]). In this section, we investigate Corollary 2.4 to stabilize the existence of solution for the system of integral inclusions.

Consider the following system of integral inclusion:

where $\kappa \in \left(-\infty, +\infty \right)$, $U$ is a bounded compact subset of $\left(-\infty, +\infty \right)$ and the operator $D\left(t, \varsigma \left(t\right) \right)$ is lower semi-continuous. Let $\hat{U} = C(I)$ be the space of all continuous real valued functions ($C(I)$ is complete with respect to the metric $d_{b}$) endowed with the $b$-metric defined by

Assume that there exists $D:\left(-\infty, +\infty \right) \times \left(-\infty, +\infty \right) \rightarrow \left(-\infty, +\infty \right)$ which is continuous on

where $\alpha _{1} = \max_{u\in U}\left\vert U\right\vert,$ $0 < \alpha _{2} < \alpha _{1}$ and $h\geq 2$ such that

where $D$ is bounded as

Moreover, let $\check{C} = \left\{ \varsigma \in C(I):\widehat{V}\left(\varsigma, \kappa \right) \leq \frac{1}{2\alpha _{2}}\right\}$ be a closed subspace of $C(I)$ and the operator $g$ be defined by

Set $V_{\hat{U}}\left(r\right) = \int_{r_{0}}^{r}V\left(t, \varsigma \left(t\right) \right) dt.$ Note that

Then

This implies that

The third one of our results is as follows:

Theorem 4.1. Let $\hat{U} = C(I)$ be the space of all continuous real valued functions and $g:\left(\check{C}, d\right) \rightarrow \left(V\left(\check{C }\right), H_{b}\right)$ be a lower semi-continuous mapping. Suppose that the following assumptions hold:

(i) $g$ is defined for all $\varsigma \in \check{C};$

(ii) $g(\varsigma \left(r\right))$ is a compact subset of $\check{C}$ for all $\varsigma \in \check{C};$

Then the integral equation $\left(4.3\right)$ has a solution on

Proof. Let $\varkappa \in I$. Then $\left\vert \varkappa -r_{0}\right\vert \leq \left[ \frac{\alpha _{1}^{h-2}}{\alpha _{1}^{h-1}}\right].$ Hence we have $\left\vert \varsigma \left(\varkappa \right) -\kappa \right\vert \leq \frac{ 1}{2}\left(\frac{\alpha _{2}}{\alpha _{1}}\right)$. If $\left(\varkappa, \varsigma \left(\varkappa \right) \right) \in \left(-\infty, +\infty \right)$, then the integral equation in (4.1) exists. Since $\kappa \in \left(-\infty, +\infty \right)$ is continuous, $\varkappa$ is defined for all $\varkappa \in \check{C}.$ Next, let $\vartheta \left(r\right) \in g(\varsigma \left(r\right))$. Then $\vartheta \left(r\right) = \kappa + \overline{u}V_{\hat{U}}\left(r\right)$ for $\overline{u}\in U$ and so

Thus $\left\vert \vartheta \left(r\right) -\kappa \right\vert \leq \frac{1 }{2}\left(\frac{\alpha _{2}}{\alpha _{1}}\right)$ for all $\vartheta \left(r\right) \in g(\varsigma \left(r\right)).$ So $g(\varsigma \left(r\right))$ is a subset of $\check{C}.$ Now, let $\left\{ \varsigma _{i}\right\} \subset g(\varsigma \left(r\right))$. Then $\varsigma = \kappa +\overline{u_{i}}D_{\hat{U}}\left(r\right)$ for $\overline{u_{i}}\in U.$ Since $U$ is compact, there exists a subsequence $\widehat{u_{i^{\ast }}}\in \widehat{u_{i}}$ such that $\{\widehat{u_{i^{\ast }}}\}$ is convergent to $\overline{u}\in U$. Let $\widehat{u} = \kappa +\widehat{u}V_{\hat{U}}\left(r\right)$. Then

Hence $g(\varsigma \left(r\right))$ is a compact subset of $\check{C}$ for all $\varsigma \in \check{C}.$ Next,

Therefore, we get

Similarly,

Hence (4.2) implies that

Taking $\varphi \left(\varsigma \right) = \varsigma$, $\varsigma > 0$ and $\xi \left(\varsigma \right) = \left(\frac{\alpha _{1}}{\alpha _{2}}\right) ^{h-2}\varsigma ^{h},$ $\varsigma \in E$ with $d_{b}\left(\varsigma _{1}, \varsigma _{2}\right) < \frac{\alpha _{2}}{\alpha _{1}}$, we get

Hence the requied conditions $\left(\rm{i}\right)$-$\left(\rm{ii} \right)$ are equivalent to (a)-(b) of Corollary 2.3. So there exists a fixed point $c^{\ast }(\in \Lambda)$ in $\check{C}$, which is a bounded solution of (4.1).

5.   Conclusions

The paper deals with the pre-existing results of fixed point for multi-valued maps satisfying $\varphi$-contraction via $b$-B-GGF in the context of $b$-metric space. Within this frame work, we introduced two related fixed point results in $b$-metric space. Afterwards, the results have been explained by rendering concrete examples and some foremost corollaries have been deduced from the main results. At the end, we have proved existence theorem for the system of multi-valued integral inclusion.

Acknowledgments

We would like to express our sincere gratitude to the anonymous referee for his/her helpful comments that will help to improve the quality of the manuscript.

Conflict of interest

The authors declare that they have no competing interests.