On α-ψ-Contractive Condition for Single-Valued and Multi-Valued Operators in Strong b-Metric Spaces

Abstract

This paper aims to establish fixed point theorems in a complete strong b-metric space under the $\alpha$-$\psi$-contractive condition imposed on single-valued mappings. Subsequently, we prove certain fixed point theorems, both locally and globally, under the ${\alpha }_{\ast }$-$\psi$-contractive condition and the $\alpha$-$\psi$-contractive condition on multi-valued mappings in a complete strong b-metric space. The theorems presented in this paper extend, generalize, and improve various existing results in the literature. To demonstrate the superiority of the results, we present multiple examples throughout this article and two applications: one in dynamic programming and another in ordinary differential equations. Moreover, the proposed results provide stronger and more general conclusions compared to several well-known fixed point theorems in the literature. In particular, our findings highlight the novelty and superiority of the $\alpha$-$\psi$-contractive framework in the setting of strong b-metric spaces, offering broader applicability and deeper insight into both theoretical and applied contexts.

1. Introduction

The Fixed Point Theorem is a crucial tool in mathematics, particularly in analysis, geometry, and topology. It states that under specific conditions, a function will have at least one point that is mapped to itself, known as a fixed point. In the literature, Banach proved the first metric fixed point theorem in 1922. The Banach contraction principle is considered to be a fundamental tool for solving certain differential equations because the existence and uniqueness of solutions to these equations are reduced to the existence and uniqueness of a fixed point. In fact, this insight is likely the primary reason that metric fixed point theory has been extensively studied. There are many generalizations of the Banach contraction principle, obtained under weaker contractive assumptions within various types of distance spaces; see, for example, [1,2,3].

The contractive conditions on single-valued and multi-valued mappings are considered essential tools that guarantee the existence of fixed points in different distance spaces. Recently, Samet et al. ([4]) introduced the concept of $\alpha$-$\psi$-contractive mapping in complete metric space depending on the concept of $\alpha$-admissible mapping. Their results involve using a supporting function $\alpha$, which satisfies specific assumptions on the contractive condition to find the fixed point in the context of a complete metric space. Following that, many scholars extended and generalized that notion in different types of distance spaces for single functions and these have been studied for multi-functions; see, for example, [5,6,7,8,9,10,11].

In another direction, Alam and Imdad [12,13] developed relation-theoretic counterparts of classical metric concepts such as contraction, continuity, and completeness, and demonstrated that, under a universal relation (i.e., when $\mathcal{R}=X^2$), these new notions coincide with their standard metric analogues. A binary relation on a nonempty set X is defined as a subset of $X\times X$ and is denoted by $\left(\mathcal{R}\right);$ we say that x is related to y under $\mathcal{R}$ if and only if $(x,y)\in \mathcal{R}.$ Inspired by their work, Prasad [14] employed the notion of comparable mappings along with several classical fixed point theorems in the setting of relational and ordered metric spaces. Notably, he investigated the relationship between such relations and the concept of $\alpha$-admissible mappings, showing that $\alpha$-admissibility and certain binary relations are closely connected. He further proved that, under suitable assumptions, a binary relation $\mathcal{R}$ and $\alpha$-admissible mappings are equivalent (see Theorem 3.12 and Remark 3.13 in [14] for more details).

In recent years, the strong b-metric space has appeared in the state-of-the-art literature by Kirk and Shahzad (see [15]), as a remedy for the shortcomings of the b-metric space. Specifically, the b-metric space lacks continuity, and not all open balls are open sets. The strong b-metric space is obtained by replacing the relaxed triangle inequality in the b-metric space with a stronger one, as follows: $d(x,z)\le d(x,y)+sd(y,z)$, where $s\ge 1$. The class of strong b-metric spaces lies between the class of b-metric spaces and the class of metric spaces. This highlights the superiority of strong b-metric spaces over general b-metric spaces, as many well-known fixed point results that hold in strong b-metric spaces either do not fully apply or may completely fail in b-metric spaces. For instance, Dontchev and Hager [16] introduced an interesting extension of Nadler’s theorem—often referred to as the DH theorem—in the setting of a metric space. An and Dung [17] later showed that this theorem does not hold under the weaker assumption of a strong b-metric space (see Example 2.1 in [17]). Consequently, the authors of [18] modified the assumptions of the DH theorem to make it applicable in strong b-metric spaces, and demonstrated that their result still fails in b-metric spaces (see Example 6 in [18]).

Another notable example is due to Ćirić [19], who proved the existence and uniqueness of fixed points for quasi-contractions in metric spaces. This result has been generalized by several authors in the context of b-metric spaces. In particular, Amini-Harandi [20] extended it using the Fatou property with a contraction constant $k<{\displaystyle \frac{1}{s}}.$ Subsequently, in 2017, He et al. [21] and Zhao et al. [22] improved this result by removing the Fatou property. In 2019, Lu et al. [23] further generalized Ćirić’s theorem by extending the admissible contraction constant to the interval $[0,1),$ under certain additional assumptions. This means that this finding is not fully extended to b-metric space.

In the same year, Mitrović and Hussain [24] proved that on a complete strong b-metric space, every quasi-contraction $T:X\to X$ with $k<1$ has a unique fixed point. This confirms that in the strong b-metric setting, the theory of quasi-contractions fully extends the classical result, in contrast to the general b-metric case, where additional restrictions are often required.

It has been noted that several fixed point theorems have been established within the framework of strong b-metric spaces; see, for example, [18,24,25,26,27].

The manuscript is structured as follows. Section 2 is dedicated to recalling the essential definitions, lemmas, and preliminary results that will be fundamental throughout the paper. Section 3 introduces the concept of the $\alpha$-$\psi$-contractive for single-valued mappings in the context of strong b-metric spaces and proves some fixed point results for it. In Section 4, the existence theorems for multi-valued mappings that satisfy the $\alpha$-$\psi$-contractive condition in strong b-metric spaces are established. At the end, in Section 5, we present two applications to illustrate the applicability of our main result.

2. Preliminaries

In this section, we review some key definitions and fundamental results in metric and b-metric spaces that are related to the concept of $\alpha$-$\psi$-contraction mapping. We start by introducing the definition of a strong b-metric space.

Definition 1

([15]). Let W be a nonempty set and $s\ge 1.$ Let $\tilde{d}:W\times W\to \mathbb{R}$ be a mapping satisfying for each ${\varpi }_1,{\varpi }_2$, and ${\varpi }_3\in W:$

• SbM1. $\tilde{d}({\varpi }_1,{\varpi }_2)\ge 0,$
• SbM2. $\tilde{d}({\varpi }_1,{\varpi }_2)=0\iff {\varpi }_1={\varpi }_2,$
• SbM3. $\tilde{d}({\varpi }_1,{\varpi }_2)=\tilde{d}({\varpi }_2,{\varpi }_1),$
• SbM4. $\tilde{d}({\varpi }_1,{\varpi }_3)\le \tilde{d}({\varpi }_1,{\varpi }_2)+s\tilde{d}({\varpi }_2,{\varpi }_3).$

Then, the pair $(W,\tilde{d})$ is called a strong b-metric space $\left(SbMS\right)$.

Example 1

([18]). Let $W=\mathbb{R},$ and $\tilde{d}({\varpi }_1,{\varpi }_2)=max\left\{\right|{\varpi }_1-{\varpi }_2|,2|{\varpi }_1-{\varpi }_2|-1\}$ for all ${\varpi }_1,{\varpi }_2\in \mathbb{R},$ then $(W,\tilde{d})$ is an SbMS with $s=2.$ $\tilde{d}$ is not metric mapping on $\mathbb{R}$ since

$$\tilde{d}(2,6)\nless \tilde{d}(2,2.5)+\tilde{d}(2.5,6).$$

We need the following proposition to prove a sequence in $\left(SbMS\right)$ is a Cauchy sequence.

Proposition 1

([15]). Let ${\left({\varpi }_n\right)}_{n\in \mathbb{N}}$ be a sequence in a strong b-metric space and suppose

$$\sum _{n=1}^{+\infty }\tilde{d}({\varpi }_n,{\varpi }_{n+1})<+\infty .$$

Then, ${\left({\varpi }_n\right)}_{n\in \mathbb{N}}$ is a Cauchy sequence.

Definition 2.

Let $(W,\tilde{d})$ be a metric space

i. 

A mapping $\varphi :[0,+\infty )\to [0,+\infty )$ is said to be a comparison function if:

1. 

ϕ is a non-decreasing function;

2. 

${lim}_{n\to +\infty }{\varphi }^n\left(\iota \right)=0,$ for $\iota \ge 0.$

The class of all the comparison functions is denoted by $\mathsf{\Phi }.$

ii. 

Ψ is defined as $\mathsf{\Psi }=$ {$\psi \in \mathsf{\Phi }$ and such that ${\sum }_{n=1}^{+\infty }{\psi }^n\left(\iota \right)<+\infty ,$ for each $\iota >0$}. It is clear that $\mathsf{\Psi }\subset \mathsf{\Phi }.$

The following functions $\varphi$ are examples of comparison functions, that is, $\varphi \in \mathsf{\Phi }.$

• $\varphi \left(\iota \right)=\lambda \iota$ where $0\le \lambda <1;$
• $\varphi \left(\iota \right)={\displaystyle \frac{\iota }{1+\iota }};$
• $\varphi \left(\iota \right)=arctan\iota .$

Lemma 1

([28,29]). If $\varphi \in \mathsf{\Phi },$ then the following hold:

i. 

ϕ is continuous at $0;$

ii. 

for any $\iota >0$, $\varphi \left(\iota \right)<\iota .$

Berinde [30] presented the notion of $\left(c\right)-$ comparison function as follows.

Definition 3

([30]). A mapping $\phi :[0,+\infty )\to [0,+\infty )$ is said to be a $\left(c\right)-$ comparison function if:

i. 

φ is increasing;

ii. 

there exist $j_0\in \mathbb{N},$$a\in (0,1)$ and a convergent series of non-negative terms ${\sum }_{j=1}^{+\infty }{w}_j$ such that ${\phi }^{j+1}\left(\iota \right)\le a[{\phi }^j\left(\iota \right)+w_j],$ for $j\ge j_0$ and any $\iota \ge 0.$

Definition 4

([10]). Let $(W,\tilde{d})$ be a metric space and $\alpha :W\times W\to [0,+\infty )$ be a function. W satisfies the condition $\left(C_{\alpha }\right)$ if for every sequence ${\left({\varpi }_n\right)}_{n\in \mathbb{N}}$ in W such that $\alpha ({\varpi }_n,{\varpi }_{n+1})\ge 1$ for all n and ${\varpi }_n\to \varpi \in W$ as $n\to +\infty$, there exists a subsequence ${\left({\varpi }_{n_k}\right)}_{k\in \mathbb{N}}$ of a sequence ${\left({\varpi }_n\right)}_{n\in \mathbb{N}}$ with $\alpha ({\varpi }_{n_k},\varpi )\ge 1$ for all $k.$

The following definitions introduce the notion of the $\alpha$-admissible and the $\alpha$-$\psi$-contractive for single-valued mappings.

Definition 5.

Let $f:W\to W$ be a map and $\alpha :W\times W\to [0,+\infty )$ be a function. Then f is said to be:

i. 

α-admissible ([4]) if $\alpha ({\varpi }_1,{\varpi }_2)\ge 1$ implies $\alpha (f{\varpi }_1,f{\varpi }_2)\ge 1.$

ii. 

Triangular α-admissible ([31]) if

1. 

f is α-admissible;

2. 

$\alpha ({\varpi }_1,z)\ge 1$ and $\alpha (z,{\varpi }_2)\ge 1,$ then $\alpha ({\varpi }_1,{\varpi }_2)\ge 1.$

iii. 

α-orbital admissible ([32]) if $\alpha (\varpi ,f\varpi )\ge 1$ implies $\alpha (f\varpi ,f^2\varpi )\ge 1.$

The following function $f:(0,+\infty )\to (0,+\infty )$ is an example on f is $\alpha$-admissible (see [4]) defined by $f\left(\varpi \right)=ln(\varpi +1)$ where $\varpi >0,$ and

$$\alpha ({\varpi }_1,{\varpi }_2)=\left\{\begin{array}{cc}2\hfill & \mathrm{if}{\varpi }_1\ge {\varpi }_2,\hfill \\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Remark 1

([32]). Every α-admissible mapping is an α-orbital admissible mapping and every triangular α-admissible mapping is a triangular α-orbital admissible mapping.

Definition 6

([4]). Let $(W,\tilde{d})$ be a metric space, $\psi \in \mathsf{\Psi }$ and $\alpha :W\times W\to [0,+\infty ).$ The mapping $T:W\to W$ is called an α-ψ- contractive mapping if for all ${\varpi }_1,{\varpi }_2\in W$ the following hold:

$$\alpha ({\varpi }_1,{\varpi }_2)\tilde{d}(T{\varpi }_1,T{\varpi }_2)\le \psi \left(\tilde{d}({\varpi }_1,{\varpi }_2)\right).$$

The following is the main fixed point result from the paper by Samet et al.

Theorem 1

(Samet [4], Theorems 2.1 and 2.2). Let $(W,\tilde{d})$ be a complete metric space. Let $T:W\to W$ be an α-ψ- contractive single-valued mapping such that

i. 

T is α-admissible;

ii. 

There exists ${\varpi }_0\in W$ such that $\alpha ({\varpi }_0,T{\varpi }_0)\ge 1;$

iii. 

T is continuous or W satisfies for all $n\in \mathbb{N},{\left({\varpi }_n\right)}_{n\in \mathbb{N}}\subseteq W$ converges to ${\varpi }^{\ast }$ in W and $\alpha ({\varpi }_n,{\varpi }_{n+1})\ge 1$ then $\alpha ({\varpi }_n,{\varpi }^{\ast })\ge 1,\forall n.$

Then, T has a fixed point in $W.$

Berinde, in [33], improved the concept of $\left(c\right)-$ comparison function to a suitable notion called $\left(b\right)-$ comparison function which is applicable in b-metric space.

Definition 7.

A mapping $\phi :[0,+\infty )\to [0,+\infty )$ is said to be a $\left(b\right)-$ comparison function for $s\ge 1$ if:

i. 

φ is monotone increasing;

ii. 

There exist $j_0\in \mathbb{N}$, $a\in (0,1)$ and a convergent series of non-negative terms ${\sum }_{j=1}^{+\infty }{w}_j$ such that $s^{j+1}{\phi }^{j+1}\left(\iota \right)\le a{s}^j{\phi }^j\left(\iota \right)+w_j,$ for $j\ge j_0$ and any $\iota \ge 0.$

Denote by ${\mathsf{\Psi }}_b$ the class of $\left(b\right)$-comparison functions.

Observe that a (b)-comparison function is a (c)-comparison function when $s=1.$ In addition, observe that any (b)-comparison function is a comparison function due to the following lemma:

Lemma 2

([34]). Let $\phi :[0,+\infty )\to [0,+\infty )$ be a $\left(b\right)$-comparison function then we have:

1. 

The series ${\sum }_{j=0}^{+\infty }{s}^j{\phi }^j\left(\iota \right)$ converges for any $\iota \ge 0;$

2. 

The function $f:[0,+\infty )\to [0,+\infty )$ defined by: $f\left(\iota \right)={\sum }_{j=0}^{+\infty }{s}^j{\phi }^j\left(\iota \right),\iota \ge 0,$ is increasing and continuous at $0.$

Example 2.

The mapping $\psi :[0,+\infty )\to [0,+\infty ),$ defined by $\psi \left(\iota \right)={\displaystyle \frac{2}{3}}\iota ,$ is a (c)-comparison function but not a (b)-comparison function for any $s\ge {\displaystyle \frac{3}{2}},$ since the series ${\sum }_{n=1}^{+\infty }{s}^n{\left({\displaystyle \frac{2}{3}}\right)}^n\iota$ diverges.

Example 3

([33]). Let $(W,\tilde{d})$ be a b-metric space with $s\ge 1.$ Then, $\psi \left(\iota \right)=k\iota$ where $\iota \ge 0$ and $0<k<{\displaystyle \frac{1}{s}}$ is a (b)-comparison function.

Theorem 1 was extended partially for (b)-comparison functions in the context of a complete b-metric space in ([35,36]) and is given as follows.

Theorem 2

(Mukheimer [36], Theorems 2.1 and 2.2). Let $(W,\tilde{d})$ be a complete b-metric space with $s>1,$ and $\alpha :W\times W\to [0,+\infty ).$ Let $T:W\to W,$ be such that the following hold:

i. 

$\alpha ({\varpi }_1,{\varpi }_2)\tilde{d}(T{\varpi }_1,T{\varpi }_2)\le \psi \left(\tilde{d}({\varpi }_1,{\varpi }_2)\right)$ where $\psi \in {\mathsf{\Psi }}_b;$

ii. 

T is α-admissible;

iii. 

There exists ${\varpi }_0\in W$ such that $\alpha ({\varpi }_0,T{\varpi }_0)\ge 1;$

iv. 

T is continuous or W satisfies for all $n\in \mathbb{N},{\left({\varpi }_n\right)}_{n\in \mathbb{N}}\subseteq W$ converges to ${\varpi }^{\ast }$ in W and $\alpha ({\varpi }_n,{\varpi }_{n+1})\ge 1$ then $\alpha ({\varpi }_n,{\varpi }^{\ast })\ge 1,\forall n.$

Then, T has a fixed point in $W.$

Similar to Theorem 2, Karapınar proved his result assuming that T is $\alpha$-orbital admissible instead of $\alpha$-admissible; see Theorems 2.3 and 2.4 in [35]. In a subsequent section, we are able to extend Theorem 1 fully in the context of a complete strong b-metric space in Theorem 7.

Later Asl and others, in [6], extended the concepts given in Definitions 5 and 6 by introducing the suitable notions for multi-functions and then established some fixed point results for these multi-valued mappings.

Definition 8

([6]). A closed valued mapping $T:W\to 2^W$ is an ${\alpha }_{\ast }$-admissible whenever $\alpha ({\varpi }_1,{\varpi }_2)\ge 1$ implies ${\alpha }_{\ast }(T{\varpi }_1,T{\varpi }_2)\ge 1.$ Where ${\alpha }_{\ast }(A,B)=inf\left\{\alpha (a,b):a\in A,b\in B\right\}.$

Definition 9

([6]). Let $(W,\tilde{d})$ be a metric space, $\psi \in \mathsf{\Psi }$ and $\alpha :W\times W\to [0,+\infty ).$ The closed valued mapping $T:W\to 2^W$ is an ${\alpha }_{\ast }$-ψ- contractive mapping if for ${\varpi }_1,{\varpi }_2\in W$ the following holds:

$${\alpha }_{\ast }(T{\varpi }_1,T{\varpi }_2)\mathcal{H}(T{\varpi }_1,T{\varpi }_2)\le \psi \left(\tilde{d}({\varpi }_1,{\varpi }_2)\right),$$

where $\mathcal{H}$ is the Hausdorff–Pompeiu metric.

Theorem 3

([6]). Let $(W,\tilde{d})$ be a complete metric space, $\alpha :W\times W\to [0,+\infty )$ be a function, $\psi \in \mathsf{\Psi }$ be a strictly increasing map, and T be a closed valued, ${\alpha }_{\ast }-$ admissible, and ${\alpha }_{\ast }$-ψ- contractive multi-function on $W.$ Suppose that there exist ${\varpi }_0\in W$ and ${\varpi }_1\in T{\varpi }_0$ such that $\alpha ({\varpi }_0,{\varpi }_1)\ge 1.$ Assume that if ${\left({\varpi }_n\right)}_{n\in \mathbb{N}}$ is a sequence in W such that $\alpha ({\varpi }_n,{\varpi }_{n+1})\ge 1$ for all n and ${\varpi }_n\to \varpi ,$ then $\alpha ({\varpi }_n,\varpi )\ge 1$ for all $n.$ Then, T has a fixed point.

In 2014, Hussain and others [8] studied the existence of the fixed point in the closed ball for ${\alpha }_{\ast }$-$\psi$- contractive multi-functions instead of the whole space.

Theorem 4

([8]). Let $(W,\tilde{d})$ be a complete metric space and $T:W\to 2^W$ be a closed valued multi-mapping and ${\alpha }_{\ast }$-ψ- contractive mapping on the closed ball $B[{\varpi }_0;r]$ where ${\varpi }_0\in W$ and such that

i. 

T is ${\alpha }_{\ast }$-admissible;

ii. 

There exists ${\varpi }_0\in W$ and ${\varpi }_1\in T{\varpi }_0$ such that $\alpha ({\varpi }_0,{\varpi }_1)\ge 1;$

iii. 

For all $n\in \mathbb{N},{\left({\varpi }_n\right)}_{n\in \mathbb{N}}\subseteq B[{\varpi }_0;r]$ converges to ${\varpi }^{\ast }$ in $B[{\varpi }_0;r]$ and $\alpha ({\varpi }_n,{\varpi }_{n+1})\ge 1$ then $\alpha ({\varpi }_n,{\varpi }^{\ast })\ge 1,\forall n;$

iv. 

For ${\varpi }_0\in W,$ there exists ${\varpi }_1\in T{\varpi }_0$ such that

${\sum }_{i=0}^n{\psi }^i\left(\tilde{d}({\varpi }_0,{\varpi }_1)\right)<r.$

Then, T has a fixed point in $B[{\varpi }_0;r].$

In [10], Mohammadi and others modified the notions of ${\alpha }_{\ast }$-admissible and ${\alpha }_{\ast }$-$\psi$- contractive mapping for multi-valued mappings as follows.

Definition 10.

The multi-function T is α-admissible whenever for each ${\varpi }_1\in W$ and ${\varpi }_2\in T{\varpi }_1$ with $\alpha ({\varpi }_1,{\varpi }_2)\ge 1,$ we have $\alpha ({\varpi }_2,{\varpi }_3)\ge 1$ for all ${\varpi }_3\in T{\varpi }_2.$

Definition 11.

Let $(W,\tilde{d})$ be a metric space, $\psi \in \mathsf{\Psi }$ and $\alpha :W\times W\to [0,+\infty ).$ The closed valued mapping $T:W\to 2^W$ is an α-ψ- contractive mapping if for ${\varpi }_1,{\varpi }_2\in W$ the following holds:

$$\alpha ({\varpi }_1,{\varpi }_2)\mathcal{H}(T{\varpi }_1,T{\varpi }_2)\le \psi \left(\tilde{d}({\varpi }_1,{\varpi }_2)\right).$$

Theorem 5

([10]). Let $(W,\tilde{d})$ be a complete metric space, $\alpha :W\times W\to [0,+\infty )$ be a function, and $\psi \in \mathsf{\Psi }$ be a strictly increasing map. Let $T:W\to \mathcal{CB}\left(W\right)$ be an $\alpha -$ admissible multi-function such that for all ${\varpi }_1,{\varpi }_2\in W,\alpha ({\varpi }_1,{\varpi }_2)\mathcal{H}(T{\varpi }_1,T{\varpi }_2)\le \psi \left(\tilde{d}({\varpi }_1,{\varpi }_2)\right)$ and there exist ${\varpi }_0\in W$ and ${\varpi }_1\in T{\varpi }_0$ with $\alpha ({\varpi }_0,{\varpi }_1)\ge 1.$ If T is continuous or W satisfies the condition $\left(C_{\alpha }\right),$ then T has a fixed point, where $\mathcal{CB}\left(W\right)$ is the family of all nonempty closed and bounded subsets of $W.$

In [7], Bota and others introduced the notion of ${\alpha }_{\ast }$-$\psi$-contractive multi-valued mapping in b-metric space for $\psi \in {\mathsf{\Psi }}_b$ as follows.

Definition 12

([7]). Let $(W,\tilde{d})$ be a b-metric space, $\psi \in {\mathsf{\Psi }}_b$ and $\alpha :W\times W\to [0,+\infty )$, then the closed valued multi-mapping $T:W\to 2^W$ be an ${\alpha }_{\ast }$-ψ- contractive mapping of type $-\left(b\right)$ if for ${\varpi }_1,{\varpi }_2\in W$ the following hold:

$${\alpha }_{\ast }(T{\varpi }_1,T{\varpi }_2)\mathcal{H}(T{\varpi }_1,T{\varpi }_2)\le \psi \left(\tilde{d}({\varpi }_1,{\varpi }_2)\right).$$

Theorem 6

([7]). Let $(W,\tilde{d})$ be a complete b-metric space with $s>1$ and $T:W\to 2^W$ be an ${\alpha }_{\ast }$-ψ- contractive closed multi-valued operator of type $-\left(b\right)$ satisfying the following:

i. 

T is ${\alpha }_{\ast }$-admissible;

ii. 

There exists ${\varpi }_0\in W$ and ${\varpi }_1\in T{\varpi }_0$ such that $\alpha ({\varpi }_0,{\varpi }_1)\ge 1;$

iii. 

For all $n\in \mathbb{N},{\left({\varpi }_n\right)}_{n\in \mathbb{N}}\subseteq W$ converges to ${\varpi }^{\ast }$ in W and $\alpha ({\varpi }_n,{\varpi }_{n+1})\ge 1$, then $\alpha ({\varpi }_n,{\varpi }^{\ast })\ge 1,\forall n;$

Then, T has a fixed point in $W.$

The above theorem extends Theorem 3 partially for (b)-comparison functions in the context of a complete b-metric space. In a subsequent section, we are able to extend Theorem 3 fully in the context of a complete strong b-metric space in Theorem 9.

3. Results on Single-Valued Mappings

This section introduces $\alpha$-$\psi$-contractive mappings for single-valued functions in strong b-metric spaces and provides a representative example that demonstrates the advantages of our results over those in the literature.

Theorem 7.

Let $(W,\tilde{d})$ be a complete strong b-metric space (with $s\ge 1)$ and $\psi \in \mathsf{\Psi }$. Let $T:W\to W$ be an α-ψ- contractive single-valued mapping such that

i. 

T is α-admissible;

ii. 

There exists ${\varpi }_0\in W$ such that $\alpha ({\varpi }_0,T{\varpi }_0)\ge 1;$

iii. 

T is continuous or W satisfies for all $n\in \mathbb{N},{\left({\varpi }_n\right)}_{n\in \mathbb{N}}\subseteq W$ converges to ${\varpi }^{\ast }$ in W and $\alpha ({\varpi }_n,{\varpi }_{n+1})\ge 1$ then $\alpha ({\varpi }_n,{\varpi }^{\ast })\ge 1,\forall n.$

Then, T has a fixed point in $W.$

Proof. 

Let ${\varpi }_0\in W$ such that $\alpha ({\varpi }_0,T{\varpi }_0)\ge 1.$ Define the sequence ${\left({\varpi }_n\right)}_{n\in \mathbb{N}}\subseteq W$ by ${\varpi }_n=T{\varpi }_{n-1},$ for all $n\in \mathbb{N}.$ If ${\varpi }_n={\varpi }_{n+1}$ for some $n\in \mathbb{N}$, then we have nothing to prove since ${\varpi }_n$ is a fixed point. Assume that ${\varpi }_n\ne {\varpi }_{n+1}$ for all $n\in \mathbb{N}.$ Since T is $\alpha$-admissible and $\alpha ({\varpi }_0,{\varpi }_1)=\alpha ({\varpi }_0,T{\varpi }_0)\ge 1,$ we get $\alpha ({\varpi }_1,{\varpi }_2)\ge 1.$ By induction we have

$$\alpha ({\varpi }_n,{\varpi }_{n+1})\ge 1,\mathrm{for}\mathrm{all}n\in \mathbb{N}.$$

(1)

Hence

$$\begin{array}{ccc}\hfill \tilde{d}({\varpi }_n,{\varpi }_{n+1})=\tilde{d}(T{\varpi }_{n-1},T{\varpi }_n)& \le & \alpha ({\varpi }_{n-1},{\varpi }_n)\tilde{d}(T{\varpi }_{n-1},T{\varpi }_n)\hfill \\ & \le & \psi \left(\tilde{d}({\varpi }_{n-1},{\varpi }_n)\right)\hfill \\ & \le & {\psi }^n\left(\tilde{d}({\varpi }_0,{\varpi }_1)\right)\mathrm{for}\mathrm{all}n\in \mathbb{N}.\hfill \end{array}$$

That leads to ${\sum }_{n=1}^{+\infty }\tilde{d}({\varpi }_n,{\varpi }_{n+1})<+\infty$ since $\psi \in \mathsf{\Psi }.$ By completeness of W and by Proposition 1 we have ${\left({\varpi }_n\right)}_{n\in \mathbb{N}}$ is a Cauchy sequence which converges to some ${\varpi }^{\ast }\in W.$ Finally, by condition (iii) we have

Case 1.

By continuity of T we obtain

$$0=\underset{n\to +\infty }{lim}\tilde{d}({\varpi }_{n+1},{\varpi }^{\ast })=\underset{n\to +\infty }{lim}\tilde{d}(T{\varpi }_n,{\varpi }^{\ast })=\tilde{d}(T\underset{n\to +\infty }{lim}{\varpi }_n,{\varpi }^{\ast })=\tilde{d}(T{\varpi }^{\ast },{\varpi }^{\ast }).$$

Hence, ${\varpi }^{\ast }=T{\varpi }^{\ast },$ that is ${\varpi }^{\ast }$ is a fixed point of $T.$

Case 2.

Since ${\left({\varpi }_n\right)}_{n\in \mathbb{N}}\subseteq W$ is a Cauchy sequence, it converges to ${\varpi }^{\ast }\in W.$ Moreover, by Relation 1, we have $\alpha ({\varpi }_n,{\varpi }_{n+1})\ge 1$ for all $n\in \mathbb{N}.$ Hence, by Assumption (iii) of the theorem, it follows that $\alpha ({\varpi }_n,{\varpi }^{\ast })\ge 1$ for all $n\in \mathbb{N}$. Then

$$\begin{array}{ccc}\hfill \tilde{d}({\varpi }^{\ast },T{\varpi }^{\ast })& \le & \tilde{d}({\varpi }_{n+1},{\varpi }^{\ast })+s\tilde{d}(T{\varpi }^{\ast },T{\varpi }_n)\hfill \\ & \le & \tilde{d}({\varpi }_{n+1},{\varpi }^{\ast })+s\alpha ({\varpi }_n,{\varpi }^{\ast })\tilde{d}(T{\varpi }_n,T{\varpi }^{\ast })\hfill \\ & \le & \tilde{d}({\varpi }_{n+1},{\varpi }^{\ast })+s\psi \left(\tilde{d}({\varpi }_n,{\varpi }^{\ast })\right)\hfill \end{array}$$

Letting $n\to +\infty$ and by continuity of $\psi$ at $0,$ we obtain that ${\varpi }^{\ast }=T{\varpi }^{\ast }.$

□

Consider the following condition to obtain a unique fixed point of $\alpha$-$\psi$- contractive mapping.

$\left(U\right)$ For all ${\varpi }_1,{\varpi }_2$ belong to $W,$ there exists ${\varpi }_3\in W$ such that $\alpha ({\varpi }_1,{\varpi }_3)\ge 1$ and $\alpha ({\varpi }_2,{\varpi }_3)\ge 1.$

Theorem 8.

Adding condition $\left(U\right)$ to the hypotheses of the previous theorem, Theorem 7, we obtain uniqueness of the fixed point of $T.$

Proof. 

Suppose that ${\varpi }^{\ast }$ and ${\varpi }^{\ast \ast }$ are two fixed points of $T.$ Choose $\beta \in W$ with $\alpha ({\varpi }^{\ast },\beta )\ge 1$ and $\alpha ({\varpi }^{\ast \ast },\beta )\ge 1.$ Since T is $\alpha$-admissible, we have for all $n\in \mathbb{N}$,

$$\alpha ({\varpi }^{\ast },T^n\beta )\ge 1\mathrm{and}\alpha ({\varpi }^{\ast \ast },T^n\beta )\ge 1.$$

Using that ${\varpi }^{\ast }=T{\varpi }^{\ast }$ (and similarly ${\varpi }^{\ast \ast }=T{\varpi }^{\ast \ast }$),

$$\begin{array}{ccc}\hfill \tilde{d}({\varpi }^{\ast },T^n\beta )=\tilde{d}(T{\varpi }^{\ast },T\left(T^{n-1}\beta \right))& \le & \alpha ({\varpi }^{\ast },T^{n-1}\beta )\tilde{d}(T{\varpi }^{\ast },T\left(T^{n-1}\beta \right))\hfill \\ & \le & \psi \left(\tilde{d}({\varpi }^{\ast },T^{n-1}\beta )\right).\hfill \end{array}$$

By successive application of the contractive condition and if $\psi$ is a non-decreasing function, we get,

$$\begin{array}{ccc}\hfill \tilde{d}({\varpi }^{\ast },T^n\beta )& \le & \psi \left(\tilde{d}({\varpi }^{\ast },T^{n-1}\beta )\right)\hfill \\ & ⋮& \\ & \le & {\psi }^n\left(\tilde{d}({\varpi }^{\ast },\beta )\right).\hfill \end{array}$$

Since $\psi \in \mathsf{\Psi }$, the iterates ${\psi }^n\left(\iota \right)\to 0$ as $n\to +\infty$ for every $\iota \ge 0$. Hence

$$\underset{n\to +\infty }{lim}\tilde{d}({\varpi }^{\ast },T^n\beta )=0,$$

so $T^n\beta \to {\varpi }^{\ast }$. Similarly,

$$\underset{n\to +\infty }{lim}\tilde{d}({\varpi }^{\ast \ast },T^n\beta )=0.$$

Finally, by relaxed triangle inequality (SbM4),

$$\tilde{d}({\varpi }^{\ast },{\varpi }^{\ast \ast })\le \tilde{d}({\varpi }^{\ast },T^n\beta )+s\tilde{d}(T^n\beta ,{\varpi }^{\ast \ast }).$$

Letting $n\to +\infty$ gives $\tilde{d}({\varpi }^{\ast },{\varpi }^{\ast \ast })=0$; hence, ${\varpi }^{\ast }={\varpi }^{\ast \ast }$. □

If we take $s=1$ in Theorem 8, we deduce the following result.

Corollary 1

(Samet [4], Theorem 2.3). Let $(W,\tilde{d})$ be a complete metric space. Let $T:W\to W$ be an α-ψ- contractive single-valued mapping such that

i. 

T is α-admissible;

ii. 

There exists ${\varpi }_0\in W$ such that $\alpha ({\varpi }_0,T{\varpi }_0)\ge 1;$

iii. 

T is continuous or W satisfies for all $n\in \mathbb{N},{\left({\varpi }_n\right)}_{n\in \mathbb{N}}\subseteq W$ converges to ${\varpi }^{\ast }$ in W and $\alpha ({\varpi }_n,{\varpi }_{n+1})\ge 1$ then $\alpha ({\varpi }_n,{\varpi }^{\ast })\ge 1,\forall n.$

iv. 

For all ${\varpi }_1,{\varpi }_2\in W,$ there exists ${\varpi }_3\in W$ such that $\alpha ({\varpi }_1,{\varpi }_3)\ge 1$ and $\alpha ({\varpi }_2,{\varpi }_3)\ge 1.$

Then, T has a unique fixed point in $W.$

Corollary 2

(Banach contraction principle in a strong b-metric space). Let $(W,\tilde{d})$ be a complete strong b-metric space. Suppose $k\in (0,1)$ and suppose $T:W\to W$ satisfies

$$\tilde{d}(T\left({\varpi }_1\right),T\left({\varpi }_2\right))\le k\tilde{d}({\varpi }_1,{\varpi }_2)$$

for all ${\varpi }_1,{\varpi }_2\in W.$ Then, T has a unique fixed point ${\varpi }^{\ast }\in W.$

Proof. 

For all ${\varpi }_1,{\varpi }_2\in W$ put $\alpha ({\varpi }_1,{\varpi }_2)=1$ and $\psi \left(\iota \right)=k\iota$ in Theorem 8. Then, T has a unique fixed point. □

Example 4.

Let $W=\mathbb{R},$ and $\tilde{d}({\varpi }_1,{\varpi }_2)=max\left\{\right|{\varpi }_1-{\varpi }_2|,2|{\varpi }_1-{\varpi }_2|-1\}$ for all ${\varpi }_1,{\varpi }_2\in \mathbb{R},$ then $(W,\tilde{d})$ is a strong b-metric space with $s=2.$ Let $T:\mathbb{R}\to \mathbb{R}$ be defined by

$$T\varpi =\left\{\begin{array}{cc}2\varpi -{\displaystyle \frac{5}{4}}\hfill & \mathrm{if}\varpi >1,\hfill \\ {\displaystyle \frac{3\varpi }{4}}\hfill & \mathrm{if}\varpi \in [0,1],\hfill \\ 0\hfill & \mathrm{if}\varpi <0.\hfill \end{array}\right.$$

and

$$\alpha ({\varpi }_1,{\varpi }_2)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}{\varpi }_1,{\varpi }_2\in [0,1],\hfill \\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Clearly, T is continuous and T is α-admissible since

$$\alpha ({\varpi }_1,{\varpi }_2)\ge 1\Rightarrow {\varpi }_1,{\varpi }_2\in [0,1]\Rightarrow T{\varpi }_1,T{\varpi }_2\in [0,1]\Rightarrow \alpha (T{\varpi }_1,T{\varpi }_2)\ge 1.$$

For ${\varpi }_1,{\varpi }_2\in [0,1],$ assume $\psi \left(\iota \right)={\displaystyle \frac{3}{4}}\iota$ where $\iota \ge 0,$ then we have

$$\begin{array}{c}\hfill \alpha ({\varpi }_1,{\varpi }_2)\tilde{d}(T{\varpi }_1,T{\varpi }_2)={\displaystyle \frac{3}{4}}|{\varpi }_1-{\varpi }_2|\le \psi \left(\tilde{d}({\varpi }_1,{\varpi }_2)\right).\end{array}$$

For ${\varpi }_1,{\varpi }_2\in \mathbb{R}\setminus [0,1],\alpha ({\varpi }_1,{\varpi }_2)=0,$ then T is an α-ψ- contractive mapping. Moreover, when ${\varpi }_0={\displaystyle \frac{1}{3}}$, we have $\alpha ({\displaystyle \frac{1}{3}},T\left({\displaystyle \frac{1}{3}}\right))=\alpha ({\displaystyle \frac{1}{3}},{\displaystyle \frac{1}{4}})\ge 1.$

All assumptions of Theorem 7 hold, so T has a fixed point $0.$

Remark 2.

In the previous example, we note the following:

• The Banach contraction principle cannot be applied since

  $$\tilde{d}(T2,T4)=2|{\displaystyle \frac{11}{4}}-{\displaystyle \frac{27}{4}}|-1=7\nless k\tilde{d}(2,4)=3k,\mathrm{for}k\in [0,1).$$
• Theorem 1 cannot be applied since $(W,\tilde{d})$ is not a metric space.
• Theorem 2 cannot be applied since $\psi \notin {\mathsf{\Psi }}_b.$

4. Results on Multi-Valued Mappings

This section states and proves some results of the fixed point of the ${\alpha }_{\ast }$-$\psi$- contractive and $\alpha$-$\psi$- contractive for multi-valued mappings in the context of a strong b-metric.

We start this section by giving the following lemmas.

Lemma 3.

Let $(W,\tilde{d})$ be a strong b-metric space and $A,B$ be bounded closed subsets of $W.$ Then, for $c>1$ and each $a\in A$ we have

i. 

With $D(a,B)>0,$ there exists $b\in B$ such that $\tilde{d}(a,b)<cD(a,B)$ where $D(x,A)=inf\{\tilde{d}(x,a):a\in A\};$

ii. 

There exists $b\in B$ such that $\tilde{d}(a,b)<c\mathcal{H}(A,B).$

Proof. 

Given $D(a,B)>0,$ assume $\delta =(c-1)D(a,B)>0,$ since B is a closed and bounded set then the definition of $D(a,B)$ guarantees that there exists $b\in B$ such that $\tilde{d}(a,b)<\delta +D(a,B),$ Thus, $\tilde{d}(a,b)<cD(a,B)\le c\mathcal{H}(A,B).$ □

It is easy to show the following lemma

Lemma 4.

Let $(W,\tilde{d})$ be a complete strong b-metric space and $T:W\to \mathcal{CB}\left(W\right)$ be such that

i. 

T is ${\alpha }_{\ast }$-admissible or T is α-admissible as Definition 10;

ii. 

There exists ${\varpi }_0\in W$ and ${\varpi }_1\in T{\varpi }_0$ such that $\alpha ({\varpi }_0,{\varpi }_1)\ge 1;$

By creating the Picard iteration process, let ${\varpi }_n\in T{\varpi }_{n-1}$ for $n\ge 1.$ Then, we have that $\alpha ({\varpi }_{n-1},{\varpi }_n)\ge 1$ for all $n\in \mathbb{N}.$

Now, we present the version of the fixed point theorem for the set-valued T that is an ${\alpha }_{\ast }$-$\psi$-contractive mapping on a complete strong b-metric space.

Assume the notions of $\mathsf{\Psi }$ and ${\alpha }_{\ast }$ as they are defined in Definitions 2 and 8, respectively.

Definition 13.

Let $(W,\tilde{d})$ be a strong b-metric space. A multi-valued mapping $T:W\to \mathcal{CB}\left(W\right)$ is called an ${\alpha }_{\ast }$-ψ- contractive mapping whenever there exist $\alpha :W\times W\to [0,+\infty )$ and $\psi \in \mathsf{\Psi }.$ such that

$${\alpha }_{\ast }(T{\varpi }_1,T{\varpi }_2)\mathcal{H}(T{\varpi }_1,T{\varpi }_2)\le \psi \left(\tilde{d}({\varpi }_1,{\varpi }_2)\right)$$

(2)

for all ${\varpi }_1,{\varpi }_2\in W,$ and $\psi \in \mathsf{\Psi }.$

Theorem 9.

Let $(W,\tilde{d})$ be a complete strong b-metric space (with $s\ge 1)$ and $\psi \in \mathsf{\Psi }$ be a strictly increasing function. Let $T:W\to \mathcal{CB}\left(W\right)$ be an ${\alpha }_{\ast }$-ψ- contractive multi-valued mapping such that

i. 

T is ${\alpha }_{\ast }$-admissible;

ii. 

There exists ${\varpi }_0\in W$ and ${\varpi }_1\in T{\varpi }_0$ such that $\alpha ({\varpi }_0,{\varpi }_1)\ge 1;$

iii. 

T is continuous or W satisfies for all $n\in \mathbb{N},{\left({\varpi }_n\right)}_{n\in \mathbb{N}}\subseteq W$ converges to ${\varpi }^{\ast }$ in W and $\alpha ({\varpi }_n,{\varpi }_{n+1})\ge 1$ then $\alpha ({\varpi }_n,{\varpi }^{\ast })\ge 1,\forall n.$

Then, T has a fixed point in $W.$

Proof. 

By Assumption (ii), there exists ${\varpi }_0\in W$ and ${\varpi }_1\in T{\varpi }_0$ such that $\alpha ({\varpi }_0,{\varpi }_1)\ge 1.$ If ${\varpi }_0={\varpi }_1$, then we have nothing to prove. Let ${\varpi }_0\ne {\varpi }_1$ and assume that ${\varpi }_1\notin T{\varpi }_1,$ since otherwise ${\varpi }_1$ is a fixed point of $T.$ Since ${\varpi }_0\ne {\varpi }_1$, let $c>1$ and there exists ${\varpi }_2\in T{\varpi }_1$ such that

$$\begin{array}{ccc}\hfill D({\varpi }_1,T{\varpi }_1)\le \tilde{d}({\varpi }_1,{\varpi }_2)& <& c\mathcal{H}(T{\varpi }_0,T{\varpi }_1)\hfill \\ & \le & c{\alpha }_{\ast }(T{\varpi }_0,T{\varpi }_1)\mathcal{H}(T{\varpi }_0,T{\varpi }_1)\hfill \\ & \le & c\psi \left(\tilde{d}({\varpi }_0,{\varpi }_1)\right).\hfill \end{array}$$

Since $\psi$ is a strictly increasing function,

$$\psi \left(\tilde{d}({\varpi }_1,{\varpi }_2)\right)<\psi \left(c\psi \left(\tilde{d}({\varpi }_0,{\varpi }_1)\right)\right)$$

After choosing ${\varpi }_2\in T{\varpi }_1$ satisfying the above inequality, and since ${\varpi }_1\notin T{\varpi }_1$, it follows that ${\varpi }_1\ne {\varpi }_2$. We then define the auxiliary constant

$$c_1:={\displaystyle \frac{\psi \left(c\psi \left(\tilde{d}({\varpi }_0,{\varpi }_1)\right)\right)}{\psi \left(\tilde{d}({\varpi }_1,{\varpi }_2)\right))}}>1.$$

Take ${\varpi }_3\in T{\varpi }_2$ with ${\varpi }_2\ne {\varpi }_3$ since otherwise ${\varpi }_2$ is a fixed point. By definition T is an ${\alpha }_{\ast }$-admissible and since $\alpha ({\varpi }_0,{\varpi }_1)\ge 1$ we have $\alpha ({\varpi }_1,{\varpi }_2)\ge {\alpha }_{\ast }(T{\varpi }_0,T{\varpi }_1)\ge 1$, and hence

$$\begin{array}{ccc}\hfill D({\varpi }_2,T{\varpi }_2)\le \tilde{d}({\varpi }_2,{\varpi }_3)& <& c_1\mathcal{H}(T{\varpi }_1,T{\varpi }_2)\hfill \\ & \le & c_1{\alpha }_{\ast }(T{\varpi }_1,T{\varpi }_2)\mathcal{H}(T{\varpi }_1,T{\varpi }_2)\hfill \\ & \le & c_1\psi \left(\tilde{d}({\varpi }_1,{\varpi }_2)\right)\hfill \\ & =& \psi \left(c\psi \left(\tilde{d}({\varpi }_0,{\varpi }_1)\right)\right).\hfill \end{array}$$

Now, $\alpha ({\varpi }_2,{\varpi }_3)\ge 1$ and $\psi \left(\tilde{d}({\varpi }_2,{\varpi }_3)\right)<{\psi }^2\left(c\psi \left(\tilde{d}({\varpi }_0,{\varpi }_1)\right)\right).$ Similarly, after choosing ${\varpi }_3\in T{\varpi }_2$, we define

$$c_2:={\displaystyle \frac{{\psi }^2\left(c\psi \left(\tilde{d}({\varpi }_0,{\varpi }_1)\right)\right)}{\psi \left(\tilde{d}({\varpi }_2,{\varpi }_3)\right))}}>1.$$

Take ${\varpi }_4\in T{\varpi }_3$ with ${\varpi }_3\ne {\varpi }_4$ since otherwise ${\varpi }_3$ is a fixed point. Since $\alpha ({\varpi }_2,{\varpi }_3)\ge 1$, then we have

$$\begin{array}{ccc}\hfill D({\varpi }_3,T{\varpi }_3)\le \tilde{d}({\varpi }_3,{\varpi }_4)& <& c_2{\alpha }_{\ast }(T{\varpi }_2,T{\varpi }_3)\mathcal{H}(T{\varpi }_2,T{\varpi }_3)\hfill \\ & \le & c_2\psi \left(\tilde{d}({\varpi }_2,{\varpi }_3)\right)\hfill \\ & =& {\psi }^2\left(c\psi \left(\tilde{d}({\varpi }_0,{\varpi }_1)\right)\right).\hfill \end{array}$$

Continuing in this manner, we obtain a sequence ${\left({\varpi }_n\right)}_{n\in \mathbb{N}}\subseteq W$ such that for all $n\in \mathbb{N},{\varpi }_n\in T{\varpi }_{n-1},{\varpi }_{n-1}\ne {\varpi }_n,\alpha ({\varpi }_{n-1},{\varpi }_n)\ge 1$ and $\tilde{d}({\varpi }_n,{\varpi }_{n+1})<{\psi }^{n-1}\left(c\psi \left(\tilde{d}({\varpi }_0,{\varpi }_1)\right)\right).$ That leads to ${\sum }_{n=1}^{+\infty }\tilde{d}({\varpi }_n,{\varpi }_{n+1})<+\infty$ since $\psi \in \mathsf{\Psi }.$ Then, by completeness of W and by Proposition 1, we have ${\left({\varpi }_n\right)}_{n\in \mathbb{N}}$ as a Cauchy sequence which converges to ${\varpi }^{\ast }\in W.$ Finally, by Condition (iii) we have

Case 1.

By continuity of T we obtain

$$D({\varpi }^{\ast },T{\varpi }^{\ast })=D(\underset{n\to +\infty }{lim}{\varpi }_{n+1},T{\varpi }^{\ast })\le \mathcal{H}(\underset{n\to +\infty }{lim}T{\varpi }_n,T{\varpi }^{\ast })=0.$$

Hence, ${\varpi }^{\ast }\in T{\varpi }^{\ast }.$

Case 2.

Since T is ${\alpha }_{\ast }-$ admissible and $\forall n,\alpha ({\varpi }_n,{\varpi }^{\ast })\ge 1$, we have ${\alpha }_{\ast }(T{\varpi }_n,T{\varpi }^{\ast })\ge 1$ for all $n,$ and therefore

$$\begin{array}{ccc}\hfill D({\varpi }^{\ast },T{\varpi }^{\ast })& \le & \tilde{d}({\varpi }^{\ast },{\varpi }_{n+1})+s\tilde{d}({\varpi }_{n+1},z),z\in T{\varpi }^{\ast }\hfill \\ & \le & \tilde{d}({\varpi }^{\ast },{\varpi }_{n+1})+sq\mathcal{H}(T{\varpi }_n,T{\varpi }^{\ast }),q>1\hfill \\ & \le & \tilde{d}({\varpi }^{\ast },{\varpi }_{n+1})+sq{\alpha }_{\ast }(T{\varpi }_n,T{\varpi }^{\ast })\mathcal{H}(T{\varpi }_n,T{\varpi }^{\ast })\hfill \\ & \le & \tilde{d}({\varpi }^{\ast },{\varpi }_{n+1})+sq\psi \left(\tilde{d}({\varpi }_n,{\varpi }^{\ast })\right).\hfill \end{array}$$

When n tends to infinity, we conclude that ${lim}_{n\to +\infty }D({\varpi }^{\ast },T{\varpi }^{\ast })=0,$ and since $T\left({\varpi }^{\ast }\right)$ is closed, then ${\varpi }^{\ast }\in T{\varpi }^{\ast }$ and ${\varpi }^{\ast }$ is a fixed point of $T.$

□

Lemma 5 explains the relation between the concept of $\alpha$-admissible and ${\alpha }_{\ast }$-admissible for the multi-function $T.$

Lemma 5.

Every ${\alpha }_{\ast }$-admissible $T:W\to \mathcal{CB}\left(W\right)$ is α-admissible.

Proof. 

Assume $\alpha ({\varpi }_1,{\varpi }_2)\ge 1$ where ${\varpi }_2\in T{\varpi }_1,$ since T is an ${\alpha }_{\ast }$-admissible, then

$$\begin{array}{c}\hfill {\alpha }_{\ast }(T{\varpi }_1,T{\varpi }_2)=inf\{\alpha (z_1,z_2):z_1\in T{\varpi }_1,z_2\in T{\varpi }_2\}\ge 1.\end{array}$$

Hence

$$\alpha ({\varpi }_2,{\varpi }_3)\ge {\alpha }_{\ast }(T{\varpi }_1,T{\varpi }_2)\ge 1$$

where ${\varpi }_2\in T{\varpi }_1$, and ${\varpi }_3\in T{\varpi }_2,$ then T is an $\alpha$-admissible. □

Similarly to proof of Theorem 9, it is easy to prove the following theorem.

Theorem 10.

Let $(W,\tilde{d})$ be a complete strong b-metric space (with $s\ge 1)$ and $\psi \in \mathsf{\Psi }$ be a strictly increasing function. Let $T:W\to \mathcal{CB}\left(W\right)$ be an α-ψ- contractive multi-valued mapping such that

i. 

T is α-admissible;

ii. 

There exists ${\varpi }_0\in W$ and ${\varpi }_1\in T{\varpi }_0$ such that $\alpha ({\varpi }_0,{\varpi }_1)\ge 1;$

iii. 

T is continuous or W satisfies for all $n\in \mathbb{N},{\left({\varpi }_n\right)}_{n\in \mathbb{N}}\subseteq W$ converges to ${\varpi }^{\ast }$ in W and $\alpha ({\varpi }_n,{\varpi }_{n+1})\ge 1$ then $\alpha ({\varpi }_n,{\varpi }^{\ast })\ge 1,\forall n.$

Then, T has a fixed point in $W.$

Inspired by Theorem 4, the following result studies fixed points on a closed ball, which is a subset of $W.$

Theorem 11.

Let $(W,\tilde{d})$ be a complete strong b-metric space (with $s\ge 1)$ and $\psi \in \mathsf{\Psi }$ be a strictly increasing function. Let $T:W\to \mathcal{CB}\left(W\right)$ be an α-ψ- contractive multi-valued mapping such that

i. 

T is α-admissible;

ii. 

There exists ${\varpi }_0\in W$ and ${\varpi }_1\in T{\varpi }_0$ such that $\alpha ({\varpi }_0,{\varpi }_1)\ge 1;$

iii. 

T is continuous or for all $n\in \mathbb{N},{\left({\varpi }_n\right)}_{n\in \mathbb{N}}\subseteq B[{\varpi }_0;r]$ converges to ${\varpi }^{\ast }$ in $B[{\varpi }_0;r]$ and $\alpha ({\varpi }_n,{\varpi }_{n+1})\ge 1$ then $\alpha ({\varpi }_n,{\varpi }^{\ast })\ge 1,\forall n.$

iv.

For ${\varpi }_0\in W,$ there exists ${\varpi }_1\in T{\varpi }_0$ such that $\forall n\in \mathbb{N}$ and $r>0,$

$\tilde{d}({\varpi }_0,{\varpi }_1)+{\sum }_{i=0}^n{\psi }^i\left(c\psi \left(\tilde{d}({\varpi }_0,{\varpi }_1)\right)\right)\le {\displaystyle \frac{r}{s}}$ where $c>1.$

Then, T has a fixed point in $B[{\varpi }_0;r].$

Proof. 

Similarly to the lines of argument given in the proof of Theorem 9, we have a sequence ${\left({\varpi }_n\right)}_{n\in \mathbb{N}}\subseteq W$ such that for all $n\in \mathbb{N},{\varpi }_n\in T{\varpi }_{n-1},{\varpi }_{n-1}\ne {\varpi }_n,\alpha ({\varpi }_{n-1},{\varpi }_n)\ge 1$ and $\tilde{d}({\varpi }_n,{\varpi }_{n+1})<{\psi }^{n-1}\left(c\psi \left(\tilde{d}({\varpi }_0,{\varpi }_1)\right)\right).$ Using (iv), we have that

$$\begin{array}{ccc}\hfill \tilde{d}({\varpi }_{n+1},{\varpi }_0)& \le & s\tilde{d}({\varpi }_0,{\varpi }_1)+s\tilde{d}({\varpi }_1,{\varpi }_2)+s\tilde{d}({\varpi }_2,{\varpi }_3)+\dots +s\tilde{d}({\varpi }_n,{\varpi }_{n+1})\hfill \\ & \le & s[\tilde{d}({\varpi }_0,{\varpi }_1)+c\psi \left(\tilde{d}({\varpi }_0,{\varpi }_1)\right)+\psi \left(c\psi \left(\tilde{d}({\varpi }_0,{\varpi }_1)\right)\right)+\dots +{\psi }^{n-1}\left(c\psi \left(\tilde{d}({\varpi }_0,{\varpi }_1)\right)\right)]\hfill \\ & \le & s[\tilde{d}({\varpi }_0,{\varpi }_1)+\sum _{i=0}^{n-1}{\psi }^i\left(c\psi \left(\tilde{d}({\varpi }_0,{\varpi }_1)\right)\right)]\mathrm{where}{\psi }^0\left(c\psi \left(\tilde{d}({\varpi }_0,{\varpi }_1)\right)\right)=c\psi \left(\tilde{d}({\varpi }_0,{\varpi }_1)\right)\hfill \\ & \le & r.\hfill \end{array}$$

Hence we have ${\left({\varpi }_n\right)}_{n\in \mathbb{N}}\subseteq B[{\varpi }_0;r].$ Then imitating the rest of proof of Theorem 9 and by completeness of $B[{\varpi }_0;r],$ we obtain the desired result. □

Using Lemma 5 and Theorem 11, we get the following corollary.

Corollary 3.

Let $(W,\tilde{d})$ be a complete strong b-metric space (with $s\ge 1)$ and $\psi \in \mathsf{\Psi }$ be a strictly increasing function. Let $T:W\to \mathcal{CB}\left(W\right)$ be an ${\alpha }_{\ast }$-ψ- contractive multi-valued mapping such that

i. 

T is ${\alpha }_{\ast }$-admissible;

ii. 

There exists ${\varpi }_0\in W$ and ${\varpi }_1\in T{\varpi }_0$ such that $\alpha ({\varpi }_0,{\varpi }_1)\ge 1;$

iii. 

T is continuous or for all $n\in \mathbb{N},{\left({\varpi }_n\right)}_{n\in \mathbb{N}}\subseteq B[{\varpi }_0;r]$ converges to ${\varpi }^{\ast }$ in $B[{\varpi }_0;r]$ and $\alpha ({\varpi }_n,{\varpi }_{n+1})\ge 1$ then $\alpha ({\varpi }_n,{\varpi }^{\ast })\ge 1,\forall n.$

iv. 

For ${\varpi }_0\in W,$ there exists ${\varpi }_1\in T{\varpi }_0$ such that $\forall n\in \mathbb{N}$ and $r>0,$

$\tilde{d}({\varpi }_0,{\varpi }_1)+{\sum }_{i=0}^n{\psi }^i\left(c\psi \left(\tilde{d}({\varpi }_0,{\varpi }_1)\right)\right)\le {\displaystyle \frac{r}{s}}$ where $c>1.$

Then, T has a fixed point in $B[{\varpi }_0;r].$

Corollary 4

(Nadler’s theorem in strong b-metric spaces, (Tassaddiq [27], Theorem 3.4)). Let $(W,\tilde{d})$ be a complete strong b-metric space (with $s\ge 1)$ and $T:W\to \mathcal{CB}\left(W\right)$ be a multi-valued mapping such that

$$\mathcal{H}(T{\varpi }_1,T{\varpi }_2)\le k\tilde{d}({\varpi }_1,{\varpi }_2)\forall {\varpi }_1,{\varpi }_2\in W\mathrm{and}k\in [0,1).$$

Then, T has a fixed point.

Proof. 

For all ${\varpi }_1,{\varpi }_2\in W$ put $\alpha ({\varpi }_1,{\varpi }_2)=1$ and $\psi \left(\iota \right)=k\iota$ in Theorem 10. Then, T has a fixed point. □

Example 5.

Let $W=\mathbb{R}$ and $\tilde{d}({\varpi }_1,{\varpi }_2)=max\left\{\right|{\varpi }_1-{\varpi }_2|,2|{\varpi }_1-{\varpi }_2|-1\}$ for all ${\varpi }_1,{\varpi }_2\in \mathbb{R}$ and let $T:\mathbb{R}\to \mathcal{CB}\left(\mathbb{R}\right)$ be defined by

$$T\varpi =\left\{\begin{array}{cc}[0,{\displaystyle \frac{\varpi }{9}}]\hfill & \mathrm{if}\varpi \in [0,1],\hfill \\ {\displaystyle \frac{1}{9}},{\displaystyle \frac{3}{9}}]\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

and

$$\alpha ({\varpi }_1,{\varpi }_2)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}{\varpi }_1,{\varpi }_2\in [0,1],\hfill \\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

For $\psi \left(\iota \right)=\iota /2$ and ${\varpi }_1,{\varpi }_2\in [0,1]$ with ${\varpi }_1\le {\varpi }_2,$ we have

$$\begin{array}{c}\hfill \alpha ({\varpi }_1,{\varpi }_2)\mathcal{H}(T{\varpi }_1,T{\varpi }_2)={\displaystyle \frac{1}{9}}\tilde{d}({\varpi }_1,{\varpi }_2)\le {\displaystyle \frac{1}{2}}\tilde{d}({\varpi }_1,{\varpi }_2)=\psi (\tilde{d}({\varpi }_1,{\varpi }_2).\end{array}$$

For ${\varpi }_1,{\varpi }_2\in \mathbb{R}\setminus [0,1],\alpha ({\varpi }_1,{\varpi }_2)=0$, then Inequality (2) holds. Hence, T is an α-ψ- contractive mapping. By the definitions of T and α, we have

• T is α-admissible;
• For ${\varpi }_0=1$ and ${\varpi }_1={\displaystyle \frac{1}{9}}$ we have $\alpha (1,{\displaystyle \frac{1}{9}})=1;$
• For all $n\in \mathbb{N},{\left({\varpi }_n\right)}_{n\in \mathbb{N}}\subseteq W$ converges to ${\varpi }^{\ast }$ in W and $\alpha ({\varpi }_n,{\varpi }_{n+1})\ge 1$, then ${\left({\varpi }_n\right)}_{n\in \mathbb{N}}\subseteq [0,1],$ so ${\varpi }^{\ast }\in [0,1]$; hence, $\alpha ({\varpi }_n,{\varpi }^{\ast })\ge 1,\forall n.$

All assumptions of Theorem 10 hold, so T has a fixed point 0.

On the other hand, let ${\varpi }_0={\displaystyle \frac{1}{2}},{\varpi }_1={\displaystyle \frac{1}{18}},c={\displaystyle \frac{3}{2}}$, and $r={\displaystyle \frac{20}{9}},$ then $B[{\displaystyle \frac{1}{2}};{\displaystyle \frac{20}{9}}]=[{\displaystyle \frac{-20}{18}},{\displaystyle \frac{38}{18}}]$, and then by Theorem 11, T has a fixed point in $B[{\displaystyle \frac{1}{2}};{\displaystyle \frac{20}{9}}].$

Finally, comparing Theorems 6, 9, and 10 leads us to ask the following question.

Open question.

Can Theorem 10 be extended to the b-metric space?

5. Applications

This section presents two important applications of our results, demonstrating the existence of solutions to nontrivial equations, supported by a numerical example.

5.1. An Application to Dynamic Programming

In this part, we illustrate our results by investigating the existence of solutions for a different functional equation, distinct from the one arising in parameterized dynamic programming problems that characterizes the optimal return function with initial state $\varpi .$ The functional equation is given by

$$u\left({\varpi }_1\right)=\underset{{\varpi }_2\in \mathfrak{D}}{sup}\{g({\varpi }_1,{\varpi }_2)+F({\varpi }_1,{\varpi }_2,u\left(T({\varpi }_1,{\varpi }_2)\right))\},$$

(3)

where ${\varpi }_1\in \mathfrak{S}$ is the state vector ($\mathfrak{S}$ is a state space), ${\varpi }_2$ is a control or decision vector ($\mathfrak{D}$ is a decision space), $g:\mathfrak{S}\times \mathfrak{D}\to \mathbb{R}$ represents the immediate reward (instant gain from action ${\varpi }_2$), $T:\mathfrak{S}\times \mathfrak{D}\to \mathfrak{S}$ represents the transformation of the process, and $F:\mathfrak{S}\times \mathfrak{D}\times \mathbb{R}\to \mathbb{R}$ represents the future reward, often recursively defined via a value function, see [37] for more details.

We work in the complete strong b-metric space $(W,\tilde{d})=(B\left(\mathfrak{S}\right),\tilde{d})$ with relaxation constant $s=2,$ where $B\left(\mathfrak{S}\right)$ denotes the family of all bounded real functions defined on nonempty set $\mathfrak{S},$ and $\tilde{d}(u,v)={sup}_{\varpi \in \mathfrak{S}}\left\{\right|u\left(\varpi \right)-v\left(\varpi \right)|,2|u\left(\varpi \right)-v\left(\varpi \right)|-1\}.$

First we need the following lemmas:

Lemma 6.

Let $G,H:\mathfrak{S}\to \mathbb{R}$ be two bounded functions, then

$$|\underset{\varpi \in \mathfrak{S}}{sup}G\left(\varpi \right)-\underset{\varpi \in \mathfrak{S}}{sup}H\left(\varpi \right)|\le \underset{\varpi \in \mathfrak{S}}{sup}|G\left(\varpi \right)-H\left(\varpi \right)|$$

Lemma 7.

Suppose the following assumptions:

1. 

$g:\mathfrak{S}\times \mathfrak{D}\to \mathbb{R}$ and $F(·,·,0):\mathfrak{S}\times \mathfrak{D}\times \left\{0\right\}\to \mathbb{R}$ are bounded functions.

2. 

There exists $M\ge 0$ such that

$$|F({\varpi }_1,{\varpi }_2,\iota )-F({\varpi }_1,{\varpi }_2,\rho )|\le M|\iota -\rho |$$

for all ${\varpi }_1\in \mathfrak{S},{\varpi }_2\in \mathfrak{D}$ and $\iota ,\rho \in \mathbb{R}.$

Then, for all $u\in B\left(\mathfrak{S}\right)$ and all ${\varpi }_1\in \mathfrak{S},$ the operator $R:B\left(\mathfrak{S}\right)\to B\left(\mathfrak{S}\right)$ given as

$$Ru\left({\varpi }_1\right)=\underset{{\varpi }_2\in \mathfrak{D}}{sup}\{g({\varpi }_1,{\varpi }_2)+F({\varpi }_1,{\varpi }_2,u\left(T({\varpi }_1,{\varpi }_2)\right))\}$$

(4)

is well defined.

Proof. 

Let $u\in B\left(\mathfrak{S}\right)$ be arbitrary, since u is bounded; then there exists $M_1>0$ such that

$$\left|u\left({\varpi }_1\right)\right|\le M_1\mathrm{for}\mathrm{all}{\varpi }_1\in \mathfrak{S}.$$

By Assumption (1), there exist two real positive constants $M_2$ and $M_3$ such that for ${\varpi }_1\in \mathfrak{S}$ and ${\varpi }_2\in \mathfrak{D},$

$$\left|g({\varpi }_1,{\varpi }_2)\right|\le M_2\mathrm{and}\left|F({\varpi }_1,{\varpi }_2,0)\right|\le M_3.$$

Hence,

$$\begin{array}{ccc}\hfill |g({\varpi }_1,{\varpi }_2)+F({\varpi }_1,{\varpi }_2,u\left(T({\varpi }_1,{\varpi }_2)\right))|& \le & \left|g({\varpi }_1,{\varpi }_2)\right|+\left|F({\varpi }_1,{\varpi }_2,u\left(T({\varpi }_1,{\varpi }_2)\right))\right|\hfill \\ & \le & M_2+|F({\varpi }_1,{\varpi }_2,u\left(T({\varpi }_1,{\varpi }_2)\right))-F({\varpi }_1,{\varpi }_2,0)|+\left|F({\varpi }_1,{\varpi }_2,0)\right|\hfill \\ & \le & M_2+M\left|u\left(T({\varpi }_1,{\varpi }_2)\right)\right|+M_3\hfill \\ & \le & M_2+M·M_1+M_3=K.\hfill \end{array}$$

Therefore, for all ${\varpi }_1\in \mathfrak{S},$ we have

$$\begin{array}{ccc}\hfill \left|Ru\right({\varpi }_1\left)\right|& \le & \underset{{\varpi }_2\in \mathfrak{D}}{sup}|g({\varpi }_1,{\varpi }_2)+F({\varpi }_1,{\varpi }_2,u\left(T({\varpi }_1,{\varpi }_2)\right))|\hfill \\ & \le & K.\hfill \end{array}$$

That means $Ru$ is a bounded function on $\mathfrak{S},$ that is, $Ru\in B\left(\mathfrak{S}\right)$ and the operator R is well defined. □

To present our result we assume the following conditions on Equation (3) as:

i. 

$g:\mathfrak{S}\times \mathfrak{D}\to \mathbb{R}$ and $F(·,·,0):\mathfrak{S}\times \mathfrak{D}\times \left\{0\right\}\to \mathbb{R}$ are bounded functions.

ii. 

There exists $M\ge 0$ such that

$$|F({\varpi }_1,{\varpi }_2,\iota )-F({\varpi }_1,{\varpi }_2,\rho )|\le M|\iota -\rho |$$

for all ${\varpi }_1\in \mathfrak{S},{\varpi }_2\in \mathfrak{D}$ and $\iota ,\rho \in \mathbb{R}.$

iii. 

There exists a function $\beta :B\left(\mathfrak{S}\right)\times B\left(\mathfrak{S}\right)\to \mathbb{R}$ such that if $\beta (u,v)\ge 0,$ for $u,v\in B\left(\mathfrak{S}\right)$, then for ${\varpi }_1\in \mathfrak{S}$ and ${\varpi }_2\in \mathfrak{D}$ we have

$$|F({\varpi }_1,{\varpi }_2,u\left(T({\varpi }_1,{\varpi }_2)\right))-F({\varpi }_1,{\varpi }_2,v\left(T({\varpi }_1,{\varpi }_2)\right))|\le {\displaystyle \frac{1}{2}}\psi \left(\tilde{d}(u,v)\right),$$

where $\psi \in \mathsf{\Psi }.$

iv. 

There exists $u_0\in B\left(\mathfrak{S}\right)$ such that for all ${\varpi }_1\in \mathfrak{S},$ we have

$$\beta \left(u_0\left({\varpi }_1\right),\underset{{\varpi }_2\in \mathfrak{D}}{sup}\{g({\varpi }_1,{\varpi }_2)+F({\varpi }_1,{\varpi }_2,u_0\left(T({\varpi }_1,{\varpi }_2)\right))\}\right)\ge 0.$$

v. 

For all ${\varpi }_1\in \mathfrak{S},u,v\in B\left(\mathfrak{S}\right),$ if $\beta (u\left({\varpi }_1\right),v\left({\varpi }_1\right))\ge 0,$ then

$$\beta \left(\underset{{\varpi }_2\in \mathfrak{D}}{sup}\{g({\varpi }_1,{\varpi }_2)+F({\varpi }_1,{\varpi }_2,u\left(T({\varpi }_1,{\varpi }_2)\right))\},\underset{{\varpi }_2\in \mathfrak{D}}{sup}\{g({\varpi }_1,{\varpi }_2)+F({\varpi }_1,{\varpi }_2,v\left(T({\varpi }_1,{\varpi }_2)\right))\}\right)\ge 0.$$

vi. 

For all $n\in \mathbb{N},{\left(u_n\right)}_{n\in \mathbb{N}}\subseteq B\left(\mathfrak{S}\right)$ converges to $u^{\ast }$ in $B\left(\mathfrak{S}\right)$ and $\beta (u_n,u_{n+1})\ge 0,$ then $\beta (u_n,u^{\ast })\ge 0,\forall n.$

Theorem 12.

Under the Assumptions $\left(i\right)$ to $\left(vi\right).$ The functional Equation (3) has a solution in $B\left(\mathfrak{S}\right).$

Proof. 

The operator $R:B\left(\mathfrak{S}\right)\to B\left(\mathfrak{S}\right)$ given in (4) is well defined by Lemma 7.

From (iii) and Lemma 6, and for all $u,v\in B\left(\mathfrak{S}\right)$ with $\beta \left(u\right(\varpi ),v(\varpi \left)\right)\ge 0,$ where ${\varpi }_1\in \mathfrak{S},$ we have

$$\begin{array}{cc}\hfill |\left(Ru\right)\left({\varpi }_1\right)-\left(Rv\right)\left({\varpi }_1\right)|& =|\underset{{\varpi }_2\in \mathfrak{D}}{sup}\left\{g({\varpi }_1,{\varpi }_2)+F({\varpi }_1,{\varpi }_2,u\left(T({\varpi }_1,{\varpi }_2)\right))\right\}-\underset{{\varpi }_2\in \mathfrak{D}}{sup}\left\{g({\varpi }_1,{\varpi }_2)+F({\varpi }_1,{\varpi }_2,v\left(T({\varpi }_1,{\varpi }_2)\right))\right\}|\hfill \\ \hfill & \le \underset{{\varpi }_2\in \mathfrak{D}}{sup}\left|F({\varpi }_1,{\varpi }_2,u\left(T({\varpi }_1,{\varpi }_2)\right))-F({\varpi }_1,{\varpi }_2,v\left(T({\varpi }_1,{\varpi }_2)\right))\right|\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}\psi \left(\tilde{d}(u,v)\right).\hfill \end{array}$$

Hence,

$$\begin{array}{ccc}\hfill |\left(Ru\right)\left({\varpi }_1\right)-\left(Rv\right)\left({\varpi }_1\right)|& \le & {\displaystyle \frac{1}{2}}\psi \left(\tilde{d}(u,v)\right)\hfill \\ & \le & \psi \left(\tilde{d}(u,v)\right).\hfill \end{array}$$

Moreover, we have,

$$\begin{array}{ccc}\hfill 2|\left(Ru\right)\left({\varpi }_1\right)-\left(Rv\right)\left({\varpi }_1\right)|-1& \le & \psi \left(\tilde{d}(u,v)\right)-1\hfill \\ & \le & \psi \left(\tilde{d}(u,v)\right).\hfill \end{array}$$

As a consequence, for all $u,v\in B\left(\mathfrak{S}\right)$ such that $\beta (u\left({\varpi }_1\right),v\left({\varpi }_1\right))\ge 0$ for all ${\varpi }_1\in \mathfrak{S},$ we have,

$$\tilde{d}(Ru,Rv)\le \psi \left(\tilde{d}(u,v)\right).$$

Define $\alpha :B\left(\mathfrak{S}\right)\times B\left(\mathfrak{S}\right)\to [0,+\infty )$ by

$$\alpha (u,v)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}\beta \left(u\right(\varpi ),v(\varpi \left)\right)\ge 0,\mathrm{for}\varpi \in \mathfrak{S}\hfill \\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

From Assumption (v), it is clear that R is $\alpha$-admissible.

For all $u,v\in B\left(\mathfrak{S}\right),$ we have

$$\alpha (u,v)d(Ru,Rv)\le \psi \left(d\right(u,v\left)\right).$$

Then, R is an $\alpha$–$\psi$-contractive mapping. From (iv), there exists $u_0\in B\left(\mathfrak{S}\right)$ such that $\alpha (u_0,R{u}_0)\ge 1.$ Finally, from (vi) and using Theorem 7, we deduce the existence of $u^{\ast }\in B\left(\mathfrak{S}\right)$ such that $u^{\ast }=R{u}^{\ast },$ that is, $u^{\ast }$ is a solution to (3). □

Corollary 5.

Suppose the following assumptions

i. 

$g:\mathfrak{S}\times D\to \mathbb{R}$ and $F(·,·,0):\mathfrak{S}\times \mathfrak{D}\times \left\{0\right\}\to \mathbb{R}$ are bounded functions.

ii. 

There exists $M\ge 0$ such that

$$|F({\varpi }_1,{\varpi }_2,\iota )-F({\varpi }_1,{\varpi }_2,\rho )|\le M|\iota -\rho |$$

for all ${\varpi }_1\in \mathfrak{S},{\varpi }_2\in D$, and $\iota ,\rho \in \mathbb{R}.$

iii. 

There exists a function $\psi \in \mathsf{\Psi }$ such that for all ${\varpi }_1\in \mathfrak{S},{\varpi }_2\in \mathfrak{D}$, and $u,v\in B\left(\mathfrak{S}\right),$

$$|F({\varpi }_1,{\varpi }_2,u\left(T({\varpi }_1,{\varpi }_2)\right))-F({\varpi }_1,{\varpi }_2,v\left(T({\varpi }_1,{\varpi }_2)\right))|\le {\displaystyle \frac{1}{2}}\psi \left(d(u,v)\right).$$

Then, the functional Equation (3) has a solution in $B\left(\mathfrak{S}\right).$

Proof. 

Put $\beta (u,v)=1$ for all $u,v\in B\left(\mathfrak{S}\right)$ in Theorem 12. □

Corollary 6.

Suppose the following assumptions

i. 

$g:\mathfrak{S}\times D\to \mathbb{R}$ and $F(·,·,0):\mathfrak{S}\times \mathfrak{D}\times \left\{0\right\}\to \mathbb{R}$ are bounded functions.

ii. 

There exists a function $\psi \in \mathsf{\Psi }$ such that for all ${\varpi }_1\in \mathfrak{S},{\varpi }_2\in \mathfrak{D}$, and $\iota ,\rho \in \mathbb{R},$

$$|F({\varpi }_1,{\varpi }_2,\iota ))-F({\varpi }_1,{\varpi }_2,\rho ))|\le {\displaystyle \frac{1}{2}}\psi \left(\right|\iota -\rho \left|\right).$$

Then, the functional Equation (3) has a solution in $B\left(\mathfrak{S}\right).$

Proof. 

We claim that Corollary 5 is applicable. Since $\psi \in \mathsf{\Psi }$, then by Lemma 1 we have $\psi \left(\iota \right)<\iota$ for any $\iota >0.$ Thus, we can take $M={\displaystyle \frac{1}{2}}$ in Assumption (ii) in Corollary 5, and we get

$$|F({\varpi }_1,{\varpi }_2,\iota )-F({\varpi }_1,{\varpi }_2,\rho )|\le {\displaystyle \frac{1}{2}}|\iota -\rho |.$$

Now, for all ${\varpi }_1\in \mathfrak{S},{\varpi }_2\in \mathfrak{D}$, and $u,v\in B\left(\mathfrak{S}\right),$ we have

$$|u\left(T({\varpi }_1,{\varpi }_2)\right)-v\left(T({\varpi }_1,{\varpi }_2)\right)|\le \underset{\varpi \in \mathfrak{S}}{sup}|u\left(\varpi \right)-v\left(\varpi \right)|\le d(u,v).$$

As $\psi$ is a non-decreasing function,

$$\psi \left(\right|u\left(T({\varpi }_1,{\varpi }_2)\right)-v\left(T({\varpi }_1,{\varpi }_2)\right)\left|\right)\le \psi \left(d(u,v)\right).$$

By (ii), we have

$$\begin{array}{ccc}\hfill |F({\varpi }_1,{\varpi }_2,u\left(T({\varpi }_1,{\varpi }_2)\right))-F({\varpi }_1,{\varpi }_2,v\left(T({\varpi }_1,{\varpi }_2)\right))|& \le & {\displaystyle \frac{1}{2}}\psi \left(\right|u\left(T({\varpi }_1,{\varpi }_2)\right)-v\left(T({\varpi }_1,{\varpi }_2)\right)\left|\right)\hfill \\ & \le & {\displaystyle \frac{1}{2}}\psi \left(d(u,v)\right)\hfill \end{array}$$

Hence, Corollary 5 guarantees that the functional Equation (3) has a solution in $B\left(\mathfrak{S}\right).$ □

Example 6.

Let us consider the following functional equation:

$$u\left({\varpi }_1\right)=\underset{{\varpi }_2\in \mathbb{R}}{sup}\{arctan({\varpi }_1+2|{\varpi }_2\left|\right)+1+{\displaystyle \frac{\left|u\right(\frac{{\varpi }_1}{1+|{\varpi }_1+{\varpi }_2|}\left)\right|}{4+|{\varpi }_1+{\varpi }_2|}}\},\mathrm{for}{\varpi }_1\in [0,1].$$

(5)

Notice that Equation (5) is a particular case of Equation (3) where

• $\mathfrak{S}=[0,1],$$D=\mathbb{R};$;
• $g:[0,1]\times \mathbb{R}\to \mathbb{R}$ is defined by $g({\varpi }_1,{\varpi }_2)=arctan({\varpi }_1+2|{\varpi }_2\left|\right);$
• $T:[0,1]\times \mathbb{R}\to [0,1]$ is defined by $T({\varpi }_1,{\varpi }_2)={\displaystyle \frac{{\varpi }_1}{1+|{\varpi }_1+{\varpi }_2|}};$
• $F:[0,1]\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ is defined by $F({\varpi }_1,{\varpi }_2,\iota )=1+{\displaystyle \frac{\left|\iota \right|}{4+|{\varpi }_1+{\varpi }_2|}}.$

It is clear that for all ${\varpi }_1\in [0,1],$ and all ${\varpi }_2\in \mathbb{R}$:

• $\left|g({\varpi }_1,{\varpi }_2)\right|\le {\displaystyle \frac{\pi }{2}};$
• $\left|F\right({\varpi }_1,{\varpi }_2,0\left)\right|=1.$

Hence, Assumption (i) of Corollary 6 is satisfied. Furthermore, consider the function $\psi \in \mathsf{\Psi }$ given by $\psi \left(\iota \right)={\displaystyle \frac{1}{2}}\iota ,$ for all $\iota \ge 0.$ Therefore, for all ${\varpi }_1\in [0,1],$ and all ${\varpi }_2,\iota ,\rho \in \mathbb{R}$, we have

$$\begin{array}{ccc}\hfill |F({\varpi }_1,{\varpi }_2,\iota )-F({\varpi }_1,{\varpi }_2,\rho )|& =& |{\displaystyle \frac{\left|\iota \right|}{4+|{\varpi }_1+{\varpi }_2|}}-{\displaystyle \frac{\left|\rho \right|}{4+|{\varpi }_1+{\varpi }_2|}}|\hfill \\ & =& {\displaystyle \frac{1}{4+|{\varpi }_1+{\varpi }_2|}}\left|\right|\iota |-\left|\rho \right||\hfill \\ & \le & {\displaystyle \frac{1}{4}}|\iota -\rho |={\displaystyle \frac{1}{2}}\psi \left(\right|\iota -\rho \left|\right)\hfill \end{array}$$

As a result, Assumption (ii) of Corollary 6 is also satisfied, which implies that Problem (5) has a solution in $B\left(\right[0,1\left]\right).$

5.2. An Application to Ordinary Differential Equations

As an application, consider the following second-order problem

$$y^{\prime \prime }\left(\iota \right)=f(\iota ,y\left(\iota \right)),\iota \in [0,\pi ],$$

(6)

with the boundary conditions,

$$y\left(0\right)=y\left(\pi \right)=0.$$

The boundary value problem can be written as the integral equation,

$$y\left(\iota \right)={\int }_0^{\pi }G(\iota ,w)f(w,y\left(w\right))dw\iota \in [0,\pi ],$$

where $G(\iota ,w)$ is the Green’s function given by

$$G(\iota ,w)=\left\{\begin{array}{cc}\left({\displaystyle \frac{w-\pi }{\pi }}\right)\iota & 0\le \iota <w\le \pi ,\\ {\displaystyle \frac{w}{\pi }}(\iota -\pi )& 0\le w<\iota \le \pi .\end{array}\right.$$

Clearly, $G(\iota ,w)$ is continuous on $[0,\pi ]\times [0,\pi ],G(0,w)=G(\pi ,w)=0.$ Note that ${\int }_0^{\pi }G(\iota ,w)dw=\{{\displaystyle \frac{-1}{2}}{\iota }^2+{\displaystyle \frac{\pi }{2}}\iota -{\displaystyle \frac{{\pi }^2}{2}}\}.$ Thus, ${max}_{\iota \in [0,\pi ]}{\int }_0^{\pi }\left|G(\iota ,w)\right|dw={\displaystyle \frac{{\pi }^2}{2}}.$

Consider $W=C[0,\pi ],$ the space of continuous functions on $[0,\pi ],$ with the distance given for all ${\varpi }_1,{\varpi }_2\in C[0,\pi ]$ by

$$\tilde{d}({\varpi }_1,{\varpi }_2)=\underset{\iota \in [0,\pi ]}{max}\left\{\right|{\varpi }_1\left(\iota \right)-{\varpi }_2\left(\iota \right)|,2|{\varpi }_1\left(\iota \right)-{\varpi }_1\left(\iota \right)|-1\}.$$

The space $(W,\tilde{d})$ is a complete strong b-metric space with $s=2.$

Assume the following conditions on Equation (6) as:

i. 

$f:[0,\pi ]\times \mathbb{R}\to \mathbb{R}$ is bounded;

ii. 

There exists a function $\beta :C[0,\pi ]\times C[0,\pi ]\to \mathbb{R}$ such that if $\beta ({\varpi }_1,{\varpi }_2)\ge 0,$ for ${\varpi }_1,{\varpi }_2\in C[0,\pi ]$ then for $w\in [0,\pi ]$ we have

$$|f(w,{\varpi }_1\left(w\right))-f(w,{\varpi }_2\left(w\right))|\le {\displaystyle \frac{1}{{\pi }^2}}\psi \left(\right|{\varpi }_1\left(w\right)-{\varpi }_2\left(w\right)\left|\right),$$

where $\psi \in \mathsf{\Psi }.$

iii. 

There exists ${\varpi }_0\in C[0,\pi ]$ such that for all $\iota \in [0,\pi ],$ we have

$$\beta \left({\varpi }_0\left(\iota \right),{\int }_0^{\pi }G(\iota ,w)f(w,{\varpi }_0\left(w\right))dw\right)\ge 0.$$

iv. 

For all $\iota \in [0,\pi ],{\varpi }_1,{\varpi }_2\in C[0,\pi ],$

$$\beta ({\varpi }_1\left(\iota \right),{\varpi }_2\left(\iota \right))\ge 0\Rightarrow \beta \left({\int }_0^{\pi }G(\iota ,w)f(w,{\varpi }_1\left(w\right))dw,{\int }_0^{\pi }G(\iota ,w)f(w,{\varpi }_2\left(w\right))dw\right)\ge 0.$$

v. 

For all $n\in \mathbb{N},{\left({\varpi }_n\right)}_{n\in \mathbb{N}}\subseteq C[0,\pi ]$ converges to ${\varpi }^{\ast }$ in $C[0,\pi ]$ and $\beta ({\varpi }_n,{\varpi }_{n+1})\ge 0$ then $\beta ({\varpi }_n,{\varpi }^{\ast })\ge 0,\forall n.$

Theorem 13.

Under the Assumptions $\left(i\right)$ to $\left(v\right).$ The second boundary problem, Equation (6), has a solution in $C[0,\pi ].$

Proof. 

Consider the operator $T:W\to W$ defined by

$$T\varpi \left(\iota \right)={\int }_0^{\pi }G(\iota ,w)f(w,\varpi \left(w\right))dw\iota \in [0,\pi ].$$

Firstly, since f is bounded, let $M={sup}_{w\in [0,\pi ]}\left|f(w,\varpi \left(w\right))\right|$ and let $\varpi \in W$ and ${\iota }_1,{\iota }_2\in [0,\pi ]$; then

$$\begin{array}{ccc}\hfill |T{\varpi }_1\left({\iota }_1\right)-T{\varpi }_1\left({\iota }_2\right)|& =& |{\int }_0^{\pi }G({\iota }_1,w)f(w,{\varpi }_1\left(w\right))dw-{\int }_0^{\pi }G({\iota }_2,w)f(w,{\varpi }_1\left(w\right))dw|\hfill \\ & \le & {\int }_0^{\pi }|G({\iota }_1,w)-G({\iota }_2,w)|\left|f(w,{\varpi }_1\left(w\right))\right|dw\hfill \\ & \le & \pi M\underset{w\in [0,\pi ]}{sup}|G({\iota }_1,w)-G({\iota }_2,w)|.\hfill \end{array}$$

Hence, for $|{\iota }_1-{\iota }_2|\to 0,$ we have $|T\left({\varpi }_1\right)\left({\iota }_1\right)-T\left({\varpi }_1\right)\left({\iota }_2\right)|\to 0.$ Therefore, T is well defined. For all ${\varpi }_1,{\varpi }_2\in W$ with $\beta ({\varpi }_1,{\varpi }_2)\ge 0,$ and for all $\iota \in [0,\pi ]$, we have

$$\begin{array}{ccc}\hfill |T{\varpi }_1\left(\iota \right)-T{\varpi }_2\left(\iota \right)|& =& |{\int }_0^{\pi }G(\iota ,w)f(w,{\varpi }_1\left(w\right))dw-{\int }_0^{\pi }G(\iota ,w)f(w,{\varpi }_2\left(w\right))dw|\hfill \\ & \le & {\int }_0^{\pi }\left|G(\iota ,w)\right||f(w,{\varpi }_1\left(w\right))-f(w,{\varpi }_2\left(w\right))|dw\hfill \\ & \le & {\displaystyle \frac{1}{{\pi }^2}}{\int }_0^{\pi }\left|G(\iota ,w)\right|\psi \left(\right|{\varpi }_1\left(w\right)-{\varpi }_2\left(w\right)\left|\right)dw\hfill \\ & \le & {\displaystyle \frac{1}{{\pi }^2}}\psi \left(\tilde{d}({\varpi }_1,{\varpi }_2)\right){\int }_0^{\pi }\left|G(\iota ,w)\right|dw\hfill \\ & \le & {\displaystyle \frac{1}{2}}\psi \left(\tilde{d}({\varpi }_1,{\varpi }_2)\right).\hfill \end{array}$$

Hence,

$$|T{\varpi }_1\left(\iota \right)-T{\varpi }_2\left(\iota \right)|\le {\displaystyle \frac{1}{2}}\psi \left(\tilde{d}({\varpi }_1,{\varpi }_2)\right).$$

We also have

$$\begin{array}{ccc}\hfill 2|T{\varpi }_1\left(\iota \right)-T{\varpi }_2\left(\iota \right)|-1& \le & \psi \left(\tilde{d}({\varpi }_1,{\varpi }_2)\right)-1\hfill \\ & \le & \psi \left(\tilde{d}({\varpi }_1,{\varpi }_2)\right).\hfill \end{array}$$

As a consequence, for all ${\varpi }_1,{\varpi }_2\in C[0,\pi ]$ such that $\beta ({\varpi }_1\left(\iota \right),{\varpi }_2\left(\iota \right))\ge 0$ and for all $\iota \in [0,\pi ],$ we have

$$\tilde{d}(T{\varpi }_1,T{\varpi }_2)\le \psi \left(\tilde{d}({\varpi }_1,{\varpi }_2)\right).$$

Now, define $\alpha :W\times W\to [0,+\infty )$ by

$$\alpha ({\varpi }_1,{\varpi }_2)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}\beta ({\varpi }_1,{\varpi }_2)\ge 0,\hfill \\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

By $\left(iv\right),$ it is clear that T is $\alpha$-admissible. Moreover, for all ${\varpi }_1,{\varpi }_2\in C[0,\pi ]$ we obtain that

$$\alpha ({\varpi }_1,{\varpi }_2)\tilde{d}(T{\varpi }_1\left(\iota \right),T{\varpi }_2\left(\iota \right))\le \psi \left(\tilde{d}({\varpi }_1,{\varpi }_2)\right).$$

Then T is an $\alpha$-$\psi$-contractive mapping. Assumption $\left(iii\right)$ guarantees the existence ${\varpi }_0\in C[0,\pi ]$ such that $\alpha ({\varpi }_0,T{\varpi }_0)\ge 1.$ Finally, by $\left(v\right)$ and Theorem 7, we deduce the existence of ${\varpi }^{\ast }$ is a solution to Equation (6). □

6. Conclusions

This paper introduced several fixed point theorems in complete strong b-metric spaces under the $\alpha$–$\psi$-contractive condition for both single-valued and multi-valued mappings. These theorems extend, generalize, and improve various existing results in the literature, as discussed in Corollaries 1, 2, and 4, and Remark 2. To demonstrate the scope of the results, multiple examples and applications have been presented throughout the paper.

In particular, the present results extend the classical work of Samet et al. [4], who established fixed point results for single-valued mappings in complete metric spaces, to the broader setting of strong b-metric spaces. Moreover, our findings generalize those of Asl et al. [6] and Mohammadi et al. [10], who investigated ${\alpha }_{\ast }$–$\psi$ and $\alpha$–$\psi$ conditions for multi-valued mappings in metric spaces, by proving analogous results in the strong b-metric framework.

The present study also extends the contributions of Mukheimer [36] and Bota et al. [7], which were formulated in b-metric spaces using the class ${\mathsf{\Psi }}_b$, to a more general setting involving the class $\mathsf{\Psi }$. Consequently, the results in this paper unify and broaden several previously established fixed point theorems.

As a possible direction for future research, one may consider extending and generalizing the present results to the setting of relational and ordered strong b-metric spaces. Such an investigation would help to unify the current framework of $\alpha$-admissible mappings with relation-theoretic and order-theoretic approaches and may lead to broader classes of fixed point results in generalized metric structures.

In addition, we would like to draw the reader’s attention to several works that have studied fixed point theorems in F-metric spaces and b-metric-like spaces endowed with binary relations (see, for example, [38,39]). These spaces can be regarded as extensions of strong b-metric spaces, and exploring their relation-theoretic counterparts may provide deeper insights into the structure and applicability of fixed point theory.