On a common fixed point theorem in vector-valued $b$-metric spaces: Its consequences and application

1.   Introduction

In the field of metric fixed point theory, the Banach contraction principle ^[3] is regarded as a key fixed point theorem. The introduction of a new contraction principle is the core part of the metric fixed-point theory. Such a new contraction principle not only contributes in theory but also ensures the existence of solutions to mathematical models. In an effort to introduce a new contraction principle, Rakotch ^[27] introduced a new contraction principle that involved a function instead of a Lipschitz constant in the Banach contraction principle. Then, Boyd and Wong ^[5] generalized the Rakotch contraction principle. In a similar effort, Kannan ^[16] introduced a contraction principle that characterizes the metric completeness and gave a new direction in metric fixed point theory that led many mathematicians to introduce various contraction principles. Among the classical contraction principles, most famous are: the Meir and Keeler contraction principle ^[19], Chatterjea contraction principle, Reich contraction principle ^[28], Hardy and Rogers contraction principle ^[14], Ćirić contraction principle ^[9] and Caristi contraction principle ^[7].

Wardowski ^[32] (2012) generalized the Banach contraction principle by using an auxiliary nonlinear function $F:(0, \infty)\rightarrow (-\infty, \infty)$ that satisfied three conditions. In the literature, this new contraction principle is known as the $F$-contraction principle. This idea proved another milestone in metric fixed point theory. The $F$-contraction principle has been revisited and generalized in many abstract spaces (see ^{[8,10,13,21,23,24,29]} and references therein).

On the other hand, metric generalization also has a significant impact on metric fixed point theory. Many mathematicians have contributed in this direction, producing many generalizations of a metric space (see ^[18]). Perov ^[26], by extending the co-domain of the metric function from $\mathbb{R}$ to $\mathbb{R}^{n}$, gave a vector version of the metric and hence produced another generalization of the Banach contraction principle. Altun, et al. (2020) ^[2] obtained an extension of Wardowski ^[32] fixed point theorem in the vector-valued metric spaces as follows:

Theorem 1.1. ^[2] Every self-mapping $T$ on a complete vector-valued metric space $(X, d)$ that satisfies the inequality

admits a unique fixed point, provided ${\bf F}$ satisfies $(AF_1)-(AF_3)$ and ${\bf I} = \left(\tau_i\right)_{i = 1}^{m} \succeq {\bf 0}$;

the operator ${\bf F}:\mathbb{P}^{m} \rightarrow \mathbb{R}^{m}$ satisfies the following conditions:

($AF_1$) $\forall$ $Q, W\in \mathbb{P}^{m}$ with $Q\preceq W$, and we have ${\bf F}(Q)\preceq {\bf F}(W)$;

($AF_2$) $\forall$ $\{{\bf v}_n: n\in \mathbb{N}\}\subset\mathbb{P}^{m}$, and we have

($AF_3$) $\exists\, \, \kappa \in (0, 1)$ satisfying $\lim_{v_i\rightarrow 0^+}(v_i)^{\kappa}u_i = 0.$

Recently, Mínak, et al.^[20] and Cosentino, et al.^[12] have established some fixed point theorems for the existence of fixed points of Ćirić type and Hardy-Rogers type $F$-contractions, respectively. The significance of the fixed point results on Ćirić type and Hardy-Rogers type $F$-contractions ^[12,20] requires more research work in generalized metric spaces. The vector valued distance function (being a column matrix) has many applications in Matrix Analysis and hence in Engineering, so, to broaden the scope of the fixed point results on Ćirić type and Hardy-Rogers type $F$-contractions, in this paper, we decided to revisit these notions in a vector-valued $b$-metric space $(G, {\bf A}, s)$ (defined in next section) and to investigate the possible conditions on the mappings $W$, and $E$ for which these mappings admit a unique common fixed point. Note that for $s = 1$, we have a vector valued metric space. We will see that the Theorem 1.1 and some results in ^[22] are special cases of results presented in this paper.

Integro-differential equations have found applications in epidemiology, the mathematical modeling of epidemics, particularly when the models contain age-structure ^[6]. The Kermack-McKendrick theory of infectious disease transmission is one particular example where age-structure in the population is incorporated into the modeling framework. Following the work in ^[25], we will show the existence of a solution to the system of delay integro-differential equations that represent an Infectious Disease Model:

where

(a) $l(t), q(t):$ the prevalence of infection at time $t$ in the population.

(b) $0 < L:$ the amount of time a person can still spread disease.

(c) $l^{'}(h), q^{'}(h) :$ the current rate of infectivity.

(d) $W(h, l(h), l^{'}(h)), E(h, q(h), q^{'}(h)):$ the rate of newly acquired infections per unit of time.

We are aimed at applying Theorem 5.1 to show the existence of a solution to a system of delay integro-differential equations.

2.   Preliminaries and related results

In this section, we present a summary of prerequisites and notations to be considered in the sequel. Let $\mathbb{R}^{m} = \left\{{\bf v} = \left(x_i\right)_{i = 1}^{m} = \left(x_1, x_2, \cdots, x_m\right)|\, \, \forall i \, \, \, x_i\in \mathbb{R}\right\}$ represents all matrices of order $m\times 1$ (which will be called vectors), and then $\left(\mathbb{R}^{m}, \oplus, \odot\right)$ is a linear space with $\oplus$ and $\odot$ defined by

Note that $+$ and $\cdot$ represent the usual addition and multiplication of scalars. By using the above operations, we can define the difference of vectors as ${\bf v}\ominus {\bf w} = {\bf v}\oplus (-1)\odot{\bf w}$. Define the relations $\preceq$ and $\prec$ on $\mathbb{R}^{m}$ by

The relation $\preceq$ defines a partial-order on $\mathbb{R}^{m}$. Let $\mathbb{P}^{m}$ denote the set of positive definite vectors, that is, if ${\bf v} = \left(x_i\right)_{i = 1}^{m}\succ {\bf 0}$ (zero vector of order $m\times 1$), and ${\bf v} = \left(x_i\right)_{i = 1}^{m}\in \mathbb{R}^{m}$, then ${\bf v} = \left(x_i\right)_{i = 1}^{m}\in \mathbb{P}^{m}$. Also, let $\mathbb{R}_{0}^{m} = \left\{{\bf v} = \left(x_i\right)_{i = 1}^{m} = \left(x_1, x_2, \cdots, x_m\right)|\, \, \forall i \, \, \, x_i\in [0, \infty)\right\}$. The two vectors are considered equal if their corresponding coordinates are equal.

Definition 2.1. ^[31] (1) Let $V = [v_{ij}]$ be an $m\times m$ complex matrix having eigenvalues $\lambda_{i}$, $1\leq i\leq n$. Then, the spectral radius $\rho(V)$ of matrix $V$ is defined by $\rho(V) = \max_{1\leq i\leq m}|\lambda_i|$.

(2) The matrix $V$ converges to zero, if the sequence $\{V^{n}; n\in \mathbb{N}\}$ converges to zero matrix $O$.

Theorem 2.1. ^[31] Let ${\bf V}$ be any complex matrix of order $m\times m$, then ${\bf V}$ is convergent if and only if $\rho(V) < 1$.

Perov ^[26] applied Theorem 2.1 to obtain the following result in vector-valued metric spaces.

Theorem 2.2. ^[26] Every self-mapping $J$ defined on a complete vector-valued metric space $(X, {\bf d})$ satisfying the inequality

admits a unique fixed point provided $\rho(A) < 1$ and that $A$ is a positive square matrix of order $m$.

By a vector-valued metric, we mean a mapping $d:X\times X\rightarrow \mathbb{R}^{m}$ obeying all the axioms of the metric. The object $d(x, y)$ is an m-tuple or a column matrix. Let

The following concepts and results will be required in the sequel.

3.   The vector-valued $b$-metric space

In light of the definitions of $b$-metric and vector-valued metric given by Czerwik ^[11] and Perov ^[2], respectively, Boriceanu ^[4] introduced a vector-valued $b$-metric as follows:

Definition 3.1. (vector-valued $b$-metric)^[4] Let $G$ be a non-empty set. The operator ${\bf A}:G\times G\rightarrow \mathbb{R}_{0}^{m}$ satisfying the axioms $({\bf A}_1)-({\bf A}_3)$ given below is known as a vector-valued $b$-metric.

$({\bf A}_1)$ $q = t$ if and only if ${\bf A}(q, t) = {\bf 0},$ for all $q, t\in G.$

$({\bf A}_2)$ ${\bf A}(q, t) = {\bf A}(t, q)$, for all $q, t\in G.$

$({\bf A}_3)$ ${\bf A}(q, g)\preceq s\odot\left[{\bf A}(q, t)\oplus{\bf A}(t, g)\right]; s\geq 1,$ for all $q, t, g\in G.$

The triplet $(G, {\bf A}, s)$ represents a vector-valued $b$-metric-space.

For $s = 1$, $(G, {\bf A}, s)$ is a vector-valued metric-space, but this is not true when $s > 1$. Thus, it can be remarked that every vector-valued metric-space is a vector-valued $b$-metric-space but not conversely.

Example 3.1. Let $G = \mathbb{R}$ and the operator ${\bf A}:G\times G\rightarrow \mathbb{R}_{0}^{m}$ be defined by

where $H = |l-q|$. Then, $(G, {\bf A}, 2^{m})$ is a vector-valued $b$-metric space. Note that it is not a vector-valued-metric space.

Example 3.2. Let $G\neq \emptyset$ and $d_{i}:G\times G\rightarrow [0, \infty)$ be a $b$-metric for each $i$ with respective constant $s_{i}\geq 1$ $(1\leq i\leq m)$ for each positive integer $i$. The mapping ${\bf A}:G\times G\rightarrow \mathbb{R}_{0}^{m}$ defined by

defines a vector-valued $b$-metric on $G$.

The axioms $({\bf A}_1)$ and $({\bf A}_2)$ hold trivially. To prove $({\bf A}_3)$, consider

Since the vector-valued $b$-metric is a discontinuous operator, fixed point theorems in the vector-valued $b$-metric spaces need to be supported by an auxiliary convergence result. We provide the following lemma (Lemma 3.1) for this purpose.

Lemma 3.1. Let $(G, {\bf A}, s)$ be a vector-valued $b$-metric space. If $l^*, g^*\in G$, and $\{l_n\}_{n\in \mathbb{N}}$ is such that $\lim_{n\rightarrow \infty}l_n = l^*$, then

Proof. By $({\bf A}_3)$, we have

By $\lim \inf$, we have

By $({\bf A}_3)$, we get

This implies

We know that

Combining (3.1)–(3.3), we get the required result.

Apart from Lemma 3.1, to fulfill the objective of this paper, the following compatibility condition is required:

($AF_4$): for every positive term sequence ${\bf v}_n = \left(x^{(i)}_{n}\right)_{i = 1}^{m}$, $\exists$ ${\bf I} = \left(\tau_i\right)_{i = 1}^{m} \succeq {\bf 0}$ satisfying

Our findings rely mostly on the class of vector-valued nonlinear operators satisfying $(AF_1)$, $(AF_3)$ and $(AF_4)$ denoted by $\Pi_{s}^b$.

Remark 3.1. The collection of vector-valued nonlinear operators $\Pi_{s}^b$ is non-empty.

Let ${\bf F}:\mathbb{P}^{m} \rightarrow \mathbb{R}^{m}$ be defined by ${\bf F}((x_i)_{i = 1}^{m}) = \left(\log_{e} (x_i+1)\right)_{i = 1}^{m}$ for all ${\bf v}\in \mathbb{P}^{m}$, and then $(AF_1)$ and $(AF_3)$ are obvious.

We establish $(AF_4)$: Let ${\bf I}\oplus{\bf F}(s{\bf v}_n) \preceq {\bf F}({\bf v}_{n-1})$, and then for m-tuple ${\bf I} = \left(\log_{e}(s^{n-1}), \log_{e}(s^{n-1}), \cdots, \log_{e}(s^{n-1})\right) = \left(\log_{e}(s^{n-1})\right)_{1}^{m}$, we have

Now, consider

Hence, ${\bf F}\in \Pi_{s}^b$.

Example 3.3. Let ${\bf F}:\mathbb{P}^{m} \rightarrow \mathbb{R}^{m}$ be defined by

(a) ${\bf F}({\bf v}) = \left(\log_{e} (x_i)\right)_{i = 1}^{m}$;

(b) ${\bf F}({\bf v}) = \left(x_i+\log_{e} (x_i)\right)_{i = 1}^{m}$;

(c) ${\bf F}({\bf v}) = \log_{e} (x_{i}^2+x_i)_{i = 1}^{m}$;

(d) ${\bf F}({\bf v}) = \left(-\frac{1}{\sqrt{x_i}}\right)_{i = 1}^{m}$;

(e) ${\bf F}({\bf v}) = \left(x_{i}^a\right)_{i = 1}^{m}; \, \, a > 0$;

(f) ${\bf F}({\bf v}) = \left(\log_{e} (x_i+1)\right)_{i = 1}^{m}$.

Note that the Definitions (a)–(d) satisfy $(AF_1)$–$(AF_3)$, while (e), (f) belong to the family $\Pi_{s}^b$.

Define ${\bf F}:\mathbb{R}^{2}_+ \rightarrow \mathbb{R}^{2}$ by ${\bf F}((g_1, g_2)) = \left(g_{1}^t, \log_{e} (g_2+1) \right), \, \, \, t > 0$, and then ${\bf F}\in \Pi_{s}^b$.

The following lemma explains the reasons to omit axiom $(AF_2)$.

Lemma 3.2. Let ${\bf F}$ satisfy $(AF_1)$, and $\{{\bf v}_n\}_{n\in \mathbb{N}}\subset \mathbb{P}^{m}$ is a decreasing sequence satisfying $\lim_{n\rightarrow \infty} u^{(i)}_{n} = -\infty$. Then $\lim_{n\rightarrow \infty} v^{(i)}_{n} = 0$ for each $i\in \left\{1, 2, \cdots, m\right\}$.

4.   Common fixed points of $(\zeta, {\bf F})$-contractions

Recently, Altun, et al. ^[2] obtained an existence theorem involving a vector-valued nonlinear operator and explained it through nontrivial examples. We will introduce and investigate the notion of $(\zeta, {\bf F})$-contractions where the operator ${\bf F}$ is taken from $\Pi_{s}^{b}$ and $\zeta$ is defined below.

Definition 4.1. Let ${\bf F}\in \Pi_{s}^{b}$ and ${\bf I}\succ {\bf 0}$. The mapping $T:(G, {\bf A}, s)\rightarrow (G, {\bf A}, s)$ is said to be an $(s, {\bf F})$-contraction, if it satisfies the following inequality:

Remark 4.1. Note that for $s = 1$, Definition 4.1 is identical to Perov's type $F$-contraction introduced by Altun, et al. ^[2]. Thus, the class of $(s, {\bf F})$-contractions (defined in Definition 4.1) is wider compared to that of Perov's type $F$-contraction introduced by Altun, et al. ^[2]. Now, we explain inequality (4.1) with the following example (Example 4.1).

Example 4.1. Let $G = \{l_n = 2^{\frac{n}{2}}n|n\in \mathbb{N}\}$. Define ${\bf A}\colon G\times G\rightarrow \mathbb{P}^{m}$ by ${\bf A}\left(l, q\right) = \left(|l-q|^2\right)_{1}^{m}$, and then $(G, {\bf A}, s = 2)$ is a vector-valued $b$-metric space. Define the mapping $\phi:G\rightarrow G$ by

Take $(1)_{1}^{m} = {\bf I}\succ {\bf 0}$ and define ${\bf F}:\mathbb{P}^{m} \rightarrow \mathbb{R}^{m}$ by ${\bf F}\left(\left(g_{i}\right)_{i = 1}^{m}\right) = \left(g_{i}\right)_{i = 1}^{m}$. Then, for every $l, q\in G$ such that $\phi(l)\neq \phi(q)$, we have

Indeed, for $l = l_{n+k}$ and $q = l_n$, consider

We notice that ${\bf F}\in \Pi_s^b$. Indeed, for ${\bf F}\left(\left(g_{i}\right)_{i = 1}^{m}\right) = \left(g_{i}\right)_{i = 1}^{m}$, axioms $(AF_1)$ and $(AF_3)$ hold. For axiom $(AF_4)$, we proceed as follows: Let ${\bf I} \oplus {\bf F}\left(s\odot\left(g^{(i)}_{n}\right)_{i = 1}^{m}\right)\preceq {\bf F}\left(\left(g^{(i)}_{n-1}\right)\right)_{i = 1}^{m}$, that is, $1 + \ell g^{(i)}_{n} \leq g^{(i)}_{n-1}$ for each $i\in \left\{1, 2, \cdots, m\right\}$. Now, consider

This shows that for ${\bf I} = (1)_{1}^{m}$, $\phi$ is an ${\bf F}$-contraction.

Remark 4.2. We observe that the function $\alpha_s$ (defined in ^[1]) is superficial because we can always have a function $\zeta :G\times G\rightarrow [0, \infty)$ defined by $\zeta(l, q) = \dfrac{\alpha_s (l, q)}{s^2}$ with the following properties:

(1) ($\phi$ is $\zeta$-admissible)

(2) the $\alpha_s$-completeness implies $\zeta$-completeness and vice versa.

Definition 4.2. Let $G$ be a non-empty set, and $\zeta:G\times G\rightarrow [0, \infty)$. The function $\phi:G\rightarrow G$ is said to be $\zeta$-admissible if

triangular $\zeta$-admissible if in addition $\zeta$ follows

Definition 4.3. The mappings $W, E:G\rightarrow G$ defined on the space $(G, {\bf A}, s)$ and satisfying the inequality

are called weakly $\zeta$-admissible. Moreover, $W, E$ are called triangular weakly $\zeta$-admissible mappings if

(1) $\zeta(l, q)\geq 1$ implies $\zeta (W(l), EW(l))\geq 1$ and $\zeta (E(q), WE(q))\geq 1$ for all $l, q\in G,$

(2) $\zeta(l, u)\geq 1$, $\zeta(u, q)\geq 1$, imply $\zeta (l, q)\geq 1$,

for all $l, q, u\in G$.

Example 4.2. Let $G = [0, \infty)$, and

Define $\zeta:G\times G\rightarrow \mathbb{R}_0^+$ by $\zeta(l, q) = \left\{ \begin{array}{ll} 1+q-l & \mbox{if } l, q\in [0, 1); \\ 0 & \mbox{if }l, q\in[1, \infty).\end{array} \right.$ If $\left\{ \begin{array}{ll}\zeta(l, q)\geq 1;\\ \beta(q, u)\geq 1, \end{array} \right.$ then $\left\{ \begin{array}{ll} l-q\leq 0;\\ q-u\leq 0, \end{array} \right.$ which implies that $l-u\leq 0$. Hence, $\zeta(l, u) = 1+u-l\geq 1$,

for all $l, q\in [0, 1).$

Definition 4.4. Let $(G, {\bf A}, s)$ be a vector-valued $b$-metric space, $\zeta:G\times G\rightarrow [0, \infty)$, $l\in G$, and sequence $\{l_n\}\subseteq G$. A mapping $q:G\rightarrow G$ is $\zeta$-continuous at $l = l_0$ if whenever

Example 4.3. Let $G = [0, \infty)$ and define ${\bf A}:G\times G\rightarrow \mathbb{R}_{0}^{m}$ by

where $H = |l-q|$, and let $q:G\rightarrow G$ be defined by

Obviously, $q$ is not continuous at $l_0 = 1$; however, $q$ is a $\zeta$-continuous mapping at this point. Indeed, the assumption $\lim_{n\rightarrow \infty}{\bf A}(l_n, l_0) = {\bf 0}$ leads us to choose, $l_n = 1-\frac{1}{n}\subseteq [0, 1]$ and $\zeta(l_n, l_{n+1})\geq 1$ directs us to choose $[0, 1]$ as the domain of mapping $q$. Thus,

Hence, $\lim_{n\rightarrow \infty}{\bf A}(q(l_n), q(l)) = {\bf 0}$.

Definition 4.5. If an arbitrary Cauchy sequence $\{l_n\}\subseteq G$ satisfying $\zeta(l_n, l_{n+1})\geq 1$ converges in $G$, the space $(G, {\bf A}, s)$ is called $\zeta$-complete.

Remark 4.3. Every complete vector-valued $b$-metric space is a $\zeta$-complete vector-valued $b$-metric space but not conversely.

Look at the following example.

Example 4.4. Let $G = (0, \infty)$ and define the vector-valued $b$-metric ${\bf A}:G\times G\rightarrow \mathbb{R}_{0}^{m}$ by

where $H = |l-q|$. Define $\zeta\colon G\times G\rightarrow [0, \infty)$ by

We observe that the space $(G, {\bf A}, s)$ is not complete, but it satisfies $\zeta$-completeness criteria.

Definition 4.6. If an arbitrary sequence $\{l_n\}\subset G$ satisfies the condition

$\forall$ $n\in \mathbb{N},$ then the space $(G, {\bf A}, s)$ is known as a $\zeta$ -regular space.

Let $G = [2, 5]$ and define vector-valued $b$-metric as in Example 1 and $\zeta$ as in Example 7. Let $l_n = 2+\frac{3}{n}$ be $n^{th}$ term of a sequence in $G$. Then, $(G, {\bf A}, s)$ is a $\zeta$-regular space.

Suzuki ^[30] established the following lemma.

Lemma 4.1. ^[30] If there is a number $C > 0$ such that the sequence $\{x_n\}\subset (G, d)$ satisfies the inequality

then $\{x_n\}$ is a Cauchy sequence.

The following Lemma extends Lemma 4.1.

Lemma 4.2. If there is a number $C > 0$ such that the sequence $\{x_n\}\subset (G, {\bf A}, s)$ satisfies the inequality

then $\{x_n\}$ is a Cauchy sequence.

Proof. Let $G$ be any non-empty set and $s = \max\{s_{i}:\quad 1\leq i \leq m\}$. Let $d_{i}:G\times G\rightarrow [0, \infty)$ be a $b$-metric for every $i\in\left\{1, 2, 3, \cdots, m\right\}$ and $s_{i}\geq 1$. Define the vector-valued $b$-metric ${\bf A}$ by

Let $\{x_n\}$ be a sequence in $G$ and assume that

Then, by definition of partial order $\preceq$ defined by (2.1), we have for each $i$

Lemma 4.1 does not depend on a particular $b$-metric; therefore, Lemma 4.1 can be applied for each $d_{i}$ $(1\leq i \leq m)$. Thus, $\{x_n\}$ is a Cauchy sequence with respect to every $d_{i}$ $(1\leq i \leq m)$. Thus,

This leads us to write that

Hence, $\{x_n\}$ is a Cauchy sequence in $(G, {\bf A}, s)$.

Now, we have an analogue of Lemma 4.2 subject to F contraction.

Lemma 4.3. Let $\{D_n\}$ be a sequence in $\mathbb{P}^{m}$ where $D_n: = (j^{(i)}_n)_{i = 1}^{m}$. Assume that there exist a mapping ${\bf F}:\mathbb{P}^{m}\rightarrow \mathbb{R}^{m}$, ${\bf I} = \left(\tau_i\right)_{i = 1}^{m}\succ O$ and $\kappa\in(0, \ell): \ell = 1/1+\log_{2}s$ satisfying $(AF_3)$ and the following:

Then, $D_n \preceq \left(Cn^{-\frac{1}{\kappa}}\right)_{1}^{m}$.

Proof. The inequality (4.2) implies $\lim_{n\rightarrow \infty}{\bf F}\left(s^{n}\odot D_n\right) = (-\infty)_{1}^{m}$, and by Lemma 3.2, we get $\lim_{n\rightarrow \infty}s^{n}\odot D_n = {\bf 0}$. By $(AF_3)$,

By (4.2), we also have the following information for each $i$.

As $n\rightarrow \infty$ in (4.3), we have

Equivalently there exists a positive integer $N_{1}$ such that $n\left(s^{n} j^{(i)}_n\right) ^{\kappa}\leq 1$ for $n\geq N_{1}$. It then follows for each $i$ that

This implies $j^{(i)}_n\leq C n^{-\frac{1}{\kappa }}\text{ for }n\geq N_{1}$ and for each $i$, where $C = s^{-1}$. Hence, $D_n\preceq \left(Cn^{-\frac{1}{\kappa}}\right)_{1}^{m}$.

5.   Fixed point theorems

Recently, Mínak et al.^[20] and Cosentino et al. ^[12] have established some fixed point theorems for the existence of fixed points of Ćirić type and Hardy-Rogers type $F$-contractions, respectively. In this section, we revisit these notions in a vector-valued $b$-metric space $(G, {\bf A}, s)$. For this purpose, we consider the operator ${\bf F}:\mathbb{P}^{m} \rightarrow \mathbb{R}^{m}$ that is an element of the collection $\Pi_{s}^{b}$. We introduce the notions of $(\zeta, {\bf F})$-Ćirić contraction and $(\zeta, {\bf F})$-Hardy-Rogers contraction defined as follows:

Definition 5.1. Let $(G, {\bf A}, s)$ be a vector-valued $b$ metric space. The mappings $W, E:G\rightarrow G$ are said to form $(\zeta, {\bf F})$-Ćirić contraction if there exist ${\bf F}\in \Pi_{s}^{b}$ and ${\bf I}\succ {\bf 0}$ such that

where

Recall that the norm on $\mathbb{P}^{m}$ is defined by

Also, we have assumed that

Definition 5.2. Let $(G, {\bf A}, s)$ be a vector-valued $b$ metric space. The mappings $W, E:G\rightarrow G$ are said to form $(\zeta, {\bf F})$-Hardy-Rogers contraction if there exist $a_i\geq 0\; (i = 1, 2, 3, 4)$ such that $a_1+a_2+a_3+2sa_4 = 1$, ${\bf F}\in \Pi_{s}^{b}$ and ${\bf I}\succ {\bf 0}$ such that

where

Remark 5.1. Every $(\zeta, {\bf F})$-Hardy-Rogers contraction implies $(\zeta, {\bf F})$-Ćirić contraction but not conversely.

The following main result states the requirements that ensure the existence of common fixed points of $(\zeta, {\bf F})$-Ćirić contraction.

Theorem 5.1. Let $W, E:G\rightarrow G$ be a pair of $\zeta$-continuous and weakly $\zeta$-admissible mappings forming $(\zeta, {\bf F})$-Ćirić contraction defined on $\zeta$-complete space $(G, {\bf A}, s)$. If $\exists$ $\kappa\in \left(0, \dfrac{1}{1+\log_2 s}\right)$ and $g_0$ in $G$ such that $\zeta(g_0, W(g_0))\geq 1$, then $h$ is a common fixed point of $W, E$ provided $\zeta(h, h)\geq 1$. Moreover, if $W, E$ are not $\zeta$-continuous, then assuming that $G$ is $\zeta$-regular space and the operator ${\bf F}$ is continuous guarantees the existence of a fixed point.

Proof. {Uniqueness of the common fixed point:} Suppose that $h$ and $v$ are two different common fixed points of $W$ and $E$. Then, $W(h) = h\neq v = E(v)$. It follows that ${\bf A}(W(h), E(v)) = {\bf A}(h, v)\succ 0$. Since $\zeta(h, v)\geq 1$, the contractive condition (5.1) implies

This shows that ${\bf I}\preceq 0$, a contradiction. Hence, the pair $(W, E)$ has at most one common fixed point.

(a) We note that for all $l\neq q$, ${\bf M}(l, q)\succ 0$. Let $g_0 \in G$ be as in (2). We construct an iterative sequence $\left\{g_{n}\right\}$ of points in $G$ such that $g_1 = W(g_0)$, $g_2 = E(g_1)$, and generally $g_{2n+1} = W(g_{2n})$, $g_{2n} = E(g_{2n-1})$ for all $n\in\{0, 1, 2, 3, ...\}$. By assumption (1), we have

and continuing on the same pattern, we have

Hence, $\zeta(g_n, g_{n+1})\geq 1$ for all $n$. If ${\bf A} (W(g_{2n}), E(g_{2n+1})) = {\bf 0}$, then $g_{2n}$ is a common fixed point of mappings $W, E$. Let ${\bf A} (W(g_{2n}), E(g_{2n+1}))\succ {\bf 0}$, and then by contractive condition (5.1), we get

for all $n = 0, 1, 2, ...$ where

If $\left\|{\bf M}(g_{2n}, g_{2n+1})\right\| = \left\|{\bf A}(g_{2n+1}, g_{2n+2})\right\|$, then

which is a contradiction to $(AF_1)$. Therefore,

for all $n\in\{0, 1, 2, 3, ...\}$. Similarly, we can have

for all $n\in\{0, 1, 2, 3, ...\}$. Hence, from (5.3) and (5.4), we have

for all $n\in\{0, 1, 2, 3, ...\}$. Let ${\bf b}_{n} = {\bf A}(g_{n}, g_{n+1})$ for each $n\in\{0, 1, 2, 3, ...\}$, and by (5.5) and $(AF_4)$, we have

Repeating the above process, we obtain

By Lemma 4.3, $\{{\bf b}_n\}\in O(n^{-\frac{1}{\kappa }})$. Since $\frac{1}{\kappa }\in (1+\log_2 s, \infty)$, by Lemma 4.1, we infer that $\{g_n\}$ is a Cauchy sequence. Since $G$ is a $\zeta$-complete vector-valued $b$-metric space, there exists (say) $h\in G$ such that $g_{2n+1}\rightarrow h$ and $g_{2n+2}\rightarrow h$ as $n\rightarrow \infty$. The $\zeta$-continuity of $E$ implies

If ${\bf A}(h, W(h))\succ {\bf 0}$, and $\zeta(h, h)\geq 1$, then by contractive condition (5.1), we have

a contradiction. Thus, ${\bf A}(W(h), h) = {\bf 0}$ and $({\bf A}_1)$ imply $h = W(h)$. Thus, we have $W(h) = E(h) = h$. Hence, $(W, E)$ has a common fixed point $h$.

(b) We have two different cases.

Case 1. if there exists a subsequence $\{g_{n_j}\}_{j\in \mathbb{N}}\subset \{g_n\}_{n\in \mathbb{N}}$ such that

then

So, we are done.

Case 2. if there is no such subsequence of $\{g_{n_j}\}_{n\in \mathbb{N}}$ then there exists a natural number $L_0$ such that for every $n\geq L_0$, we have ${\bf A}(W(g_{2n}), E(h))\succ {\bf 0}$ and ${\bf A}(E(g_{2n+1}), W(h))\succ {\bf 0}$. It is given that the space $G$ is $\zeta$-regular, and thus $\zeta(g_{2n+1}, h)\geq 1$, $\zeta(g_{2n}, h)\geq 1$. By contractive condition (5.1), we have

We show that ${\bf A}(h, E(h)) = 0$. Suppose on the contrary that ${\bf A}(h, E(h)) = {\bf u}\succ {\bf 0}$. Put ${\bf y}_n = {\bf A}(g_{n}, h)$ for all $n\in \mathbb{N}$. Since $\lim_{n\rightarrow \infty}g_n = h$, there exists $L_1\in \mathbb{N}$ such that for every $n\geq L_1$ both ${\bf y}_n\prec \frac{{\bf u}}{2}$ and ${\bf b}_n < \frac{{\bf u}}{2}$ hold. Consequently, by (5.7), we have

Thus, for every $n\geq \max\{L_0, L_1\}$, we obtain

Since ${\bf F}$ is continuous and increasing, by Lemma 3.1 and inequality (5.8), we have

The above inequality shows that ${\bf I}\preceq 0$, which is a contradiction. Thus, ${\bf A}(E(h), h) = {\bf 0}$, and hence $h = E(h)$. Similarly, we can prove that $h = W(h)$, and consequently, $h$ is a common fixed point of $W$ and $E$.

Note that for $W\equiv E$, $s = 1$ and $\zeta(l, q) = 1, \, \forall \, l, q\in G$, Theorem 5.1 reduces to Altun's fixed point theorem ^[2].

The following example illustrates Theorem 5.1.

Example 5.1. Let $G = [0, \infty)$ and define ${\bf A}\colon G\times G\rightarrow \mathbb{R}^{m}$ by ${\bf A}\left(l, q\right) = \left(|l-q|^j\right)_{j = 2}^{m+1}$.

so, $\left(G, {\bf A}, s\right)$ is a $\zeta$-complete vector-valued $b$-metric space with $s = 2^{m}$. Define the mappings $W, E\colon G\rightarrow G$, for all $g\in G$, by

$W$ is $\zeta$-continuous self-mapping: Indeed, consider the sequence $l_{n} = \frac{K}{n^2}$ for all positive integers $n$, $\frac{K}{n^2}\geq \frac{K}{(n+1)^{2}}$, so $\zeta(l_n, l_{n+1})\geq 1$, and $\lim_{n\rightarrow \infty} {\bf A}(l_n, l) = {\bf 0}$ implies $(l^2, l^3, \cdots, l^{m+1}) = {\bf 0}$. This is true for $l = 0$.

Thus, whenever $\zeta(l_n, l_{n+1})\geq 1$, and $\lim_{n\rightarrow \infty} {\bf A}(l_n, l) = {\bf 0}$, we have $\lim_{n\rightarrow \infty} {\bf A}(W(l_n), W(l)) = {\bf 0}_{W}$. Similarly, $E$ is a $\zeta$-continuous self-mapping. To prove that $(W, E)$ is a weakly $\zeta$-admissible pair of mappings, let $l, q\in G$ be such that $q = W(l)$, and thus, we have $q = \log_{e}\left(1+\frac{l}{6}\right)$.

Thus, $\zeta(Wl, EW(l))\geq 1$. Again, let $q, t\in G$ be such that $t = E(q)$, and thus, we have $t = \log_{e}\left(1+\frac{q}{7}\right)$.

Thus, $\zeta(Eq, WE(q))\geq 1$. Hence, $(W, E)$ is a weakly $\zeta$-admissible pair of mappings. Now, for each $l, q\in G$ with $l\geq q$ and choosing $z$ such that $\frac{z}{2e^{\left\|{\bf A}\left(l, q\right)\right\|}} > 1+\log_2 s$, we have

Define the function ${\bf F}:\mathbb{P}^{m}\rightarrow \mathbb{R}^{m}$ by ${\bf F}({\bf v}) = \left(\log_{e} (v_{i}+1)\right)_{i = 2}^{m+1}$, for all ${\bf v} = \left(v_i\right)_{i = 2}^{m+1}\in \mathbb{P}^{m}$, and then ${\bf F}\in \Pi_s^b$ (as shown above). Hence, for all $l, q\in G$ such that ${\bf A}(W(l), E(q))\succ {\bf 0}$, ${\bf I} = \left(\log_{e}\left(\frac{z}{se^{\left\|{\bf A}\left(l, q\right)\right\|}}\right)\right)_{1}^{m}$, and we obtain

This shows that the inequality (5.1) holds true for all $l, q\in G$. Thus, the mappings $W, E$ fulfill all the requirements of Theorem 5.1; moreover, $W, E$ have a unique common fixed point $h = 0$.

Corollary 5.1. Let $(G, {\bf A}, s)$ be a $\zeta$-complete vector-valued $b$-metric space. Suppose that $W, E:G\rightarrow G$ are self-mappings such that

for all $l, q\in G$, $\rho(Q)\in \left(0, \dfrac{1}{1+\log_2 s}\right)$. If $W$ or $E$ is continuous, then $W, E$ have a unique common fixed point in $G$.

Proof. Define $\zeta(l, q) = s^2$ for all $l, q \in G$ and let ${\bf I}\succ {\bf 0}$ be such that $Q$ is a scalar matrix of order $m$ having every non-zero entry equal to $e^{-t}, \, t > 0$. Then, for ${\bf F}({\bf v}) = \left(\log_{e}(v_i)\right)_{i = 2}^{m+1}$ and applying Theorem 5.1, we have the required result. Note that for $s = 1$ and $W\equiv E$, Corollary 5.1 reduces to Perov's fixed point theorem ^[26].

The weak $\zeta$-admissibility.

Definition 5.3. The mapping $W:G\rightarrow G$ defined on a space $(G, {\bf A}, s)$ and satisfying the inequality

is called weakly $\zeta$-admissible.

Definition 5.4. The weakly $\zeta$-admissible mapping $W:G\rightarrow G$ defined on a space $(G, {\bf A}, s)$ is called a $(\zeta, {\bf F})$-weak contraction, if there exist ${\bf F}\in \Pi_s^b$, ${\bf I}\succ {\bf 0}$ such that

for all $l, q\in G$, whenever $\min \left\{{\bf A}(W(l), W(q)), {\bf M}(l, q)\right\}\succ {\bf 0}$.

The following corollary extends the results in Mínak et al.^[20].

Corollary 5.2. Let $W:G\rightarrow G$ be a $\zeta$-continuous and $(\zeta, {\bf F})$-weak contraction defined on $\zeta$-complete space $(G, {\bf A}, s)$. If $\exists$ $\kappa\in \left(0, \dfrac{1}{1+\log_2 s}\right)$ and $g_0$ in $G$ such that $\zeta(g_0, W(g_0))\geq 1$, then $h$ is a fixed point of $W$ provided $\zeta(h, h)\geq 1$. Moreover, if $W$ is not $\zeta$-continuous, then assuming that $G$ is $\zeta$-regular space and the operator ${\bf F}$ is continuous guarantees the existence of a fixed point.

Proof. Set $E\equiv W$ in Theorem 5.1.

The following theorem is on the $(\zeta, {\bf F})$-Hardy-Rogers contraction.

Theorem 5.2. Let $W, E:G\rightarrow G$ be a pair of $\zeta$-continuous and weakly $\zeta$-admissible mappings forming a $(\zeta, {\bf F})$-Hardy-Rogers contraction defined on $\zeta$-complete space $(G, {\bf A}, s)$. If $\exists$ $\kappa\in \left(0, \dfrac{1}{1+\log_2 s}\right)$ and $g_0$ in $G$ such that $\zeta(g_0, W(g_0))\geq 1$, then $h$ is a common fixed point of $W, E$ provided $\zeta(h, h)\geq 1$. Moreover, if $W, E$ are not $\zeta$-continuous, then assuming that $G$ is $\zeta$-regular space and the operator ${\bf F}$ is continuous, guarantees the existence of a fixed point.

Proof. Since

the inequality (5.2) implies the inequality (5.1), so the proof of Theorem 5.2 follows from Theorem 5.1.

6.   Results subject to binary relation

Let ${\bf M}(l, q)$ and ${\bf H}(l, q)$ represent the same vector as in the above section. Let $R$ represent a binary relation over $G$. We need the following definitions.

Definition 6.1. We say the mappings $W, E$ weakly increasing subject to $R$ if for all $l, q\in G$, we have

Let $\zeta:G\times G\rightarrow \mathbb{R}_0^{+}$ be given by

Note that Definition 6.1 seems to be a particular case of Definition 4.2.

Definition 6.2. The vector-valued $b$-metric space $(G, {\bf A}, s)$ is said to be $R$-regular if for any sequence, $\{g_n\}\subset G$ such that $g_n R g_{n+1}$ and $g_n\rightarrow h$ as $n\rightarrow \infty$, we have $g_n R h$ for all $n\in \mathbb{N}.$

Now, we are able to revisit Theorems 5.1 and 5.2.

Theorem 6.1. Let $W, E:G\rightarrow G$ be $R$-continuous and $R$-weakly increasing mappings defined on $R$-complete space $(G, {\bf A}, s, \preceq)$. If there exist $g_0 \in G$ such that $g_0 R W(g_0)$, ${\bf F}\in \pi_s^b$, ${\bf I}\succ {\bf 0}$ and $\kappa\in \left(0, \dfrac{1}{1+\log_2 s}\right)$ such that

for all $l, q \in G$ with $l R q$, whenever $\min\{{\bf A}(W(l), E(q)), {\bf M}(l, q)\}\succ {\bf 0}$, then $W$ and $E$ admit a common fixed point. Moreover, if $W, E$ are not $R$-continuous, then assuming that $G$ is $R$-regular space and the operator ${\bf F}$ is continuous guarantees the existence of a common fixed point.

Proof. Define

and follow the proof of Theorem 5.1.

Theorem 6.2. Let $W, E:G\rightarrow G$ be $R$-continuous and $R$-weakly increasing mappings defined on $R$-complete space $(G, {\bf A}, s, \preceq)$. If there exist $g_0 \in G$ such that $g_0 R W(g_0)$, ${\bf F}\in \pi_s^b$, ${\bf I}\succ {\bf 0}$ and $\kappa\in \left(0, \dfrac{1}{1+\log_2 s}\right)$ such that

for all $l, q \in G$ with $l R q$, whenever $\min\{{\bf A}(W(l), E(q)), {\bf H}(l, q)\}\succ {\bf 0}$, then $W$ and $E$ admit a common fixed point. Moreover, if $W, E$ are not $R$-continuous, then assuming that $G$ is $R$-regular space and the operator ${\bf F}$ is continuous guarantees the existence of a common fixed point.

Proof. Define

and follow the proof of Theorem 5.2.

7.   Results in graph theory

Jachymski ^[15] recently presented a very intriguing approach to the theory of fixed points in some generic structures utilizing the context of metric spaces furnished with a graph. Let $(G, {\bf A}, s)$ be a vector-valued $b$-metric space, and $\delta$ denotes the diagonal of the Cartesian product $G\times G$. Assume that the set $V(B)$ of the vertices in a directed graph $B$ coincides with $G$, where $e(B)$ is the set of edges in the graph. Assume that $B$ does not have any parallel edges. Then, $B$ can be associated with the pair $(V(B), e(B))$. Now we give some results that generalize the results in Jachymski ^[15].

Definition 7.1. Let $W$ and $E$ be two self-mappings on a vector-valued $b$-metric space $(V(B), {\bf A}, s)$ endowed with graph $B$. A pair $[W, E]$ is said to be,

(ⅰ) weakly $B$-connected if $(W(g), EW(g))\in e(B)$ and $(E(g), WE(g))\in e(B)$ for all $g\in G$,

(ⅱ) partially weakly $B$-connected if $(W(g), EW(g))\in e(B)$ for all $g\in G$.

Let $(V(B), {\bf A}, s)$ be a vector-valued $b$-metric space associated with graph $B$ and let

Then, the above definitions are special cases of the definition of weak $\zeta$-admissibility and partially weak $\zeta$-admissibility.

Definition 7.2. Let $(V(B), {\bf A}, s)$ be a vector-valued $b$-metric space associated with graph $B$. We say it is $B$-complete if and only if every Cauchy sequence $\{g_n\}$ in $G$ satisfying $(g_n, g_{n+1})\in e(B)$ $\forall$ $n\in \mathbb{N}$ converges in $G$.

Definition 7.3. Let $(V(B), {\bf A}, s)$ be a vector-valued $b$-metric space associated with graph $B$ and $W:G\rightarrow G$ be a mapping. We say that $W$ is a $B$-continuous mapping on $(G, {\bf A}, s)$ if for given $g\in G$ and sequence $\{g_n\}$, $\lim_{n\rightarrow \infty}{\bf A}(g_n, g) = {\bf 0}$ and $(g_n, g_{n+1})\in e(B)$ $\forall$ $n\in\mathbb{N}$ imply $\lim_{n\rightarrow \infty}{\bf A}(W(g_n), W(g)) = {\bf 0}$.

Definition 7.4. Let $(V(B), {\bf A}, s)$ be a vector-valued $b$-metric space associated with graph $B$. The pair $[W, E]$ is said to be an $B$-compatible if and only if $\lim_{n\rightarrow \infty}{\bf A}(WE(g_n), EW(g_n)) = {\bf 0}$, whenever $\{g_n\}$ is a sequence in $G$ satisfying $(g_n, g_{n+1})\in e(B)$ and

Now we are able to revisit Theorems 5.1 and 5.2 in the framework of vector-valued $b$-metric space associated with graph $B$

Theorem 7.1. Let $W, E:V(B)\rightarrow V(B)$ be $B$-continuous and weakly $B$-increasing self-mappings defined on $B$-complete vector-valued $b$-metric space $(V(B), {\bf A}, s)$ endowed with graph. If there exist $g_0 \in V(B)$ such that $(g_0, W(g_0))\in e(B)$, ${\bf F}\in \Pi_s^b$, ${\bf I}\succ {\bf 0}$ and $\kappa\in \left(0, \dfrac{1}{1+\log_2 s}\right)$ such that

for all $l, q \in V(B)$ with $(l, q)\in e(B)$, whenever $\min\{{\bf A}(W(l), E(q)), {\bf M}(l, q)\}\succ {\bf 0}$, then $W$ and $E$ admit a common fixed point.

Proof. Define

and follow the proof of Theorem 5.1.

Theorem 7.2. Let $W, E:V(B)\rightarrow V(B)$ be $B$-continuous and weakly $B$-increasing self-mappings defined on $B$-complete vector-valued $b$-metric space $(V(B), {\bf A}, s)$ endowed with graph. If there exist $g_0 \in V(B)$ such that $(g_0, W(g_0))\in e(B)$, ${\bf F}\in \Pi_s^b$, ${\bf I}\succ {\bf 0}$ and $\kappa\in \left(0, \dfrac{1}{1+\log_2 s}\right)$ such that

for all $l, q \in V(B)$ with $(l, q)\in e(B)$, whenever $\min\{{\bf A}(W(l), E(q)), {\bf H}(l, q)\}\succ {\bf 0}$, then, $W$ and $E$ admit a common fixed point.

Proof. Define

and follow the proof of Theorem 5.2.

8.   Application of Theorem 5.1

A tool used to explore the processes of disease transmission, forecast the trajectory of an outbreak, and assess epidemic control measures is infectious disease modeling. The progression of infectious diseases can be predicted mathematically to demonstrate the likely course of an epidemic (even in plants) and to guide public health and plant health actions. In order to determine parameters for different infectious diseases and utilize those parameters to calculate the impact of various treatments, such mass vaccination programs, models use basic assumptions or collected statistics together with mathematics. The models can assist in selecting the interventions to try and those to avoid, or it can forecast future growth trends. In this section, we apply Theorem 5.1 to the following system of delay integro-differential equations that represents an infectious model:

where

(a) $l(t), q(t):$ the prevalence of infection at time $t$ in the population.

(b) $0 < L:$ the amount of time a person can still spread disease.

(c) $l^{'}(h), q^{'}(h) :$ the current rate of infectivity.

(d) $W(h, l(h), l^{'}(h)), E(h, q(h), q^{'}(h)):$ the rate of newly acquired infections per unit of time.

Now, we look for existence and uniqueness of the positive, periodic solution to (8.1) and (8.2) by the application of Theorem 5.1.

Let $\exists$ $p > 0$ and $W, E\in C(\mathbb{R}\times \mathbb{R}_{+} \times\mathbb{R})$ satisfying

Let us define the functional spaces by

Let $V = \mathcal{F}_{+}(p)\times \mathcal{F}(p)$, and define a metric ${\bf A}: V\times V\rightarrow \mathbb{R}^{2}$ by

where $\left\|{\bf l}\right\| = \max\{|l(t)|: \quad t\in [0, p], \quad {\bf l}\in \mathcal{F}(p)\}$. The function $\zeta:V\times V\rightarrow [1, \infty)$ is defined by $\zeta(l, q) = K^{2}$ for all $l, q\in V$. Then, $(V, {\bf A}, s)$ is a $\zeta$-complete vector-valued $b$-metric space. Now, we develop the structure to apply Theorem 5.1. Let $g(t) = l^{'}(t)$, and $h(t) = q^{'}(t)$. Then, we have

Thus, Eqs (8.1) and (8.2) can be written as

Let $\mathbb{Y}: V\rightarrow C(\mathbb{R})\times C(\mathbb{R})$ be a mapping defined by

where $\mathbb{Y}_1$ and $\mathbb{Y}_2$ are defined by the following matrix equation:

Similarly, define the mapping $\mathbb{X}: V\rightarrow C(\mathbb{R})\times C(\mathbb{R})$ by

where $\mathbb{X}_1$ and $\mathbb{X}_2$ are defined by the following matrix equation:

$(I_{1})$ Suppose that the mappings $W$, $E$ are bounded and periodic having period $p$ and there exist $\omega, K > 0$ so that $\forall t\in \mathbb{R}, l(t), u(t)\in \mathbb{R}_{+} \text{ and } q(t), \epsilon(t)\in \mathbb{R}$

$(I_{2})$ there exists $l_0\in V$ such that $\zeta(l_0, \mathbb{Y}(l_0)) = K^2$.

Theorem 8.1. Let the mappings $W, E\in C(\mathbb{R}\times \mathbb{R}_{+} \times\mathbb{R})$ satisfy conditions $(I_{1})-(I_{2})$ and $\frac{e^{-2\omega}}{s^{2}K^{2}} < \frac{1}{4+L^{2}}$. Then, Eqs (8.1) and (8.2) admit a common solution in $\mathcal{F}_{+}(p)$.

Proof. We note that the conditions (1) and (2) of Theorem 5.1 can be verified by using $(I_{2})$ and continuity of mappings $W, E$ respectively. In the following, we prove the contractive condition (5.1). By definition,

This shows that $\mathbb{Y}_{1}(V)\subseteq \mathcal{F}_{+}(p)$. Similarly, we have that $\mathbb{Y}_{2}(V), \mathbb{X}_{1}(V)$ and $\mathbb{X}_{2}(V)$ are subsets of $\mathcal{F}_{+}(p)$. Let $(l_1, q_1), (l_2, q_2) \in V$ and consider

and

Consequently, we obtain the following matrix inequality:

Now, define the mappings ${\bf F}: \mathbb{R}_{+}^{2}\rightarrow \mathbb{R}^{2}$ by ${\bf F}(l, q) = (\log_{e} (l), \log_{e} (q))$ and ${\bf I} = \left(\frac{s}{e^{-2\omega} L^{2}}, \frac{s}{4e^{-2\omega}}\right)$, and we obtain

Finally, keeping in mind the definition of mapping $\zeta$ and the above inequality, we say that the mappings $\mathbb{Y}, \mathbb{X}$ satisfy all the requirements of Theorem 5.1 and hence admit a common fixed point in their domain. Consequently, the Eqs (8.1) and (8.2) have a positive, periodic solution.

9.   Conclusions

The findings and analysis discussed here could inspire further investigation into this topic by interested academics. A fundamental finding in vector-valued $b$-metric space, the main result (Theorem 5.1) concerns an $F$-contraction in vector-valued $b$-metric spaces. In order to demonstrate the presence of solutions to various linear and nonlinear equations reflecting models of the associated real-world issues, the application approach is also discussed. We conclude that: (1) the classical approach only addresses the functions while the new approach involves operators (linear and nonlinear). (2) The classical approach only involves real numbers on both sides of inequalities (called $F$-contractions). However, this approach also allows matrices and vectors on both sides of inequalities and hence makes it applicable in engineering and optimization problems.

Use of AI tools declaration

The authors declare they have not used artificial intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors would like to thank the King Abdulaziz University Deanship of Scientific research (DSR) throughout this project IFPIP: 401-130-1443.

Conflict of interest

The authors declare that they have no competing interests.