Solving a System of Differential Equations with Infinite Delay by Using Tripled Fixed Point Techniques on Graphs

Abstract

In this manuscript, some similar tripled fixed point results under certain restrictions on a $b-$metric space endowed with graphs are established. Furthermore, an example is provided to support our results. The obtained results extend, generalize, and unify several similar significant contributions in the literature. Finally, to further extend our results, the existence of a solution to a system of ordinary differential equations with infinite delay is derived.

1. Introduction and Basic Concepts

One of the most crucial methods for comprehending the world around us is mathematics. With the help of the various fields of mathematics, other sciences can be analyzed. The use of integral and differential equations is crucial for creating patterns for better understanding. Integral and differential equations likewise heavily rely on the fixed point theory.

In 2011, Berinde and Borcut [1] defined the notion of a tripled fixed point (TFP) for self-mappings and established some interesting consequences in partially ordered metric spaces. The (TFP) theory has a large number of significant applications that have been successfully employed to address a wide variety of issues. Researchers have focused on these issues to examine possible solutions, as seen in [2,3,4,5,6,7].

In 2008, Jachymski [8] proposed considering partial order sets as graphs in metric spaces. He obtained novel contraction mappings using this concept, which generalized many of the prior contractions. Moreover, in a metric space endowed with a graph, some results of the fixed points under these contractions were successfully deduced. Several authors have used this contribution in various applications. See the series of papers [9,10,11,12].

As a continuation of this approach, the results of coupled fixed points and TFPs for edge-preserving mappings with applications in abstract spaces have been investigated. For more details, see [13,14,15,16,17].

Czerwik [18] introduced the concept of $b-$metric spaces as a generalization of ordinary metric spaces as follows:

Definition 1.

Let $\chi \ne \varnothing$ be a set and $s\ge 1$ be a real number. A function $\varpi :\chi \times \chi \to {\mathbb{R}}^{+}$ is said to be a $b-$metric on $\chi ,$ if for each $z,d,r\in \chi ,$ the hypotheses below hold:

• $\varpi \left(z,d\right)=0\iff z=d;$
• $\varpi \left(z,d\right)=\varpi \left(d,z\right)\iff z=d;$
• $\varpi \left(z,d\right)\le s\left[\varpi \left(z,r\right)+\varpi \left(r,d\right)\right].$

The pair $\left(\chi ,\varpi \right)$ is known as $b-$metric space.

In the context of a metric space $\left(\chi ,\varpi \right),$ let $\nabla =\left\{\right(z,z):z\in \chi \}$ be the set of self loops and $\mho =\left(\vee (\mho ),\Xi \left(\mho \right)\right)$ be a directed graph where $\vee (\mho )$ represents the set of vertices and $\Xi \left(\mho \right)$ refers to the set of edges, so $\Xi \left(\mho \right)\supseteq \nabla$ and $\mho$ has no parallel edges.

Consider $z,d\in \vee (\mho ),$ a path from z to d is a finite sequence ${\left\{z_t\right\}}_{t=0}^N\subseteq \mho$, where $z_0=z,$ $z_t=d$, and $(z_t,z_{t-1})\in \Xi \left(\mho \right),$ $t=1,2,\cdots ,N$. For simplicity, we write

$${\left[z\right]}_{\mho }=\{d\in \chi \mathrm{there}\mathrm{is}\mathrm{a}\mathrm{path}\mathrm{from}z\mathrm{to}d\}.$$

If $\vee (\mho )={\left[z\right]}_{\mho },$ then $\mho$ is said to be connected for all $l\in \chi .$

By reversing the directions of the edges on a directed graph $\mho ,$ we may obtain the directed graph ${\mho }^{-1}$, i.e., $\vee \left({\mho }^{-1}\right)=\vee (\mho )$ and

$$\Xi \left({\mho }^{-1}\right)=\{(d,z):(z,d)\in \Xi \left(\mho \right)\}.$$

Moreover, by neglecting the direction of edges, we have the indirect graph $\tilde{\mho }$, i.e., $\vee \left(\tilde{\mho }\right)=\vee (\mho )$ and

$$\Xi \left(\tilde{\mho }\right)=\Xi \left(\mho \right)\cup \Xi \left({\mho }^{-1}\right).$$

Herein, we assume that $\left(\chi ,\varpi \right)$ is a $b-$metric space, and $\mho$ is a directed graph, so $\vee (\mho )=\chi$ and $\Xi \left(\mho \right)\supseteq \nabla$. Further, we define another graph $\mho$ on the product $\chi \times \chi \times \chi$ as follows:

$$\left(\left(z,d,r\right),\left(\overline{z},\overline{d},\overline{r}\right)\right)\in \Xi \left(\mho \right)\iff \left(z,\overline{z}\right)\in \Xi \left(\mho \right),\left(\overline{d},d\right)\in \Xi \left(\mho \right)\mathrm{and}\left(r,\overline{r}\right)\in \Xi \left(\mho \right),$$

for all $\left(z,d,r\right),\left(\overline{z},\overline{d},\overline{r}\right)\in {\chi }^3.$

Definition 2

([1]). A trio $\left(z,d,r\right)\in {\chi }^3$ is called a TFP of the mapping $\Omega :{\chi }^3\to \chi$ if

$$z=\Omega \left(z,d,r\right),d=\Omega \left(d,r,z\right),\mathrm{and}r=\Omega \left(r,z,d\right).$$

Definition 3

([15]).Let $\Omega :\chi \times \chi \to \chi$ be a given mapping defined on a complete metric space $\left(\chi ,\varpi \right)$ equipped with a directed graph $\mho .$ We say that $\Omega$ has the mixed $\mho -$monotone property if for all $z,z_1,z_2,d,d_1,d_2\in \chi ,$

$$\left(z_1,z_2\right)\in \Xi \left(\mho \right)\mathit{\text{implies}}\left(\Omega \left(z_1,d\right),\Omega \left(z_2,d\right)\right)\in \Xi \left(\mho \right),$$

and

$$\left(d_1,d_2\right)\in \Xi \left(\mho \right)\mathit{\text{implies}}\left(\Omega \left(z,d_2\right),\Omega \left(z,d_1\right)\right)\in \Xi \left(\mho \right).$$

In a similar vein, our work seeks to create a new generalization of TFP results in the context of a $b-$metric space with a graph. Our results extend and unify the results of Alfuraidan and Khamsi [15], Luong and Thuan [19], and Işik and Türkoğlu [20] in partially ordered metric spaces. Our theoretical findings have been used to show that a system of ordinary differential equations with infinite delay has a solution.

2. Main Results

This section starts with a generalization of Definition 3 as follows:

Definition 4.

Let $\Omega :{\chi }^3\to \chi$ be a function defined on a complete metric space $(\chi ,\varpi )$ with a directed graph. We say that $\Omega$ has the mixed $\mho$-monotone property if for all $z,z_1,z_2,d,d_1,d_2,r,r_1,r_2\in \chi ,$

$$\begin{array}{ccc}\hfill \left(z_1,z_2\right)& \in & \Xi \left(\mho \right)\mathit{\text{implies}}\left(\Omega \left(z_1,d,r\right),\Omega \left(z_2,d,r\right)\right)\in \Xi \left(\mho \right),\hfill \\ \hfill \left(d_1,d_2\right)& \in & \Xi \left(\mho \right)\mathit{\text{implies}}\left(\Omega \left(z,d_1,r\right),\Omega \left(z,d_2,r\right)\right)\in \Xi \left(\mho \right),\hfill \end{array}$$

and

$$\left(r_1,r_2\right)\in \Xi \left(\mho \right)\mathit{\text{implies}}\left(\Omega \left(z,d,r_1\right),\Omega \left(z,d,r_2\right)\right)\in \Xi \left(\mho \right).$$

In order to facilitate our study, we denote by $\Gamma$ the set of pairs of functions $\left(\theta ,\vartheta \right),$ where $\theta ,\vartheta :[0,\infty )\to [0,\infty )$ fulfilling the constraints below:

(c${}_1$)

$\theta$ is non-decreasing and continuous;

(c${}_2$)

$\theta \left(a\right)=0$, if and only if $a=0;$

(c${}_3$)

$\vartheta$ is continuous;

(c${}_4$)

for all $a>0$, $\theta \left(a\right)>\vartheta \left(a\right).$

The lemma below is useful for our main results.

Lemma 1.

Assume that $(\chi ,\varpi )$ is a $b-$metric space with $s\ge 1.$ Suppose that $\left\{{\ell }_k\right\}$, $\left\{{\delta }_k\right\}$, and $\left\{{\lambda }_k\right\}$ are three sequences in χ, and there is $\sigma \in [0,\frac{1}{s})$, justifying

$$\varpi \left({\ell }_k,{\ell }_{k+1}\right)+\varpi \left({\delta }_k,{\delta }_{k+1}\right)+\varpi \left({\lambda }_k,{\lambda }_{k+1}\right)\le \sigma \left(\varpi \left({\ell }_{k-1},{\ell }_k\right)+\varpi \left({\delta }_{k-1},{\delta }_k\right)+\varpi \left({\lambda }_{k-1},{\lambda }_k\right)\right),$$

(1)

for any $k\in \mathbb{N}.$ Then, $\left\{{\ell }_k\right\},$ $\left\{{\delta }_k\right\}$, and $\left\{{\lambda }_k\right\}$ are Cauchy sequences.

Proof. 

Let $j,k\in \mathbb{N}$, and $j<k.$ Then,

$$\begin{array}{ccc}\hfill \varpi \left({\ell }_j,{\ell }_k\right)+\varpi \left({\delta }_j,{\delta }_k\right)+\varpi \left({\lambda }_j,{\lambda }_k\right)& \le & s\left(\varpi \left({\ell }_j,{\ell }_{j+1}\right)+\varpi \left({\ell }_{j+1},{\ell }_k\right)\right)+s\left(\varpi \left({\delta }_j,{\delta }_{j+1}\right)+\varpi \left({\delta }_{j+1},{\delta }_k\right)\right)\hfill \\ & & +s\left(\varpi \left({\lambda }_j,{\lambda }_{j+1}\right)+\varpi \left({\lambda }_{j+1},{\lambda }_k\right)\right)\hfill \\ & \le & s\left(\varpi \left({\ell }_j,{\ell }_{j+1}\right)+\varpi \left({\delta }_j,{\delta }_{j+1}\right)+\varpi \left({\lambda }_j,{\lambda }_{j+1}\right)\right)\hfill \\ & & +s^2\left(\varpi \left({\ell }_{j+1},{\ell }_{j+2}\right)+\varpi \left({\delta }_{j+1},{\delta }_{j+2}\right)+\varpi \left({\lambda }_{j+1},{\lambda }_{j+2}\right)\right)\hfill \\ & & +s^2\left(\varpi \left({\ell }_{j+2},{\ell }_k\right)+\varpi \left({\delta }_{j+2},{\delta }_k\right)+\varpi \left({\lambda }_{j+2},{\lambda }_k\right)\right)\hfill \\ & \le & \cdots \hfill \\ & \le & s\left(\varpi \left({\ell }_j,{\ell }_{j+1}\right)+\varpi \left({\delta }_j,{\delta }_{j+1}\right)+\varpi \left({\lambda }_j,{\lambda }_{j+1}\right)\right)\hfill \\ & & +s^2\left(\varpi \left({\ell }_{j+1},{\ell }_{j+2}\right)+\varpi \left({\delta }_{j+1},{\delta }_{j+2}\right)+\varpi \left({\lambda }_{j+1},{\lambda }_{j+2}\right)\right)+\cdots \hfill \\ & & +s^{k-j-1}\left(\varpi \left({\ell }_{k-2},{\ell }_{k-1}\right)+\varpi \left({\ell }_{k-1},{\ell }_k\right)+\varpi \left({\delta }_{k-2},{\delta }_{k-1}\right)+\varpi \left({\delta }_{k-1},{\delta }_k\right)\right)\hfill \\ & & +s^{k-j-1}\left(\varpi \left({\lambda }_{k-2},{\lambda }_{k-1}\right)+\varpi \left({\lambda }_{k-1},{\lambda }_k\right)\right)\hfill \\ & \le & s\left(\varpi \left({\ell }_j,{\ell }_{j+1}\right)+\varpi \left({\delta }_j,{\delta }_{j+1}\right)+\varpi \left({\lambda }_j,{\lambda }_{j+1}\right)\right)\hfill \\ & & +s^2\left(\varpi \left({\ell }_{j+1},{\ell }_{j+2}\right)+\varpi \left({\delta }_{j+1},{\delta }_{j+2}\right)+\varpi \left({\lambda }_{j+1},{\lambda }_{j+2}\right)\right)\hfill \\ & & +s^{k-j-1}\left(\varpi \left({\ell }_{k-2},{\ell }_{k-1}\right)+\varpi \left({\delta }_{k-2},{\delta }_{k-1}\right)+\varpi \left({\lambda }_{k-2},{\lambda }_{k-1}\right)\right)\hfill \\ & & +s^{k-j}\left(\varpi \left({\ell }_{k-1},{\ell }_k\right)+\varpi \left({\delta }_{k-1},{\delta }_k\right)+\varpi \left({\lambda }_{k-1},{\lambda }_k\right)\right).\hfill \end{array}$$

From the fact that $s\sigma <1$, and using (1), we have

$$\begin{array}{ccc}& & \varpi \left({\ell }_j,{\ell }_k\right)+\varpi \left({\delta }_j,{\delta }_k\right)+\varpi \left({\lambda }_j,{\lambda }_k\right)\hfill \\ & \le & \left(s{\sigma }^j+s^2{\sigma }^{j+1}+\cdots +s^{k-j-1}{\sigma }^{k-2}+s^{k-j}{\sigma }^{k-1}\right)\left(\varpi \left({\ell }_0,{\ell }_1\right)+\varpi \left({\delta }_0,{\delta }_1\right)+\varpi \left({\lambda }_0,{\lambda }_1\right)\right)\hfill \\ & =& s{\sigma }^j\left(1+s\sigma +\cdots +s^{k-j-2}{\sigma }^{k--j-2}+s^{k-j-1}{\sigma }^{k-j-1}\right)\left(\varpi \left({\ell }_0,{\ell }_1\right)+\varpi \left({\delta }_0,{\delta }_1\right)+\varpi \left({\lambda }_0,{\lambda }_1\right)\right)\hfill \\ & =& \frac{s{\sigma }^j}{1+s\sigma }\left(\varpi \left({\ell }_0,{\ell }_1\right)+\varpi \left({\delta }_0,{\delta }_1\right)+\varpi \left({\lambda }_0,{\lambda }_1\right)\right).\hfill \end{array}$$

It follows that

$$\underset{j\to \infty }{lim}\left(\varpi \left({\ell }_j,{\ell }_k\right)+\varpi \left({\delta }_j,{\delta }_k\right)+\varpi \left({\lambda }_j,{\lambda }_k\right)\right)=0.$$

Hence, $\left\{{\ell }_k\right\},$ $\left\{{\delta }_k\right\}$, and $\left\{{\lambda }_k\right\}$ are Cauchy sequences. □

Now, we formulate and prove the first main result.

Theorem 1.

On $(\chi ,\Xi ,\varpi )$, let $(\chi ,\varpi )$ be a complete $b-$metric space with $s\ge 1$ and $\Omega :{\chi }^3\to \chi$ be a continuous mapping that has the mixed $\mho -$monotone property on χ for which there is a pair $\left(\theta ,\vartheta \right)\in \Gamma$, so that

$$\theta \left(s^2\varpi \left(\Omega \left(z,d,r\right),\Omega \left(z^{\ast },d^{\ast },r^{\ast }\right)\right)\right)\le \frac{1}{3}\vartheta \left(\varpi \left(z,z^{\ast }\right)+\varpi \left(d,d^{\ast }\right)+\varpi \left(r,r^{\ast }\right)\right),$$

(2)

for all $\left(z,d,r\right),\left(z^{\ast },d^{\ast },r^{\ast }\right)\in {\chi }^3,$ where $\left(\left(z,d,r\right),\left(z^{\ast },d^{\ast },r^{\ast }\right)\right)\in \Xi \left(\mho \right)$. If there are $z_0,d_0,r_0\in \chi$ so that

$$\left(\left(z_0,d_0,r_0\right),\left(\Omega \left(z_0,d_0,r_0\right),\Omega \left(d_0,r_0,z_0\right),\Omega \left(r_0,z_0,d_0\right)\right)\right)\in \Xi \left(\mho \right);$$

then, $\Omega$ owns a TFP $\left(\widehat{z},\widehat{d},\widehat{r}\right)\in {\chi }^3.$

Proof. 

Put $z_{k+1}=\Omega \left(z_k,d_k,r_k\right),$ $d_{k+1}=\Omega \left(d_k,r_k,z_k\right)$, and $r_{k+1}=\Omega \left(r_k,z_k,d_k\right).$ Based on our assumption, we have

$$\left(\left(z_0,d_0,r_0\right),\left(z_1,d_1,r_1\right)\right)\in \Xi \left(\mho \right),$$

which leads to

$$\begin{array}{ccc}\hfill \theta \left(s^2\varpi \left(z_2,z_1\right)\right)& =& \theta \left(s^2\varpi \left(\Omega \left(z_1,d_1,r_1\right),\Omega \left(z_0,d_0,r_0\right)\right)\right)\hfill \\ & \le & \frac{1}{3}\vartheta \left(\varpi \left(z_1,z_0\right)+\varpi \left(d_1,d_0\right)+\varpi \left(r_1,r_0\right)\right).\hfill \end{array}$$

Analogously, since $\left(\left(d_0,r_0,z_0\right),\left(d_1,r_1,z_1\right)\right)\in \Xi \left(\mho \right),$ one can obtain

$$\theta \left(s^2\varpi \left(d_2,d_1\right)\right)\le \frac{1}{3}\vartheta \left(\varpi \left(d_1,d_0\right)+\varpi \left(r_1,r_0\right)+\varpi \left(z_1,z_0\right)\right).$$

Similarly, since $\left(\left(r_0,z_0,d_0\right),\left(r_1,z_1,d_1\right)\right)\in \Xi \left(\mho \right),$ we can write

$$\theta \left(s^2\varpi \left(r_2,r_1\right)\right)\le \frac{1}{3}\vartheta \left(\varpi \left(r_1,r_0\right)+\varpi \left(z_1,z_0\right)+\varpi \left(d_1,d_0\right)\right).$$

Because $\Omega$ has the mixed $\mho -$monotone property, we have for $k\ge 1,$

$$\begin{array}{ccc}\hfill \left(\left(z_k,d_k,r_k\right),\left(z_{k+1},d_{k+1},r_{k+1}\right)\right)& \in & \Xi \left(\mho \right),\hfill \\ \hfill \left(\left(d_k,r_k,z_k\right),\left(d_{k+1},r_{k+1},z_{k+1}\right)\right)& \in & \Xi \left(\mho \right),\hfill \end{array}$$

and

$$\left(\left(r_k,z_k,d_k\right),\left(r_{k+1},z_{k+1},d_{k+1}\right)\right)\in \Xi \left(\mho \right).$$

Then,

$$\theta \left(s^2\varpi \left(z_{k+1},z_k\right)\right)\le \frac{1}{3}\vartheta \left(\varpi \left(z_k,z_{k-1}\right)+\varpi \left(d_k,d_{k-1}\right)+\varpi \left(r_k,r_{k-1}\right)\right),$$

(3)

$$\theta \left(s^2\varpi \left(d_{k+1},d_k\right)\right)\le \frac{1}{3}\vartheta \left(\varpi \left(d_k,d_{k-1}\right)+\varpi \left(r_k,r_{k-1}\right)+\varpi \left(z_k,z_{k-1}\right)\right),$$

(4)

and

$$\theta \left(s^2\varpi \left(r_{k+1},r_k\right)\right)\le \frac{1}{3}\vartheta \left(\varpi \left(r_k,r_{k-1}\right)+\varpi \left(z_k,z_{k-1}\right)+\varpi \left(d_k,d_{k-1}\right)\right).$$

(5)

Adding (3)–(5), we obtain

$$\theta \left(s^2\varpi \left(z_{k+1},z_k\right)\right)+\theta \left(s^2\varpi \left(d_{k+1},d_k\right)\right)+\theta \left(s^2\varpi \left(r_{k+1},r_k\right)\right)\le \vartheta \left(\varpi \left(z_k,z_{k-1}\right)+\varpi \left(d_k,d_{k-1}\right)+\varpi \left(r_k,r_{k-1}\right)\right).$$

It follows from the properties of $(\theta ,\vartheta )$ that

$$\theta \left(s^2\left(\varpi \left(z_{k+1},z_k\right)+\varpi \left(d_{k+1},d_k\right)+\varpi \left(r_{k+1},r_k\right)\right)\right)\le \vartheta \left(\varpi \left(z_k,z_{k-1}\right)+\varpi \left(d_k,d_{k-1}\right)+\varpi \left(r_k,r_{k-1}\right)\right);$$

again, from the properties of $(\theta ,\vartheta ),$ we have

$$\theta \left(s^2\left(\varpi \left(z_{k+1},z_k\right)+\varpi \left(d_{k+1},d_k\right)+\varpi \left(r_{k+1},r_k\right)\right)\right)\le \theta \left(\varpi \left(z_k,z_{k-1}\right)+\varpi \left(d_k,d_{k-1}\right)+\varpi \left(r_k,r_{k-1}\right)\right);$$

since $\theta$ is non-decreasing, we obtain

$$s^2\left(\varpi \left(z_{k+1},z_k\right)+\varpi \left(d_{k+1},d_k\right)+\varpi \left(r_{k+1},r_k\right)\right)\le \varpi \left(z_k,z_{k-1}\right)+\varpi \left(d_k,d_{k-1}\right)+\varpi \left(r_k,r_{k-1}\right),$$

which leads to

$$\varpi \left(z_{k+1},z_k\right)+\varpi \left(d_{k+1},d_k\right)+\varpi \left(r_{k+1},r_k\right)\le \frac{1}{s^2}\left(\varpi \left(z_k,z_{k-1}\right)+\varpi \left(d_k,d_{k-1}\right)+\varpi \left(r_k,r_{k-1}\right)\right).$$

Because $0\le \frac{1}{s^2}<\frac{1}{s},$ then by Lemma 1, we observe that $\left\{z_k\right\},$ $\left\{d_k\right\}$, and $\left\{r_k\right\}$ are Cauchy sequences. The completeness of $\chi$ implies that there are $\widehat{z},\widehat{d},\widehat{r}\in \chi$, so that

$$\underset{k\to \infty }{lim}{z}_k=\widehat{z},\underset{k\to \infty }{lim}{d}_k=\widehat{d}m\mathrm{and}\underset{k\to \infty }{lim}{r}_k=\widehat{r}.$$

Since $\Omega$ is continuous, we obtain

$$\begin{array}{ccc}\hfill \widehat{z}& =& \underset{k\to \infty }{lim}{z}_k=\underset{k\to \infty }{lim}\Omega \left(z_{k-1},d_{k-1},r_{k-1}\right)=\Omega \left(\underset{k\to \infty }{lim}{z}_{k-1},\underset{k\to \infty }{lim}{d}_{k-1},\underset{k\to \infty }{lim}{r}_{k-1}\right)=\Omega \left(\widehat{z},\widehat{d},\widehat{r}\right),\hfill \\ \hfill \widehat{d}& =& \underset{k\to \infty }{lim}{d}_k=\underset{k\to \infty }{lim}\Omega \left(d_{k-1},r_{k-1},z_{k-1}\right)=\Omega \left(\underset{k\to \infty }{lim}{d}_{k-1},\underset{k\to \infty }{lim}{r}_{k-1},\underset{k\to \infty }{lim}{z}_{k-1}\right)=\Omega \left(\widehat{d},\widehat{r},\widehat{z}\right),\hfill \\ \hfill \widehat{r}& =& \underset{k\to \infty }{lim}{r}_k=\underset{k\to \infty }{lim}\Omega \left(r_{k-1},z_{k-1},d_{k-1}\right)=\Omega \left(\underset{k\to \infty }{lim}{r}_{k-1},\underset{k\to \infty }{lim}{z}_{k-1},\underset{k\to \infty }{lim}{d}_{k-1}\right)=\Omega \left(\widehat{r},\widehat{z},\widehat{d}\right).\hfill \end{array}$$

This proves that $\left(\widehat{z},\widehat{d},\widehat{r}\right)$ is a TFP of $\Omega .$ □

In the case of the non continuity of $\Omega ,$ we can state another sufficient condition for the existence of TFP by giving the following postulate on the trio $(\chi ,\Xi ,\varpi ):$

(p)

for any sequence ${\left\{z_k\right\}}_{k\in \mathbb{N}}$ in $\chi$, so that $\left(z_k,z_{k+1}\right)\in \Xi \left(\mho \right)$, $\left(z_{k+1},z_k\right)\in \Xi \left(\mho \right)$, and ${lim}_{k\to \infty }{z}_k=z,$ we have $\left(z_k,z,\right)\in \Xi \left(\mho \right)$ and $\left(z,z_k\right)\in \Xi \left(\mho \right).$

Now, our second theoretical result is as follows:

Theorem 2.

On $(\chi ,\Xi ,\varpi )$, suppose that $(\chi ,\varpi )$ is a complete $b-$ms with $s\ge 1$, and $(\chi ,\Xi ,\varpi )$ satisfies Postulate (p). Suppose also the mapping $\Omega :{\chi }^3\to \chi$ has the mixed $\mho -$monotone property on $\chi .$ Assume that $\left(\theta ,\vartheta \right)\in \Gamma$, so that the contractive condition (2) holds. If there are $z_0,d_0,r_0\in \chi$ so that

$$\left(\left(z_0,d_0,r_0\right),\left(\Omega \left(z_0,d_0,r_0\right),\Omega \left(d_0,r_0,z_0\right),\Omega \left(r_0,z_0,d_0\right)\right)\right)\in \Xi \left(\mho \right),$$

then $\Omega$ possesses a TFP $\left(\widehat{z},\widehat{d},\widehat{r}\right)\in {\chi }^3.$

Proof. 

By the same line proof of Theorem 1 and since

$$\begin{array}{ccc}\hfill \underset{k\to \infty }{lim}{z}_{k+1}& =& \underset{k\to \infty }{lim}\Omega \left(z_k,d_k,r_k\right)=\widehat{z},\hfill \\ \hfill \underset{k\to \infty }{lim}{d}_{k+1}& =& \underset{k\to \infty }{lim}\Omega \left(d_k,r_k,z_k\right)=\widehat{d},\hfill \\ \hfill \underset{k\to \infty }{lim}{r}_{k+1}& =& \underset{k\to \infty }{lim}\Omega \left(r_k,z_k,d_k\right)=\widehat{r},\hfill \end{array}$$

and

$$\left(z_k,z_{k+1}\right)\in \Xi \left(\mho \right),\left(d_k,d_{k+1}\right)\in \Xi \left(\mho \right)\mathrm{and}\left(r_k,r_{k+1}\right)\in \Xi \left(\mho \right),$$

then, by Postulate (p), one can write

$$\left(z_k,\widehat{z}\right)\in \Xi \left(\mho \right),\left(d_k,\widehat{d}\right)\in \Xi \left(\mho \right)\mathrm{and}\left(r_k,\widehat{r}\right)\in \Xi \left(\mho \right).$$

Then,

$$\left(\left(z_k,d_k,r_k\right),\left(\widehat{z},\widehat{d},\widehat{r}\right)\right)\in \Xi \left(\mho \right).$$

Hence, we obtain

$$\theta \left(s^2\varpi \left(\Omega \left(z_k,d_k,r_k\right),\Omega \left(\widehat{z},\widehat{d},\widehat{r}\right)\right)\right)\le \frac{1}{3}\vartheta \left(\varpi \left(z_k,\widehat{z}\right)+\varpi \left(d_k,\widehat{d}\right)+\varpi \left(r_k,\widehat{r}\right)\right).$$

(6)

Analogously, we obtain

$$\theta \left(s^2\varpi \left(\Omega \left(d_k,r_k,z_k\right),\Omega \left(\widehat{d},\widehat{r},\widehat{z}\right)\right)\right)\le \frac{1}{3}\vartheta \left(\varpi \left(d_k,\widehat{d}\right)+\varpi \left(r_k,\widehat{r}\right)+\varpi \left(z_k,\widehat{z}\right)\right),$$

(7)

and

$$\theta \left(s^2\varpi \left(\Omega \left(d_k,r_k,z_k\right),\Omega \left(\widehat{d},\widehat{r},\widehat{z}\right)\right)\right)\le \frac{1}{3}\vartheta \left(\varpi \left(d_k,\widehat{d}\right)+\varpi \left(r_k,\widehat{r}\right)+\varpi \left(z_k,\widehat{z}\right)\right).$$

(8)

Taking the limit as $k\to \infty$ in (6)–(8), we have

$$\begin{array}{ccc}\hfill \underset{k\to \infty }{lim}\varpi \left(\Omega \left(z_k,d_k,r_k\right),\Omega \left(\widehat{z},\widehat{d},\widehat{r}\right)\right)& =& 0,\underset{k\to \infty }{lim}\varpi \left(\Omega \left(d_k,r_k,z_k\right),\Omega \left(\widehat{d},\widehat{r},\widehat{z}\right)\right)=0\hfill \\ \hfill \mathrm{and}\underset{k\to \infty }{lim}\varpi \left(\Omega \left(d_k,r_k,z_k\right),\Omega \left(\widehat{d},\widehat{r},\widehat{z}\right)\right)& =& 0.\hfill \end{array}$$

This implies that

$$\underset{k\to \infty }{lim}{z}_{k+1}=\Omega \left(\widehat{z},\widehat{d},\widehat{r}\right),\underset{k\to \infty }{lim}{d}_{k+1}=\Omega \left(\widehat{d},\widehat{r},\widehat{z}\right)\mathrm{and}\underset{k\to \infty }{lim}{r}_{k+1}=\Omega \left(\widehat{r},\widehat{z},\widehat{d}\right),$$

which yields that

$$\widehat{z}=\Omega \left(\widehat{z},\widehat{d},\widehat{r}\right),\widehat{d}=\Omega \left(\widehat{d},\widehat{r},\widehat{z}\right)\mathrm{and}\widehat{r}=\Omega \left(\widehat{r},\widehat{z},\widehat{d}\right);$$

that is, $\left(\widehat{z},\widehat{d},\widehat{r}\right)$ is a TFP of $\Omega$ on $\chi .$ □

Next, we shall state some contributions of Theorems 1 and 2 in the literature.

The results of Alfuraidan and Khamsi [15] can be generalized if we let $\theta \left(a\right)=a$ and $\vartheta \left(a\right)=\ell a$ in Theorems 1 and 2 with $b=1$ as follows:

Corollary 1.

Let $(\chi ,\varpi )$ be a complete metric space with a direct graph $\Xi$ and the mapping $\Omega :{\chi }^3\to \chi$ has the mixed $\mho -$monotone property on χ for which there exists $\ell \in [0,1)$ such that

$$\theta \left(\varpi \left(\Omega \left(z,d,r\right),\Omega \left(z^{\ast },d^{\ast },r^{\ast }\right)\right)\right)\le \frac{\ell }{3}\theta \left(\varpi \left(z,z^{\ast }\right)+\varpi \left(d,d^{\ast }\right)+\varpi \left(r,r^{\ast }\right)\right),$$

for all $\left(z,d,r\right),\left(z^{\ast },d^{\ast },r^{\ast }\right)\in {\chi }^3$ with $\left(\left(z,d,r\right),\left(z^{\ast },d^{\ast },r^{\ast }\right)\right)\in \Xi \left(\mho \right)$. Assume that either $\Omega$ is a continuous mapping or the triple $(\chi ,\Xi ,\varpi )$ has the property (p). If there are $z_0,d_0,r_0\in \chi$ so that

$$\left(\left(z_0,d_0,r_0\right),\left(\Omega \left(z_0,d_0,r_0\right),\Omega \left(d_0,r_0,z_0\right),\Omega \left(r_0,z_0,d_0\right)\right)\right)\in \Xi \left(\mho \right),$$

then, $\Omega$ has a TFP $\left(\widehat{z},\widehat{d},\widehat{r}\right)\in {\chi }^3.$

It should be noted that if $\left(\theta ,\vartheta \right)\in \Gamma$ and ${\vartheta }_1\left(a\right)=\theta \left(a\right)-3\vartheta \left(\frac{a}{3}\right),$ then $\left(\theta ,{\vartheta }_1\right)\in \Gamma .$ Based on this notion, the results of Luong and Thuan [19] in a metric space endowed with a graph can be re-formulated as follows:

Corollary 2.

Let $(\chi ,\varpi )$ be a complete metric space with a direct graph $\Xi$, and the mapping $\Omega :{\chi }^3\to \chi$ has the mixed $\mho -$monotone property. Let $\left(\theta ,\vartheta \right)\in \Gamma$, so that

$$\begin{array}{ccc}\hfill \theta \left(\varpi \left(\Omega \left(z,d,r\right),\Omega \left(z^{\ast },d^{\ast },r^{\ast }\right)\right)\right)& \le & \frac{1}{3}\theta \left(\varpi \left(z,z^{\ast }\right)+\varpi \left(d,d^{\ast }\right)+\varpi \left(r,r^{\ast }\right)\right)\hfill \\ & & -\vartheta \left(\frac{\varpi \left(z,z^{\ast }\right)+\varpi \left(d,d^{\ast }\right)+\varpi \left(r,r^{\ast }\right)}{3}\right)\hfill \end{array}$$

for all $\left(z,d,r\right),\left(z^{\ast },d^{\ast },r^{\ast }\right)\in {\chi }^3$ with $\left(\left(z,d,r\right),\left(z^{\ast },d^{\ast },r^{\ast }\right)\right)\in \Xi \left(\mho \right)$. Assume either the mapping $\Omega$ is continuous or a trio $(\chi ,\Xi ,\varpi )$ satisfies the postulate (p). If there are $z_0,d_0,r_0\in \chi$, so that

$$\left(\left(z_0,d_0,r_0\right),\left(\Omega \left(z_0,d_0,r_0\right),\Omega \left(d_0,r_0,z_0\right),\Omega \left(r_0,z_0,d_0\right)\right)\right)\in \Xi \left(\mho \right),$$

then, $\Omega$ has a TFP $\left(\widehat{z},\widehat{d},\widehat{r}\right)\in {\chi }^3.$

In the following, we discuss the uniqueness of a TFP of the mapping $\Omega$.

Theorem 3.

In addition to the assumptions of Theorems 1 and 2, assume that for any $\left(z,d,r\right)$ $,\left(z^{\ast },d^{\ast },r^{\ast }\right)\in {\chi }^3$, there is $\left(\widehat{\ell },\tilde{\ell },\overline{\ell }\right)\in {\chi }^3$, so that

$$\begin{array}{c}\hfill \left(\left(z,d,r\right),\left(\widehat{\ell },\tilde{\ell },\overline{\ell }\right)\right)\in \Xi \left(\mho \right)and\left(\left(z^{\ast },d^{\ast },r^{\ast }\right),\left(\widehat{\ell },\tilde{\ell },\overline{\ell }\right)\right)\in \Xi \left(\mho \right).\end{array}$$

Then, $\Omega$ has a unique TFP.

Proof. 

Assume that there are two TFPs $\left(z,d,r\right)$ and $\left(z^{\ast },d^{\ast },r^{\ast }\right)$ of $\Omega .$ By our hypothesis, there is $\left(\varkappa ,\eta ,\zeta \right)\in {\chi }^3$, so that $\left(\left(z,d,r\right),\left(\varkappa ,\eta ,\zeta \right)\right)\in \Xi \left(\mho \right)$, and $\left(\left(z^{\ast },d^{\ast },r^{\ast }\right),\left(\varkappa ,\eta ,\zeta \right)\right)\in \Xi \left(\mho \right).$ Define three sequences $\left\{{\varkappa }_k\right\},$ $\left\{{\eta }_k\right\}$, and $\left\{{\zeta }_k\right\}$ by

$$\varkappa ={\varkappa }_0,\eta ={\eta }_0,\zeta ={\zeta }_0,{\varkappa }_{k+1}=\Omega \left({\varkappa }_k,{\eta }_k,{\zeta }_k\right),{\eta }_{k+1}=\Omega \left({\eta }_k,{\zeta }_k,{\varkappa }_k\right)\mathrm{and}{\zeta }_{k+1}=\Omega \left({\zeta }_k,{\varkappa }_k,{\eta }_k\right),\mathrm{for}\mathrm{all}n.$$

Since $\left(\left(z,d,r\right),\left(\varkappa ,\eta ,\zeta \right)\right)\in \Xi \left(\mho \right)$ and $\Omega$ has a mixed $\mho -$monotone property, we can show that $\left(\left(z,d,r\right),\left({\varkappa }_k,{\eta }_k,{\zeta }_k\right)\right)\in \Xi \left(\mho \right).$ Then,

$$\theta \left(s^2\varpi \left(z,{\varkappa }_{k+1}\right)\right)=\theta \left(s^2\varpi \left(\Omega \left(z,d,r\right),\Omega \left({\varkappa }_k,{\eta }_k,{\zeta }_k\right)\right)\right)\le \frac{1}{3}\vartheta \left(\varpi \left(z,{\varkappa }_k\right)+\varpi \left(d,{\eta }_k\right)+\varpi \left(r,{\zeta }_k\right)\right).$$

(9)

Similarly, we can write

$$\theta \left(s^2\varpi \left(d,{\eta }_{k+1}\right)\right)=\theta \left(s^2\varpi \left(\Omega \left(d,r,z\right),\Omega \left({\eta }_k,{\zeta }_k,{\varkappa }_k\right)\right)\right)\le \frac{1}{3}\vartheta \left(\varpi \left(d,{\eta }_k\right)+\varpi \left(r,{\zeta }_k\right)+\varpi \left(z,{\varkappa }_k\right)\right),$$

(10)

and

$$\theta \left(s^2\varpi \left(r,{\zeta }_{k+1}\right)\right)=\theta \left(s^2\varpi \left(\Omega \left(r,z,d\right),\Omega \left({\zeta }_k,{\varkappa }_k,{\eta }_k\right)\right)\right)\le \frac{1}{3}\vartheta \left(\varpi \left(r,{\zeta }_k\right)+\varpi \left(z,{\varkappa }_k\right)+\varpi \left(d,{\eta }_k\right)\right).$$

(11)

Combining (9)–(11) and using the properties of $\theta$ and $\vartheta ,$ we have

$$\theta \left(s^2\left(\varpi \left(z,{\varkappa }_{k+1}\right)+\varpi \left(d,{\eta }_{k+1}\right)+\varpi \left(r,{\zeta }_{k+1}\right)\right)\right)\le \vartheta \left(\varpi \left(z,{\varkappa }_k\right)+\varpi \left(d,{\eta }_k\right)+\varpi \left(r,{\zeta }_k\right)\right).$$

(12)

Because $\theta$ is non-decreasing function, and $\theta \left(a\right)>\vartheta \left(a\right)$ for $a>0,$ we have

$$s^2\left(\varpi \left(z,{\varkappa }_{k+1}\right)+\varpi \left(d,{\eta }_{k+1}\right)+\varpi \left(r,{\zeta }_{k+1}\right)\right)\le \varpi \left(z,{\varkappa }_k\right)+\varpi \left(d,{\eta }_k\right)+\varpi \left(r,{\zeta }_k\right).$$

Since $s\ge 1,$ we obtain

$$\varpi \left(z,{\varkappa }_{k+1}\right)+\varpi \left(d,{\eta }_{k+1}\right)+\varpi \left(r,{\zeta }_{k+1}\right)\le \varpi \left(z,{\varkappa }_k\right)+\varpi \left(d,{\eta }_k\right)+\varpi \left(r,{\zeta }_k\right).$$

This leads to $\{\varpi \left(z,{\varkappa }_k\right)+\varpi \left(d,{\eta }_k\right)+\varpi \left(r,{\zeta }_k\right)\}$ being a nonnegative decreasing sequence; consequently, there is $\rho \ge 0$, so that

$$\underset{k\to \infty }{lim}\left(\varpi \left(z,{\varkappa }_k\right)+\varpi \left(d,{\eta }_k\right)+\varpi \left(r,{\zeta }_k\right)\right)=\rho .$$

As the functions $\theta$ and $\vartheta$ are continuous, and by taking $k\to \infty$ in (12), one can write

$$\theta \left(s^2\rho \right)\le \vartheta \left(\rho \right).$$

It follows from the properties of $\theta$ and $\vartheta$ that $\rho =0.$ Hence,

$$\underset{k\to \infty }{lim}\left(\varpi \left(z,{\varkappa }_k\right)+\varpi \left(d,{\eta }_k\right)+\varpi \left(r,{\zeta }_k\right)\right)=0;$$

that is,

$$\underset{k\to \infty }{lim}\varpi \left(z,{\varkappa }_k\right)=0,\underset{k\to \infty }{lim}\varpi \left(d,{\eta }_k\right)=0,\mathrm{and}\underset{k\to \infty }{lim}\varpi \left(r,{\zeta }_k\right)=0.$$

Following the same scenario, we have

$$\underset{k\to \infty }{lim}\varpi \left(z^{\ast },{\varkappa }_k\right)=0,\underset{k\to \infty }{lim}\varpi \left(d^{\ast },{\eta }_k\right)=0\mathrm{and}\underset{k\to \infty }{lim}\varpi \left(r^{\ast },{\zeta }_k\right)=0.$$

Let $k\to \infty$ in the following inequalities

$$\begin{array}{ccc}\hfill \varpi \left(z,z^{\ast }\right)& \le & s\left(\varpi \left(z,{\varkappa }_k\right)+\varpi \left({\varkappa }_k,z^{\ast }\right)\right),\hfill \\ \hfill \varpi \left(d,d^{\ast }\right)& \le & s\left(\varpi \left(d,{\varkappa }_k\right)+\varpi \left({\varkappa }_k,d^{\ast }\right)\right),\hfill \\ \hfill \varpi \left(r,r^{\ast }\right)& \le & s\left(\varpi \left(r,{\varkappa }_k\right)+\varpi \left({\varkappa }_k,r^{\ast }\right)\right).\hfill \end{array}$$

Thus, $\varpi \left(z,z^{\ast }\right)=0,$ $\varpi \left(d,d^{\ast }\right)=0$, and $\varpi \left(r,r^{\ast }\right)=0.$ Hence, $z=z^{\ast },$ $d=d^{\ast }$, and $r=r^{\ast }.$ □

Theorem 4.

Assume that $\left(\left(\widehat{z},\widehat{d}\right),\left(\widehat{d},\widehat{r}\right),\left(\widehat{r},\widehat{z}\right)\right)\in \Xi \left(\mho \right)$ and the assumptions of Theorems 1 and 2 are true. If $\left(\widehat{z},\widehat{d},\widehat{r}\right)$ is a TFP of $\Omega ,$ then $\widehat{z}=\widehat{d}=\widehat{r}.$

Proof. 

Because $\left(\left(\widehat{z},\widehat{d}\right),\left(\widehat{d},\widehat{r}\right),\left(\widehat{r},\widehat{z}\right)\right)\in \Xi \left(\mho \right)$, we have

$$\begin{array}{ccc}\hfill \theta \left(s^2\varpi \left(\widehat{z},\widehat{d}\right)\right)& =& \theta \left(s^2\varpi \left(\Omega \left(\widehat{z},\widehat{d},\widehat{r}\right),\Omega \left(\widehat{d},\widehat{r},\widehat{z}\right)\right)\right)\hfill \\ & \le & \frac{1}{3}\vartheta \left(\varpi \left(\widehat{z},\widehat{d}\right)+\varpi \left(\widehat{d},\widehat{r}\right)+\varpi \left(\widehat{r},\widehat{z}\right)\right).\hfill \end{array}$$

Similarly, we can write

$$\theta \left(s^2\varpi \left(\widehat{d},\widehat{r}\right)\right)\le \frac{1}{3}\vartheta \left(\varpi \left(\widehat{d},\widehat{r}\right)+\varpi \left(\widehat{r},\widehat{z}\right)+\varpi \left(\widehat{z},\widehat{d}\right)\right),$$

and

$$\theta \left(s^2\varpi \left(\widehat{r},\widehat{z}\right)\right)\le \frac{1}{3}\vartheta \left(\varpi \left(\widehat{r},\widehat{z}\right)+\varpi \left(\widehat{z},\widehat{d}\right)+\varpi \left(\widehat{d},\widehat{r}\right)\right).$$

Combining the above three inequalities, we have

$$\begin{array}{ccc}\hfill \theta \left(s^2\left[\varpi \left(\widehat{z},\widehat{d}\right)+\varpi \left(\widehat{d},\widehat{r}\right)+\varpi \left(\widehat{r},\widehat{z}\right)\right]\right)& \le & \vartheta \left(\varpi \left(\widehat{z},\widehat{d}\right)+\varpi \left(\widehat{d},\widehat{r}\right)+\varpi \left(\widehat{r},\widehat{z}\right)\right)\hfill \\ & <& \theta \left(\varpi \left(\widehat{z},\widehat{d}\right)+\varpi \left(\widehat{d},\widehat{r}\right)+\varpi \left(\widehat{r},\widehat{z}\right)\right).\hfill \end{array}$$

Since the function $\theta$ is non-decreasing, we obtain

$$s^2\left(\varpi \left(\widehat{z},\widehat{d}\right)+\varpi \left(\widehat{d},\widehat{r}\right)+\varpi \left(\widehat{r},\widehat{z}\right)\right)<\varpi \left(\widehat{z},\widehat{d}\right)+\varpi \left(\widehat{d},\widehat{r}\right)+\varpi \left(\widehat{r},\widehat{z}\right).$$

Hence, $\varpi \left(\widehat{z},\widehat{d}\right)+\varpi \left(\widehat{d},\widehat{r}\right)+\varpi \left(\widehat{r},\widehat{z}\right)=0;$ that is, $\varpi \left(\widehat{z},\widehat{d}\right)=0,$ $\varpi \left(\widehat{d},\widehat{r}\right)=0$, and $\varpi \left(\widehat{r},\widehat{z}\right)=0.$ So, $\widehat{z}=\widehat{d}=\widehat{r}.$ This completes the proof. □

In the end of this part, we present the following example to support our theoretical results.

Example 1.

Assume that $\chi =\mathbb{R},$ $\varpi \left(z,d\right)={\left|z-d\right|}^2$ is a $b-$metric space with $s=2$. Define a directed graph $\mho$ on χ by

$$\left(\left(z,d,r\right),\left(z^{\ast },d^{\ast },r^{\ast }\right)\right)\in \Xi \left(\mho \right),ifandonlyifz\le z^{\ast },d^{\ast }\le dandr\le r^{\ast }.$$

Describe the mapping $\Omega :{\chi }^3\to \chi$ as $\Omega \left(z,d,r\right)=\frac{1}{6}\left(z+d+r\right),$ $\left(z^{\ast },d^{\ast },r^{\ast }\right)\in {\chi }^3.$ It is clear that $\Omega$ has a $\mho -$monotone property. For any $\left(z,d,r\right),\left(z^{\ast },d^{\ast },r^{\ast }\right)\in {\chi }^3$ with $\left(\left(z,d,r\right),\left(z^{\ast },d^{\ast },r^{\ast }\right)\right)\in \Xi \left(\mho \right)$, we have

$$\begin{array}{ccc}\hfill \theta \left(s^2\varpi \left(\Omega \left(z,d,r\right),\Omega \left(z^{\ast },d^{\ast },r^{\ast }\right)\right)\right)& =& \frac{1}{4}\left(2^2{\left(\frac{z+d+r}{6}-\frac{z^{\ast }+d^{\ast }+r^{\ast }}{6}\right)}^2\right)\hfill \\ & =& \frac{1}{36}{\left(\left(z-z^{\ast }\right)+\left(d-d^{\ast }\right)+\left(r-r^{\ast }\right)\right)}^2\hfill \\ & \le & \frac{1}{9}\left({\left(z-z^{\ast }\right)}^2+{\left(d-d^{\ast }\right)}^2+{\left(r-r^{\ast }\right)}^2\right)\hfill \\ & =& \frac{1}{3}\vartheta \left(\varpi \left(z,z^{\ast }\right)+\varpi \left(d,d^{\ast }\right)+\varpi \left(r,r^{\ast }\right)\right).\hfill \end{array}$$

Hence, the condition (2) is satisfied with $\theta \left(a\right)=\frac{1}{4}a$ and $\vartheta \left(a\right)=\frac{1}{3}a.$ Clearly, $(\theta ,\vartheta )\in \Gamma .$ Therefore, all requirements of Theorem 1 are fulfilled. Moreover, $\left(\left(0,0,0\right),\left(0,0,0\right)\right)\in \Xi \left(\mho \right)$ So, by Theorems 1 and 3, the point $(0,0,0)$ is a unique TFP of the mapping $\Omega .$

3. Solving a System of Ordinary Differential Equations

This section is the mainstay of our paper in which the existence and uniqueness of the solution to a system of ordinary differential equations is investigated. This system is given as follows:

$$\left\{\begin{array}{c}{z}^{{}^{\prime }}\left(\nu \right)=\wp \left(\nu ,z_{\nu },u_{\nu },r_{\nu }\right),\\ u^{{}^{\prime }}\left(\nu \right)=\wp \left(\nu ,u_{\nu },r_{\nu },z_{\nu }\right),\\ r^{{}^{\prime }}\left(\nu \right)=\wp \left(\nu ,r_{\nu },z_{\nu },u_{\nu }\right),\end{array}\right.\nu \in \chi ,$$

(13)

under the conditions

$$z\left(\nu \right)={\varpi }_1\left(\nu \right),u\left(\nu \right)={\varpi }_2\left(\nu \right)\mathrm{and}r\left(\nu \right)={\varpi }_3\left(\nu \right),\nu \in (-\infty ,0],$$

(14)

where $\chi =[0,b],$ $\wp :\chi \times {⅁}^3\to {\mathbb{R}}^k,$ (where ${⅁}^3=⅁\times ⅁\times ⅁)$ ${\varpi }_1,{\varpi }_2,{\varpi }_3\in ⅁$, and $z_{\nu },u_{\nu },r_{\nu }$ are the history of the state from $-\infty$ to the time $\nu .$ Let the histories $z_{\nu },u_{\nu },r_{\nu }\in ⅁,$ where $\left(⅁,{∥.∥}_{⅁}\right)$ is a seminormed linear space of functions mapping $z:(-\infty ,0]\to {\mathbb{R}}^k,$ $k\in \mathbb{N}$ and satisfying the hypotheses below that were presented by Hale and Kato [21] for the ODE.

(i)

If $z:(-\infty ,b]\to {\mathbb{R}}^k,$ $b>0$ is continuous on $\chi$ and $z_0\in ⅁,$ then there are constant $\tau ,\xi >0$; so, for each $a\in [0,b)$, the following assumptions are satisfied:

(1)

$z_a\in ⅁;$

(2)

$∥z∥\le {∥z_a∥}_{⅁};$

(3)

${∥z_a∥}_{⅁}\le \tau sup\left\{∥z\left(c\right)∥:0\le c\le a\right\}+\xi {∥z_0∥}_{⅁}.$

(ii)

The function $z_a$ is a $⅁-$valued continuous function on $[0,b),$ where $z(.)$ is the function defined in $\left(i\right)$.

(iii)

The space $⅁$ is complete.

Now, we consider the following space to define a solution for Problems (13) and (14):

$$\Theta =\left\{z,z:(-\infty ,0]\to {\mathbb{R}}^k,z\in C\left(\chi ,{\mathbb{R}}^k\right),k\in \mathbb{N},z\left(\nu \right)={\varpi }_1\left(\nu \right),\nu \in (-\infty ,0],{\varpi }_1\in ⅁\right\},$$

equipped with the following seminorm

$${∥z∥}_{\Theta }={∥z_0∥}_{⅁}+\underset{0\le c\le b}{sup}∥z\left(c\right)∥.$$

It should be noted that the function $\left(z,u,r\right)\in {\Theta }^3$ (where ${\Theta }^3=\Theta \times \Theta \times \Theta )$ is a solution of (13) and (14), if $\left(z,u,r\right)$ fulfills (13) and (14).

Describe the operator ${\rm Y}:{\Theta }^3\to \Theta$ as

$$\begin{array}{ccc}\hfill {\rm Y}\left(z,u,r\right)& =& \left\{\begin{array}{cc}{\varpi }_1\left(\nu \right)& \mathrm{if}\nu \in (-\infty ,0)\\ {\varpi }_1\left(\nu \right)+\underset{0}{\overset{\nu }{\int }}\wp \left(\hslash ,z_{\hslash },u_{\hslash },r_{\hslash }\right)d\hslash & \mathrm{if}\nu \in \chi \end{array}\right.,\hfill \\ \hfill {\rm Y}\left(u,r,z\right)& =& \left\{\begin{array}{cc}{\varpi }_2\left(\nu \right)& \mathrm{if}\nu \in (-\infty ,0)\\ {\varpi }_2\left(\nu \right)+\underset{0}{\overset{\nu }{\int }}\wp \left(\hslash ,u_{\hslash },r_{\hslash },z_{\hslash }\right)d\hslash & \mathrm{if}\nu \in \chi \end{array}\right.,\hfill \end{array}$$

and

$${\rm Y}\left(r,z,u\right)=\left\{\begin{array}{cc}{\varpi }_3\left(\nu \right)& \mathrm{if}\nu \in (-\infty ,0)\\ {\varpi }_3\left(\nu \right)+\underset{0}{\overset{\nu }{\int }}\wp \left(\hslash ,r_{\hslash },z_{\hslash },u_{\hslash }\right)d\hslash & \mathrm{if}\nu \in \chi \end{array}\right..$$

Assume that ${\tilde{\varpi }}_1,{\tilde{\varpi }}_2,{\tilde{\varpi }}_3:(-\infty ,b)\to {\mathbb{R}}^k$ are functions defined by

$${\tilde{\varpi }}_1\left(\nu \right)=\left\{\begin{array}{cc}{\varpi }_1\left(\nu \right)& \mathrm{if}\nu \in (-\infty ,0)\\ {\varpi }_1\left(0\right)& \mathrm{if}\nu \in \chi \end{array}\right.,{\tilde{\varpi }}_2\left(\nu \right)=\left\{\begin{array}{cc}{\varpi }_2\left(\nu \right)& \mathrm{if}\nu \in (-\infty ,0)\\ {\varpi }_2\left(0\right)& \mathrm{if}\nu \in \chi \end{array}\right.,$$

and

$${\tilde{\varpi }}_3\left(\nu \right)=\left\{\begin{array}{cc}{\varpi }_3\left(\nu \right)& \mathrm{if}\nu \in (-\infty ,0)\\ {\varpi }_3\left(0\right)& \mathrm{if}\nu \in \chi \end{array}\right..$$

Then, ${\tilde{\varpi }}_1^0={\varpi }_1,$ ${\tilde{\varpi }}_2^0={\varpi }_2$, and ${\tilde{\varpi }}_3^0={\varpi }_3.$ For each ${\delta }_1,{\delta }_2,{\delta }_3\in C\left([0,b],{\mathbb{R}}^k\right)$ with ${\delta }_1\left(0\right)=0,$ ${\delta }_2\left(0\right)=0$, and ${\delta }_3\left(0\right)=0.$ Describe the functions ${\widehat{\delta }}_1,$ ${\widehat{\delta }}_2$, and ${\widehat{\delta }}_3$ as

$${\widehat{\delta }}_1\left(\nu \right)=\left\{\begin{array}{cc}0& \mathrm{if}\nu \in (-\infty ,0)\\ {\delta }_1\left(\nu \right)& \mathrm{if}\nu \in \chi \end{array}\right.,{\widehat{\delta }}_2\left(\nu \right)=\left\{\begin{array}{cc}0& \mathrm{if}\nu \in (-\infty ,0)\\ {\delta }_2\left(\nu \right)& \mathrm{if}\nu \in \chi \end{array}\right.,$$

and

$${\widehat{\delta }}_3\left(\nu \right)=\left\{\begin{array}{cc}0& \mathrm{if}\nu \in (-\infty ,0)\\ {\delta }_3\left(\nu \right)& \mathrm{if}\nu \in \chi \end{array}\right..$$

If $z(.),$ $u(.)$, and $r(.)$ satisfy the integral equations

$$\begin{array}{ccc}\hfill z\left(\nu \right)& =& {\varpi }_1\left(\nu \right)+\underset{0}{\overset{\nu }{\int }}\wp \left(\hslash ,z_{\hslash },u_{\hslash },r_{\hslash }\right)d\hslash ,\hfill \\ \hfill u\left(\nu \right)& =& {\varpi }_2\left(\nu \right)+\underset{0}{\overset{\nu }{\int }}\wp \left(\hslash ,u_{\hslash },r_{\hslash },z_{\hslash }\right)d\hslash ,\hfill \end{array}$$

and

$$r\left(\nu \right)={\varpi }_3\left(\nu \right)+\underset{0}{\overset{\nu }{\int }}\wp \left(\hslash ,r_{\hslash },z_{\hslash },u_{\hslash }\right)d\hslash ,$$

we can decompose $z(.),$ $u(.)$, and $r(.)$ as $z\left(\nu \right)={\widehat{\delta }}_1\left(\nu \right)+{\tilde{\varpi }}_1\left(\nu \right),$ $u\left(\nu \right)={\widehat{\delta }}_2\left(\nu \right)+{\tilde{\varpi }}_2\left(\nu \right)$, and $r\left(\nu \right)={\widehat{\delta }}_3\left(\nu \right)+{\tilde{\varpi }}_3\left(\nu \right)$ for every $0\le \nu \le b.$ In addition, the functions ${\delta }_1,$ ${\delta }_2$, and ${\delta }_3$ satisfy

$$\begin{array}{ccc}\hfill {\delta }_1\left(\nu \right)& =& \underset{0}{\overset{\nu }{\int }}\wp \left(\hslash ,{\widehat{\delta }}_1\left(\hslash \right)+{\tilde{\varpi }}_1\left(\hslash \right),{\widehat{\delta }}_2\left(\hslash \right)+{\tilde{\varpi }}_2\left(\hslash \right),{\widehat{\delta }}_3\left(\hslash \right)+{\tilde{\varpi }}_3\left(\hslash \right)\right)d\hslash ,\hfill \\ \hfill {\delta }_2\left(\nu \right)& =& \underset{0}{\overset{\nu }{\int }}\wp \left(\hslash ,{\widehat{\delta }}_2\left(\hslash \right)+{\tilde{\varpi }}_2\left(\hslash \right),{\widehat{\delta }}_3\left(\hslash \right)+{\tilde{\varpi }}_3\left(\hslash \right),{\widehat{\delta }}_1\left(\hslash \right)+{\tilde{\varpi }}_1\left(\hslash \right)\right)d\hslash ,\hfill \end{array}$$

and

$${\delta }_3\left(\nu \right)=\underset{0}{\overset{\nu }{\int }}\wp \left(\hslash ,{\widehat{\delta }}_3\left(\hslash \right)+{\tilde{\varpi }}_3\left(v\right),{\widehat{\delta }}_1\left(\hslash \right)+{\tilde{\varpi }}_1\left(\hslash \right),{\widehat{\delta }}_2\left(\hslash \right)+{\tilde{\varpi }}_2\left(\hslash \right)\right)d\hslash .$$

Put $C_0=\left\{\delta \in C\left([0,b],{\mathbb{R}}^k\right):\delta \left(0\right)=0\right\}$ equipped with a $b-$metric $\varpi \left(z,u\right)={\left({sup}_{\nu \in \chi }∥z\left(\nu \right)-u\left(\nu \right)∥\right)}^2$ with $s=2$.

Consider the following partial order relation on $C_0^3$ (where $C_0^3=C_0\times C_0\times C_0):$

$$(z_1,u_1,r_1)\le (z_2,u_2,r_2)\iff z_1\left(a\right)\le z_2\left(a\right),u_1\left(a\right)\ge u_2\left(a\right),\mathrm{and}{r}_1\left(a\right)\le r_2\left(a\right),a\in \chi .$$

Now, Problems (13) and (14) will be considered under the following hypotheses:

Hypothesis 1 (H1).

The function $\wp :\chi \times {⅁}^3\to {\mathbb{R}}^k,$ $k\in \mathbb{N}$ is continuous.

Hypothesis 2 (H2).

For all $z,u,r,z_1,u_1,r_1\in {\mathbb{R}}^k$ with $z\le z_1,$ $u_1\le u$ and $r\le r_1,$

$$\wp \left(\nu ,z,u,r\right)\le \wp \left(\nu ,z_1,u_1,r_1\right).$$

Hypothesis 3 (H3).

For each $\nu \in [0,b],$ $z,u,r,z_1,u_1,r_1\in {\mathbb{R}}^k$, $z\le z_1,$ $u_1\le u$, and $r\le r_1,$ we have

$${∥\wp \left(\nu ,z,u,r\right)-\left(\nu ,z_1,u_1,r_1\right)∥}^2\le \frac{1}{12b^2}ln\left(1+\frac{1}{\tau }{∥z-z_1∥}_{⅁}^2+{∥u-u_1∥}_{⅁}^2+{∥r-r_1∥}_{⅁}^2\right).$$

Theorem 5.

Consider Problems (13) and (14) under the hypotheses $\left(H_1\right)$–$\left(H_3\right).$ If there are $(e,f,g)\in C_0^3$, so that

$$\begin{array}{ccc}\hfill e\left(\nu \right)& \ge & \underset{0}{\overset{\nu }{\int }}\wp \left(\hslash ,\widehat{e}\left(\hslash \right)+{\tilde{\varpi }}_1\left(\hslash \right),\widehat{f}\left(\hslash \right)+{\tilde{\varpi }}_2\left(\hslash \right),\widehat{g}\left(\hslash \right)+{\tilde{\varpi }}_3\left(\hslash \right)\right)d\hslash ,\hfill \\ \hfill f\left(\nu \right)& \le & \underset{0}{\overset{\nu }{\int }}\wp \left(\hslash ,\widehat{f}\left(\hslash \right)+{\tilde{\varpi }}_1\left(\hslash \right),\widehat{g}\left(\hslash \right)+{\tilde{\varpi }}_2\left(\hslash \right),\widehat{e}\left(\hslash \right)+{\tilde{\varpi }}_3\left(\hslash \right)\right)d\hslash ,\hfill \end{array}$$

and

$$g\left(\nu \right)\ge \underset{0}{\overset{\nu }{\int }}\wp \left(\hslash ,g\left(\hslash \right)+{\tilde{\varpi }}_1\left(\hslash \right),\widehat{e}\left(\hslash \right)+{\tilde{\varpi }}_2\left(\hslash \right),\widehat{f}\left(\hslash \right)+{\tilde{\varpi }}_3\left(\hslash \right)\right)d\hslash .$$

Then, there is at least one solution to the problem (13) and (14).

Proof. 

Let $\aleph :C_0^3\to C_0$ be an operator defined by

$$\aleph \left({\delta }_1,{\delta }_2,{\delta }_3\right)=\underset{0}{\overset{\nu }{\int }}\wp \left(\hslash ,{\widehat{\delta }}_1\left(\nu \right)+{\tilde{\varpi }}_1\left(\nu \right),{\widehat{\delta }}_2\left(\nu \right)+{\tilde{\varpi }}_2\left(\nu \right),{\widehat{\delta }}_3\left(\nu \right)+{\tilde{\varpi }}_3\left(\nu \right)\right)d\hslash .$$

It is clear that if ${\rm Y}$ has a TFP, then ℵ has a TFP and vice versa. So the existence solution of Problems (13) and (14) is equivalent to finding a TFP of the mapping ℵ. To achieve this, we demonstrate that ℵ fulfills the requirements of Theorems 1 or 2.

Define the graph $\mho$ with $\vee (\mho )=C_0^3$ and

$$\Xi \left(\mho \right)=\left\{\left(\left(z,u,r\right),\left(z^{\ast },u^{\ast },r^{\ast }\right)\right)\in C_0^3\times C_0^3:z\le z^{\ast },u^{\ast }\ge u\mathrm{and}r\le r^{\ast }\right\}.$$

It follows that

$$\left(\left(z,u,r\right),\left(z^{\ast },u^{\ast },r^{\ast }\right)\right)\in \Xi \left(\mho \right)\iff \left(z,z^{\ast }\right)\in \Xi \left(\mho \right),\left(u^{\ast },u\right)\in \Xi \left(\mho \right)\mathrm{and}\left(r,r^{\ast }\right)\in \Xi \left(\mho \right),$$

for all $\left(\left(z,u,r\right),\left(z^{\ast },u^{\ast },r^{\ast }\right)\right)\in C_0^3.$

Consider $z,u,r,z_1,u_1,r_1,z_2,u_2,r_2\in C_0.$ If $(z_1,z_2)\in \Xi \left(\mho \right),$ then, from $\left(H_2\right),$ we can write

$$\begin{array}{ccc}\hfill \aleph \left(z_1,u,r\right)& =& \underset{0}{\overset{\nu }{\int }}\wp \left(\hslash ,{\widehat{z}}_1\left(\hslash \right)+{\tilde{\varpi }}_1\left(\hslash \right),\widehat{u}\left(\hslash \right)+{\tilde{\varpi }}_2\left(\hslash \right),\widehat{r}\left(\hslash \right)+{\tilde{\varpi }}_3\left(\hslash \right)\right)d\hslash \hfill \\ & \le & \underset{0}{\overset{\nu }{\int }}\wp \left(\hslash ,{\widehat{z}}_2\left(\hslash \right)+{\tilde{\varpi }}_1\left(\hslash \right),\widehat{u}\left(\hslash \right)+{\tilde{\varpi }}_2\left(\hslash \right),\widehat{r}\left(\hslash \right)+{\tilde{\varpi }}_3\left(\hslash \right)\right)d\hslash \hfill \\ & =& \aleph \left(z_2,u,r\right),\hfill \end{array}$$

which implies that $\left(\aleph \left(z_1,u,r\right),\aleph \left(z_2,u,r\right)\right)\in \Xi \left(\mho \right).$ Moreover, if $(u_1,u_2)\in \Xi \left(\mho \right),$ we can write

$$\begin{array}{ccc}\hfill \aleph \left(z,u_2,r\right)& =& \underset{0}{\overset{\nu }{\int }}\wp \left(\hslash ,{\widehat{z}}_1\left(\hslash \right)+{\tilde{\varpi }}_1\left(\hslash \right),{\widehat{u}}_2\left(\hslash \right)+{\tilde{\varpi }}_2\left(\hslash \right),\widehat{r}\left(\hslash \right)+{\tilde{\varpi }}_3\left(\hslash \right)\right)d\hslash \hfill \\ & \le & \underset{0}{\overset{\nu }{\int }}\wp \left(\hslash ,{\widehat{z}}_2\left(\hslash \right)+{\tilde{\varpi }}_1\left(\hslash \right),{\widehat{u}}_1\left(\hslash \right)+{\tilde{\varpi }}_2\left(\hslash \right),\widehat{r}\left(\hslash \right)+{\tilde{\varpi }}_3\left(\hslash \right)\right)d\hslash \hfill \\ & =& \aleph \left(z,u_1,r\right),\hfill \end{array}$$

which leads to $\left(\aleph \left(z,u_2,r\right),\aleph \left(y,u_1,r\right)\right)\in \Xi \left(\mho \right).$ Analogously, we obtain $\left(\aleph \left(z,u,r_1\right),\aleph \left(y,u,r_2\right)\right)\in \Xi \left(\mho \right).$ Hence, ℵ has the mixed ℧-monotone property. In order to prove the contractive condition of Theorem 1, assume that $\left(\left(z,u,r\right),\left(z^{\ast },u^{\ast },r^{\ast }\right)\right)\in C_0^3$, so that

$$\left(\left(z,u,r\right),\left(z_1,u_1,r_1\right)\right)\in \Xi \left(\mho \right)\iff \left(z,z_1\right)\in \Xi \left(\mho \right),\left(u_1,u\right)\in \Xi \left(\mho \right),\mathrm{and}\left(r,r_1\right)\in \Xi \left(\mho \right);$$

then, by using the assumptions $\left(H_1\right),$ $\left(H_1\right)$, and $\left(H_3\right),$ we have

$$\begin{array}{ccc}& & {∥\aleph \left(\left(z,u,r\right)\right)-\aleph \left(z_1,u_1,r_1\right)∥}^2\hfill \\ & =& ∥\underset{0}{\overset{\nu }{\int }}\wp \left(\hslash ,\widehat{z}\left(\hslash \right)+{\tilde{\varpi }}_1\left(\hslash \right),\widehat{u}\left(\hslash \right)+{\tilde{\varpi }}_2\left(\hslash \right),\widehat{r}\left(\hslash \right)+{\tilde{\varpi }}_3\left(\hslash \right)\right)d\hslash \hfill \\ & & {-\underset{0}{\overset{\nu }{\int }}\wp \left(\hslash ,{\widehat{z}}_1\left(\hslash \right)+{\tilde{\varpi }}_1\left(\hslash \right),{\widehat{u}}_1\left(\hslash \right)+{\tilde{\varpi }}_2\left(\hslash \right),{\widehat{r}}_1\left(\hslash \right)+{\tilde{\varpi }}_3\left(\hslash \right)\right)d\hslash ∥}^2\hfill \\ & \le & b\underset{0}{\overset{\nu }{\int }}∥\wp \left(\hslash ,\widehat{z}\left(\hslash \right)+{\tilde{\varpi }}_1\left(\hslash \right),\widehat{u}\left(\hslash \right)+{\tilde{\varpi }}_2\left(\hslash \right),\widehat{r}\left(\hslash \right)+{\tilde{\varpi }}_3\left(\hslash \right)\right)\hfill \\ & & -{\wp \left(\hslash ,{\widehat{z}}_1\left(\hslash \right)+{\tilde{\varpi }}_1\left(\hslash \right),{\widehat{u}}_1\left(\hslash \right)+{\tilde{\varpi }}_2\left(\hslash \right),{\widehat{r}}_1\left(\hslash \right)+{\tilde{\varpi }}_3\left(\hslash \right)\right)∥}^{2}d\hslash \hfill \\ & \le & \frac{1}{12b}\underset{0}{\overset{\nu }{\int }}ln\left(1+\frac{1}{\tau }{∥\widehat{z}\left(d\hslash \right)-{\widehat{z}}_1\left(\hslash \right)∥}_{⅁}^2+\frac{1}{\tau }{∥\widehat{u}\left(\hslash \right)-{\widehat{u}}_1\left(\hslash \right)∥}_{⅁}^2+\frac{1}{\tau }{∥\widehat{r}\left(\hslash \right)-{\widehat{r}}_1\left(\hslash \right)∥}_{⅁}^2\right)d\hslash \hfill \\ & \le & \frac{1}{12}ln\left(\underset{\hslash \in \chi }{sup}{∥\widehat{z}\left(\hslash \right)-{\widehat{z}}_1\left(\hslash \right)∥}_{⅁}^2+\underset{\hslash \in \chi }{sup}{∥\widehat{u}\left(\hslash \right)-{\widehat{u}}_1\left(\hslash \right)∥}_{⅁}^2+\underset{\hslash \in \chi }{sup}{∥\widehat{r}\left(\hslash \right)-{\widehat{r}}_1\left(\hslash \right)∥}_{⅁}^2\right),\hfill \end{array}$$

which yields

$$\theta \left(s^2\varpi \left(\aleph \left(\left(z,u,r\right)\right)-\aleph \left(z_1,u_1,r_1\right)\right)\right)\le \frac{1}{3}\vartheta \left(\varpi \left(z,z_1\right),\varpi \left(u,u_1\right),\varpi \left(r,r_1\right)\right),$$

where $\theta \left(a\right)=a$, and $\vartheta \left(a\right)=ln(1+a).$ Obviously, the pair $\left(\theta ,\vartheta \right)\in \Gamma .$ Hence, by our assumptions, we conclude that

$$\left(\left(z,u,r\right),\left(\Omega \left(z,u,r\right),\Omega \left(u,r,z\right),\Omega \left(r,z,u\right)\right)\right)\in \Xi \left(\mho \right).$$

The operator ℵ is continuous, and the triple $\left(C_0,\Xi ,\varpi \right)$ satisfy the property (p). Hence, all requirements of Theorems 1 and 3 are fulfilled. Hence, there is a TFP of the mapping $\Omega$ in $C_0^3,$ which represents a solution to the problem (13) and (14). □

4. Conclusions

There has been much development of the theory of delay differential equations. This was connected to a variety of practical issues whose study required the resolution of delay equations. Equations of this kind are necessary to describe processes whose rate depends on their prior states. Such processes are commonly described as “delay processes” or “processes with aftereffects.” The present paper was dedicated to the study of the existence and uniqueness of tripled fixed points in a $b-$metric space with a directed graph. Common tripled fixed point results were also provided. Moreover, some applications of the main results in solving different types of tripled equation systems were presented. Then, using our main results, we studied the existence and uniqueness of a solution to a system of ordinary differential equations with infinite delay. Our results help to improve some results from the related literature and provide new directions in the study of economic phenomena, using the tripled fixed point technique.