A new type of three dimensional metric spaces with applications to fractional differential equations

1.   Introduction and preliminaries

The importance of fixed point theory is due to its application in many fields, for example the existence and uniqueness of a solution of system of equations or fractional differential equations, integral equations. Also, it has many applications in economics, engineering and many more fields. The very first proof of existence and uniqueness of a fixed point was given by Banach ^[1] which was an inspiration to many researchers around the world to work in the field of fixed point theory See ^{[2,3,4,5,6,7,8,9,10]}. Generalizing Banach result is the focus of researchers now a days ^{[11,12,13,14,15,16,17,18,19,20,21,22]}. Lately, Beg et al. ^[23,24] introduced the concept of $S^{JS}$-metric spaces, which is a three dimension metric space. However, given there defined triangle inequality, we do not see the point of the third component due to the fact that there is no use for it in their inequality, so basically it is a two dimension space. In this manuscript, we define $J$-metric spaces which are three dimension spaces where $S^{JS}$-metric spaces is a special case. Also, we present an application of our result to fractional differential equations along with an application to system of linear equations. First, we remind the reader of the definition of $S^{JS}$-metric spaces.

Definition 1.1. ^[23] Consider a nonempty set $\Omega$ and a function $J:\Omega^3\rightarrow [0, \infty)$. Let us define the set

for all $\delta\in \Omega.$

Definition 1.2. ^[23] Let $\Omega$ be a nonempty set and, $J:\Omega^3\rightarrow [0, \infty)$ satisfy the following hypothesis:

(i) $J(\delta, \xi, \nu) = 0$ implies $\delta = \xi = \nu$ for any $\delta, \xi, \nu \in \Omega$;

(ii) There exists some $b > 0$ such that, for any $(\delta, \xi, \nu) \in \Omega^3$ and $\{\nu_n\} \in S(J, \Omega, \nu)$,

Then the pair $(\Omega, J)$ is called an $S^{JS}$-metric space.

Moreover, if $J$ also satisfies $J(\delta, \delta, \xi) = J(\xi, \xi, \delta)$ for all $\delta, \xi \in \Omega$, then we call it a symmetric $S^{JS}$-metric space.

2.   Main result

In this section, we introduce the notion of $J-$metric spaces, and prove fixed point theorems for self mappings in this new space.

Definition 2.1. Consider a nonempty set $\Omega$ and a function $J:\Omega^3\rightarrow [0, \infty)$. Let us define the set

for all $\nu\in \Omega$

Definition 2.2. Let $\Omega$ be a nonempty set and, $J:\Omega^3\rightarrow [0, \infty)$ satisfy the following hypothesis:

(i) $J(\tau, \upsilon, \zeta) = 0$ implies $\tau = \upsilon = \zeta$ for any $\tau, \upsilon, \zeta \in \Omega$;

(ii) There exists some $b > 0$ such that, for any $(\tau, \upsilon, \zeta) \in \Omega^3$ and $\{\nu_n\} \in S(J, \Omega, \nu)$,

Then the pair $(\Omega, J)$ is called a $J$-metric space. Moreover, if $J(\tau, \tau, \upsilon) = J(\upsilon, \upsilon, \tau)$ for all $\tau, \upsilon \in \Omega,$ then the pair $(\Omega, J)$ is called a symmetric $J$-metric space.

Remark 2.3. Note that, the following condition is not necessary true

Now, we present some of the topological properties of $J-$metric spaces.

Definition 2.4. Let $(\Omega, J)$ be an $J$-metric space. A sequence $\{\tau_n\} \subset \Omega$ is said to be convergent to an element $\tau \in \Omega$ if $\{\tau_n\} \in S(J, \Omega, \tau)$.

Proposition 2.5. In a $J$-metric space $(\Omega, J)$, if $\{\tau_n\}$ converges to both $\tau_1$ and $\tau_2$, then $\tau_1 = \tau_2$.

Proof. Assume that $\{\tau_n\}$ converges to both $\tau_1$ and $\tau_2.$ Hence,

Thus,

Definition 2.6. Let $(\Omega, J_{1})$ and $(\Gamma, J_{2})$ be two $J$-metric spaces and $\sigma : \Omega \rightarrow \Gamma$ be a mapping. Then $\sigma$ is called continuous at $a_0 \in \Omega$ if, for any $\varepsilon > 0$, there exists $\xi > 0$ such that, for any $\tau \in \Omega$, $J_{2}(\sigma a_0, \sigma a_0, \sigma \tau) < \varepsilon$ whenever $J_{1}(a_0, a_0, \tau) < \xi$.

Definition 2.7. (1) Let $(\Omega, J)$ be a $J$-metric space. A sequence ${\tau_n} \subset \Omega$ is said to be Cauchy if $\lim_{n, m\rightarrow \infty} J(\tau_n, \tau_n, \tau_m) = 0$.

(2) A $J$-metric space is said to be complete if every Cauchy sequence in $\Omega$ is convergent.

(3) In a $J$-metric space $(\Omega, J)$ if $\sigma$ is continuous at $a_0 \in \Omega$ then for any sequence ${\tau_n} \in S(J, \Omega, a_0)$ implies $\{\sigma \tau_n\} \in S(J, \Omega, \sigma a_0)$.

Remark 2.8. Note that $S(J, \Omega, \delta)$ in some cases can be empty. The following example presents a nonempty set of $S(J, \Omega, \delta)$.

Example 2.9. Let $\Omega = \mathbb{R}$ and, $J:\Omega^3\rightarrow [0, \infty)$ defined by $J(\delta, \xi, \nu) = |\delta - \xi|+|\xi -\nu|$ for all $\delta, \xi, \nu \in \mathbb{R}$. Let $\nu \in \mathbb{R}$ and the sequence $(\nu_n)$ such that $\nu_n = \nu +\dfrac{1}{n}$.

It is easy to see that $\lim\limits_{n\rightarrow \infty}J(\nu, \nu +\dfrac{1}{n}, \nu +\dfrac{1}{n}) = 0$. Therefore, for every $\nu \in \mathbb{R}$ there exists a sequence $\nu_n = \nu +\dfrac{1}{n}$ such that $S(J, \Omega, \nu)\neq \emptyset$.

Next, we present two examples of $J$-metric spaces.

Example 2.10. Let $X = \mathbb{R}$ and $J(\tau, \upsilon, \zeta) = |\tau|+|\upsilon|+2|\zeta|$ for all $\tau, \upsilon, \zeta \in X$.

We have $J(\tau, \upsilon, \zeta) = 0$ imply that $|\tau|+|\upsilon|+2|\zeta| = 0$ which gives us $|\tau| = |\upsilon| = |\zeta| = 0$ then the first condition of the Definition 2.2 is satisfied. Also the symmetry of $J$ is satisfied since we have $J(\tau, \tau, \upsilon) = 2|\tau|+2|\upsilon| = J(\upsilon, \upsilon, \tau)$. Now, let's verfiy the triangle inequality. Let $\tau, \upsilon, \zeta \in X$ and $\nu_{n}$ a convergent sequence in $X$ such that $\lim_{n\rightarrow \infty}J(\nu, \nu, \nu_{n}) = 0$, we have

Then, all the assumptions of Definition 2.2 are satisfied. Hence, $J$ is a $J$-metric with $b = 2.$

Example 2.11. Let $X = [0, \infty)$ and $J(\tau, \upsilon, \zeta) = |\tau-\upsilon|+|\tau-\zeta|$ for all $\tau, \upsilon, \zeta \in X$. We have

● $J(\tau, \upsilon, \zeta) = 0$ imply that $|\tau| = |\upsilon| = |\zeta| = 0$.

● $J(\tau, \tau, \upsilon) = |\tau-\tau|+|\tau-\upsilon| = |\upsilon-\tau| = J(\upsilon, \upsilon, \tau)$.

● Let $\nu_{n}$ a convergent sequence in $X$ such that $\lim_{n\rightarrow \infty}J(\nu, \nu, \nu_{n}) = 0$, we have

Then, all the assumptions of Definition 2.2 are satisfied. Hence, $J$ is a $J$-metric with $b = 2.$

Theorem 2.12. Let $(\Omega, J)$ be a $J$-complete symmetric metric space and $\sigma:\Omega \rightarrow \Omega$ be a continuous mapping satisfying

where $\psi :[0, +\infty) \rightarrow [0, +\infty)$ is an increasing function such that, $\lim_{n\rightarrow \infty} \psi^{n}(t) = 0$ for each fixed $t > 0$.

Then $\sigma$ has a unique fixed point in $\Omega$.

Proof. Let $\tau_{0}$ be an arbitrary element in $\Omega$. We define the sequence $\{\tau_{n}\}_{n \geq 0}\subset \Omega$ as follows

We will prove that $\{\tau_{n}\}$ is a Cauchy sequence in $\Omega$. Let $n, m \in \mathbb{N}$, using (2.1) we obtain

We assume w.l.o.g that $m = n+p$ for some constant $p\in \mathbb{N}$ we get

By taking the limit in (2.3) as $n\longrightarrow \infty$ we obtain

Therefore, $\{\tau_{n}\}$ is a Cauchy sequence in $\Omega$ and due to its completeness, there exists $\tau\in \Omega$ such that $\tau_k\rightarrow \tau$ as $k\rightarrow \infty$.

In addition, $\tau = \lim\limits_{k\rightarrow \infty}\tau_k = \lim\limits_{k\rightarrow \infty}\tau_{k+1} = \lim\limits_{k\rightarrow \infty}\sigma\tau_k = \sigma \tau$. Thus, $\sigma$ has $\tau$ as a fixed point.

Let $\tau_1$ and $\tau_2$ be two fixed points of $\sigma$.

Since, $\psi ^{n}(t) < t$ for any $t > 0$, we obtain from (2.5), $J(\tau_1, \tau_1, \tau_2) < J(\tau_1, \tau_1, \tau_2)$, then $J(\tau_1, \tau_1, \tau_2) = 0$ and $\tau_1 = \tau_2$, and $\sigma$ has a unique fixed point in $\Omega$.

Theorem 2.13. Let $(\Omega, J)$ be a $J$- complete symmetric metric space and $\sigma :\Omega\rightarrow \Omega$ be a mapping satisfying

where $\lambda \in A = \big\{ \lambda : \Omega^3 \rightarrow (0, 1), \lambda(f(\tau, \upsilon, \zeta)) \leq \lambda (\tau, \upsilon, \zeta)$ and $f:\Omega \rightarrow \Omega$ a given mapping$\big\}$. Then $\sigma$ has a unique fixed point.

Proof. Let $\tau_{0}$ be an arbitrary element in $\Omega$. We construct the sequence $\{\tau_{n}$ as follows $\{\tau_{n} = \sigma ^{n}\tau_{0}\}$. Let's prove that $\{\tau_{n}\}$ is a Cauchy sequence. For all natural numbers $n, m$, we suppose w.l.o.g that $n < m$ and assume that these exists a constant $p \in N$ such that $m = n+p$. By using (2.6) we have

Taking the limit as $n\rightarrow \infty$ and considering the property of $\lambda$ into view, we obtain that $\lim\limits_{n, m\rightarrow \infty} J(\tau_{n}, \tau_{n}, \tau_{m}) = 0$, that is $\{\tau_{n}\}$ is a Cauchy sequence. Then, by completeness of $\Omega$, there exists $u\in \Omega$ such that

We claim that $u$ is a fixed point of $\sigma$. From (2.7), we deduce that $\tau_{n} \in S(J, \Omega, u)$ and

and

By using the triangle inequality and taking into account (2.8) we get:

Using (2.9) and (2.10) we obtain that $J(\sigma u, \sigma u, u) = 0$, that is $\sigma u = u$. Therefore $u$ is a fixed point of $\sigma$.

Let, $\xi_1, \xi_2 \in \Omega$ be two fixed points of $\sigma$ such that $\xi_1\neq \xi_2$ that is $\sigma \xi_1 = \xi_1$ and $\sigma \xi_2 = \xi_2$

Then, $J(\xi_1, \xi_1, \xi_2) = 0$ which implies that $\xi_1 = \xi_2$.

Theorem 2.14. Let $(\Omega, J)$ be a complete symmetric $J$-metric space and $\sigma :\Omega\rightarrow \Omega$ be a continuous mapping such that:

for all $\tau, \upsilon, \nu \in \Omega$ where

Then, there exists a unique fixed point of $\sigma$.

Proof. Let $\tau_{0}\in \Omega$ be an arbitrary point of $\Omega$ and $\{ \tau_{n} = \sigma ^{n}\tau_{0}\}$ be a sequence in $\Omega$. From (2.11) we have

Then

Taking $\lambda = \dfrac{\alpha+\beta}{1-\gamma -\tau}$, then from (2.12) we have $0 < \lambda < 1$. By induction we get $J(\tau_{n}, \tau_{n+1}, \tau_{n+1}) \leq \lambda^n J(\tau_{0}, \tau_{1}, \tau_{1})$ which gives that

We denote $J_{n} = J(\tau_{n}, \tau_{n+1}, \tau_{n+1})$. For all $n, m \in N, n < m$ we assume w.l.o.g that there exists a fixed $p \in N$ such that $m = n+p$. we have

By taking the limit in (2.15) as $n \rightarrow \infty$ and using (2.13) and (2.14), we obtain

Then, $\{\tau_{n}\}$ is a Cauchy sequence in $\Omega$. By completeness, there exists $u \in \Omega$ such that $\tau_{n} \rightarrow u$ as $n\rightarrow \infty$ and

In addition, $u = \lim\limits_{k\rightarrow \infty}\tau_k = \lim\limits_{k\rightarrow \infty}\tau_{k+1} = \lim\limits_{k\rightarrow \infty}\sigma\tau_k = \sigma u$. Therefore, $\sigma$ has $u$ as a fixed point.

Let $\xi_1, \xi_2 \in \Omega$ be two fixed point of $\sigma$, $\xi_1\neq \xi_2$ that is $\sigma \xi_1 = \xi_1$, $\sigma \xi_2 = \xi_2$.

Then, $(1-\alpha)J(\xi_1, \xi_1, \xi_2) \leq 0$. Using (2.13) we conclude that $J(\xi_1, \xi_1, \xi_2) = 0$ that is $\xi_1 = \xi_2$.

3.   Applications

3.1.   Fractional differential equation

There has been many applications of fixed point to fractional differential equations see ^[25]. In this section, we discuss the existence of a solution to the following problem:

where $T > 0$ and $f:I\times \mathbb{R}\longrightarrow\mathbb{R}$ is a continuous function, $I = [0, T]$ and $D^{\lambda}x$ denotes a Riemann-Liouville fractional derivative of $x$ with $\lambda\in (0, 1)$.

Let $C_{1-\lambda}(I, \mathbb{R}) = \{f\in C((0, T], \mathbb{R}): t^{1-\lambda}\in C(I, \mathbb{R})\}$. We define the following weighted norm

Theorem 3.1. Let $\lambda\in(0, 1)$, $f\in C(I\times \mathbb{R}, \mathbb{R})$ increasing and $0 < \alpha < 1$. In addition, we assume the following hypothesis:

Then the problem $\mathcal{(P)}$ has a unique solution.

Proof. Problem $\mathcal{(P)}$ is equivalent to the problem $\mathcal{M}x(t) = x(t)$ where

In fact, proving that the operator $\mathcal{M}$ has a fixed point is sufficient to say that problem $\mathcal{P}$ has a unique solution. Indeed, $(A = C_{1-\lambda}(I, \mathbb{R}), J)$ is a complete $J-$metric space if we consider

The mapping $\mathcal{M}$ is increasing since $f$ is increasing.

Now, we must prove that $\mathcal{M}$ is a contraction map. Let $x, y, z\in C_{1-\lambda}(J, \mathbb{R})$, $0 < \lambda < 1$.

Subsequently, by the hypothesis of the theorem we have

From the Riemann-Liouville fractional integral we have

Therefore, we have

Therefore, $\sigma$ satisfies the conditions of Theorem 2.12 with $\psi(t) = \alpha t$.

Thus, by Theorem 2.12, we deduce that $\mathcal{M}$ has a unique fixed point which leads us to conclude that equation $(\mathcal{(P)})$ has a unique solution as required.

3.2.   Fredholm type integral equation

Let $\Omega = C([0, 1], \mathbb{R})$ and consider the following Fredholm type integral equation

where $\chi: [0, 1]^3\longrightarrow \mathbb{R}$ is a continuous function.

Let $J:\Omega^3 \longrightarrow [0, \infty)$ defined by $J(f, g, h) = |\dfrac{\sup_{t\in[0, 1]}(f(t), g(t))-h(t)}{2}|$. It is not diffuclt to see that $(\Omega, J)$ is a complete $J-$metric space.

Theorem 3.2. If $f, g, h\in \Omega$ satisfy the following condition

then Eq (3.1) has a unique solution.

Proof. Let $\sigma: \Omega \longrightarrow \Omega$ defined by $\sigma f(\nu) = \int_0^1 \chi(\delta, \xi, f(\nu)) ds$ then

Assume w.l.o.g that $\chi(\delta, \xi, f(\nu)) > \chi(\delta, \xi, g(\nu))$, then we obtain

Therefore, $\sigma$ satisfies the conditions of Theorem 2.12 with $\psi(t) = \lambda t$.

Thus, by Theorem 2.12, we deduce that $\sigma$ has a unique fixed point which leads us to conclude that Eq (3.1) has a unique solution as desired.

3.3.   Linear system of equations

Consider the set $\Omega = \mathbb{R}^{n}$ where $\mathbb{R}$ is the set of real numbers and $n$ a positive integer. Now, consider the symmetric $J-$metric space $(\Omega, J)$ defined by

for all $\delta = (\delta_1, ..., \delta_n), \xi = (\xi_1, ..., \xi_n), \nu = (\nu_1, ..., \nu_n) \in \Omega.$

Theorem 3.3. Consider the following system

if $\theta = \max\limits_{1\leq i \leq n}\Big(\sum\limits_{j = 1, j\neq i}^{n}|s_{ij}|+|1+s_{ii}|\Big) < 1,$ then the above linear system has a unique solution.

Proof. Consider the map $\sigma:\Omega\rightarrow \Omega$ defined by $\sigma\delta = (B+I_{n})\delta-r$ where

$\delta = (\delta_{1}, \delta_{2}, \cdots, \delta_{n}); \xi = (\xi_{1}, \xi_{2}, \cdots, \xi_{n}) \ \text{and} \ \nu = (\nu_{1}, \nu_{2}, \cdots, \nu_{n})\in \mathbb{R}^{n}$, $I_{n}$ is the identity matrix for $n\times n$ matrices and $r = (r_{1}, r_{2}, \cdots, r_{n})\in \mathbb{C}^{n}.$ Let us prove that $J(\sigma\delta, \sigma\xi, \sigma\nu)\leq \theta J(\delta, \xi, \nu)$, $\forall \delta, \xi, \nu\in \mathbb{R}^n$.

We denote by

with

Hence,

On the other hand, for all $i = 1, ..., n$, we have

Therefore, using (3.2) and (3.3) we get

where, $\Phi(t) = \theta t$, $\forall t\geq 0$. Note that, all the hypotheses of Theorem 2.12 are satisfied. Thus, $\sigma$ has a unique fixed point. Therefore, the above linear system has a unique solution as desired.

4.   Conclusions

In this manuscript, we introduced a new type of metric spaces called $J-$metric spaces, which is a three dimension metric space, we proved the existence and uniqueness of a fixed point for self mapping in such space under different types of metric spaces. Moreover, we presented an application of our results to solving system of linear equations and a fractional differential equation using fixed point theory approach.

Acknowledgments

The authors N. Mlaiki, S. Haque and W. Shatanawi would like to thank Prince Sultan University for paying the publication fees for this work through TAS LAB.

Conflict of interest

The authors declare no conflicts of interest.