Fixed point approach to solve nonlinear fractional differential equations in orthogonal $\mathcal{F}$-metric spaces

1.   Introduction

In different fields of pure and applied sciences, the analysis of metric spaces has played an essential role. We can discover various effective and impressive applications of metric spaces in different fields of sciences such as mathematics, computer science, medicine, economics, physics and biology ^[1,2,3]. Various researchers generalized, improved and extended the notion of metric spaces ^[4] to $b$-metric spaces of Czerwik ^[5], generalized metric spaces of Branciari ^[6], $\mathcal{F}$-metric spaces of Jleli et al. ^[7], orthogonal metric spaces of Gordji et al. ^[8], orthogonal $\mathcal{F}$-metric spaces of Kanwal et al. ^[9] and others.

The famous extensions of the concept of metric spaces have been done by Bakhtin ^[10] and were formally defined by Czerwik ^[5] in 1993. Czerwik ^[5] gave the idea of a $b$-metric space, which broadens the notion of a metric space by improving the triangle inequality metric axiom by putting a constant $s\geq 1$ multiplied on the right-hand side, and it is one of the enormous applied extensions for metric spaces. Khamsi et al. ^[11] reintroduced this notion under the name metric-type and proved some fixed point results in this newly introduced space. In ^[6], Branciari gave the notion of a rectangular metric space and generalized the classical metric space by replacing the triangle inequality with a more general inequality that is called a rectangular inequality. This inequality involves distances of four points. In 2018, Jleli et al. ^[7] gave an important extension of metric space, $b$-metric space and rectangular metric space which is known as $\mathcal{F}$-metric space. Subsequently, Kanwal et al. ^[9] initiated the theory of orthogonal $\mathcal{F}$-metric spaces and established some common fixed point results. For more details in this direction, we refer the readers to ^{[12,13,14,15,16,17,18,19,20,21,22,23,24,25]}.

On the other hand, Stefan Banach ^[26] introduced the concept of a contraction in the background of complete metric spaces and proved a fixed point result which is known as the Banach contraction principle. It has been extremely convenient in numerous fields, such as optimization problems, differential equations, control theory and many other fields. A number of research articles in this field have been devoted to the generalization and improvement of this result in different directions (see ^[27,28]).

In 2014, Jleli et al. ^[29] generalized the concept of a contraction by introducing a new type of contraction named $\Theta$-contraction in the framework of generalized metric spaces. Ahmad et al. ^[30] replaced the third condition of a $\Theta$-contraction with a weaker condition and established some theorems in the setting of metric spaces. Later on, Hussain et al. ^[31] extended this notion of $\Theta$-contraction by adding a general condition in it and obtained some fixed point results. For more characteristics in this way, we refer the readers to ^{[32,33,34,35,36]}.

Motivated and inspired by the results of Kanwal et al. ^[9] and Hussain et al. ^[31], we give the concept of a generalized $(\alpha, \Theta _{\mathcal{F}})$-contraction in the context of an orthogonal $\mathcal{F }$-complete metric space and obtain contemporary common fixed point theorems which enable us to show the uniqueness and existence of the solution for a fractional differential equation.

2.   Preliminaries

Frechet ^[4] introduced the concept of a metric space in this way.

A metric on $\aleph \not = \emptyset$ is a mapping $\nu :\aleph \times \aleph\rightarrow \lbrack 0, +\infty)$ satisfying these properties:

(ⅰ) $\nu (\rho, \varsigma)\geq 0$, and $\nu (\rho, \varsigma) = 0$ if and only if $\rho = \varsigma$,

(ⅱ) $\nu (\rho, \varsigma) = \nu (\varsigma, \rho)$,

(ⅲ) $\nu (\rho, \omega)\leq \nu (\rho, \omega)+\nu (\omega, \varsigma)$,

for all $\rho, \varsigma \in \aleph.$ If $\nu$ is a metric, then ($\aleph, \nu$) is called a metric space.

In 1993, Czerwik ^[5] gave the notion of a $b$-metric by take into consideration this assertion instead of the triangular inequality:

For all $\rho, \varsigma \in \aleph$ and $b\geq 1,$

In 2017, Gordji et al. ^[8] introduced the notion of the orthogonal set ($\mathcal{O}$-set).

Definition 1. A non empty $\aleph$ is called an orthogonal set if there exists a binary relation $\bot \subseteq \aleph \times \aleph$ satisfying the condition

Furthermore, $\rho _{0}$ is called an orthogonal point. We denote this $\mathcal{O}$-set by ($\aleph$, $\bot$).

Gordji et al. ^[8] considered $\aleph$ as an $\mathcal{O}$-set in metric space and gave the concept of an orthogonal metric space.

Definition 2. (see ^[8]) Let ($\aleph$, $\bot$) be an $\mathcal{O}$-set. A sequence $\{ \rho _{n}\}$ is said to be an orthogonal sequence if

We denote an orthogonal sequence by $\mathcal{O}$-sequence.

Recently, Jleli et al. ^[7] introduced the notion of an $\mathcal{F}$ -metric space in such manner.

Let $\mathcal{F}$ be the family of mappings $\xi :(0, +\infty)\rightarrow \mathbb{R}$ satisfying

($\mathcal{F}_{1}$) $0 < \rho _{1} < \rho _{2}\Rightarrow \xi (\rho _{1})\leq \xi (\rho _{2}).$

($\mathcal{F}_{2}$) For all $\{ \rho _{n}\} \subseteq \mathbb{R} ^{+}$, $\lim_{n\rightarrow \infty }$ $\rho _{n} = 0$ if and only if $\lim_{n\rightarrow \infty }\xi (\rho _{n}) = -\infty.$

Definition 3. (see ^[7]) Let $\aleph \not = \emptyset$ and $\nu :\aleph \times \aleph \rightarrow \lbrack 0, +\infty)$. Assume that there exists $(\xi, \alpha)\in \mathcal{F}\times \lbrack 0, +\infty)$ such that

(D$_{1}$) $(\rho, \varsigma)\in \aleph \times \aleph$, $\nu (\rho, \varsigma) = 0$ if and only if $\rho = \varsigma$.

(D$_{2}$) $\nu (\rho, \varsigma) = \nu (\varsigma, \rho)$, for all $\rho, \varsigma \in \aleph.$ with $(\rho _{1}, \rho _{N}) = (\rho, \varsigma)$, we have

(D$_{3}$) For all $(\rho, \varsigma)\in \aleph \times \aleph$, and $(\rho _{i})_{i = 1}^{N}\subset \aleph,$

for all $N\geq 2.$ Then, $(\aleph, \nu)$ is named as an $\mathcal{F}$-metric space.

Example 1. (see ^[7]) Let $\aleph = \mathbb{R}.$ Then, $\nu :\aleph \times \aleph \rightarrow \lbrack 0, +\infty)$ is an $\mathcal{F}$-metric defined by

with $\xi (t) = \ln (t)$ and $\alpha = \ln (3)$.

Definition 4. (see ^[7]) Let $(\aleph, \nu)$ be an $\mathcal{F}$-metric space.

(i) A sequence $\{ \rho _{n}\}$ $\subseteq \aleph$ is said to be $\mathcal{F}$-convergent if

(ii) A sequence $\{ \rho _{n}\}$ is $\mathcal{F}$-Cauchy, if

Subsequently, Kanwal et al. ^[9] integrated both notions of $\mathcal{ O}$-set and $\mathcal{F}$-metric space and gave the concept of an orthogonal $\mathcal{F}$-metric space ($\mathcal{OF}$-metric space) as follows.

Definition 5. (see ^[9]) Let ($\aleph$, $\bot$) be an $\mathcal{O}$ -set and $\nu :\aleph \times \aleph \rightarrow \lbrack 0, +\infty)$ be an $\mathcal{F}$-metric; then, $(\aleph, \bot, \nu)$ is named as an orthogonal $\mathcal{F}$-metric space ($\mathcal{OF}$-metric space).

Example 2. (see ^[9]) Let $\aleph = [0, 1]$. Define $\mathcal{F}$-metric $\nu$ by

for all $\rho, \varsigma \in \aleph \mathfrak{, }$ $\xi (t) = -\frac{1}{t},$ $t > 0,$ and $\alpha = 1.$ Define $\rho \bot \varsigma$ if $\rho \varsigma \leq \rho$ or $\rho \varsigma \leq \varsigma.$ Then, for all $\rho \in \aleph \mathfrak{, }$ 0$\bot \varsigma,$ so ($\aleph \mathfrak{, }\bot$) is an $\mathcal{O}$-set. Then, $(\aleph, \nu, \bot)$ is an $\mathcal{OF}$ -metric space.

Definition 6. (see ^[9]) Let $(\aleph, \nu, \bot)$ be an $\mathcal{OF}$-metric space, and $\Re :(\aleph, \nu, \bot)\rightarrow (\aleph, \nu, \bot).$ Then, $\Re$ is called $\bot$-continuous at $\rho \in \aleph$ if for each $\mathcal{O}$-sequence $equence \{ \rho _{n}\}$ in $\aleph$ if $\rho _{n}\rightarrow \rho,$ then $\Re \rho _{n}\rightarrow \Re \rho.$ Also, $\Re$ is $\bot$ -continuous on $\aleph$ if $\Re$ is $\bot$-continuous at each $\rho \in$ $\aleph$.

Definition 7. (see ^[9]) If every Cauchy $\mathcal{O}$-sequence in $\mathcal{OF}$ -metric space $(\aleph, \nu, \bot)$ is $\mathcal{F}$-convergent, then $(\aleph, \nu, \bot)$ is called $\mathcal{O}$-$\mathcal{F}$-complete.

Samet et al. ^[22] began the thought of being $\alpha$-admissible in this manner.

Definition 8. A mapping $\Re :\aleph \times \aleph \rightarrow \lbrack 1, \infty)$ is said to be $\alpha$-admissible if

Ramezani ^[23] gave the notion of being orthogonally $\alpha$-admissible in this way.

Definition 9. A mapping $\Re :\aleph \times \aleph \rightarrow \lbrack 1, \infty)$ is called an orthogonally $\alpha$-admissible if

Recently, Ahmad et al. ^[12] gave the following property for an orthogonally $\alpha$-admissible mapping: ($\beth$) $\alpha (\rho, \varsigma)\geq 1$ for any $\rho, \varsigma \in \left \{ \rho ^{\ast }\in \aleph \mathfrak{:}\rho ^{\ast } = \Re \rho ^{\ast } \text{ }\right \}$ and $\rho \bot \varsigma.$

In 2014, Jleli and Samet ^[29] started a state of the art contraction which is called a $\Theta$-contraction along these lines.

Definition 10. Let $\Theta : \mathbb{R} ^{+}\rightarrow \lbrack 1, \infty)$ be a function such that

($\mathcal{\dagger }_{1}$) $\Theta (\rho) < \Theta (\varsigma)$ for $\rho < \varsigma;$

($\mathcal{\dagger }_{2}$) for all $\{ \rho _{n}\} \subseteq \lbrack 0, +\infty)$, $\lim_{n\rightarrow \infty }(\rho _{n}) = 0$ $\Leftrightarrow$ $\lim_{n\rightarrow \infty }\Theta (\rho _{n}) = 1$;

($\mathcal{\dagger }_{3}$) there exist $0 < h < 1$ and $0 < \sigma \leq +\infty$ such that $\lim_{\rho \rightarrow 0^{+}}\frac{\Theta (\rho)-1}{ \rho ^{h}} = \sigma.$

A mapping $\Re :(\aleph, \nu)\rightarrow (\aleph, \nu)$ is called a $\Theta$-contraction if there exist some constant $\varrho \in (0, 1)$ and a mapping $\Theta : \mathbb{R} ^{+}\rightarrow \lbrack 1, \infty)$ satisfying ($\mathcal{\dagger }_{1}$)–($\mathcal{\dagger }_{3}$) such that

for all $\rho, \varsigma \in \aleph$. They proved a result for such contraction in this way.

Theorem 1. (see ^[29]) If the mapping $\Re$ is a $\Theta$-contraction on a complete metric space $(\aleph, \nu)$, then there exists $\rho ^{\ast }\in \aleph$ such that $\rho ^{\ast } = \Re \rho ^{\ast }$.

Later on, Hussain et al. ^[31] added another condition,

($\mathcal{\dagger }_{4}$) $\Theta (\rho +\varsigma)\leq \Theta (\rho)\Theta (\varsigma),$

and extended the above result of Jleli and Samet ^[7] in complete metric spaces. To be steady with Hussain et al. ^[31], we represent by $\Psi$ the family of all functions $\Theta : \mathbb{R} ^{+}\rightarrow (1, \infty)$ satisfying ($\mathcal{\dagger }_{1}$)–($\mathcal{ \dagger }_{4}$).

3.   Results and discussions

We define the notion of a generalized ($\alpha$, $\Theta _{\mathcal{F}})$ -contraction as follows:

Definition 11. Let $(\aleph, \nu, \bot)$ be an $\mathcal{OF}$-metric space. A mapping $\Re :\aleph \rightarrow \aleph$ is called a generalized ($\alpha$, $\Theta _{ \mathcal{F}})$-contraction if there exist $\Theta \in \Psi,$ $\alpha :\aleph \mathfrak{\times }\aleph \mathfrak{\longrightarrow \lbrack 1, \infty) }$ and nonnegative real numbers $\varrho _{1}, \varrho _{2}, \varrho _{3}, \varrho _{4}$ with $\varrho _{1}+\varrho _{2}+\varrho _{3}+2\varrho _{4} < 1$ such that

Theorem 2. Let $(\aleph, \nu, \bot)$ be an $\mathcal{O}$-complete $\mathcal{ OF}$-metric space, and $\Re :\aleph \rightarrow \aleph$ is $\perp$-continuous, $\perp$-preserving, orthogonally $\alpha$-admissible and a generalized ($\alpha$, $\Theta _{\mathcal{F}})$-contraction. If there exists $\rho _{0}\in \aleph$ such that $\rho _{0}\perp \Re \rho _{0}$ and $\alpha \left(\rho _{0}, \Re \rho _{0}\right) \geq 1,$ then there exists $\rho ^{\ast }\in \aleph$ such that $\Re \rho ^{\ast } = \rho ^{\ast }$. Moreover, if $\aleph$ has the property ($\beth$), then $\rho ^{\ast }$ is unique.

Proof. Let there exist $\rho _{0}\in$ $\aleph$ such that $\rho _{0}\perp \Re \rho _{0}$ and $\alpha \left(\rho _{0}, \Re \rho _{0}\right) \geq 1,$ and define the sequence $\{\rho _{n}\}$ as

for all $n\geq 0.$ Now, using the orthogonal $\alpha$-admissibility of $\Re$, we have

for all $n\geq 0$. If $\rho _{n} = \rho _{n+1},$ for any $n\in \mathbb{N} \cup \{0\},$ then clearly $\rho _{n}$ is a fixed point of $\Re$. Now, we assume that $\rho _{n}\not = \rho _{n+1},$ for all $n\in \mathbb{N} \cup \{0\}.$ Thus, we get

for all $n\in \mathbb{N} \cup \{0\}.$ Since $\Re$ is $\perp$-preserving, so we have

for all $n\in \mathbb{N} \cup \{0\}.$ This means that $\{ \rho _{n} \}$ is an $\mathcal{O}$-sequence. Hence, we presume that

for all $n\in \mathbb{N} \cup \{0\}.$ From (3.1), (3.2) and ($\mathcal{\dagger }_{4}$), we get

which implies

for all $n\in \mathbb{N} \cup \{0\},$ where $\varrho = \frac{\varrho _{1}+\varrho _{2}+\varrho _{4}}{ 1-\varrho _{3}-\varrho _{4}} < 1.$ Consequently, we have

for all $n\in \mathbb{N} \cup \{0\}.$ Now, taking $n\rightarrow \infty$ and by ($\mathcal{\dagger } _{2}$), we get

By ($\mathcal{\dagger }_{3}$), there exist $0 < h < 1$ and $l\in (0, \infty ]$ such that

Let $l < \infty,$ and then we take $\beta = \frac{l}{2} > 0.$ By definition of the limit, there exists $n_{1}\in \mathbb{N}$ such that

for all $n > n_{1}.$ It yields

Then,

for all $n > n_{1},$ where $\gamma = \frac{1}{\beta }.$ Now, when $l = \infty,$ suppose that $\beta > 0$. By definition of the limit, there exists $n_{1}\in \mathbb{N}$ such that

It yields

Therefore, in all ways, there exists $\gamma > 0$ and $n_{1}\in \mathbb{N}$ such that

for all $n > n_{1}.$ Hence, by (3.4) and (3.8), we have

Taking $n\rightarrow \infty$, we have

Hence, there exists $n_{2}\in \mathbb{N}$ such that

for all $n > n_{2}.$ This yields

for all $m > n.$ As $\sum_{i = n}^{\infty }\frac{1}{i^{1/r}}$ converges, there exists $n_{2}\in \mathbb{N}$ such that

for $n > n_{2}.$ Hence, by (3.10) and ($\mathcal{F}_{1}$), we get

$m > n\geq n_{2}.$ Using (D$_{3}$) and (3.11), we get

which, from ($\mathcal{F}_{1}$), gives that

for all $m > n\geq n_{2}.$ Therefore, $\{ \rho _{n}\}$ is a Cauchy $\mathcal{O}$ -sequence in $(\aleph, \perp, \nu)$. Now, since $(\aleph, \perp, \nu)$ is $\mathcal{O}$-complete, there exists $\rho ^{\ast }\in \aleph$ such that $\lim_{n\rightarrow \infty }\rho _{n}\rightarrow \rho ^{\ast }.$ Now, we show that $\rho ^{\ast } = \Re \rho ^{\ast }$. Since $\Re$ is $\perp$ -continuous, we have $\Re \rho _{n}\rightarrow \Re \rho ^{\ast }$ as $n\rightarrow \infty.$ Thus,

Now, we suppose that $\rho ^{/} = \Re \rho ^{/}$ is another fixed point of $\Re$ such that $\rho ^{/}\not = \rho ^{\ast }.$ From ($\beth$), we have $\rho ^{\ast }\bot \rho ^{/}$ or $\rho ^{/}\perp \rho ^{\ast }$ and $\alpha \left(\rho ^{\ast }, \rho ^{/}\right) \geq 1.$ Thus, from (3.1), we have

which implies that

which is a contradiction because $\varrho _{1}+2\varrho _{4} < 1$. Thus $\rho ^{/} = \rho ^{\ast }.$ Hence the fixed point is unique.

Corollary 1. Let $(\aleph, \nu, \bot)$ be an $\mathcal{O}$-complete $\mathcal{OF}$ -metric space, and $\Re :(\aleph, \nu, \bot)\rightarrow (\aleph, \nu, \bot)$ is $\perp$-continuous and $\perp$-preserving$.$ Suppose there exist $\Theta \in \Psi$ and nonnegative real numbers $\varrho _{1}, \varrho _{2}, \varrho _{3}, \varrho _{4}$ with $\varrho _{1}+\varrho _{2}+\varrho _{3}+2\varrho _{4} < 1$ such that for all

Then, there exists a unique point $\rho ^{\ast }\in \aleph$ such that $\Re \rho ^{\ast } = \rho ^{\ast }$.

Proof. Take $\alpha :\aleph \mathfrak{\times }\aleph \mathfrak{\longrightarrow } [1, \infty)$ by $\alpha \left(\rho, \varsigma \right) = 1,$ for all $\rho, \varsigma \in \aleph$ in Theorem 2.

Corollary 2. Let $(\aleph, \nu, \bot)$ be an $\mathcal{O}$-complete $\mathcal{ OF}$-metric space, and $\Re :(\aleph, \nu, \bot)\rightarrow (\aleph, \nu, \bot)$ is $\perp$-continuous, $\perp$-preserving and orthogonally $\alpha$ -admissible$.$ Suppose there exist $\Theta \in \Psi,$ $\alpha :\aleph \mathfrak{\times }\aleph \mathfrak{\longrightarrow }[1, \infty)$ and some nonnegative real number $\varrho \in (0, 1)$ such that

for all $\rho, \varsigma \in \aleph \mathfrak{, }\rho \perp \varsigma.$ If there exists $\rho _{0}\in \aleph$ such that $\rho _{0}\perp \Re \rho _{0}$ and $\alpha \left(\rho _{0}, \Re \rho _{0}\right) \geq 1,$ then, there exists $\rho ^{\ast }\in \aleph$ such that $\Re \rho ^{\ast } = \rho ^{\ast }$. Moreover, if $\aleph$ has the property ($\beth$), then $\rho ^{\ast }$ is unique.

Proof. Take $\varrho _{1} = \varrho < 1$ and $\varrho _{2} = \varrho _{3} = \varrho _{4} = 0$ in Theorem 2.

Corollary 3. (see ^[9]) Let $(\aleph, \nu, \bot)$ be an $\mathcal{O}$ -complete $\mathcal{OF}$-metric space, and $\Re :(\aleph, \nu, \bot)\rightarrow (\aleph, \nu, \bot)$ is $\perp$-continuous and $\perp$-preserving. Suppose there exist $\Theta \in \Psi$ and some nonnegative real number $\varrho \in (0, 1)$ such that

holds; then, there exists a unique point $\rho ^{\ast }\in \aleph$ such that $\Re \rho ^{\ast } = \rho ^{\ast }$.

Proof. Take $\alpha :\aleph \mathfrak{\times }\aleph \mathfrak{\longrightarrow } [1, \infty)$ by $\alpha \left(\rho, \varsigma \right) = 1,$ for all $\rho, \varsigma \in \aleph$ in Corollary 2.

Example 3. Define the sequence $\{ \rho _{n}\}$ as follows:

for all $n\geq 1.$ Let $\aleph \mathfrak{ = }\left \{ \rho _{n}:n\in \mathbb{N} \right \}$ be provided with $\nu :\aleph \times \aleph \rightarrow \lbrack 0, +\infty),$ defined by

with $\xi \left(t\right) = \frac{-1}{t}$ and $a = 1\mathfrak{.}$ For all $\rho _{n}, \rho _{m}\in \aleph$, define $\rho _{n}\bot \rho _{m}$ if and only if $\left(m\geq 2\wedge n = 1\right)$. Thus, $(\aleph, \nu, \bot)$ is an $\mathcal{O}$-complete $\mathcal{OF}$-metric space. Define $\Re :(\aleph, \nu, \bot)\rightarrow (\aleph, \nu, \bot)$ by

and $\alpha :\aleph \mathfrak{\times }\aleph \mathfrak{\rightarrow \lbrack 1, +\infty)}$ by

Clearly,

and then $\Re$ is not a contraction.

It is very simple to show that $\Re$ is $\bot$-preserving and $\bot$ -continuous. Define $\Theta :(0, \infty)\rightarrow \mathbb{R} ^{+}$ by

Then, $\Theta \in \Psi.$ Now, we prove that $\Re$ is a generalized ($\alpha, \Theta)$-contraction, i.e.,

for some $\varrho \in (0, 1).$ The above condition is equivalent to

So, we have to check that

For $m\in \mathbb{N},$ and $m\geq 2,$ we get

Thus, the inequality (3.1) is satisfied. Hence, $\Re$ is a generalized ($\alpha, \Theta)$-contraction. Hence, by Theorem 2, $\rho$ $= ln(1)$ is a unique fixed point of $\Re$.

For particular choices of $\Theta$, we get some noteworthy results. If we take $\Theta (t) = e^{\sqrt{t}}$ in ($3.1$), we get an extension of Ćirić result ^[37].

Theorem 3. Let $(\aleph, \nu, \bot)$ be an $\mathcal{O}$-complete $\mathcal{OF}$ -metric space and $\Re :(\aleph, \nu, \bot)\rightarrow (\aleph, \nu, \bot)$ is $\perp$-continuous, $\perp$-preserving and orthogonally $\alpha$ -admissible. Suppose that there exists nonnegative real numbers $\varrho _{1}, \varrho _{2}, \varrho _{3}, \varrho _{4}$ with $\varrho _{1}+\varrho _{2}+\varrho _{3}+2\varrho _{4} < 1$ such that these conditions hold:

If there exists $\rho _{0}\in \aleph$ such that $\rho _{0}\perp \Re \rho _{0}$ and $\alpha \left(\rho _{0}, \Re \rho _{0}\right) \geq 1,$ then there exists $\rho ^{\ast }\in \aleph$ such that $\rho ^{\ast } = \Re \rho ^{\ast }$. Moreover, if $\aleph$ has the property ($\beth$), then $\rho ^{\ast }$ is unique.

Remark 1. If we take the square on both sides, then that condition ($3.13$) is equivalent to

Next, in view of Remark $1$, by taking $\varrho _{1} = \varrho _{4}$ $= 0$ in Theorem 2, we obtain this result, which is an extension of Kannan's result ^[38].

Theorem 4. Let $(\aleph, \nu, \bot)$ be an $\mathcal{O}$-complete $\mathcal{OF}$ -metric space and $\Re :(\aleph, \nu, \bot)\rightarrow (\aleph, \nu, \bot)$ is $\perp$-continuous, $\perp$-preserving and orthogonally $\alpha$ -admissible. Suppose that there exist nonnegative real numbers $\varrho _{2}, \varrho _{3}$ with $0\leq \varrho _{2}+\varrho _{3} < 1$ such that these conditions hold: for all $\rho, \varsigma \in \aleph \mathfrak{, }\; \rho \perp \varsigma,$ $\nu (\Re \rho, \Re \varsigma)\not = 0$ implies

If there exists $\rho _{0}\in \aleph$ such that $\rho _{0}\perp \Re \rho _{0}$ and $\alpha \left(\rho _{0}, \Re \rho _{0}\right) \geq 1,$ then, there exists $\rho ^{\ast }\in \aleph$ such that $\rho ^{\ast } = \Re \rho ^{\ast }$. Moreover, if $\aleph$ has the property ($\beth$), then $\rho ^{\ast }$ is unique.

On the other hand, by taking $\varrho _{1} = \varrho _{2} = \varrho _{3}$ = 0 in Theorem 2, we obtain this theorem, which is an expansion of Chatterjea's result ^[39].

Theorem 5. Let $(\aleph, \nu, \bot)$ be an $\mathcal{O}$-complete $\mathcal{OF}$ -metric space and $\Re :(\aleph, \nu, \bot)\rightarrow (\aleph, \nu, \bot)$ is $\perp$-continuous, $\perp$-preserving and orthogonally $\alpha$ -admissible. Suppose that there exists nonnegative real number $\varrho _{4}\in \left[ 0, \frac{1}{2}\right)$ such that for all $\rho, \varsigma \in \aleph \mathfrak{, } \ \rho \perp \varsigma,$ $\nu (\Re \rho, \Re \varsigma)\not = 0$ implies

If there exists $\rho _{0}\in \aleph$ such that $\rho _{0}\perp \Re \rho _{0}$ and $\alpha \left(\rho _{0}, \Re \rho _{0}\right) \geq 1,$ then, there exists $\rho ^{\ast }\in \aleph$ such that $\rho ^{\ast } = \Re \rho ^{\ast }$. Moreover, if $\aleph$ has the property ($\beth$), then $\rho ^{\ast }$ is unique.

From Theorem 2, by taking $\varrho _{4} = 0$, we obtain the extension of Reich contraction ^[40].

Corollary 4. Let $(\aleph, \nu, \bot)$ be an $\mathcal{O}$-complete $\mathcal{OF}$ -metric space and $\Re :(\aleph, \nu, \bot)\rightarrow (\aleph, \nu, \bot)$ is $\perp$-continuous, $\perp$-preserving and orthogonally $\alpha$ -admissible. Suppose that there exist nonnegative real numbers $\varrho _{1}, \varrho _{2}, \varrho _{3}$ with $0\leq \varrho _{1}+\varrho _{2}+\varrho _{3} < 1$ such that for all $\rho, \varsigma \in \aleph \mathfrak{, } \ \rho \perp \varsigma,$ $\nu (\Re \rho, \Re \varsigma)\not = 0$ implies

If there exists $\rho _{0}\in \aleph$ such that $\rho _{0}\perp \Re \rho _{0}$ and $\alpha \left(\rho _{0}, \Re \rho _{0}\right) \geq 1,$ then there exists $\rho ^{\ast }\in \aleph$ such that $\rho ^{\ast } = \Re \rho ^{\ast }$. Moreover, if $\aleph$ has the property ($\beth$), then $\rho ^{\ast }$ is unique.

Eventually, if we take $\Theta (t) = e^{\sqrt[n]{t}}$ in (3.1), then we derive this corollary.

Corollary 5. Let $(\aleph, \nu, \bot)$ be an $\mathcal{O}$-complete $\mathcal{OF}$ -metric space and $\Re :(\aleph, \nu, \bot)\rightarrow (\aleph, \nu, \bot)$ is $\perp$-continuous, $\perp$-preserving and orthogonally $\alpha$ -admissible. Suppose that there exists nonnegative real numbers $\varrho _{1}, \varrho _{2}, \varrho _{3}, \varrho _{4}$ with $\varrho _{1}+\varrho _{2}+\varrho _{3}+2\varrho _{4} < 1$ such that for all $\rho, \varsigma \in \aleph \mathfrak{, } \ \rho \perp \varsigma,$ $\nu (\Re \rho, \Re \varsigma)\not = 0$ implies

If there exists $\rho _{0}\in \aleph$ such that $\rho _{0}\perp \Re \rho _{0}$ and $\alpha \left(\rho _{0}, \Re \rho _{0}\right) \geq 1,$ then, there exists $\rho ^{\ast }\in \aleph$ such that $\rho ^{\ast } = \Re \rho ^{\ast }$. Moreover, if $\aleph$ has the property ($\beth$), then $\rho ^{\ast }$ is unique.

4.   Consequences

Now, we consider some special cases, where in our result we deduce several well-known fixed point theorems of the existing literature.

Theorem 6. Let $(\aleph, \nu)$ be an $\mathcal{F}$-complete $\mathcal{F}$ -metric space and $\Re :\aleph \rightarrow \aleph$ be an $\alpha$-admissible mapping, and there exist $\Theta \in \Psi$, $\alpha :$$\aleph \mathfrak{\times }\aleph \mathfrak{\longrightarrow }[1, \infty \mathfrak{)}$ and nonnegative real numbers $\varrho _{1}, \varrho _{2}, \varrho _{3}, \varrho _{4}$ with $\varrho _{1}+\varrho _{2}+\varrho _{3}+2\varrho _{4}$$< 1$ such that for all

for all $\rho, \varsigma \in \aleph.$ Assume that there exists $\rho _{0}\in \aleph$ such that $\alpha \left(\rho _{0}, \Re \rho _{0}\right) \geq 1;$ then, there exists $\rho ^{\ast }\in \aleph$ such that $\Re \rho ^{\ast } = \rho ^{\ast }$. Moreover, if $\aleph$ has the property ($\beth$), then $\rho ^{\ast }$ is unique.

Proof. Assume that

Fix $\rho _{0}\in \aleph \mathfrak{.}$ Since $\Re$ satisfies the inequality (4.1), for all $\varsigma \in \aleph \mathfrak{, }$ $\rho _{0}\bot \varsigma,$ it yields that $(\aleph, \perp)$ is an $\mathcal{O}$-set. Then, $(\aleph, \nu)$ is $O$-complete. It is very obvious to prove that $\Re$ is $\perp$-preserving and $\perp$-continuous. Hence, by result 3.1, there exists a unique point $\rho ^{\ast }\in \aleph$ such that $\Re \rho ^{\ast } = \rho ^{\ast }$.

Corollary 6. Let $(\aleph, \nu)$ be an $\mathcal{F}$-complete $\mathcal{F}$ -metric space, $\Re :\aleph \rightarrow \aleph,$ and have nonnegative real numbers $\varrho _{1}, \varrho _{2}, \varrho _{3}, \varrho _{4}$ with $\varrho _{1}+\varrho _{2}+\varrho _{3}+2\varrho _{4} < 1$ such that

for all $\rho, \varsigma \in \aleph$ and $\Theta \in \Psi$; then, there exists a unique point $\rho ^{\ast }\in \aleph$ such that $\Re \rho ^{\ast } = \rho ^{\ast }$.

Proof. Take $\alpha :\aleph \mathfrak{\times }\aleph \mathfrak{\longrightarrow } [1, \infty \mathfrak{)}$ by $\alpha \left(\rho, \varsigma \right) = 1$ in Theorem 4.1.

Theorem 7. Let $(\aleph, \nu, \bot)$ be an $\mathcal{O}$-complete metric space, and $\Re :(\aleph, \nu, \bot)\rightarrow (\aleph, \nu, \bot)$ is $\perp$-continuous, $\perp$-preserving and orthogonally $\alpha$-admissible. Assume that there exist $\Theta \in \Psi$, $\alpha :\aleph \mathfrak{\times }\aleph \mathfrak{\longrightarrow }[1, \infty \mathfrak{)}$ and nonnegative real numbers $\varrho _{1}, \varrho _{2}, \varrho _{3}, \varrho _{4}$ with $\varrho _{1}+\varrho _{2}+\varrho _{3}+2\varrho _{4} < 1$ such that

If there exists $\rho _{0}\in \aleph$ such that $\rho _{0}\perp \Re \rho _{0}$ and $\alpha \left(\rho _{0}, \Re \rho _{0}\right) \geq 1,$ then, there exists $\rho ^{\ast }\in \aleph$ such that $\Re \rho ^{\ast } = \rho ^{\ast }.$ Moreover, if $\aleph$ has the property ($\beth$), then $\rho ^{\ast }$ is unique.

Proof. Take $\xi (t) = \ln (t),$ for $t > 0$ and $\alpha = 1$ in Definition 5, and then $\mathcal{OF}$-metric space reduced to $\mathcal{O}$- metric space. It follows directly from Theorem 2.

Corollary 7. (see ^[31]) Let $(\aleph, \nu)$ be an complete metric space, $\Re :\aleph \rightarrow \aleph$ and nonnegative real numbers $\varrho _{1}, \varrho _{2}, \varrho _{3}, \varrho _{4}$ with $\varrho _{1}+\varrho _{2}+\varrho _{3}+2\varrho _{4} < 1$ such that

for all $\rho, \varsigma \in \aleph$ and $\Theta \in \Psi$; then, there exists a unique point $\rho ^{\ast }\in \aleph$ such that $\Re \rho ^{\ast } = \rho ^{\ast }$.

Proof. Assume that

It follows from Theorem 7 by considering $\alpha :\aleph \mathfrak{ \times }\aleph \mathfrak{\longrightarrow }[1, \infty \mathfrak{)}$ as $\alpha \left(\rho, \varsigma \right) = 1.$

Remark 2. By using the remark 1 and equating nonnegative real numbers $\varrho _{1}, \varrho _{2}, \varrho _{3}$ and $\varrho _{4}$ to zero appropriately in Theorems 6, 7 and Corollaries 6, 7 one can derive a number of results which are more general results in the context of $\mathcal{F}$-metric spaces, orthogonal metric spaces and metric spaces.

Remark 3. By taking the functions $\alpha :\aleph \mathfrak{\times }\aleph \mathfrak{ \longrightarrow }[1, \infty \mathfrak{)}$ and $\Theta : \mathbb{R} ^{+}\rightarrow \lbrack 1, \infty)$ in different ways in above results, one can obtain various results in different generalized metric spaces.

5.   Applications

The field of fractional differential equations has been subjected to a comprehensive evolution of theory and applications (^[41,42,43] and references therein). In the present section, we give an application of result 3 to investigate the existence of a solution for a nonlinear fractional differential equation considered in ^[9,44,45].

Consider a nonlinear differential equation of fractional order

$\left(0 < t < 1, \text{ }1 < \eta \leq 2\right)$ via the integral boundary conditions

where $^{C}D^{\eta }$ denotes the Caputo fractional derivative of order $\eta$ defined by

$\left(j-1 < \eta < j, \text{ }j = \left[ \eta \right] +1\right)$ and $f$ is a continuous mapping. We take

with supremum norm $\left \Vert \rho \right \Vert _{\infty } = \sup_{t\in \lbrack 0, 1]}\left \vert \rho (t)\right \vert.$ Thus, $\left(\aleph \mathfrak{, }\left \Vert \rho \right \Vert _{\infty }\right)$ is a Banach space. Remember that

is a Riemann-Liouville fractional integral.

Lemma 1. (see ^[9]) The Banach space $\left(\aleph \mathfrak{, }\left \Vert \cdot \right \Vert _{\infty }\right)$ endowed with the $\mathcal{F}$ -metric $d$ defined by

and orthogonal relation $\rho \bot \varsigma$ if and only if $\rho \varsigma \geq 0,$ where $\rho, \varsigma \in \aleph \mathfrak{, }$ is an orthogonal $\mathcal{F}$-metric space.

Theorem 8. Assume that $f$ is a continuous mapping satisfying

(i) there exists a constant $\vartheta$ such that

for $t\in \lbrack 0, 1]$ and for all $\rho, \varsigma \in \aleph$ such that $\rho (t)\varsigma (t)\geq 0$ and with $\vartheta \beta < 1$, where

(ii) there exists $\mathcal{L}:\left(\aleph \mathfrak{, }\bot, d\right) \rightarrow \left(\aleph \mathfrak{, }\bot, d\right),$ which is defined by

for all $\rho, \varsigma \in \aleph$ such that $\rho (t)\varsigma (t)\geq 0,$ where ($0 < \lambda < 1$). Also, $\Re$ is orthogonally $\alpha$-admissible, and there exists $\rho _{0}(t)\in \left(\aleph \mathfrak{, }\bot, d\right)$ such that $\rho _{0}(t)\perp \Re \rho _{0}(t)$ and $\alpha \left(\rho _{0}(t), \Re \rho _{0}(t)\right) \geq 1$ Then, (5.1) has a unique solution.

Proof. It is conventional that $\rho \in$ $\aleph$ is a solution of (5.1) iff $\rho \in$ $\aleph$ is a solution of the integral equation

Then, problem (5.1) is equivalent to finding $\rho \in$ $\aleph$ that is a fixed point of $\Re$. Assume that $\bot \subseteq \aleph \times \aleph$ is defined by

for all $t\in \lbrack 0, 1].$ Then, $\aleph$ is orthogonal under this relation $\bot$, since for $\rho \in \aleph \mathfrak{, }$ there exists $\varsigma (t) = 0,$ for all $t\in \lbrack 0, 1]$ such that $\rho (t)\varsigma (t) = 0.$ Now, define $d:\aleph \times \aleph \rightarrow \lbrack 0, +\infty)$ by

for all $\rho, \varsigma \in \aleph \mathfrak{, }$ and then $(\aleph, d, \bot)$ is a complete $\mathcal{OF}$-metric space. This is quite simple from the definition that $\Re$ is $\bot$-continuous. We first show that $\Re$ is $\bot$-preserving. Let $\rho (t)\bot \varsigma (t),$ for all $t\in \lbrack 0, 1].$ Now, we have

which yields that $\Re \rho (t)\bot \Re \varsigma (t),$ that is, $\Re$ is $\bot$-preserving. Subsequently, for all $t\in \lbrack 0, 1],$ $\rho (t)\bot \varsigma (t),$ we have

which yields

Take $\varrho = \vartheta \beta < 1.$ By the definition of $d$, we have

Then,

where $\varrho \in (0, 1).$ Now, if $\Theta (t) = e^{\sqrt{t}},$ for all $t > 0$, then $\Theta \in \Psi$. Hence, from above,

for all $\rho, \varsigma \in \aleph$ and $d(\Re \rho, \Re \varsigma) > 0.$ Thus, all the conditions of result 3 are satisfied, and thus Eq (5.1) has a unique solution.

6.   Conclusions

In this work, we defined the notion of a generalized ($\alpha$, $\Theta _{ \mathcal{F}})$-contraction in the background of an orthogonal $\mathcal{F}$ -metric space and proved some certain fixed point results. As outcomes of the leading results, we obtained some fixed point theorems in $\mathcal{F}$ -metric spaces and orthogonal metric spaces. Moreover, a non trivial example is also furnished to validate the originality of the obtained results. We investigated the existence and uniqueness of a solution for the fractional differential equation as application of our main results.

For future work, the notion of an $\mathcal{F}$-metric space can be extended to a graphical $\mathcal{F}$-metric space, and the results proved in this article can be extended to multivalued mappings and fuzzy set valued mappings. Moreover, we can solve differential and integral inclusions as applications of fixed point results for multivalued mappings in the setting of $\mathcal{F}$ -metric space.

Acknowledgments

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors acknowledge with thanks DSR, KAU for financial support.

Conflict of interest

The authors declare that they have no conflicts of interests.