Fixed point results with applications to nonlinear fractional differential equations

1.   Introduction

The most fascinating and vital theory in the growth of nonlinear analysis is the fixed point theory. In this extent, the Banach fixed point theorem ^[1] was pioneer result for investigators over the past hundred years. This theorem plays a significant and essential role in the existence and uniqueness of solutions to different nonlinear integrals, functional equations and differential equations. In 2008, Berinde ^[2,3] gave the notions of almost and generalized almost contractions and obtained a fixed point result as an extension and generalization of the Banach fixed point theorem. Samet et al. ^[4] initiated the notions of $\rho$-admissible mappings and ($\rho, \psi$)-contractions to prove certain fixed point results for such contractive mappings in the framework of complete metric spaces. Later on, Salimi et al. ^[5] generalized the notion of $\rho$-admissible mappings by introducing the idea of twisted ($\rho$, $\mu)$-admissible mappings. They also defined the concept of ($\rho$, $\mu, \psi)$-contractions and obtained some fixed point theorems in this context of metric spaces.

On the other hand, Jleli et al. ^[6] gave a new family of contractions, named $\Theta$-contractions, and established a result associated with these contractions in the framework of complete metric spaces. Later on, Ahmad et al. ^[7] defined generalized $\Theta$-contractions to obtain additional generalized fixed point results. Recently, Abbas et al. ^[8] proved some fixed point theorems for a Suzuki type multivalued ($\Theta, \mathfrak{R}$)-contraction in the context of a complete metric space equipped with a binary relation. They employed their results to prove the existence of solutions of nonlinear fractional differential equations.

In this article, we combined the concepts of generalized almost contractions, twisted ($\rho$, $\mu)$-admissible mappings, and $\Theta$ -contractions to define Berinde type ($\rho$, $\mu)$-$\Theta$ contractions and obtain some fixed point results for self mappings in complete metric spaces. We derive new fixed point results, which enhance and upgrade some results in the literature. We also supply a non trivial example to support the obtained results. Finally, we discuss the solutions for the nonlinear fractional differential equations.

2.   Preliminaries

In 1922, Banach ^[1] established a theorem that is, known as Banach's contraction principle (or Banach's fixed point theorem), which is one of the decisive results of nonlinear analysis, which states that if $\mathcal{L}$ is self mapping on a complete metric space $(\mathcal{U}, \mathcal{\varrho })$ such that, for some $\Game \in \lbrack 0, 1)$,

for all $\mathcal{\varkappa }, \varpi \in \mathcal{U}$, then there exists a unique point $\mathcal{\varkappa }^{\ast }$ $\in \mathcal{U}$ such that $\mathcal{L\varkappa }^{\ast } = \mathcal{\varkappa }^{\ast }.$ Because of its significance and importance, many researchers have proved lots of fascinating enhancements, development, and generalizations of Banach's contraction principle.

Berinde ^[2,3] introduced the following notions of almost contraction and generalized almost contractions and proved a fixed point theorem.

Definition 1. (^[2]) Let $(\mathcal{U}, \mathcal{\varrho })$ be a metric space. A mapping $\mathcal{L}:(\mathcal{U}, \mathcal{\varrho })\rightarrow (\mathcal{ U}, \mathcal{\varrho })$ is said to be an almost contraction if there exists a constant $\Game \in \lbrack 0, 1)$ and some $L\geq 0$ such that

for all $\mathcal{\varkappa }, \varpi \in \mathcal{U}$.

Definition 2. (^[3]) Let $(\mathcal{U}, \mathcal{\varrho })$ be a metric space. A mapping $\mathcal{L}:(\mathcal{U}, \mathcal{\varrho })\rightarrow (\mathcal{U}, \mathcal{\varrho })$ is called a generalized almost contraction if there exists a constant $\Game \in \lbrack 0, 1)$ and some $L\geq 0$ such that

for all $\mathcal{\varkappa }, \varpi \in \mathcal{U}.$

Theorem 1. (^[3]) Let $(\mathcal{U}, \mathcal{\varrho })$ be a complete metric space and $\mathcal{L}:(\mathcal{U}, \mathcal{\varrho })\rightarrow (\mathcal{ U}, \mathcal{\varrho })$ is a generalized almost contraction, then $\mathcal{L}$ has a unique fixed point.

In 2015, Jleli ^[6] presented new class of contractions named $\Theta$-contractions. He presented a new result associated with $\Theta$ -contractions in the framework of complete metric spaces.

Definition 3. Let $(\mathcal{U}, \mathcal{\varrho })$ be a complete metric space. A mapping $\mathcal{L}:(\mathcal{U}, \mathcal{\varrho })\rightarrow (\mathcal{ U}, \mathcal{\varrho })$ is said to be a $\Theta$-contraction if there exists $\Game \in (0, 1)$ such that $\mathcal{\varrho }(\mathcal{L}(\mathcal{ \varkappa }), \mathcal{L}(\varpi)) > 0$ implies

for all $\mathcal{\varkappa }, \varpi \in \mathcal{U}$, where $\Theta: \mathbb{(}0, +\infty \mathbb{)}\rightarrow \mathbb{(}1, +\infty \mathbb{)}$ is a mapping satisfying the following conditions:

($\Theta _{1}$) $\Theta (\mathcal{\varkappa }) < \Theta (\varpi)$ for $0 < \mathcal{\varkappa } < \varpi$;

($\Theta _{2}$) for $\{ \mathcal{\varkappa }_{r}\} \subseteq \mathbb{(}0, +\infty \mathbb{)}$, $\lim_{r\rightarrow \infty }\mathcal{\varkappa } _{r} = 0$ $\Leftrightarrow$ $\lim_{r\rightarrow \infty }\Theta (\mathcal{ \varkappa }_{r}) = 1;$

($\Theta _{3}$) there exists $0 < h < 1$ and $l\in \mathbb{(}0, +\infty \mathbb{]}$ such that $\lim_{\mathcal{\varkappa }\rightarrow 0^{+}}\frac{ \Theta (\mathcal{\varkappa })-1}{\mathcal{\varkappa }^{h}} = l.$

Consistent with Jleli et al. ^[6], we represent the class of all mappings $\Theta :\mathbb{(}0, +\infty \mathbb{)}\rightarrow \mathbb{(}1, +\infty \mathbb{)}$ by $\Delta _{\Theta }$ satisfying (2.4) and ($\Theta _{1}$)–($\Theta _{3}$). For more details in this direction, we refer the readers to ^{[7,8,9,10,11,12,13,14,15,16,17,18,19]}.

On the other hand, Samet et al. ^[4] initiated the notion of $\rho$ -admissible mappings in 2012.

Definition 4. (^[4]) Let $\mathcal{L}:\mathcal{U}\rightarrow \mathcal{U}$ and $\rho :\mathcal{U}\times \mathcal{U}\rightarrow \lbrack 0, +\infty)$. Then $\mathcal{L}$ is said to be $\rho$-admissible if the following condition $\quad$

for all $\mathcal{\varkappa }, \varpi \in \mathcal{U},$ holds.

In this manner, Salimi et al.^[5] gave the idea of twisted ($\rho$, $\mu)$-admissible mappings.

Definition 5. (^[5]) Let $\mathcal{L}:\mathcal{U}$ $\rightarrow \mathcal{U}$ and $\rho, \mu :\mathcal{U}\times \mathcal{U}\rightarrow \lbrack 0, +\infty)$. Then $\mathcal{L}$ is said to be twisted ($\rho$, $\mu)$-admissible if $\quad$

for all $\mathcal{\varkappa }, \varpi \in \mathcal{U}.$

3.   Results

We define the notion of a Berinde type ($\rho$, $\mu)$-$\Theta$ contraction in this way.

Definition 6. Let $(\mathcal{U}, \mathcal{\varrho })$ be a metric space. A mapping $\mathcal{L}:(\mathcal{U}, \mathcal{\varrho })\rightarrow (\mathcal{U}, \mathcal{\varrho })$ is said to be a Berinde type ($\rho$, $\mu)$-$\Theta$ contraction if there exist $\Game \in (0, 1)$, $\Theta \in \Delta _{\Theta }$, $L\geq 0$ and $\rho, \mu :\mathcal{U}\times \mathcal{U}$ $\rightarrow \lbrack 0, +\infty)$ such that

for all $\mathcal{\varkappa }, \varpi \in \mathcal{U}$ and $\mathcal{\varrho } (\mathcal{L}(\mathcal{\varkappa }), \mathcal{L}(\varpi)) > 0.$

Theorem 2. Let $(\mathcal{U}, \mathcal{\varrho })$ be a complete metric space and $\mathcal{L}:(\mathcal{U}, \mathcal{\varrho })\rightarrow (\mathcal{U}, \mathcal{\varrho })$ be a Berinde type ($\rho$, $\mu)$-$\Theta$ contraction. If these assertions are satisfied:

(a) $\mathcal{L}$ is twisted ($\rho$, $\mu)$-admissible;

(b) there exists $\mathcal{\varkappa }_{0}\in \mathcal{U}$ such that $\rho (\mathcal{\varkappa }_{0}, \mathcal{L}(\mathcal{\varkappa }_{0}))\geq 1$ and $\mu (\mathcal{\varkappa }_{0}, \mathcal{L}(\mathcal{\varkappa }_{0}))\geq 1;$

(c) $\mathcal{L}$ is continuous$.$

Then there exists $\mathcal{\varkappa }^{\ast }\in \mathcal{U}$ such that $\mathcal{L\varkappa }^{\ast } = \mathcal{\varkappa }^{\ast }$.

Proof. Let $\mathcal{\varkappa }_{0}\in \mathcal{U}$ such that $\rho (\mathcal{ \varkappa }_{0}, \mathcal{L}(\mathcal{\varkappa }_{0}))\geq 1$ and $\mu (\mathcal{\varkappa }_{0}, \mathcal{L}(\mathcal{\varkappa }_{0}))\geq 1.$ Define $\{ \mathcal{\varkappa }_{r}\}$ in $\mathcal{U}$ by $\mathcal{ \varkappa }_{r+1} = \mathcal{L}(\mathcal{\varkappa }_{r}),$ for all $r\in \mathbb{N}.$ If $\mathcal{\varkappa }_{r+1} = \mathcal{\varkappa }_{r}$ for some $r\in \mathbb{N}$, then $\mathcal{\varkappa }^{\ast } = \mathcal{\varkappa }_{r}$ is fixed point of $\mathcal{L}$. Now we suppose that $\mathcal{\varkappa }_{r+1}\not = \mathcal{\varkappa }_{r},$ for all $r\in \mathbb{N}.$ As $\mathcal{L}$ is twisted ($\rho, \mu)$-admissible, we obtain $\rho (\mathcal{\varkappa }_{0}, \mathcal{\varkappa }_{1}) = \rho (\mathcal{\varkappa } _{0}, \mathcal{L}(\mathcal{\varkappa }_{0}))\geq 1$ $\Longrightarrow$ $\rho (\mathcal{\varkappa }_{1}, \mathcal{\varkappa }_{2}) = \rho (\mathcal{L}(\mathcal{\varkappa }_{0}), \mathcal{L}(\mathcal{\varkappa }_{1}))\geq 1$ and $\mu (\mathcal{\varkappa }_{1}, \mathcal{\varkappa }_{2}) = \mu (\mathcal{L}(\mathcal{\varkappa }_{0}), \mathcal{L}(\mathcal{\varkappa }_{1}))\geq 1.$ By mathematical induction, we obtain $\rho (\mathcal{\varkappa }_{r}, \mathcal{ \varkappa }_{r+1})\geq 1$ and $\mu (\mathcal{\varkappa }_{r}, \mathcal{ \varkappa }_{r+1})\geq 1,$ for all $r\in \mathbb{N}.$ From (3.1), we have

Thus, we have

Therefore

Thus, we have

for all $r\in \mathbb{N}.$ Since $\Theta \in \Delta _{\Theta },$ so if we let $r\rightarrow \infty$ in (3.3) and by ($\Theta _{2}$), we get

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

Assume that $\mathcal{\aleph }_{1} < \infty,$ then, let $\mathcal{\aleph } _{2} = \frac{\mathcal{\aleph }_{1}}{2} > 0.$ By the concept of the limit, there exists $r_{1}\in \mathbb{N}$ such that

for all $r > r_{1}.$ It yields

for all $r > r_{1}.$ Then

for all $r > r_{1},$ where $\mathcal{\aleph }_{3} = \frac{1}{\mathcal{\aleph } _{2}}.$ Now we assume that $\mathcal{\aleph }_{1} = \infty$ and $\mathcal{ \aleph }_{2} > 0$. Therefore, there exists $r_{1}\in \mathbb{N}$ such that

for all $r > r_{1}.$ It yields

for all $r > r_{1},$ where $\mathcal{\aleph }_{3} = \frac{1}{\mathcal{\aleph } _{2}}.$ Thus, in all cases, there exist $\mathcal{\aleph }_{3} > 0$ and $r_{1}\in \mathbb{N}$ such that

for all $r > r_{1}.$ Thus, by (3.3) and (3.6), we get

If we let, $r\rightarrow \infty$, we get

Thus, there exists $r_{2}\in \mathbb{N}$ such that

for all $r > r_{2}.$

For $m, r\in \mathbb{N}$ with $m > r\geq r_{1}$, we have

If we let, $r\rightarrow \infty$ in above inequality and utilizing the fact that the series $\sum_{i = r}^{\infty }\frac{1}{i^{\frac{1}{h}}}$ is convergent, we have $\lim_{r, m\rightarrow \infty }\mathcal{\varrho }(\mathcal{\varkappa }_{r}, \mathcal{\varkappa }_{m}) = 0.$ Hence $\{ \mathcal{ \varkappa }_{r}\}$ is a Cauchy sequence in $\mathcal{U}$. As $\mathcal{U}$ is complete, there exists $\mathcal{\varkappa }^{\ast }\in \mathcal{U}$ such that $\{ \mathcal{\varkappa }_{r}\} \rightarrow$ $\mathcal{\varkappa } ^{\ast }$. As $\mathcal{L}$ is continuous, we get $\mathcal{L}\mathcal{ (\varkappa }^{\ast }\mathcal{) = }\lim \limits_{r\rightarrow \infty }\mathcal{ L(\varkappa }_{r}\mathcal{) = }\lim \limits_{r\rightarrow \infty }\mathcal{ \varkappa }_{r+1} = \mathcal{\varkappa }^{\ast }.$ Thus, $\mathcal{\varkappa } ^{\ast }$ is a fixed point of $\mathcal{L}$. □

Now we omit the continuity condition on $\mathcal{L}$ but use an adjunctive condition on $\mathcal{U}$ and obtain the same conclusion.

Theorem 3. Let $(\mathcal{U}, \mathcal{\varrho })$ be a complete metric space and $\mathcal{L}:(\mathcal{U}, \mathcal{\varrho })\rightarrow (\mathcal{U}, \mathcal{\varrho })$ be a Berinde type ($\rho$, $\mu)$-$\Theta$ contraction. Suppose that the following assertions hold:

(a) $\mathcal{L}$ is twisted ($\rho$, $\mu)$-admissible;

(b) there exists $\mathcal{\varkappa }_{0}\in \mathcal{U}$ such that $\rho (\mathcal{\varkappa }_{0}, \mathcal{L}(\mathcal{\varkappa }_{0}))\geq 1$ and $\mu (\mathcal{\varkappa }_{0}, \mathcal{L}(\mathcal{\varkappa }_{0}))\geq 1;$

(c) if $\{ \mathcal{\varkappa }_{r}\} \subseteq$ $\mathcal{U}$ such that $\rho (\mathcal{\varkappa }_{r}, \mathcal{\varkappa }_{r+1})\geq 1$ and $\mu (\mathcal{\varkappa }_{r}, \mathcal{\varkappa }_{r+1})\geq 1,$ for all $r$ and $\mathcal{\varkappa }_{r}\rightarrow \mathcal{\varkappa }^{\ast }\in \mathcal{ U}$ as $r\rightarrow \infty,$ then $\rho (\mathcal{\varkappa }_{r}, \mathcal{ \varkappa }^{\ast })\geq 1$ and $\mu (\mathcal{\varkappa }_{r}, \mathcal{ \varkappa }^{\ast })\geq 1,$ for all $r\in \mathbb{N}.$

Then $\mathcal{L}$ has a fixed point $\mathcal{\varkappa }^{\ast }\in \mathcal{U}$.

Proof. Let $\mathcal{\varkappa }_{0}\in \mathcal{U}$ such that $\rho (\mathcal{ \varkappa }_{0}, \mathcal{L}(\mathcal{\varkappa }_{0}))\geq 1$ and $\mu (\mathcal{\varkappa }_{0}, \mathcal{L}(\mathcal{\varkappa }_{0}))\geq 1.$ Proceeding in a similar manner to the proof of Theorem 2, we have $\rho (\mathcal{ \varkappa }_{r}, \mathcal{\varkappa }_{r+1})\geq 1$ and $\mu (\mathcal{ \varkappa }_{r}, \mathcal{\varkappa }_{r+1})\geq 1$ and $\{ \mathcal{\varkappa }_{r}\}$ is Cauchy in $\mathcal{U}$ that converges to $\mathcal{\varkappa } ^{\ast }$, i.e.,

By hypothesis (c), we have $\rho (\mathcal{\varkappa }_{r}, \mathcal{ \varkappa }^{\ast })\geq 1$ and $\mu (\mathcal{\varkappa }_{r}, \mathcal{ \varkappa }^{\ast })\geq 1$ for all $r\in \mathbb{N}.$ On the contrary, we assume that $\mathcal{L}(\mathcal{\varkappa } ^{\ast })\not = \mathcal{\varkappa }^{\ast }$, and there exists $r_{0}\in \mathbb{N}$ such that $\mathcal{\varkappa }_{r+1}\not = \mathcal{L}(\mathcal{\varkappa } ^{\ast }),$ for all $r\geq r_{0},$ i.e., $\mathcal{\varrho }(\mathcal{L}(\mathcal{\varkappa }_{r}), \mathcal{L}(\mathcal{\varkappa }^{\ast })) > 0,$ for all $r\geq r_{0}.$ By (3.1), ($\Theta _{1}$) and the triangle inequality, we have

If $\mathcal{\varrho }(\mathcal{\varkappa }^{\ast }, \mathcal{L}(\mathcal{ \varkappa }^{\ast })) > 0,$ then from the following fact

and $\Theta$ is continuous, taking the limit as $r\rightarrow \infty$ in above inequality, we get

which is a contradiction. Therefore $\mathcal{\varrho }(\mathcal{\varkappa } ^{\ast }, \mathcal{L}(\mathcal{\varkappa }^{\ast })) = 0,$ i.e., $\mathcal{L}(\mathcal{\varkappa }^{\ast }) = \mathcal{\varkappa }^{\ast }$ and $\mathcal{ \varkappa }^{\ast }$ is a fixed point of $\mathcal{L}$. □

For the uniqueness of the fixed point, we take the property:

(P) $\rho (\mathcal{\varkappa }, \varpi)\geq 1$ and $\mu (\mathcal{\varkappa }, \varpi)\geq 1$ for all fixed points $\mathcal{\varkappa }, \varpi \mathfrak{\in }\mathcal{U}$ of $\mathcal{L}$.

Theorem 4. In addition to the assumptions of Theorem 2, if we take property (P), then we obtain the uniqueness of the fixed point.

Proof. Let $\mathcal{\varkappa }^{\ast }, \widehat{\mathcal{\varkappa }}\in \mathcal{ U}$ be such that $\mathcal{L}(\mathcal{\varkappa }^{\ast }) = \mathcal{ \varkappa }^{\ast }\not = \widehat{\mathcal{\varkappa }} = \mathcal{L}(\widehat{ \mathcal{\varkappa }}).$ Then, from property (P), $\rho (\mathcal{\varkappa }^{\ast }, \widehat{\mathcal{\varkappa }})\geq 1$ and $\mu (\mathcal{\varkappa }^{\ast }, \widehat{\mathcal{\varkappa }})\geq 1.$ Then

a contradiction because $\Game \in (0, 1)$. Hence, $\mathcal{L}$ has a unique fixed point in $\mathcal{U}$. □

Example 1. Consider the sequence $\{\mathcal{\varkappa }_{n}\}$ as follows:

$\mathcal{\varkappa }_{1} = 1,$

$\mathcal{\varkappa }_{2} = 1+2,$

$\cdot \cdot \cdot$

$\mathcal{\varkappa }_{n} = 1+2+...+n = \frac{n(n+1)}{2},$ for $n\in \mathbb{N}.$

Let $\mathcal{U} = \left \{ \mathcal{\varkappa }_{n}:n\in \mathbb{N} \right \}$

and $\mathcal{\varrho }(\mathcal{\varkappa }, \varpi) = \left \vert \mathcal{\varkappa }-\varpi \right \vert.$ Then $\left(\mathcal{U}, \mathcal{ \varrho }\right)$ is a complete metric space.

Define the mapping $\mathcal{L }:\mathcal{U}\rightarrow \mathcal{U}$

by

First, we show that $\mathcal{L}$ is not the Banach contraction

Now, if we consider the mapping $\Theta:\mathbb{(}0, +\infty \mathbb{)} \rightarrow \mathbb{(}1, +\infty \mathbb{)}$ by $\Theta (t) = 2^{t},$ for $t > 0$. Now, we show that the mapping $\mathcal{L}$ is not the $\Theta$ -contraction for $\Game = \frac{1}{2}\in (0, 1),$ that is, $\Theta (\mathcal{ \varrho }\left(\mathcal{L}\left(\mathcal{\varkappa }_{n}\right), \mathcal{L }\left(\mathcal{\varkappa }_{m}\right) \right))\geq \left[ \Theta (\mathcal{\varrho }\left(\mathcal{\varkappa }_{n}, \mathcal{\varkappa } _{m}\right))\right] ^{\frac{1}{2}}.$

Indeed, for $n = 1$ and $m = 4$, we get

that is

because

Now, we show that the mapping $\mathcal{L}$ is the Berinde type ($\rho$, $\mu)$-$\Theta$ contraction for $\rho, \mu:\mathcal{U}\times \mathcal{U}$ $\rightarrow \lbrack 0, +\infty)$ defined by

for all $\mathcal{\varkappa }, \varpi \in \mathcal{U}$ and some $\Game = \frac{ 1}{2}\in (0, 1)$ and $L = 2 > 0.$

To see this, we discuss our main result for $(1 < n < m).$ Now since,

Therefore, for $1 < n < m,$ we have

Hence, all the conditions of Theorem 4 are satisfied and $\mathcal{\varkappa }_{1}$ is a unique fixed point of mapping $\mathcal{L}.$

The following results are a direct consequence of Theorem 2 and Theorem 3.

Corollary 1. Let $(\mathcal{U}, \mathcal{\varrho })$ be a complete metric space and $\mathcal{L}:(\mathcal{U}, \mathcal{\varrho })\rightarrow (\mathcal{U}, \mathcal{\varrho })$ be twisted ($\rho$, $\mu)$-admissible mapping such that

for all $\mathcal{\varkappa }, \varpi \in \mathcal{U}$ and $\mathcal{\varrho } (\mathcal{L}(\mathcal{\varkappa }), \mathcal{L}(\varpi)) > 0.$ If these assertions are satisfied:

(a) there exists $\mathcal{\varkappa }_{0}\in \mathcal{U}$ such that $\rho (\mathcal{\varkappa }_{0}, \mathcal{L}(\mathcal{\varkappa }_{0}))\geq 1$ and $\mu (\mathcal{\varkappa }_{0}, \mathcal{L}(\mathcal{\varkappa }_{0}))\geq 1,$

(b) $\mathcal{L}$ is continuous or if $\{ \mathcal{\varkappa }_{r}\}$ $\subseteq$ $\mathcal{U}$ such that $\rho (\mathcal{\varkappa }_{r}, \mathcal{ \varkappa }_{r+1})\geq 1$ and $\mu (\mathcal{\varkappa }_{r}, \mathcal{ \varkappa }_{r+1})\geq 1$ and $\mathcal{\varkappa }_{r}\rightarrow \mathcal{ \varkappa }^{\ast }\in \mathcal{U}$ as $r\rightarrow \infty,$ then $\rho (\mathcal{\varkappa }_{r}, \mathcal{\varkappa }^{\ast })\geq 1$ and $\mu (\mathcal{\varkappa }_{r}, \mathcal{\varkappa }^{\ast })\geq 1,$ $\forall$ $r\in \mathbb{N},$

then there exists $\mathcal{\varkappa }^{\ast }\in \mathcal{U}$ such that $\mathcal{\varkappa }^{\ast } = \mathcal{L\varkappa }^{\ast }$.

Proof. Take $L = 0$ in Theorem 3.1 and Theorem 3.2. □

Corollary 2. Let $(\mathcal{U}, \mathcal{\varrho })$ be a complete metric space and $\mathcal{L}:\mathcal{U}\rightarrow \mathcal{U}$ be such that

If these assertions are satisfied:

(a) $\mathcal{L}$ is $\rho$-admissible,

(b) there exists $\mathcal{\varkappa }_{0}\in \mathcal{U}$ such that $\rho (\mathcal{\varkappa }_{0}, \mathcal{L}(\mathcal{\varkappa }_{0}))\geq 1,$

(c) $\mathcal{L}$ is continuous or if $\{ \mathcal{\varkappa }_{r}\}$ $\subseteq$ $\mathcal{U}$ such that $\rho (\mathcal{\varkappa }_{r}, \mathcal{ \varkappa }_{r+1})\geq 1$ and $\mathcal{\varkappa }_{r}\rightarrow \mathcal{ \varkappa }^{\ast }\in \mathcal{U}$ as $r\rightarrow \infty,$ then $\rho (\mathcal{\varkappa }_{r}, \mathcal{\varkappa }^{\ast })\geq 1,$ $\forall$ $r\in \mathbb{N},$ then there exists a unique point $\mathcal{\varkappa }^{\ast }\in \mathcal{U}$ such that $\mathcal{\varkappa }^{\ast } = \mathcal{L\varkappa }^{\ast }.$

Proof. Taking $\mu (\mathcal{\varkappa }, \varpi) = 1,$ for all $\mathcal{\varkappa }, \varpi \in \mathcal{U}$ in Theorem 2 and Theorem 3. □

Corollary 3. (^[6]) Let $(\mathcal{U}, \mathcal{\varrho })$ be a complete metric space and $\mathcal{L}:(\mathcal{U}, \mathcal{\varrho })\rightarrow (\mathcal{U}, \mathcal{\varrho })$. If there exists $\Game \in (0, 1)$ and $\Theta \in \Delta _{\Theta }$ such that

then there exists a unique point $\mathcal{\varkappa }^{\ast }\in \mathcal{U}$ such that $\mathcal{\varkappa }^{\ast } = \mathcal{L\varkappa }^{\ast }.$

Proof. Take $\rho (\mathcal{\varkappa }, \varpi) = \mu (\mathcal{\varkappa }, \varpi) = 1,$ for all $\mathcal{\varkappa }, \varpi \in \mathcal{U}$ and $L = 0$ in Theorem 2 and Theorem 3. □

4.   Applications

On the other hand, the fixed point theory is a very strong mathematical tool to establish the existence and uniqueness of nearly all problems modeled by nonlinear relations. Consequently, the existence and uniqueness problems of fractional differential equations are studied by means of the fixed point theory. Recently, the existence of solutions of fractional differential equations have been studied, see ^[8,21,22].

In the present section, we discuss the existence of a solution of the following nonlinear fractional differential equation:

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

where $\mathcal{\varkappa }\in C([0, 1], \mathbb{R})$. Here, $^{C}\mathcal{\varrho }^{\eta }$ represents the Caputo fractional derivative of order $\eta$ defined by

$\left(j-1 < \eta < j, \text{ }j = \left[ \eta \right] +1\right)$ and $I^{\eta }f$ represents the Riemann-Liouville fractional integral of order $\eta$ of a continuous function $f: \mathbb{R} ^{+}\rightarrow \mathbb{R}$, given by

Let $\mathcal{U} = C[0, 1]$ be the space of all continuous functions defined on $[0, 1]$. The metric $\mathcal{\varrho }$ on $\mathcal{U}$ is given by

for all $\mathcal{\varkappa }, \varpi \in \mathcal{U}$, then the space $\mathcal{U} = (C[0, 1], \mathcal{\varrho })$ is complete metric space.

Theorem 5. Consider the nonlinear fractional differential Eq (4.1). Let $\zeta :(-\infty, +\infty)\times (-\infty, +\infty)\rightarrow \mathbb{R}$. Assume that these assertions hold:

(ⅰ) The function $f: [0, 1]\times (-\infty, +\infty)\rightarrow \mathbb{R}$ is continuous,

(ⅱ) $\forall \mathcal{\varkappa }, \varpi \in \mathcal{U}$ and $\pi \in \lbrack 1, \infty)$ such that

$\forall$ $t\in \lbrack 0, 1]$,

(ⅲ) there exists $\mathcal{\varkappa }_{0}\in C([0, 1], \mathbb{R})$ such that $\zeta (\mathcal{\varkappa }_{0}(t), \mathcal{L\varkappa } _{0}(t)) > 0,$ for all $t\in \lbrack 0, 1],$ where $\mathcal{L}:C([0, 1], \mathbb{R})\rightarrow C([0, 1], \mathbb{R})$ is defined by

for $t\in \left[ 0, 1\right]$,

(ⅳ) for each $t\in \lbrack 0, 1]$ and $\mathcal{\varkappa }, \varpi \in C([0, 1], \mathbb{R})$, $\zeta (\mathcal{\varkappa }(t), \varpi (t)) > 0$ $\Longrightarrow \zeta (\mathcal{L\varkappa }(t), \mathcal{L}\varpi (t)) > 0,$

(ⅴ) for $\mathcal{\varkappa }_{r}\}$ $\subseteq$ $C([0, 1], \mathbb{R})$ such that $\mathcal{\varkappa }_{r}\rightarrow \mathcal{\varkappa }$ in $C([0, 1], \mathbb{R})$ and $\zeta (\mathcal{\varkappa }_{r}, \mathcal{\varkappa }_{r+1}) > 0,$ for all $r$ $\in \mathbb{N}$, then $\zeta (\mathcal{\varkappa }_{r}, \mathcal{\varkappa }) > 0,$ for all $r$ $\in \mathbb{N}$.

Then, (4.1) has at least one solution.

Proof. It is very simple to show that $\mathcal{\varkappa }\in \mathcal{U}$ is a solution of (4.1) if and only if $\mathcal{\varkappa }\in \mathcal{U}$ is a solution of

for $t\in \left[ 0, 1\right].$ Now, let $\mathcal{\varkappa }, \varpi \in \mathcal{U}$ such that $\zeta (\mathcal{\varkappa }(t), \varpi (t)) > 0$ for all $t\in \lbrack 0, 1]$. By (iii), we have

which implies that

where $B$ is the beta function. From the above inequality, we get

which implies

Taking the exponential, we obtain

that is

where $\Game = \sqrt{e^{-\pi }} < 1.$ Consider $\Theta : \mathbb{R} ^{+}\rightarrow \mathbb{R}$ defined by $\Theta (u) = e^{\sqrt{u}}$, $\forall$ $u > 0$, then $\Theta \in \Delta _{\Theta }$. Additionally, we define $\rho, \mu :\mathcal{U}\times \mathcal{U} \rightarrow \lbrack 0, +\infty)$ by

and for all $\mathcal{\varkappa }, \varpi \in \mathcal{U}$. Thus,

for all $\mathcal{\varkappa }, \varpi \in \mathcal{U}$ and $\mathcal{\varrho } (\mathcal{L\varkappa }, \mathcal{L}\varpi) > 0.$ Now, by using condition (iv), we have

implies

for all $\mathcal{\varkappa }, \varpi \in \mathcal{U}$. Hence, $\mathcal{L}$ is a twisted ($\rho$, $\mu$)-admissible. Additionally, from (ⅲ), there exists $\mathcal{\varkappa }_{0}\in \mathcal{U}$ such that $\rho (\mathcal{\varkappa }_{0}, \mathcal{L\varkappa }_{0})\geq 1$ and $\mu (\mathcal{\varkappa }_{0}, \mathcal{L\varkappa }_{0})\geq 1.$ Finally, we conclude that the assertion (v) of Theorem 3 is satisfied. Hence, as application of Theorem 3, we obtain $\mathcal{\varkappa }^{\ast }\in \mathcal{U}$ such that $\mathcal{\varkappa }^{\ast } = \mathcal{L\varkappa }^{\ast }$. Thus, $\mathcal{ \varkappa }^{\ast }$ is a solution of (4.1). □

5.   Conclusions

In this paper, we have defined Berinde type ($\rho$, $\mu$)-$\vartheta$ contractions and obtained some generalized fixed point results. In practice, we discussed the solutions for the nonlinear fractional differential equation. We derived new fixed point results, which upgraded and enhanced some results in the literature. We also supplied a non trivial example to support the obtained results.

Our future work will focus on studying the common fixed points of multivalued mappings and fuzzy mappings for Berinde type ($\rho$, $\mu)$-$\Theta$ contractions in the context of complete metric spaces. Fractional differential inclusions and fractional integral inclusions can be solved as applications of these results.

Use of AI tools declaration

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

Acknowledgments

Saleh Abdullah Al-Mezel would like to thank Deanship of Scientific Research at Majmaah University for supporting this research work.

Conflict of interest

The authors declare no conflict of interest.