Fixed point theorems via auxiliary functions with applications to two-term fractional differential equations with nonlocal boundary conditions

1.   Introduction

One of the most interesting areas in scientific study is regarding natural phenomena, which many researchers have previously investigated in mathematical models through differential operators, see for instance ^{[1,2,3,4,5,6]}. Currently, mathematicians are paying a lot of attention to the fractional differential operator since it has been widely used in a variety of fields, including the risk-controlled financial market ^[7,8,9], engineering and scientific fields ^{[10,11,12,13,14]}. To be more precise, recent advances in multi-term fractional differential equations can be found in ^{[15,16,17,18,19,20,21,22,23]}. Moreover, research in fractional derivatives of type Caputo and integral operators was performed in ^[24]. Additionally, the nonlinear two-term fractional differential equations were intensively studied with some scientific publications related to the nonlocal BVPs equations, which are pertinent to the developing topic in ^{[19,20,25,26,27]}. It is worth mentioning that the existence and uniqueness of a solution to a differential equation are frequently obtained by using the concepts of fixed point theorem, as seen in ^{[28,29,30,31,32,33]}. Therefore, fixed point theory has played an important role in the study of fractional differential operators.

The idea of a fixed point theorem for metric spaces endowed with graphs was initially proposed by Jachymski ^[34] in 2008. Since then, other researchers have focused on this concept in a variety of spaces endowed with graphs, see ^{[35,36,37,38,39]} for instance. One of the most significant consequences of this generalization is the extension of the well-known Banach contraction principle to the case of metric spaces endowed with graphs, see ^[35,38].

There are various ways that mathematicians could investigate fixed point theory. One can consider contractions with Geraghty functions, which are among the most influential ideas in this area, see ^{[40,41,42,43,44,45,46,47,48]} for further information. In 2017, Charoensawan and Atiponrat ^[47] introduced a new class of contractions, namely $\theta$-$\phi$-contractions, which are of Geraghty's type. Now, let us recall essential concepts that will be considered throughout this paper below.

Definition 1.1. ^[47] Suppose that $(X, d)$ is a metric space endowed with a directed graph $\mathcal{G} = \left(V(\mathcal{G}), E(\mathcal{G})\right)$, and $\mu, \delta :X\to X$ are functions. Let us define the following sets:

and

We note that $C(\mu, \delta)$ is the set of all coincidence points of $\mu$ and $\delta$. Additionally, $Cm(\mu, \delta)$ is the set of all common fixed points of $\mu$ and $\delta$.

Lemma 1.2. ^[47] Let $(X, d)$ be a metric space endowed with a directed graph $\mathcal{G} = \left(V(\mathcal{G}), E(\mathcal{G})\right)$, and let $\mu, \delta :X\to X$ be functions. If $C(\mu, \delta) \neq \varnothing$, then $X(\mu, \delta) \neq \varnothing$.

Definition 1.3. ^[47] Let $\mathcal{G} = \left(V(\mathcal{G}), E(\mathcal{G})\right)$ be a directed graph, and let $\mu, \delta :X\to X$ be functions. We say that $\mu$ is $\delta$-edge preserving with respect to $\mathcal{G}$ whenever for each $x \in X$,

Definition 1.4. ^[47] Let $(X, d)$ and $(Y, d')$ be metric spaces, and let $\mu :X \rightarrow Y$ and $\delta :X\rightarrow X$ be functions. We say that $\mu$ is $\delta$-Cauchy on $X$ whenever for any sequence $\{x_n\}$ in $X$ with $\{\delta x_n\}$ being Cauchy in $(X, d)$, the sequence $\{ \mu x_n\}$ is Cauchy in $(Y, d')$.

On the other hand, Martinez-Moreno et al. ^[48] demonstrated fascinating results on common fixed point theorems for Geraghty's type contraction mappings employing the monotone property with two metrics as a consequence of $d$-compatibility and $\delta$-uniform continuity. This motivates us to investigate metric spaces equipped with two distance functions in our work.

Due to their numerous scientific applications, fractional differential equations have garnered a lot of attention from mathematicians in recent years. As can be seen, for example, in ^{[49,50,51,52]}, that fixed point theory has strongly contributed to the knowledge of fractional differential equations. In addition, it is worth emphasizing that our recent work is inspired by Karapınar's investigation of the fixed point theorem using auxiliary functions in ^[52], which provided insight into its usefulness for fractional differential equations. Therefore, in this research, we replace $\theta$-$\phi$ contraction mappings with auxiliary functions to improve the outcomes in ^[31,47]. This enables us to derive actual criteria for the existence of common fixed points in the setting of auxiliary functions endowed with two metrics and a directed graph. Subsequently, we provide the applications for a class of nonlinear two-term fractional differential equations in our third section.

2.   Main results

Study results about the existence of common fixed points for auxiliary functions with two metrics endowed with a directed graph are presented in this section. Let us first define the classes of functions that will be taken into account in this task.

Assume that $\varphi:[0, \infty)\rightarrow [0, \infty)$ is a function with the properties listed below:

● $\varphi$ is increasing and continuous,

● $\varphi(r) = 0$ if and only if $r = 0$.

The set of all functions $\varphi$ satisfying the aforementioned constraints will be referred to as $\Phi$ going forward.

In spired by ^[52], we define the class $\mathcal{A }(X)$ consisting all auxiliary functions $h : X\times X \to [0, 1]$ such that

for all sequences $\{x_n\}$ and $\{y_n\}$ in $X$ with $\{d(x_n, y_n)\}$ is decreasing, where $(X, d)$ is a metric space.

Example 2.1. ^[52] Let $h_1, h_2 : \mathbb{R}\times \mathbb{R} \to [0, 1]$, for all $x, y \in \mathbb{R}$, defined by

(1) $h_1(x, y) = c$ for some $c\in (0, 1)$,

(2) $h_2(x, y) = \dfrac{1}{1+x+y}$.

Then $h_1, h_2\in\mathcal{A }(X)$.

Lemma 2.2. For a sequence $\{x_{n}\}$ in a metric space $(X, d)$ and a function $\delta :X\to X$ such that

if $\{\delta x_{n}\}$ is not a Cauchy sequence, then there exists $\epsilon > 0$ such that, for all $k \in \mathbb{N}$, there are $n_k, m_k \in \mathbb{N}$ with $n_k > m_k \geq k$ satisfying $n_k$ is the smallest number such that

Then, we obtain

Proof. Suppose that $\{\delta x_{n}\}$ is not Cauchy. By definition, there is a positive number $\epsilon > 0$ such that for all $k \in \mathbb{N}$, there are $n_k, m_k \in \mathbb{N}$ with $n_k > m_k \geq k$ satisfying $n_k$ is the smallest number such that

This means

Letting $k \to \infty$ and applying the fact that $\lim\limits_{n \to \infty} d(\delta x_{n}, \delta x_{n+1}) = 0$, we receive

Consider, by the triangle inequality, that

and

Then, we obtain

By

and taking $k$ to $\infty$ in (2.2), we obtain

Thus,

Here, we are ready to define a new category of contractions defined as follows. Let $(X, d, \mu, \delta, \mathcal{G})$ refer to a structure throughout this work that has the properties listed below:

● $X\neq \varnothing$ and $(X, d)$ is a metric space,

● $X$ is endowed with a directed grahp $\mathcal{G} = (V(\mathcal{G}), E(\mathcal{G}))$,

● $\mu$ and $\delta$ are self mappings,

● $\mu$ is $\delta$-edge preserving with respect to $\mathcal{G}$.

Lemma 2.3. On $(X, d, \mu, \delta, \mathcal{G})$. Let a sequence $\{x_{n}\}$ in $X$ such that

where $u \in X$ and $(\delta x_{n-1}, \delta x_n)\in E(\mathcal{G})$. If $\mu$ is $\mathcal{G}$-continuous, with $\mu$ and $\delta$ being $d$-compatible, then $u\in C(\mu, \delta)$.

Proof. Let $\{x_{n}\}$ in $X$ such that

where $u \in X$, and $(\delta x_{n-1}, \delta x_n)\in E(\mathcal{G})$. Additionally, we conclude that

due to $\mu$ and $\delta$ being $d$-compatible. Finally, we consider

By combining the continuity of $\delta$ with the notion that $\mu$ is $\mathcal{G}$-continuous, $(\delta x_{n-1}, \delta x_n)\in E(\mathcal{G})$ and using (2.4), it can be concluded that $d(\delta u, \mu u) = 0$ when $n \to \infty$. As a result, $\delta u = \mu u$, which indicates that $u$ is a coincidence point of $\mu$ and $\delta$. Thus, $u\in C(\mu, \delta)$.

Definition 2.4. On $(X, d, \mu, \delta, \mathcal{G})$. If the following criteria are satisfied, the pair $(\mu, \delta)$ will be referred to as an $(h$-$\varphi)_R$-contraction with regard to $d$. There exists $h \in \mathcal{A}(X)$ and $\varphi\in\Phi$ with $(\delta x, \delta y)\in E(\mathcal{G})$ for $x, y\in X$, we have

where $R : X\times X \to [0, \infty)$ is defined by

for $x, y\in X$.

The result in ^[47] can be applied to the case of auxiliary functions owing to the aforementioned definition. In actuality, we are now prepared to demonstrate and present our key findings. The motivation for the following theorem comes from ^[53] and incorporates two metrics.

Theorem 2.5. On $(X, d', \mu, \delta, \mathcal{G})$, let $(X, d')$ be a complete metric space, and let $d$ be another metric on $X$. Assume that $(\mu, \delta)$ is an $(h$-$\varphi)_R$-contraction with respect to $d$ and that the following criteria are satisfied.

(1) $\delta :(X, d')\to (X, d')$ is continuous, and $\delta (X)$ is $d'$-closed,

(2) $\mu (X)\subseteq \delta (X)$,

(3) The transitivity property of $E(\mathcal{G})$ holds,

(4) If $d\ngeq d'$, suppose that $\mu :(X, d)\to (X, d')$ is $\delta$-Cauchy on $X$,

(5) $\mu :(X, d')\to (X, d')$ is $\mathcal{G}$-continuous, and $\mu$ and $\delta$ are $d'$-compatible.

Consequently, it can be seen that

Proof. $(\Leftarrow)$ This derives from Lemma 1.2.

$(\Rightarrow)$ Assume that $X(\mu, \delta)\neq \varnothing$ and $x_0\in X$ with $(\delta x_0, \mu x_0)\in E(\mathcal{G})$. According to the assumption that $\mu (X)\subseteq \delta (X)$ and $\mu (x_0)\in X$, we could establish a sequence $\{x_{n}\}$ in $X$ such that $\delta x_{n} = \mu x_{n-1}$ for every $n\in \mathbb{N}$. If there exists $n_0\in\mathbb{N}$ such that $\delta x_{n_0} = \delta x_{n_0-1}$, then $x_{n_0-1}$ is a coincidence point of $\mu$ and $\delta$. We may therefore now assume that $\delta x_n\ne \delta x_{n-1}$ for all $n\in\mathbb{N}$.

Because $(\delta x_0, \mu x_0) = (\delta x_0, \delta x_1)\in E(\mathcal{G})$ and $\mu$ is $\delta$-edge preserving with respect to $\mathcal{G}$, it is precise to state that $(\mu x_0, \mu x_1) = (\delta x_1, \delta x_2)\in E(\mathcal{G})$. We obtain $(\delta x_{n-1}, \delta x_n)\in E(\mathcal{G})$ for every $n\in\mathbb{N}$ through mathematical induction. As $(\mu, \delta)$ is an $(h$-$\varphi)_R$-contraction with respect to $d$, for each $n \geq 0$,

Additionally, a straightforward calculation demonstrates that

If we denote by

As $d(\delta x_{n}, \delta x_{n+2})\le d(\delta x_{n}, \delta x_{n+1})+d(\delta x_{n+1}, \delta x_{n+2})$, we get

Now, suppose that $\delta _n$ is not decreasing, then there exists $C\in\mathbb{N}$ such that $\delta _C\le \delta _{C+1}$ so we have

By the inequality (2.5) and the property of $\varphi$, we obtain

Since for every $n\in\mathbb{N}$, $\delta x_n\ne \delta x_{n-1}$, we have $\delta _{C+1} = d(\delta x_{C+1}, \delta x_{C+2}) > 0$ which follows from the above inequality, we get $h(\delta x_{C}, \delta x_{C+1}) = 1$. By the fact that $h \in \mathcal{A}(X)$, we obtain $d(\delta x_{C}, \delta x_{C+1}) = 0$. It is a contradiction. Therefore, $\delta _n$ is decreasing, $\delta _n > \delta _{n+1}$ for all $n\ge 0$, we have

Since $\delta _n$ is bounded below, the sequence converges. Let

Contrarily, suppose that $L > 0$. Due to the property of $\varphi$, $\lim\limits_{n \to \infty} \varphi(\delta _n) = \varphi(L) > 0$. By (2.5), it is demonstrated that

In the inequality above, when we take $n \to \infty$, we obtain

Therefore, $\lim\limits_{n \to \infty} h(\delta x_{n}, \delta x_{n+1}) = 1$. According to the notion of auxiliary functions,

which contradicts to the assumption. So, $\lim\limits_{n \to \infty} d(\delta x_{n}, \delta x_{n+1}) = 0$.

The sequence $\{\delta x_{n}\}$ has to be Cauchy, as we will demonstrate next. Contrarily, suppose that $\{\delta x_{n}\}$ is not Cauchy. Then there exists $\epsilon > 0$ such that, for all $k \in \mathbb{N}$, there are $n_k, m_k \in \mathbb{N}$ with $n_k > m_k \geq k$ satisfying $n_k$ is the smallest number such that

By Lemma 2.2, this implies

We determine that $(\delta x_{m_k}, \delta x_{n_k})\in E(\mathcal{G})$ for each $k \in \mathbb{N}$ owing to the transitivity property of $E(\mathcal{G})$. As a result,

where

Since $\lim\limits_{n \to \infty} d(\delta x_{n}, \delta x_{n+1}) = 0$, applying the preceding equality with $k \to \infty$ means that

Combining the aforementioned fact with the inequality (2.6), we obtain

As a result, $\lim\limits_{k \to \infty}h(\delta x_{m_k}, \delta x_{n_k}) = 1.$ Then, $\lim\limits_{k \to \infty}d(\delta x_{m_k}, \delta x_{n_k}) = 0$, which contradicts to (2.3). That $\{\delta x_{n}\}$ is Cauchy in $(X, d)$ must therefore be true.

In the following part, we demonstrate that in the metric space $(X, d')$, $\{\delta x_{n}\}$ is also Cauchy. The proof is simple when $d \geq d'$. The case $d \ngeq d'$ is therefore taken into consideration. Let $\varepsilon > 0$. We conclude that $\{ \mu x_n\}$ is Cauchy in $(X, d')$ since $\{\delta x_n\}$ is Cauchy in $(X, d)$ and $\mu$ is $\delta$-Cauchy on $X$. So, there exists $N_{0} \in \mathbb{N}$ such that

The sequence $\{\delta x_{n}\}$ is therefore Cauchy in $(X, d')$.

Since $\delta (X)$ is a $d'$-closed subset of $(X, d')$, which is complete, it follows that $u = \delta x \in \delta (X)$ exists satisfying

Since assumption $(5)$, by Lemma 2.3 which indicates that $u$ is a coincidence point of $\mu$ and $\delta$. Thus, $u\in C(\mu, \delta)$.

We analyze the scenario in which the two metrics $d$ and $d'$ coincide in our next theorem.

Definition 2.6. On $(X, d, \mu, \delta, \mathcal{G})$. If the following criteria are satisfied, the pair $(\mu, \delta)$ will be referred to as an {$(h$-$\varphi)_M$-contraction with regard to $d$}. There exists $h \in \mathcal{A}(X)$ and $\varphi\in\Phi$ with $(\delta x, \delta y)\in E(\mathcal{G})$ for $x, y\in X$, we have

where $M : X\times X \to [0, \infty)$ is defined by

for $x, y \in X$.

Theorem 2.7. On $(X, d, \mu, \delta, \mathcal{G})$, let $(X, d)$ be a complete metric space with an $(h$-$\varphi)_M$-contraction $(\mu, \delta)$. Suppose that the following criteria are satisfied.

(1) $\delta$ is continuous, and $\delta (X)$ is closed.

(2) $\mu (X)\subseteq \delta (X)$.

(3) The transitivity property $E(\mathcal{G})$ holds. $(4)$ At least one of the statements below is satisfied.

(a) $\mu$ is $\mathcal{G}$-continuous, and $\mu$ and $\delta$ are $d$-compatible,

(b) $(X, d, \mathcal{G})$ has the property $A$ in ^[34], and

Consequently, we obtain that

Proof. $(\Leftarrow)$ This derives from Lemma 1.2.

$(\Rightarrow)$ Assume that $X(\mu, \delta)\neq \varnothing$ and $x_0\in X$ with $(\delta x_0, \mu x_0)\in E(\mathcal{G})$. According to the assumption that $\mu (X)\subseteq \delta (X)$ and $\mu (x_0)\in X$, we could establish a sequence $\{x_{n}\}$ in $X$ such that $\delta x_{n} = \mu x_{n-1}$ for every $n\in \mathbb{N}$. If there exists $n_0\in\mathbb{N}$ such that $\delta x_{n_0} = \delta x_{n_0-1}$, then $x_{n_0-1}$ is a coincidence point of $\mu$ and $\delta$. We may therefore now assume that $\delta x_n\ne \delta x_{n-1}$ for all $n\in\mathbb{N}$.

Because $(\delta x_0, \mu x_0) = (\delta x_0, \delta x_1)\in E(\mathcal{G})$ and $\mu$ is $\delta$-edge preserving with respect to $\mathcal{G}$, it is precise to state that $(\mu x_0, \mu x_1) = (\delta x_1, \delta x_2)\in E(\mathcal{G})$. We obtain $(\delta x_{n-1}, \delta x_n)\in E(\mathcal{G})$ for every $n\in\mathbb{N}$ through mathematical induction. As $(\mu, \delta)$ is an $(h$-$\varphi)_M$-contraction with respect to $d$, for each $n \geq 0$,

Additionally, a straightforward calculation demonstrates that

If we denote by

We have

Now, suppose that $\delta _n$ is not decreasing, then there exists $C\in\mathbb{N}$ such that $\delta _C\le \delta _{C+1}$, we have

then by the inequality (2.7), we have

Since for every $n\in\mathbb{N}$, $\delta x_n\ne \delta x_{n-1}$, we have $\delta _{C+1} = d(\delta x_{C+1}, \delta x_{C+2}) > 0$ which follows from the above inequality, we get $h(\delta x_{C}, \delta x_{C+1}) = 1$. By the fact that $h \in \mathcal{A}(X)$, we obtain $d(\delta x_{C}, \delta x_{C+1}) = 0$. It is a contradiction. Therefore, $\delta _n$ is decreasing, $\delta _n > \delta _{n+1}$ for all $n\ge 0$, we have

Since $\delta _n$ is bounded below, the sequence converges. Let

Contrarily, suppose that $C > 0$. Due to the property of $\varphi$, $\lim\limits_{n \to \infty} \varphi(\delta _n) = \varphi(C) > 0$. By (2.7), it is demonstrated that

In the inequality above, when we take $n \to \infty$, we obtain

Therefore, $\lim\limits_{n \to \infty} h(\delta x_{n}, \delta x_{n+1}) = 1$. According to the notion of auxiliary functions,

which contradicts to the assumption. So, $\lim\limits_{n \to \infty} d(\delta x_{n}, \delta x_{n+1}) = 0$.

The sequence $\{\delta x_{n}\}$ has to be Cauchy, as we will demonstrate next. Contrarily, suppose that $\{\delta x_{n}\}$ is not Cauchy. Then there exists $\epsilon > 0$ such that, for all $k \in \mathbb{N}$, there are $n_k, m_k \in \mathbb{N}$ with $n_k > m_k \geq k$ satisfying $n_k$ is the smallest number such that

By Lemma 2.2, this implies

We determine that $(\delta x_{m_k}, \delta x_{n_k})\in E(\mathcal{G})$ for each $k \in \mathbb{N}$ owing to the transitivity property of $E(\mathcal{G})$. As a result,

where

Since $\lim\limits_{n \to \infty} d(\delta x_{n}, \delta x_{n+1}) = 0$, applying the preceding equality with $k \to \infty$ means that

Combining the aforementioned fact with the inequality (2.8), we obtain

As a result, $\lim\limits_{k \to \infty}h(\delta x_{m_k}, \delta x_{n_k}) = 1.$ Then, $\lim\limits_{k \to \infty}d(\delta x_{m_k}, \delta x_{n_k}) = 0$, which contradicts to (2.3). That $\{\delta x_{n}\}$ is Cauchy in $(X, d)$ must therefore be true.

Since $\delta (X)$ is a $d$-closed subset of $(X, d)$, which is complete, it follows that there exists $x, u\in X$ such that $u = \delta x \in \delta (X)$ satisfying

Since assumption $(a)$, by Lemma 2.3 which indicates that $u$ is a coincidence point of $\mu$ and $\delta$. Thus $u\in C(\mu, \delta)$.

Assume that the statement (b) is satisfied. Because of (2.9), we assert that $x$ must be a coincidence point of $\mu$ and $\delta$. On the other hand, suppose that $x$ is not a coincidence point of $\mu$ and $\delta$. As a result, $\mu x \neq \delta x$ and thus $d(\mu x, \delta x) > 0$. Since the triple $(X, d, \mathcal{G})$ has the property $A$, $(\delta x_n, \delta x)\in E(\mathcal{G})$ for all $n\in\mathbb{N}$. Consequently,

Hence,

The definition of $\varphi$ actually proves that

where

In the equation above, when we take $n \to \infty$ and use (2.9), we obtain

By the attribute of $\varphi$, we get

Then, taking $k \to \infty$ in (2.10) gives us that $\lim\limits_{k \to \infty}h(\delta x_{n_k}, \delta x) = 1$. This implies

which is a contradiction. As a result, $\mu x = \delta x$, and we can derive that $\mu$ and $\delta$ have $x$ as one of their coincidence points.

By applying an additional assumption, as in the following theorem, we could reach a stronger conclusion on the presence of a common fixed point.

Theorem 2.8. Let us apply all the notations and requirements from Theorem 2.5. Moreover, suppose additionally that

(6) It is precise to state that $(\delta x, \delta y)$ is in $E(\mathcal{G})$ for any $x, y\in C(\mu, \delta)$ with $\delta x\neq \delta y$.

Consequently, we obtain

Proof. By proving Theorem 2.5, it is sufficient to account for the only if case with the assumption that the statement (6) above holds. There exists an element $x\in X$ such that $\delta x = \mu x$, according to Theorem 2.5.

Initially, let us assume that $y\in X$ is also a coincidence point, i.e., $\delta y = \mu y$. We will show that $\delta x = \delta y$. Contrarily, suppose that $\delta x\neq \delta y$, we have $d(\delta x, \delta y) > 0.$ By statement (6) above, $(\delta x, \delta y)\in E(\mathcal{G})$, which concludes

Due to the property of $\varphi$, $h(\delta x, \delta y) = 1$, we have $d(\delta x, \delta y) = 0.$ It is a contradiction. Therefore, $\delta x = \delta y$.

The next step is to put $x_0 = x$ and utilize the statement (2) from Theorem 2.5 to establish a sequence $\{x_n\}$ such that $\delta x_n = \mu x_{n-1}$ for every $n\in\mathbb{N}$. As $x$ is a coincidence point, we could suppose that $x_n = x$, then $\delta x_n = \mu x$ for each $n\in\mathbb{N}$.

In order for $\delta z = \delta \delta x = \delta \mu x$, allow $z = \delta x$. Note also that $\delta x_n = \mu x = \mu x_{n-1}$ for any $n\in \mathbb{N}$. Therefore,

in $(X, d')$. Furthermore,

because $\mu$ and $\delta$ are $d'$-compatible. This indicates that $\delta \mu x = \mu \delta x$. Thus, $\delta z = \delta \mu x = \mu \delta x = \mu z$ so $z \in C(\mu, \delta)$. Following the proof above, we have $\mu z = \delta z = \delta x = z$. Hence, $z \in Cm(\mu, \delta).$

Theorem 2.9. Let us apply all the notations and requirements from Theorem 2.7. Moreover, suppose additionally that

(6) It is precise to state that $(\delta x, \delta y)$ is in $E(\mathcal{G})$ for any $x, y\in C(\mu, \delta)$ with $\delta x\neq \delta y$.

Consequently, we obtain

To support our main findings, we provide an example.

Example 2.10. Let $X = [0, \infty)\subseteq \mathbb{R}$, and $d, d':X \times X \rightarrow [0, \infty)$ be such that

for all $x, y \in X$ with a constant $L\in (1, \infty)$. We note that $d$ and $d'$ are metrics. Additionally, it is obvious that $d < d'$ by the way we specify our metrics. Then, assume

Moreover, let $\mu :X\to X$ and $\delta :X\to X$ be given by

for all $x\in X$. In order for the pair $(\mu, \delta)$ to be an $(h$-$\varphi)_R$-contraction with regard to $d$, the conditions (1) and (2) must be satisfied, which we shall demonstrate.

First, let $(\delta x, \delta y)\in E(\mathcal{G})$. It is noticeable that $(\mu x, \mu y)\in E(\mathcal{G})$ if $x = y$. In contrast, if $(\delta x, \delta y) \in E(\mathcal{G})$ and $\delta x \leq \delta y$, then $\delta x = x^2$, $\delta y = y^2 \in [0, 1]$ and $x^2 = \delta x \leq \delta y = y^2$. Hence,

and $\mu x, \mu y\in [0, 1]$. Thus, $(\mu x, \mu y)\in E(\mathcal{G})$.

Second, set $\phi(t) = 7t$, and define $h : X\times X \to [0, 1)$ by the following equation.

where

We first note that the function

is positive for $x, y > 0$. In the case of $x > y$, it is easy to see that $\psi(x, y) > 0$. On the other hand, we observe that

Since, the function $\gamma(x) = x-\ln\left(1+\dfrac{x}{7}\right)$ is an increasing function. As a result, we conclude that $\psi(x, y)$ is a positive function, therefore, $h(x, y)$ is also a positive function. It is straightforward to prove that $\phi \in \Phi$ and $h \in \mathcal{A}(X)$. The profile of the function $h(x, y)$ is plotted in Figure 1.

Next, let $x, y \in X$ such that $(\delta x, \delta y)\in E(\mathcal{G})$. If $\delta x = \delta y$, then $x = y$ so that the requirement (2) holds. In the case of $x^2 = \delta x < \delta y = y^2$, it follows that

To obtain the requirements, using $x < y$, we see that

which yields

Consequently, the pair $(\mu, \delta)$ satisfies condition (2).

We will demonstrate that the requirements (1) through (5) of Theorem 2.5 attained in the final part of this example.

(1) $\delta :(X, d') \rightarrow (X, d')$ is obviously continuous, and $\delta (X) = [0, \infty)$ is also $d'$-closed,

(2) It is observable that $\mu (X) = \delta (X) = X$,

(3) The transitivity property $E(\mathcal{G})$ holds,

(4) Because $d < d'$, we shall demonstrate that $\mu :(X, d) \rightarrow (X, d')$ is $\delta$-Cauchy. Assuming $\epsilon > 0$ and a sequence $\{x_n\}$ in $X$ where $\{\delta x_n\}$ is Cauchy in $(X, d)$, there exists $N \in \mathbb{N}$ such that $d(\delta x_n, \delta x_m) < \frac{\epsilon}{L}$ for any $n, m \geq N$. Therefore,

This pertains to $\mu :(X, d) \rightarrow (X, d')$ being $\delta$-Cauchy.

(5) $\mu :(X, d') \rightarrow (X, d')$ is obviously $\mathcal{G}$-continuous. In addition, $\mu$ and $\delta$ are $d'$-compatible since for every sequence $\{x_n\}$ in $X$ with

it has the consequence that $\ln \left(1+\dfrac{x}{7}\right) = x$. This concludes $x = 0$. As $n\to \infty$,

Finally, it is noticeable that $(\delta 0, \mu 0) = (0, 0)\in E(\mathcal{G})$ so $X(\mu, \delta)$ is nonempty. From Theorem 2.5, $C(\mu, \delta)$ is nonempty. In actuality, it is clear that $0 \in C(\mu, \delta)$.

3.   Application to nonlinear two-term fractional differential equations with nonlocal boundary conditions

Numerous scientific studies state that the theory of fractional differential equations has become more popular as a result of its applications in a variety of engineering and scientific fields, for instance, see ^{[10,11,12,13,14]}. Therefore, we apply our findings in this section to investigate the existence of any solutions to curtain Caputo fractional boundary value problems with nonlocal boundary conditions. Multi-term fractional differential equations have recently made significant contributions ^{[15,16,17,18,19,20,21,22,23]}. Motivated by ^[21,22,23], we study the nonlinear two-term fractional differential equations in the following form:

with the nonlocal boundary conditions

where $\alpha,$ $\beta$ are arbitrary real constants with $0\le \beta\le 1 < \alpha \le 2$, and $f:[0, 1]\times \mathbb{R}\to \mathbb{R}$ is continuous. It is worth noting that nonlocal BVPs appear to be more intriguing than local ones due to their greater naturalness and the variety of applications they offer. Additionally, the local conditions $y(0) = 0$ and $y'(1) = 0$ can be considered as the limit case of (3.2) when $\eta\to 1^-$. Here, we provide some scientific publications related to the nonlocal BVPs equations, which are relevant to the developing topic in ^{[19,20,25,26,27]}.

Recalling the definition of the Caputo fractional derivative and its related definitions is necessary before moving on to the outcomes of existence. Let $\alpha$ be a positive real number. The Caputo derivative of fractional order $\alpha$ is defined as follows for a continuous function $y(t)$:

where $\lceil\alpha \rceil$ is the smallest integer which is greater than $\alpha$ and $I^{\alpha}$ is the Riemann-Liouville integral operator of order $\alpha\ge 0$ defined by

Noting that when $\alpha = 0$, the operator $I^0$ is referred to the identity operator and the gamma function $\Gamma$ is defined by

The fractional integral satisfies the following equalities:

Additionally, according to the $\alpha-$order Caputo fractional derivative and its integer-ordered, we get

In order to obtain our goal, we suppose that $y:[0, 1]\to \mathbb{R}$ is a solution of the systems (3.1) and (3.2). We see that

Consequently, we have

This implies that the initial value problem (BVP) (3.1) and (3.2) is equivalent to the Volterra integral equation in a specific type. We have $a_0 = 0$ by applying the boundary conditions $y(0) = 0$. The solution is consequently condensed to

Applying the boundary condition $y(1) = y(\eta)$ allows us to have the coefficient

where the function $G:\mathbb{R}\times\mathbb{R}\to \mathbb{R}$ is defined by

Substituting the value of $a_{1}$ in the expressions for $y(t)$, we get the solution of the BVP (3.1) and (3.2) as the solution of the Volterra integral equation in the following form:

Next, an integral operator is typically used to establish a fixed point problem. In our case, we consider the integral operator $T:C[0, 1]\to C[0, 1]$ defined by

We can observe that the solution of BVP (3.1) and (3.2) is given by $Ty = y$. In order to achieve the existence of the solutions, we let $\xi: \mathbb{R}^2\to \mathbb{R}$, $E(\mathcal{G}) = \{ (u, v)\in \mathbb{R}^2 : \xi(u, v)\ge 0\}$ and consider the following conditions:

(H$_1$) There exists $u_0\in C[0, 1]$ such that $\xi(u_0, T(u_0))\ge 0$ for all $t\in [0, 1]$.

(H$_2$) For all $t \in [0, 1]$ and $u, v\in C[0, 1]$,

(H$_3$) For all $v, u, w\in C[0, 1]$ and $t \in [0, 1]$,

(H$_4$) For any $t\in[0, 1]$ and for all $u, v\in \mathbb{R}$ with $\xi(u, v)\ge 0$, there is a positive constant $L$ such that

Here, we provide the following useful lemma related to the conditions that appeared in the main theorem of this section. The results can be verified straightforwardly, therefore, we leave the proof.

Lemma 3.1. Assume that (H$_1$)–(H$_3$) hold. If $E(\mathcal{G}) = \{ (u, v)\in \mathbb{R}^2 : \xi(u, v)\ge 0\}$, then we have the following:

(1) There exists $u_0\in C[0, 1]$ such that $(u_0, T(u_0))\in E(\mathcal{G})$ for all $t\in [0, 1]$,

(2) For all $t \in [0, 1]$ and $u, v\in C[0, 1]$,

(3) The transitivity property of $E(\mathcal{G})$ holds.

Before going though the existence theorem of the BVP (3.1) and (3.2), we introduce the solution space $C([0, 1])$ equipped by the metric

We note that the metric space $(C[0, 1], d_{\sigma})$ is complete.

Theorem 3.2. Assume that the conditions (H$_1$)–(H$_4$) hold. If $\sigma$ is sufficiently large such that

then $T$ has at least one fixed point $u^*\in (C[0, 1], d_{\sigma})$, which means the BVP (3.1) and (3.2) has at least one solution $u^*\in (C[0, 1], d_{\sigma})$.

Proof. Let $E(\mathcal{G}) = \{ (u, v)\in \mathbb{R}^2 : \xi(u, v)\ge 0\}$. From Lemma 3.1, We have $X(f, g)\neq \varnothing$, $T$ is edge-preserving with regard to $\mathcal{G}$ and $E(\mathcal{G})$ satisfies the transitivity property. Now, in order to demonstrate this, we concentrate on the actual contraction property of $T$. As a result, we begin by the condition (H$_4$) that, for $u, v\in C[0, 1]$ such that $(u, v)\in E(\mathcal{G})$,

By applying the fact that

for $\gamma > 0$, we have

Consequently, we get

which implies

Here, we let $\varphi(t) = t/2$ which is $\varphi\in\Phi$. Hence,

Therefore, by applying the $\sigma$ is sufficiently large such that

then we reach

To this end, we define $h:C[0, 1]\times C[0, 1]\to [0, 1]$ by

Finally, by utilizing Theorem 2.7, we then have $T$ is $(h, \phi)_M$ contraction. It follows that $u^*$ exists in $C[0, 1]$ such that $Tu^* = u^*$ as desired.

Additionally, one can observe the following for $E(\mathcal{G}) = \mathbb{R}^2$:

(H$_1^*$) There is a positive constant $L$ such that

for each $t\in[0, 1]$ and $u, v\in \mathbb{R}$.

The following corollary is provided by Theorem 3.2.

Corollary 3.3. If (H$_1^*$) holds, then the BVP (3.1) and (3.2) has at least one solution $u^*\in C[0, 1]$.

Example 3.4. For $0\le \beta\le 1 < \alpha \le 2$, we consider the following fractional differential equation

with the boundary conditions

Observe that $f(t, y(t)) = L{\sqrt{t} } \left(y(t)-g(t)\right)$, we can have

which yields the confirmation of the condition (H$_1^*$). Consequently, Corollary 3.3 conclusion is applicable, and then the BVP (3.8) and (3.9) has at least one solution on $(C[0, 1], d_{\sigma})$, where

4.   Conclusions

In this study, we investigated the $(h$-$\varphi)_R$ and $(h$-$\varphi)_M$ contractions with two metrics endowed with a directed graph and established the requirements that guarantee the existence of some common fixed points. The obtained results extend and generalize the theorems given in the literature, including ^[31,47,52]. Furthermore, by applying our main results, the existence of solutions to a class of nonlinear two-term fractional differential equations is successfully acquired. The nonlocal boundary conditions are used in the problems, giving new consequences to study and analyze the existence of a solution to the fractional BVPs. Additionally, some examples pertaining to the fixed point theorems and the nonlocal BVPs equations are provided to support our theoretical results. Based on these findings, we shall extend the fixed-point techniques and use them to investigate the existence of solutions to nonlinear fractional equations in other types.

Acknowledgments

This research project was supported by the Fundamental Fund 2023, Chiang Mai University. The authors are grateful to Dr. Phakdi Charoensawan and Dr. Ben Wongsaijai, Department of Mathematics, Chiang Mai University, for their kind help in mathematics correction.

Conflict of interest

No conflict of interest exists.