Solving singular coupled fractional differential equations with integral boundary constraints by coupled fixed point methodology

1.   Background materials

The science of fractional differentiation is related to many engineering disciplines because its basis is based on differential equations that have a long history in chemistry, polymer rheology electrodynamics, physics and aerodynamics. Derivatives of fractional order are also included in mathematical simulations of structures and processes ^[1,2,3]. More broadly, differential equations of fractional order often become means of multiple perspectives on control systems, fluid dynamics, and so on.

Another reason for the importance of studying fractional order differential equations is that the fractional order models are more accurate than the correct order models and they also seem to have a greater degree of freedom. To learn more recent results about this branch, we cite ^{[4,5,6,7,8,9,10,11,12,13]}.

The integral boundary stipulations play a prominent role in many applications such as thermo-elasticity, population dynamics, problems with blood flow and underground water supply. To obtain a full and comprehensive explanation of the terms of integral boundaries, we direct the reader to certain recent publications ^{[14,15,16,17,18,19,20,21,22,23,24]}.

The fixed point technique is one of the final modeling methods for many fields. In engineering, it is used to achieve solutions or search for more effective results. In general, this method has become one of the best methods used in modern mathematics, especially functional analysis. This method relates to the existence, uniqueness and characteristics of a specified operator's fixed points.

One of the very important discoveries of this technique is the Banach contraction principle ^[25], as it contributed greatly to spread after exploring generalized metric areas that were greatly enamored by the authors in the field of fractional differential equations. For further clarification, see ^{[7,10,26,27,28,29]}.

The notion of coupled fixed points (CFPs) was introduced in 1987 by Guo and Lakshmikantham ^[30] and applications to it were recounted by Bhaskar and Lakshmikantham ^[31] who were able to study the monotone property and applied the theoretical results to find a unique solution to periodic boundary value problems (BVPs). In abstract spaces, a lot of authors generalized this concept and obtained pivotal results and more applications. For more details, see ^{[32,33,34,35,36,37]}.

Coupled fixed points are not only an abstract definition but have many vital applications in some models of economics such as equilibrium in duopoly markets and variational principle, for instance, see ^[1,38].

In the framework of b-metric space (bMS), our main aim of this paper is to establish some common CFP results for two rational contractive mappings under mild conditions. The theoretical results are applied to discuss the existence and uniqueness of the solution for a singular coupled fractional differential equation (CFDE) of order $\nu$ in the form of:

where $\Lambda \in C[0, 1]\times C[0, 1]$ and given by $\Lambda (\tau) = \left(z(\tau), w(\tau)\right),$ $\nu \in (3, 4),$ $\alpha \in (0, 2)$, $\ ^{c}\Theta ^{\nu }$ is the Caputo fractional derivative and $\Xi$ may be singular at $z = 0$ and $w = 0.$

2.   Preliminaries

The concept of bMSs initiated by Czerwik ^[40] in the year 1993, as a generalization of ordinary metric spaces. Just it's multiplying the constant $b$ at the right-hand side of the triangle inequality.

Definition 2.1. A $b$-metric on a nonempty set $\Omega$ is a function $\mu _{b}:\Omega \times \Omega \rightarrow \mathbb{R} ^{+}$ such that for all $a_{1}, a_{2}, a_{3}\in \Omega$ and a constant $b\geq 1,$ the hypotheses below hold:

i. $\mu _{b}(a_{1}, a_{2}) = 0$ if and only if $a_{1} = a_{2};$

ii. $\mu _{b}(a_{1}, a_{2}) = \mu _{b}(a_{2}, a_{1});$

iii. $\mu _{b}(a_{1}, a_{3})\leq b\left[\mu _{b}(a_{1}, a_{2})+\mu _{r}(a_{2}, a_{3})\right].$

The pair $\left(\Omega, \mu _{b}\right)$ is called bMS with parameter $b.$

Example 2.1. ^[41] Let $\ell _{p}(0 < p < 1) = \{\{a_{i}\}\in \mathbb{R} :\sum\limits_{i = 1}^{+\infty }\left\vert a_{i}\right\vert ^{p} < \infty \}$ and $\mu _{b}:\ell _{p}\times \ell _{p}\rightarrow \mathbb{R} ^{+}$ be a function described as $\mu _{b}(a_{1}, a_{2}) = \left(\sum\limits_{i = 1}^{+\infty }\left\vert a_{1}^{i}-a_{2}^{i}\right\vert ^{p}\right) ^{\frac{1}{p}},$ where $a_{1} = \{a_{1}^{i}\};$ $a_{2} = \{a_{2}^{i}\}\in \ell _{p}.$ Then $\left(\ell _{p}, \mu _{b}\right)$ is a bMS with $b = 2^{\frac{1}{p}}.$

Example 2.2. ^[41] Let $L_{p}(0 < p < 1)\ $be the space of all real continuous functions $a(\tau),$ $\tau \in \lbrack 0, 1]$ so that $\int\limits_{0}^{1} \left\vert a(\tau)\right\vert ^{p}d\tau \leq \infty$ and $\mu _{b}:L_{p}\times L_{p}\rightarrow \mathbb{R} ^{+}$ be a function described as $\mu _{b}(a_{1}, a_{2}) = \left(\int\limits_{0}^{1}\left\vert a_{1}(\tau)-a_{2}(\tau)\right\vert ^{p}d\tau \right) ^{\frac{1}{p}},$ for each $a_{1}, a_{2}\in L_{p}.$ Then $\left(L_{p}, \mu _{b}\right)$ is a bMS with $b = 2^{\frac{1}{p}}.$

Definition 2.2. ^[41] Let $\left(\Omega, \mu _{b}\right)$ be a bMS, the sequence $\{a_{i}\}$ in $\Omega$ is called:

(i) convergent to $a\in \Omega$ if for each $\varepsilon > 0,$ there exists $Q(\varepsilon)\in \mathbb{N}$ so that $\mu _{r}(a_{i}, a) < \varepsilon$ for all $i\geq Q(\varepsilon)$ and we write $\lim_{i\rightarrow +\infty }a_{i} = a;$

(ii) a Cauchy sequence if for each $\varepsilon > 0,$ there exists $Q(\varepsilon)\in \mathbb{N}$ so that $\mu _{r}(a_{i}, a_{j}) < \varepsilon$ for all $i, j\geq Q(\varepsilon).$

If every Cauchy sequence in $\Omega$ converges in $\Omega,$ then a bMS is called complete.

It should be noted that in a bMS, a convergent sequence has a unique limit and every convergent sequence is Cauchy.

Definition 2.3. ^[42] Assume that $\Upsilon, \Xi :\Omega \times \Omega \rightarrow \Omega$ are two mappings on a bMS $\left(\Omega, \mu _{b}\right),$ the pair $(a_{1}, a_{2})\in \Omega \times \Omega$ is called:

i. a CFP of $\Upsilon$ if $a_{1} = \Upsilon \left(a_{1}, a_{2}\right)$ and $a_{2} = \Upsilon \left(a_{2}, a_{1}\right);$

ii. a coupled coincidence point of $\Upsilon$ and $\Xi$ if $\Upsilon \left(a_{1}, a_{2}\right) = \Xi \left(a_{1}, a_{2}\right)$ and $\Xi \left(a_{2}, a_{1}\right) = \Upsilon \left(a_{2}, a_{1}\right);$

iii. a common CFP of $\Upsilon$ and $\Xi$ if $a_{1} = \Upsilon \left(a_{1}, a_{2}\right) = \Xi \left(a_{1}, a_{2}\right)$ and $a_{2} = \Xi \left(a_{2}, a_{1}\right) = \Upsilon \left(a_{2}, a_{1}\right).$

For the convenience of the reader, we present some definitions and necessary lemmas from the theory of fractional analysis.

Definition 2.4. ^[43] The Caputo derivative of fractional order $\nu > 0,$ $n-1 < \nu < n,$ $n\in \mathbb{N},$ for the function $z(\tau):[0, \infty)\rightarrow \mathbb{R}$ is described as

where $[\nu]$ represents the integer part of the real number $\nu.$

Definition 2.5. ^[43] For a function $z(\tau):[0, \infty)\rightarrow \mathbb{R},$ the Riemann–Liouville fractional integral of order $\nu$ is described as

provided that an integral exists.

Lemma 2.1. ^[44] Consider $\nu > 0,$ then

where $C_{i}\in \mathbb{R},$ $i = 0, 1, 2, ..., n-1,$ $n = [\nu]+1$.

Lemma 2.2. ^[1] If $\nu > 0$ and $\alpha > 0,$ then

$i.$ $^{c}\Theta ^{\nu }\tau ^{\alpha -1} = \frac{\Gamma \left(\alpha \right) }{\Gamma \left(\alpha -\nu \right) }\tau ^{\alpha -\nu -1},$ for $\alpha > n;$

$ii.$ $^{c}\Theta ^{\nu }\tau ^{k} = 0,$ for $k = 0, 1, ..., n-1.$

3.   Common coupled fixed point results

Theorem 3.1. Assume that $\left(\Omega, \mu _{r}\right)$ is a complete bMS with a coefficient $r( = b)\geq 1$ and let the mappings $\Upsilon, \Xi :\Omega \times \Omega \rightarrow \Omega$ satisfy

for all $\rho, a, \sigma, b\in \Omega$ and $\lambda, \tau, \zeta \geq 0$ with $r\lambda +\tau < 1$ and $\lambda +\zeta < 1.$

Then there is a unique common CFP of $\Upsilon$ and $\Xi.$

Proof. Assume that $\rho _{0}, a_{0}\in \Omega$ is arbitrary points. Describe the sequences $\{\rho _{2i+1}\}_{i = 0}^{+\infty }$, $\{a_{2i+1}\}_{i = 0}^{+\infty },$ $\{\rho _{2i+2}\}_{i = 0}^{+\infty }$ and $\{a_{2i+2}\}_{i = 0}^{+\infty }$ as

for all $i = 0, 1, 2, ...$, then by (3.1), we have

this implies that

By following the same approach, we can write

Adding (3.2) and (3.3), we get

where $0 < \rho = \frac{\lambda }{(1-\tau)} < 1.$

Also, we can write

Similarly,

Adding (3.4) and (3.5), we have

Continuing with the same previous approach, we find repeatedly that, for all $i\geq 0,$

Now, if $\mu _{r}(\rho _{i}, \rho _{i+1})+\mu _{r}(a_{i}, a_{i+1}) = \Lambda ^{i},$ then (3.6) is reduces to

For $j > i,$ we get

This proves that $\{\rho _{i}\}$ and $\{a_{i}\}$ are Cauchy sequences in $\Omega.$ The completeness of $\Omega$ leads to there are $\rho, a\in \Omega$ so that $\lim_{i\rightarrow +\infty }\rho _{i} = \rho$ and $\lim_{i\rightarrow +\infty }a_{i} = a.$

Now, we claim that $\rho = \Upsilon \left(\rho, a\right)$ and $a = \Upsilon \left(a, \rho \right).$ Suppose that the contradiction, that is $\rho \neq \Upsilon \left(\rho, a\right)$ and $a\neq \Upsilon \left(a, \rho \right)$ so that $\mu _{r}\left(\rho, \Upsilon \left(\rho, a\right) \right) = \ell _{1} > 0$ and $\mu _{r}\left(a, \Upsilon \left(a, \rho \right) \right) = \ell _{2} > 0.$

Consider

Passing $i\rightarrow +\infty$ in (3.7), we have $\ell _{1}\leq 0,$ which is a contradiction. This conclude that $\mu _{r}\left(\rho, \Upsilon \left(\rho, a\right) \right) = 0,$ i.e., $\rho = \Upsilon \left(\rho, a\right),$ similarly, one can obtain that $a = \Upsilon \left(a, \rho \right).$ It follows similarly that $\rho = \Xi \left(\rho, a\right)$ and $a = \Xi \left(a, \rho \right).$

For uniqueness: Assume that $(\widetilde{\rho }, \widetilde{a})\in \Omega \times \Omega$ is a different common CFP of $\Upsilon$ and $\Xi.$ Then

this yields

By the same manner, one can write

Adding (3.8) and (3.9), one sees that

this leads to, $\left(2-2\lambda -2\zeta \right) \left(\mu _{r}\left(\rho, \widetilde{\rho }\right) +\mu _{r}\left(a, \widetilde{a}\right) \right) \leq 0,$ since $\lambda +\zeta < 1,$ then we have $\left(\mu _{r}\left(\rho, \widetilde{\rho }\right) +\mu _{r}\left(a, \widetilde{a}\right) \right) = 0.$ This is only achieved when $\rho = \widetilde{\rho }$ and $a = \widetilde{a}.$ Therefore, $(\rho, a)$ is a unique common CFP of $\Upsilon$ and $\Xi.$

If we put $\Upsilon = \Xi$ in the above theorem, we get the result below.

Corollary 3.1. Assume that $\left(\Omega, \mu _{r}\right)$ is a complete bMS with a coefficient $r\geq 1$ and let the mapping $\Upsilon :\Omega \times \Omega \rightarrow \Omega$ verifies

for all $\rho, a, \sigma, b\in \Omega$ and $\lambda, \tau, \zeta \geq 0$ with $r\lambda +\tau < 1$ and $\lambda +\zeta < 1.$

Then there is a unique CFP of $\Upsilon.$

The following Corollary is very important in the next section (applications).

Corollary 3.2. Let $\left(\Omega, \mu _{r}\right)$ be a complete bMS with a coefficient $r\geq 1$ and let the mapping $\Upsilon :\Omega \times \Omega \rightarrow \Omega$ verifies

for all $\rho, a, \sigma, b\in \Omega$ and $\lambda \geq 0$ with $r\lambda < 1.$Then $\Upsilon$ has a unique CFP.

Proof. Just put $\Upsilon = \Xi$ and $\tau = \zeta = 0$ in Theorem 3.1, we get the proof.

Theorem 3.2. Suppose that $\left(\Omega, \mu _{r}\right)$ is a complete bMS with a coefficient $r\geq 1$ and let the mappings $\Upsilon, \Xi :\Omega \times \Omega \rightarrow \Omega$ verify

for all $\rho, a, \sigma, b\in \Omega,$ where

and $\lambda, \tau, \zeta \geq 0$ with $r\left(\lambda +\tau +\zeta \right) < 1.$ Then $\Upsilon$ and $\Xi$ have a unique common coupled fixed point.

Proof. Let $\rho _{0}, a_{0}\in \Omega$ be an arbitrary points. Define the sequences $\{\rho _{2i+1}\}_{i = 0}^{+\infty }$, $\{a_{2i+1}\}_{i = 0}^{+\infty },$ $\{\rho _{2i+2}\}_{i = 0}^{+\infty }$ and $\{a_{2i+2}\}_{i = 0}^{+\infty }$ by

for all $i = 0, 1, 2, ...$. Consider

Then

Hence, one can write

Similarly, one can easily prove via assumption $Q_{2}\neq 0$ that

By adding (3.11) to (3.12), we find that

Put $Q_{3} = Q\left(\rho _{2i+2}, a_{2i+2}, \rho _{2i+1}, a_{2i+1}\right) \neq 0,$ then we get

this leads to

Similarly, one can easily prove via assumption $Q_{4} = Q\left(a_{2i+2}, \rho _{2i+2}, a_{2i+1}, \rho _{2i+1}\right) \neq 0$ that

By adding (3.13) to (3.14), we can write

Continuing with the same scenario for all $i\geq 0,$ we get

where $\sigma = \tau +\lambda < 1.$ Now if, $\mu _{r}(\rho _{i}, \rho _{i+1})+\mu _{r}(a_{i}, a_{i+1}) = \varpi _{i},$ then

Suppose that for $j.i\in \mathbb{N} \cup \{0\}$ so that $j > i,$ then

this proves that the two sequences $\{\rho _{i}\}$ and $\{a_{i}\}$ are Cauchy. Because $\Omega$ is complete, then there exist $\rho, a\in \Omega$ so that $\lim_{i\rightarrow +\infty }\rho _{i} = \rho$ and $\lim_{i\rightarrow +\infty }a_{i} = a.$

Now, we show that $\rho = \Upsilon \left(\rho, a\right)$ and $a = \Upsilon \left(a, \rho \right).$ Suppose the opposite, that is $\rho \neq \Upsilon \left(\rho, a\right)$ and $a\neq \Upsilon \left(a, \rho \right)$ so that $\mu _{r}\left(\rho, \Upsilon \left(\rho, a\right) \right) = \ell _{1} > 0$ and $\mu _{r}\left(a, \Upsilon \left(a, \rho \right) \right) = \ell _{2} > 0.$

Consider

Letting $i\rightarrow +\infty$ in (3.15), we get $\ell _{1}\leq r\zeta \mu _{r}\left(\rho, \Upsilon \left(\rho, a\right) \right) = r\zeta \ell _{1},$ this leads to $(1-r\zeta)\ell _{1}\leq 0,$ so either $r\zeta \geq 1,$ this contradicts the condition $r\left(\lambda +\tau +\zeta \right) < 1,$ or $\ell _{1}\leq 0$ and this contradicts the condition $\ell _{1} > 0.$ Thus, in both cases we have a contradiction. This implies that $\mu _{r}\left(\rho, \Upsilon \left(\rho, a\right) \right) = 0,$ i.e., $\rho = \Upsilon \left(\rho, a\right),$ similarly, one can obtain that $a = \Upsilon \left(a, \rho \right).$ It follows similarly that $\rho = \Xi \left(\rho, a\right)$ and $a = \Xi \left(a, \rho \right).$

For uniqueness: Assume that $(\widetilde{\rho }, \widetilde{a})\in \Omega \times \Omega$ is a different common CFP of $\Upsilon$ and $\Xi.$ Then

this implies that

By the same method, one can obtain

By adding (3.16) to (3.17), one can write

this implies that, $\left(2-2\lambda -2\zeta \right) \left(\mu _{r}\left(\rho, \widetilde{\rho }\right) +\mu _{r}\left(a, \widetilde{a}\right) \right) \leq 0,$ since $\lambda +\zeta < 1,$ then we have $\left(\mu _{r}\left(\rho, \widetilde{\rho }\right) +\mu _{r}\left(a, \widetilde{a} \right) \right) = 0.$ This is only holds when $\rho = \widetilde{\rho }$ and $a = \widetilde{a}.$ Therefore, $(\rho, a)$ is a unique common CFP of $\Upsilon$ and $\Xi.$

If we set $\Upsilon = \Xi$ in Theorem 3.2, we get the following result:

Corollary 3.3. Suppose that $\left(\Omega, \mu _{r}\right)$ is a complete bMS with a coefficient $r\geq 1$ and let the mappings $\Upsilon :\Omega \times \Omega \rightarrow \Omega$ verifies

for all $\rho, a, \sigma, b\in \Omega,$ where

and $\lambda, \tau, \zeta \geq 0$ with $r\left(\lambda +\tau +\zeta \right) < 1.$ Then $\Upsilon$ has a unique common CFP.

4.   The existence solution of singular CFDEs

In this part, we discuss the existence and uniqueness of the solution of the nonlinear singular CFDE in the setting of bMSs.

Here, we begin with the proof of the following lemma which demonstrate that Green's function of a FDE with integral boundary conditions.

Lemma 4.1. Given the pair $\left(a, q\right) \in \left(C\left(0, 1\right) \cap L\left(0, 1\right) \right) \times \left(C\left(0, 1\right) \cap L\left(0, 1\right) \right),$ $\nu \in (3, 4),$ $\alpha \in (0, 2),$ such that $\nu \neq \alpha,$ the unique solution of

is

where $\Lambda (\tau) = \left(z(\tau), w(\tau)\right)$ and

Proof. Based on Lemma 2.1, problem (4.1) can be reduced to the equivalent integral equation

Since $(z^{\prime \prime }(0), w^{\prime \prime }(0)) = \left(z^{\prime \prime \prime }(0), w^{\prime \prime \prime }(0)\right) = (0, 0),$ so we get $\Lambda ^{\prime \prime }(0) = \Lambda ^{^{\prime \prime \prime }}(0)$, thus, $C_{2} = C_{3} = 0,$ and we can write

Because $\left(z^{\prime }(0), w^{\prime }(0)\right) = \left(z(1), w(1)\right) = \left(\alpha \int_{0}^{1}z(\theta)d\theta, \alpha \int_{0}^{1}w(\theta)d\theta \right),$ hence $\Lambda ^{^{\prime }}(0) = \Lambda (1) = \alpha \int_{0}^{1}\Lambda (\theta)d\theta,$ then one sees that

So, by (4.1), we conclude that

From the previous equality, we obtain that

By integrating both sides of Eq (4.5) from $0$ to $1,$ one can get

From (4.4) and (4.6), one can write

Thus, we get

Substituting the values of the constants in (4.5) we obtain

and this completes the proof.

The following lemma estimates the green's function $\Im (\tau, \theta)$ of FDE with integral boundary stipulations described in (4.1) on $L_{2}(0, 1).$

Lemma 4.2. Suppose that $\nu \in (3, 4),$ $\alpha \in (0, 2),$ such that $\nu \neq \alpha,$ then for all $\tau, \theta \in (0, 1),$ the Green's functions $\Im (\tau, .)\in L_{2}$ verifies

Proof. When $\nu \in (3, 4),$ $\alpha \in (0, 2),$ such that $\nu \neq \alpha,$ there are two clarifications:

(i) For $0\leq \theta \leq \tau \leq 1,$

(ii) For $0\leq \tau \leq \theta \leq 1,$

It follow from (4.7) and (4.8) that

which yields

Assume that $\Xi (., z(.), w(.))\in L_{2}$ for any $z, w\in C[0, 1]$ and describe a mapping $\Upsilon :C[0, 1]\times C[0, 1]\rightarrow C[0, 1]$ as follows:

where $\theta \rightarrow \Im (\tau, \theta)$ is continuous from $[0, 1]$ to $L_{2}.$ Suppose that $\tau _{n}\in \lbrack 0, 1]$ with $\tau _{n}\rightarrow \tau.$ Because $\Im (\tau, .), \Xi (., z(.), w(.))\in L_{2}$ for any $z, w\in C[0, 1]$ and $\tau \in \lbrack 0, 1],$ therefore $\Im (\tau, .)\Xi (., z(.), w(.))$ is integrable function. Hence, using the Lebesgue dominated convergence theorem, one can write

Similarly

this shows that $\Upsilon \in C[0, 1].$ Thus, the mapping $\Upsilon :C[0, 1]\times C[0, 1]\rightarrow C[0, 1]$ is well-defined.

Suppose that $\mu _{r}:C[0, 1]\times C[0, 1]\rightarrow \mathbb{R} ^{+}$ is described as

Then the pair $\left(\Omega, \mu _{r}\right)$ is complete bMS with coefficient $r = 2.$

Lemma 4.3. Assume that $\Upsilon$ is a mapping described as (4.8) and $z, w\in C[0, 1].$ The pair $\left(z\left(\tau \right), w\left(\tau \right) \right)$ is a solution of BVP (1.1) iff it is a CFP of $\Upsilon.$

Proof. Assume that the form of a solution of BVP (1.1) is $\left(z\left(\tau \right), w\left(\tau \right) \right).$ Then from Lemma 4.1, the unique solution can be described as

and

where $\Im (\tau, \theta)$ defined in (4.2). Thus, $\left(z\left(\tau \right), w\left(\tau \right) \right)$ is a CFP of $\Upsilon.$

Conversely, suppose that $\left(z\left(\tau \right), w\left(\tau \right) \right)$ is a CFP of $\Upsilon.$ Since $\alpha < n-1,$ by Lemma 2.2, we get

Similarly, one can prove that

Hence $(z(\tau), w(\tau))$ verifies problem (1.1), further, it is easy to verify that $\Lambda "(0) = \Lambda "(0) = 0,$ $\Lambda ^{^{\prime }} = \Lambda (1) = \alpha \int_{0}^{1}\Lambda (\theta)d\theta,$ which implies that $(z(\tau), w(\tau))$ is a solution to problem (1.1). Because $\Upsilon$ is continuous and has a CFP $(z(\tau), w(\tau)),$ thus $(z(\tau), w(\tau))$ is a continuous solution for the given BVP.

In order to study the existence and uniqueness solution for the BVP (1.1), we propose the following theorem.

Theorem 4.1. Suppose that $\nu \in (3, 4)$ and

holds foe any $\alpha \in (0, 2),$ $\alpha \neq \nu.$ Let $\Xi (., z(.), w(.))$ be a function in $L_{2}$ for any $z, w\in C[0, 1]$ and for any $z^{\ast }, w^{\ast }\in C[0, 1]$ the inequality below holds

Then the mapping $\Upsilon$ has a unique CFP, which is a unique solution to the BVP (1.1).

Proof. Based on the Cauchy–Schwarz inequality, the mapping $\Upsilon$ given in (4.9) and Lemma 4.2, one can write

Taking the supremum over $[0, 1],$ we have

Hence the contractive stipulation (3.10) of Corollary 3.2 is fulfilled, then the mapping $\Upsilon$ have a unique CFP. Thus, by Lemma 4.3 the BVP (1.1) has a unique solution in $C[0, 1]$.

5.   Supportive examples

This part is devoted to support the theoretical results, where some illustrative examples are presented.

Example 5.1. Assume that $\Omega = \mathbb{R}$ and $\Upsilon, \Xi : \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ are two mappings defined by

for all $a_{1}, a_{2}\in \mathbb{R},$ then $(0, 0)$, $(1, 2)$ and $(2, 1)$ are coupled coincidence points of $\Upsilon$ and $\Xi.$

Example 5.2. Let $\Omega = \mathbb{R}$ and $\Upsilon, \Xi : \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ be two mappings described as

for all $a_{1}, a_{2}\in \mathbb{R},$ then $(0, \frac{\pi }{4})$ and $(\frac{\pi }{4}, 0)$ are coupled coincidence points of $\Upsilon$ and $\Xi.$

Example 5.3. Let $\Omega = \mathbb{R}$ and $\Upsilon, \Xi : \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ be two mappings described as

for all $a_{1}, a_{2}\in \mathbb{R},$ then $(0, 0)$ and $(1, 1)$ are common CFP of $\Upsilon$ and $\Xi.$

Example 5.4. Let $\Omega = [0, \infty).$ Define $\mu _{r}:[0, \infty)\times \lbrack 0, \infty)\rightarrow \mathbb{R} ^{+}$ by

Then $(\Omega, \mu _{r})$ is bMS with $r = 2.$ Define the mappings $\Upsilon, \Xi :\Omega \times \Omega \rightarrow \Omega$ as follows:

To fulfill the rational contractive condition (3.1) of Theorem 3.1, we consider the cases below:

(i) If $\rho > a > \sigma > b,$ then $\Upsilon \left(\rho, \sigma \right) = \frac{\rho -\sigma }{4}$ and $\Xi \left(\rho, \sigma \right) = \frac{\rho -\sigma }{5}.$ Consider

Hence for any value of $\tau$ and $\zeta$ with $\lambda = \frac{1}{8}$ so that $\lambda +\tau < 1$ and $\lambda +\zeta < 1,$ we find that the condition (3.1) holds.

(ii) If $\rho \leq a\leq \sigma \leq b,$ then $\Upsilon \left(\rho, \sigma \right) = 0$ and $\Xi \left(\rho, \sigma \right) = 0.$ It is a trivial case. Thus, all requirements of Theorem 3.1 are fulfilled and $(0, 0)$ is a unique common CFP of $\Upsilon$ and $\Xi.$

Example 5.5. Let $\Omega = \{0, 1\}.$ Consider a $b-$metric $\mu _{r}:\Omega \times \Omega \rightarrow \mathbb{R} ^{+}$ by

Then $(\Omega, \mu _{r})$ is bMS with parameter $r = 2.$ Define the mappings $\Upsilon, \Xi :\Omega \times \Omega \rightarrow \Omega$ by $\Upsilon \left(\rho, \sigma \right) = \frac{\rho \sigma }{4}$ and $\Xi \left(\rho, \sigma \right) = \frac{\rho \sigma }{3},$ for all $\rho, \sigma \in \Omega.$ It is easy to conclude that the stipulation (3.1) of Theorem 3.1 is fulfilled with $\lambda = \frac{3}{8},$ $\tau = \frac{1}{5}$ and $\zeta = \frac{ 2}{5}.$ Hence $(0, 0)$ is a unique common CFP of $\Upsilon$ and $\Xi.$

Example 5.6. Consider the BVP of fractional order below:

where $\Xi$ described as

which is a singular at $z = 0 = w,$ with the stipulations

for each $\Lambda \in \lbrack 0, 1]\times \lbrack 0, 1].$ It is clear that the solution of the differential equation of fractional order (5.1) can be satisfied to fulfill the integral equations below:

and

where $\Im (\tau, \theta)$ is given by

here $\alpha = \frac{1}{3}$ and $\nu = \frac{7}{2},$ which verify the assumption (4.11)$.$ It follows from Lemma 4.2 that

Based on Green's funtion (5.2) and the the related mapping $\Upsilon$ described in (4.9), one can write

where $\lambda < 1$ and

Now, for $z, w, z^{\ast }, w^{\ast }\in \lbrack -1, 1),$ one sees that

By taking the supremum over $\tau \in \lbrack 0, 1]$ and put into account the metric distance (4.10), one can write

By the same manner, one can show the other selections. Thus, by Corollary 3.2, we conclude that the mapping $\Upsilon$ described in (4.9) has a unique CFP. So we expect a unique solution to the BVP (1.1) in $C[0, 1].$

Conflict of interest

The authors declare that they have no conflict of interest.

Acknowledgments

This research project was supported by the Thailand Science Research and Innovation Fund and the University of Phayao (Grant No. FF64-UoE002).

H. A. Hammad would like to thank Sohag University, Egypt. W. Cholamjiak would like to thank University of Phayao, Thailand.