On a coupled system under coupled integral boundary conditions involving non-singular differential operator

1.   Introduction

Fractional calculus has been given much attention in the last three decades. Because fractional differential equations (FDEs) and fractional differential systems are important tools to describe physical phenomena arising in engineering, physics, economics, and other fields. It is remarkable that arbitrary order derivative is global in nature as compared to classical integer order. This is a great merit of the said area. Because, ordinary differential equations with respect to time cannot describe processes with dynamic long memory. Due to this reason, to describe dynamical memory, it is possible to use the theory of fractional calculus with derivatives and integrals of fractional orders. Also, the concerned field has a variety of significant applications in many other scientific and engineering fields. For instance, author ^[1] has employed parameter estimation for fractional dynamical models arising in biology, modeling multiple electrochemical processes with fractional differential equations ^[2]. The physical understanding of the fractional derivative, the most famous and known one is the continuous time random walk ^[3], the reader should see the detailed theory about the area and applications in ^[4]. Keeping in mind the above mentioned details, it was proved that fractional order derivative in comparison with the classical derivative, has an advantage that the initial values take the same form as that for classical integer order differential equations which is more applicable for mathematical modeling. Mathematicians can create mathematical equations called partial and ordinary differential equations using the theory of the rate of change. In recent decades, certain kinds of differential calculus have received a lot of attention. In order to solve those equations, numerical and analytic methods have been suggested. These mathematical equations have proven to be extremely effective in simulating real-world phenomena. However, these mathematical domains of integro differential calculus have consistently failed to recreate the physical phenomena multiple times due to the complexity of several real-world problems. In addition, fractional derivatives include the memory and genetic effects that play a crucial part in investigations of many real world dynamical problems.Due to these significant applications, researchers have given much attention for investigate various real-world problems/phenomena under the concept of the fractional calculus. Recently some interesting work has been done in the engineering side ^[5,6], rheology ^[7], epidemiology ^[8], physical sciences ^[9], signal and image processing ^[10], etc.

Due to the importance of the said area, scientists and research people have given their attention to studying FDEs from different scenarios including qualitative theory, numerical, and stability analysis. In this regards plenty of work has been published. We can refer only few here like the existence theory of FDEs by fixed point theory ^[11], qualitative results of FDEs via degree theory ^[12], numerical analysis of FDEs via wavelet ^[13], spectral analysis of FDEs ^[14], decomposition technique for FDEs ^[15], etc. Here we remark that derivative with non-integer order has not a unique definition. The concerned operators have been defined in numbers of ways. The two major groups of operators are called the derivatives with powerlaw kernel and the operators with non-singular kernel. It should be kept in mind that from the said definitions the field of fractional calculus has gotten much more popularity from researchers. Because, researchers use freely different operators for the investigations of their models with different methodology and analysis. The first notable definition was given by Reimann-Liuoville which has given proper attention and still is using very well. In 1967, Caputo gave another definition which has gotten much more popularity and in plenty of research work the said operator used. Besides from the two mentioned operators, other definitions have also been given. Caputo and Fabrizio introduced a new definition in 2016 with exponential kernel ^[16]. The said definition has also attracted the attention very well. Atangana and his co-author ^[17] extended the definition of Caputo and Fabrizio further by using Mittag-Leffler type kernel instead of exponential. The said operator has also used very well. The two mentioned operators have used in many papers just for mathematical models or simple boundary value problems.

The qualitative theory is an important aspect to be investigated for FDEs by using different operators. For instance authors ^[18] have studied the existence mild solutions of coupled hybrid fractional order system with Caputo-Hadamard derivatives. In the same way, authors have derived existence theory for fractional order Volterra integro-differential equation with Mittag-Leffler kernel in ^[19]. Authors ^[20], established a detailed analysis for fractional-order nonlinear dynamical systems with general analytic kernels. Also, authors ^[21,22] studied stability analysis for fractional order problems using different analysis. In addition, assessment of the performance of the hyperbolic-NILT method has been used to solve FDEs in ^[23]. In all these mentioned references, authors have applied fixed point theory and different concept of fractional order derivatives to study the qualitative results. Moreover, some recent work we refer also as ^[24,25].

Here it is remarkable that boundary value problems (BVPs) play a significant role in modeling various phenomena in mass heat transfer and mechanical engineering. Therefore, the area of BVPs under the concept of fractional calculus has been explored very well see ^[26]. BVPs of FDEs containing integrals in their conditions have also been studied very well. Because, integral BVPs have numerous applications in applied fields including chemical engineering, blood flow problems, dynamics due to population, underground water flow, and so forth see ^[27]. It should be kept in mind that mentioned studies have been considered under the concept of the usual Caputo or Reimann-Liouville fractional derivative. As we know that fractional order differential and integral operators have numerous definitions from singular to non-singular. Researchers have significantly used the said operators in different studies. Here we refer to some published work as Caputo-Power law operator in^[28], Caputo-Fabrizio in ^[29] and Mittag-Leffler type in ^[30], and other operators in ^[31].

So far we know the area devoted to a coupled system of FDEs under coupled boundary conditions has very rarely been considered for existence and iterative analysis. Although ordinary BVPs under CFD have been considered very well. For instance, authors ^[32] have considered the following Dirichlet boundary conditions under CFD denoted by ${^{CF}\mathbf{D}}$ as

while $\mathrm{f}_1:[c, d]\times\mathrel{R}^2\rightarrow \mathrel{R}$ is a continuous function. By using nonlinear analysis tools, authors have developed sufficient conditions for the existence of a solution to the given problem.

Since the coupled system of coupled BVPs under CFD has not been discussed yet properly. Therefore, it is worth mentioning to derive the existence result for BVPs for fractional differential systems. So the present paper studies a class of coupled systems of FDEs under coupled boundary conditions. We will consider the following coupled system under coupled boundary conditions with $\xi\in [0, 1] = \mathscr{J}$ as

where $\delta_i\in (1, 2]$, and $\mathrm{f}_i:\mathscr{J}\times \mathrel{R}^2\rightarrow \mathrel{R},$ for $i = 1, 2$ are continuous. Also, the boundary functions $\phi, \psi \in L[0, 1].$ By using Perov's and Schauder theorems, sufficient conditions are developed for the uniqueness and existence of a solution. Furthermore, to compute the extremal solution, we establish a monotonic iterative scheme. Two sequences of upper and lower solutions are established in which one is increasing and converges to the upper bound while the other decreases and converges to the lower bound of the iterative solution. Pertinent examples are given to illustrate the results. Some graphical presentations are given to illustrate the behavior of upper and lower solutions.

2.   Basic results

Here we recollect some definitions and results which are needed for our analysis.

Definition 2.1. ^[33] Let $\mathrm{u}$ be a absolutely integrable function on $[0, 1]$, then the CFD for order $\delta_i\in (0, 1]$ is defined as

such that $M(\delta_i)(0) = M(\delta_i)(1) = 1$ satisfying. Further, the derivative for higher order is defined with $\delta\in [0, 1], \ n\geq 1$ for $\mathrm{u}$ a absolutely integrable function on $[0, 1]$ as in ^[34]

Definition 2.2. ^[33] Let $\mathrm{u}$ be absolutely integrable function on $[0, 1]$, then the integral with non-singular kernel for order $\delta_i\in (0, 1]$ is defined as

Lemma 2.1. ^[35] Let $\mathrm{h}$ be absolutely integrable function on $[0, 1],$ if right side vanishes at $t = 0$ of

then solution is given by

Here, we know that a real nonnegative matrix works as a linear operator which acts on suitable Euclidean space. Such a matrix plays an important role in the theory of vector-valued metric spaces and needed in the proof of Perov's fixed point theorem ^[36].

Definition 2.3. If $\mathscr{X}\neq\emptyset$, then a real valued mapping $\rho:\mathscr{X}\times\mathscr{X}\rightarrow \mathrm{R}^n$ is called a matric of vector valued type on $\mathscr{X}$ if the given hypothesis hold:

(1) $\rho(\mathrm{u}, \mathrm{v}) = 0,$ with every $\mathrm{u}, \mathrm{v} \in \mathscr{X},$ and $\rho(\mathrm{u}, \mathrm{v}) = 0$ if and only if $\mathrm{u} = \mathrm{v}.$

(2) $\rho(\mathrm{u}, \mathrm{v}) = \rho(\mathrm{v}, \mathrm{u}),$ with every $\mathrm{u}, \mathrm{v} \in \mathscr{X}.$

(3) $\rho(\mathrm{u}, \mathrm{v}) \leq \rho(\mathrm{u}, \mathrm{w}) + \rho(\mathrm{w}, \mathrm{v}),$ with every $\mathrm{u}, \mathrm{v}, \mathrm{w} \in \mathscr{X}.$

Theorem 2.4. ^[37] Let $(\mathscr{X}, \rho)$ be a complete generalized metric space with metric $\rho$ and let $\mathscr{Q} : \mathscr{X} \rightarrow \mathscr{X}$ be the operator such that

where $\mathscr{M}$ is a square matrix of nonnegative entries such that the spectral radius $\varrho(\mathscr{M}) < 1,$ then $\mathscr{Q}$ has a unique fixed point $\mathrm{u}^*$ and

for every $\mathrm{u}\in \mathscr{X}, \ k\geq 1.$

Remark 1. ^[38] Consider a square matrix $\mathscr{M}$ with four nonnegative entries as

Now if $\mathrm{m}_{11} < 1, \ \mathrm{m}_{22} < 1,$ and using famous Gelfand formula of numerical functional which state that

Then obviously, one has $\varrho(\mathscr{M}) < 1.$

Let $\mathscr{X} = C[0, 1]\times C[0, 1]$ be a Banach space with norm $\|(\mathrm{u}, \mathrm{v})\| = \|\mathrm{u}\|+\|\mathrm{v}\|,$ where $\|\mathrm{u}\| = \max_{\xi\in\mathscr{J}}|\mathrm{u}(\xi)|.$

Theorem 2.5. ^[39] If $\mathscr{X}$ be a Banach space and $\mathbf{S}\neq\emptyset$ be closed, bounded, and convex subset of $\mathscr{X}$, then any operator $\mathscr{Q}: \mathbf{S}\rightarrow \mathscr{S}$ has at least one fixed point.

3.   Main results

Here we present one of our main result.

Lemma 3.1. Consider $\mathrm{x}_1, \mathrm{x}_2\in L[0, 1]$, then the solution of the following linear coupled system under coupled boundary condition

is computed as

Proof. On applying ${_{0}^{CF}\mathbf{I}_{\xi}^{\delta_1}}, \ {_{0}^{CF}\mathbf{I}_{\xi}^{\delta_2}}$ to (3.1), we have

Using $\mathrm{u}(0) = \mathrm{v}(0) = 0,$, we have from (3.3) that $\mathrm{c}_0 = \mathrm{d}_0 = 0.$, Hence (3.3) yields

Now, inview of boundary conditions $\mathrm{u}(1) = \int_0^1 \phi(\eta)\mathrm{v}(\eta)d\eta, \ \mathrm{v}(1) = \int_0^1 \psi(\eta)\mathrm{u}(\eta)d\eta,$ one has from (3.4)

Hence using values from (3.5) in (3.2), we get the required solution.

Corollary 1. Reference to Lemma 3.1, the solution of (1.1) is given by

The given hypothesis need to be hold:

$(\mathrm{H}_1)$ For any $(\mathrm{u}, \mathrm{v}), (\mathrm{\bar{u}}, \mathrm{\bar{v}})\in \mathscr{X},$ and constant $\mathrm{L}_{i} > 0, \ i = 1, 2,$ one has

$(\mathrm{H}_2)$ For $(\mathrm{u}, \mathrm{v})\in \mathscr{X},$ and constant $\mathrm{a}_i, \mathrm{b}_i, \mathrm{c}_i > 0, i = 1, 2,$ one has

Note: For convince, we use $\Omega_1 = \int_0^1 |\phi(\eta)|d\eta, \ \Omega_2 = \int_0^1 |\psi(\eta)|d\eta.$

Theorem 3.1. Under the hypothesis $(\mathrm{H}_1)$, the proposed system (1.1) has at most one solution if there exists a $\mathscr{M}\mathrm{R}^2$ with nonnegative entries given by

such that $\varrho(\mathscr{M}) < 1.$

Proof. Consider a metric $\rho$ on $\mathscr{X}$ as

clearly, $\rho$ is a vector-valued metric on $\mathscr{X}.$ Further, let define the operator $\mathscr{Q}:\mathscr{X}\rightarrow \mathscr{X}$ as

by

Then one has from (3.7) for $(\mathrm{u}, \mathrm{v}), (\mathrm{\bar{u}}, \mathrm{\bar{v}})\in \mathscr{X}$

on rearrangement, from (3.8), one has

and in same fashion, one has

Let use

Therefore in view of (3.11), from (3.9), (3.10), one has

Hence one has from (3.12)

Since in (3.13), $\varrho(\mathscr{M}) < 1$, therefore, all conditions of Theorem 2.4 are received. So the considered problem (1.1) has at most one solution.

Theorem 3.2. Under the hypothesis $(\mathrm{H}_1, \mathrm{H}_2)$, if the operator $\mathscr{Q} = (\mathscr{Q}_1, \mathscr{Q}_2)$ is compact and completely continuous, then the considered problem (1.1) has at least one solution.

Proof. Since the functions $\mathrm{f}_i (i = 1, 2)$ are continuous, therefore the operators $\mathscr{Q}_1, \mathscr{Q}_2$ are also continuous. Hence the operator $\mathscr{Q}$ is also continuous. Assume $(\mathrm{u}_n, \mathrm{v}_n)\in \mathscr{X},$ with $(\mathrm{u}_n, \mathrm{v}_n)\rightarrow (\mathrm{u}, \mathrm{v}), \ \text{as}\ n\rightarrow \infty.$ Hence, for every $\xi \in \mathscr{J},$ one has

Which further implies that

Since $|\mathrm{u}_n-\mathrm{u}|\rightarrow 0,$ and $|\mathrm{v}_n-\mathrm{v}|\rightarrow 0,$ as $n\rightarrow \infty.$ Therefore, by using Lebesgue dominated convergence theorem ^[40], right side of (3.14) goes to zero as $n\rightarrow \infty,$ hence, one has

Also with the same process one has

Therefore, inview of (3.15) and (3.16), one has

Thus $\mathscr{Q}$ is continuous. Now to show that $\mathscr{Q}$ is bounded, let there exists a bounded set

Now using $(\mathrm{H}_2),$ and consider $(\mathrm{u}, \mathrm{v})\in \mathbf{S}$, we have

where we have used $1 < \delta_1 \leq 2$, we have $1\leq 3-\delta_1 < 2.$ Hence one has from (3.18), that

Similarly repeating the same process, one has

Thus, from (3.19), (3.20), one has

Thus (3.21) yields that $\mathscr{Q}$ is bounded. Now the said operator is bounded and continuous. Hence it is uniformly continuous. Now to derive equi-continuity, we take $\xi_1 < \xi_2 \in \mathscr{J}$, and consider

We see in (3.22) that as $\xi_2\rightarrow \xi_1$, the right side goes to zero. Hence, one has

Since $\mathscr{Q}_1$ is continuous bounded so also uniformly continuous, that is

Also, one has

Therefore, we have

Thus $\mathscr{Q}$ fulfills all conditions of Arzelá-Ascoli theorem, thus $\mathscr{Q}$ is compact. Hence, we conclude that problem (1.1) has at least one solution.

4.   Monotone iterative procedure

This part is related to develop the monotone iterative procedure for the problem (1.1).

Definition 4.1. $(\mathrm{u}_0, \mathrm{v}_0)\in \mathscr{X}$ is the lower solution of (1.1), such that

In the same way $(\mathrm{u}^*_0, \mathrm{v}^*_0)\in \mathscr{X}$ is the upper solutions of (1.1), if

Let

Here we define sector as

Recalling, the following comparison theorem:

Lemma 4.1. ^[41] If $\sigma\in C[0, 1], \ 1 < \delta_i\leq 2$ and there exist $\varphi \in C([0, 1], \mathrm{R}),$ such that

then $\mathrm{u}(\xi) \leq 0,$ for all $\xi \in [0, 1].$

Inview of Lemma 4.1, we state the following theorem following ^[42].

Theorem 4.2. If $\varphi_1, \varphi_2 \in C([0, 1], \mathrm{R})$. Assume that $(\mathbf{u}, \mathbf{v}) \in \mathscr{X},$ such that

then $\mathrm{u}(\xi) \leq 0, \ \mathrm{v}(\xi)\leq 0,$ for every $\xi \in \mathscr{J}.$

We need the following assumptions:

$(\mathrm{H}_3)$ If $\mathrm{f}_1(\xi, \mathrm{u}, \mathrm{v})$ is nondecreasing in $\mathrm{v}$ and there exist $\varphi_1 \in C(\mathscr{J}, \mathrm{R}),$ such that

$(\mathrm{H}_4)$ If $\mathrm{f}_2(\xi, \mathrm{u}, \mathrm{v})$ is nondecreasing in $\mathrm{u},$ such that $\varphi_2 \in C(\mathscr{J}, \mathrm{R}),$ that is

Theorem 4.3. Reference to the hypothesis $(\mathrm{H}_1, \mathrm{H}_2, \mathrm{H}_3)$ together with the initial approximation $(\mathrm{u}_0, \mathrm{v}_0)$ and $(\mathrm{u}^*_0, \mathrm{v}^*_0)$ are lower and upper solutions for (1.1) in the sector $\mathbf{G}$, then one has two sequences of monotonic type as $\{(\mathrm{u}_n, \mathrm{v}_n)\}$ and $\{(\mathrm{u}^*_n, \ \mathrm{v}^*_n)\},$ such that $(\mathrm{u}_n, \mathrm{v}_n) \rightarrow (\mathrm{u}_0, \mathrm{v}_0), \ (\mathrm{u}^*_n, \mathrm{v}^*_n)\rightarrow (\mathrm{u}^*_0, \mathrm{v}^*_0) \ {as} \ n\rightarrow \infty$.

Proof. Consider a sequence describe by

As from Theorem 3.2, we know that system (4.4) has a unique solutions $(\mathrm{u}_{n+1}, \mathrm{v}_{n+1}).$ On induction let $n = 0$ in (4.4), one has

For proving $(\mathrm{u}_0, \mathrm{v}_0)\leq (\mathrm{u}_1, \mathrm{v}_1)\leq (\mathrm{u}^*_0, \mathrm{v}^*_0)$ setting $p(\xi) = \mathrm{u}_0(\xi)-\mathrm{u}_1(\xi), \ q(\xi) = \mathrm{v}_0(\xi)-\mathrm{v}_1(\xi)$ in (4.5), one can get

Inview of Theorem 4.2,

which yields that $p(\xi)\leq 0, \ q(\xi)\leq 0.$ Therefore, $\mathrm{u}_0(\xi) \leq \mathrm{u}_1(\xi), \ \mathrm{v}_0(\xi) \leq \mathrm{v}_1(\xi).$ So $(\mathrm{u}_0, \mathrm{v}_0) \leq (\mathrm{u}_1, \mathrm{v}_1), \ \xi \in \mathscr{J}.$ On the same manner, $p(\xi) = \mathrm{u}_1(\xi)-\mathrm{u}^*_1(\xi), \ q(\xi) = \mathrm{v}_1(\xi)-\mathrm{v}^*_1(\xi)$ in (4.5), one can get $(\mathrm{u}_1, \mathrm{v}_1)\leq (\mathrm{u}^*_1, \mathrm{v}^*_1), \ \xi\in\mathscr{J}.$ Moreover, to prove that $(\mathrm{u}^*_1, \mathrm{v}^*_1)\leq (\mathrm{u}^*_0, \mathrm{v}^*_0)$. Let $p(\xi) = \mathrm{u}^*_1(\xi)-\mathrm{u}^*_0(\xi), \ q(\xi) = \mathrm{v}^*_1(\xi)-\mathrm{v}^*_0(\xi)$, obviously one has $\mathrm{u}^*_1(\xi)\leq \mathrm{u}^*_0(\xi), \ \mathrm{v}^*_1(\xi) \leq \mathrm{v}^*_0(\xi),$ which yields $(\mathrm{u}^*_1, \mathrm{v}^*_1)\leq (\mathrm{u}^*_0, \mathrm{v}^*_0).$ Therefore,

Also, inview of induction, for $k > 1$ and set $p(\xi) = \mathrm{u}_k-\mathrm{u}_{k+1}, \ q(\xi) = \mathrm{v}_k-\mathrm{v}_{k+1}$ in the system (4.4), one has

Using Theorem 4.2, one has $p(\xi)\leq 0, \ q(\xi) \leq 0$. Hence, $\mathrm{u}_k \leq \mathrm{u}_{k+1}, \ \mathrm{v}_k \leq \mathrm{v}_{k+1}$ which yields that $(\mathrm{u}_k, \mathrm{v}_k) \leq (\mathrm{u}_{k+1}, \mathrm{v}_{k+1}),$ on the same fashion one can prove that $(\mathrm{u}_{k+1}, \mathrm{v}_{k+1})\leq (\mathrm{u}^*_{k+1}, \mathrm{v}^*_{k+1}).$ The solution of (1.1) is given as system of integral equations as

As $(\mathrm{u}, \mathrm{v})$ is the system of solutions of (4.8). We prove that $(\mathrm{u}_0, \mathrm{v}_0), (\mathrm{u}^*_0, \mathrm{v}^*_0)$ are upper/lower solutions of (1.1). So, let $(\mathbf{u}, \mathbf{v})$ be result different from $(\mathrm{u}_0, \mathrm{v}_0), (\mathrm{u}^*_0, \mathrm{v}^*_0),$ then for some positive integer $n = k$, we take $(\mathrm{u}_k, \mathrm{v}_k)\leq (\mathbf{u}_k, \mathbf{v}_k)\leq(\mathrm{u}^*_k, \mathrm{v}^*_k), \ \xi\in\mathscr{J}.$ Sitting $p(\xi) = \mathrm{u}_{k+1}-\mathbf{u}_k, \ q(\xi) = \mathrm{v}_{k+1}-\mathbf{v}_k,$ then due to Theorem 4.2, one has $p(\xi)\leq 0, q(\xi) \leq 0$, so $\mathrm{u}_{k+1} \leq \mathbf{u}_k, \ \mathrm{v}_{k+1} \leq \mathbf{v}_k$, on $\mathscr{J}$. Analogously, one can $\mathbf{u}_k \leq \mathrm{u}^*_{k+1}, \ \mathbf{v}_k\leq \mathrm{v}^*_{k+1},$ for all positive integer. Thus, $(\mathrm{u}_k, \mathrm{v}_k)\leq (\mathbf{u}_k, \mathbf{v}_k)\leq (\mathrm{u}^*_k, \mathrm{v}^*_k),$ on the application of limit $k\rightarrow \infty$ yields that $(\mathrm{u}_0, \mathrm{v}_0)\leq (\mathbf{u}_0, \mathbf{v}_0)\leq (\mathrm{u}^*_0, \mathrm{v}^*_0),$ which implies that extremal solutions lies in the sector $\mathbf{G}$.

5.   Illustrative examples

Here to testify our results, we give some test problems.

Example 1. Taking the given system under CFD as

From the above system(5.1), we see

Also, $\phi(\xi) = \psi(\xi) = \xi$ are non-negative on $\mathscr{J}.$ Let $(-1, -1) = (\mathrm{u}_0, \mathrm{v}_0),$ $(1, 1) = (\mathrm{u}^*_0, \mathrm{v}^*_0)$ be initial approximation of system of lower and upper solutions respectively. For $(-1, -1) = (\mathrm{u}_0, \mathrm{v}_0)$, we have

Similarly by taking $(1, 1) = (\mathrm{u}^*_0, \mathrm{v}^*_0)$, one can get

Thus it relies that $(-1, -1), \ (1, 1)$ are lower and upper solutions of problem (5.1) respectively. Also,

where $\mathrm{u}_0(\xi)\leq \mathrm{u}_1(\xi)\leq \mathrm{u}_2(\xi)\leq \mathrm{u}^*_0(\xi), \ \mathrm{v}_0(\xi)\leq \mathrm{v}_1(\xi)\leq \mathrm{v}_2(\xi)\leq \mathrm{v}^*_0(\xi)$, thus all the hypothesis of Theorem 4.3 are satisfied. Thus system (5.1) has an extremal system of solutions. Further, in Figure 1, the behavior of extremal solutions has been displayed using the initial iterations.

Moreover, it is easy to verify that $(0, 0)$ is a unique solution of the system (5.1).

Example 2. Consider another test problem as

From the above system (5.2), we see

Note that $\phi(\xi) = \psi(\xi) = \xi^2,$ are non-negative on $\mathscr{J}.$

Taking $(-2, -2) = (\mathrm{u}_0, \mathrm{v}_0)$ and $(2, 2) = (\mathrm{u}^*_0, \mathrm{v}^*_0)$ be initial approximation of lower and upper solutions. Let $(-2, -2) = (\mathrm{u}_0, \mathrm{v}_0)$, then one has

In the same way, from $(2, 2) = (\mathrm{u}^*_0, \mathrm{v}^*_0)$, we get

Hence, $(-2, -2), \ (2, 2)$ justify the conditions for lower and upper solutions of (5.2). Moreover,

where

If we take $(\mathrm{u}_0, \mathrm{v}_0) = (-2, -2), (\mathrm{u}_1, \mathrm{v}_1) = (-1, -1), (\mathrm{u}_2, \mathrm{v}_2) = (1, 1), (\mathrm{u}^*_0, \mathrm{v}^*_0) = (2, 2),$ then clearly condition (5.5) holds. Thus all the conditions of Theorem 4.3 are satisfied. Thus system (5.2) has an extremal system of solutions. Also, it is simple to verify that $(0, 0)$ is a unique exact solution of (5.2). Further, in Figure 2, the behavior of extremal solutions has been displayed using the initial iteration.

Example 3. Consider another test example as

From the above system (5.4), one has

Note that $\phi(\xi) = \psi(\xi) = \sqrt{\xi},$ are non-negative on $\mathscr{J}.$

Taking $(-1, -1) = (\mathrm{u}_0, \mathrm{v}_0)$ and $(1, 1) = (\mathrm{u}^*_0, \mathrm{v}^*_0)$ be initial approximations of lower and upper solutions. Let $(-1, -1) = (\mathrm{u}_0, \mathrm{v}_0)$, then one has

In the same way, from $(1, 1) = (\mathrm{u}^*_0, \mathrm{v}^*_0)$, we get

Hence, $(-1, -1), \ (1, 1)$ satisfy the conditions for lower and upper solutions of (5.4). Moreover,

where

If we take $(\mathrm{u}_0, \mathrm{v}_0) = (-1, -1), \ (\mathrm{u}_2, \mathrm{v}_2) = (1, 1),$ then clearly condition (5.3) holds. Thus all the conditions of Theorem 4.3 are satisfied. Thus system (5.4) has an extremal system of solutions. Also, it is simple to verify that $(0, 0)$ is a unique solution of (5.4). Further, in Figure 3, the behavior of extremal solutions has been displayed using the initial iteration.

6.   Conclusions

This manuscript is related to some mathematical analysis of the coupled system of FDEs under coupled boundary conditions involving integration. The presented system has been studied by using CFD of non-singular nature. We have constructed adequate conditions addressing the existence and uniqueness of at least one solution by using Perov's and Schauder's fixed point theorems. Since exact solutions for nonlinear problems with fractional derivatives are difficult tasks. Up till now, no such a powerful procedure has yet been established to compute exact solutions to every problem of FDEs. Therefore, sophisticated tools are needed use to compute the best approximate solutions to nonlinear problems. Hence, on the application of the monotone iterative technique coupled with the upper and lower solution method, two sequences of upper and lower solutions have been constructed. In this way, extremal solutions have been developed. Also, the procedure has been testified by two pertinent examples. It should be kept in mind that the system under coupled integral BVPs with CFD has not been studied before this using the aforesaid analysis. Also, the behavior of upper and lower solutions has been presented graphically. In the future, such huge analysis can be extended to more BVPs involving fractal-fractional order derivatives and Mittag-Leffler type operators with integral boundary conditions.

Acknowledgements

Authors are thankful to the Prince Sultan University for paying the APC and support through TAS research lab.

Conflict of interest

There does not exist any kind of conflicts of interest.