Coupled systems of $\psi$-Hilfer generalized proportional fractional nonlocal mixed boundary value problems

1.   Introduction

In the last years, there has been a significant development in ordinary and partial differential equations involving fractional derivatives. These kind of equations have gained considerable importance due to their application in various sciences, such as physics, biology, economics, mechanics, chemistry, control theory, engineering, signal and image processing, etc ^{[1,2,3,4,5,6,7]}.

Nonlinear coupled systems of fractional order differential equations appear often in investigations connected with disease models ^[8], anomalous diffusion ^[9] and ecological models ^[10]. Unlike the classical derivative operator, one can find a variety of its fractional counterparts such as Riemann-Liouville, Caputo, Hadamard, Erdelyi-Kober, Hilfer, Caputo-Hadamard, etc. Recently, a new class of fractional proportional derivative operators was introduced and discussed in ^[11,12,13]. Then, the concept of Hilfer type generalized proportional fractional derivative operators was proposed in ^[14]. For the detailed advantages of the Hilfer derivative, see ^[15] and a recent application in calcium diffusion in ^[16].

Many researchers studied initial and boundary value problems for differential equations and inclusions including different kinds of fractional derivative operators, for instance, see ^{[17,18,19,20,21]}. In ^[22], the authors studied a nonlocal-initial value problem of order in $(0, 1)$ involving a $\overline{\psi}_{*}$-Hilfer generalized proportional fractional derivative of a function with respect to another function. Recently, in ^[23], the authors investigated the existence and uniqueness of solutions for a nonlocal mixed boundary value problem for Hilfer fractional $\overline{\psi}_{*}$-proportional type differential equations and inclusions of order in $(1, 2]$ of the form

where $\mathbb{D}^{\chi, \beta, \sigma, \psi}_{c+},$ denotes the $\psi$-Hilfer generalized proportional fractional derivative operator of order $\chi\in\{\alpha, \delta_k\},$ $\alpha, \delta_k \in (1, 2]$ and type $\beta \in [0, 1]$, respectively, $\sigma\in (0, 1],$ $\eta_j, \zeta_i, \lambda_k \in \mathbb{R}$ are given constants, $\Upsilon$: $[\mathtt{w}_1, \mathtt{w}_2]\times \mathbb{R} \to \mathbb{R}$ is a given continuous function, $I^{\phi_i; \sigma, \psi}_{\mathtt{w}_1+}$ is the generalized proportional fractional integral operator of order $\phi_i > 0$ and $\xi_j, \theta_i, \mu_k\in (\mathtt{w}_1, \mathtt{w}_2)$, $j = 1, 2, \cdots, m$, $i = 1, 2, \cdots, n$, $k = 1, 2, \cdots, r$, are given points.

In ^[24], the authors discussed the existence of solutions for a nonlinear coupled system of $(k, \psi)$-Hilfer fractional differential equations of different orders in $(1, 2],$ complemented with coupled $(k, \psi)$-Riemann-Liouville fractional integral boundary conditions. In ^[25] a coupled system of Hilfer type generalized proportional fractional differential equations with nonlocal multi-point boundary conditions of the form

is investigated, in which $\mathbb{D}_{\mathtt{w}_1^{+}}^{\delta_{1}, \eta, \sigma}$ and $\mathbb{D}_{\mathtt{w}_1^{+}}^{\delta_{2}, \eta, \sigma}$ are the fractional derivatives of Hilfer generalized proportional type of order $1 < \delta_{1}, \delta_{2} < 2$, the Hilfer parameter $0\leq \eta \leq 1$, $\sigma\in(0, 1]$, $k, k_{1}\in\mathbb{R}$, $\Upsilon_{1}, \Upsilon_{2}$: $[\mathtt{w}_1, \mathtt{w}_2]\times \mathbb{R}\to \mathbb{R}$ are continuous functions, $\mathtt{w}_1\geq 0$, $\theta_{j}, \varepsilon_{i}\in \mathbb{R}$, $\xi_{j}, \lambda_{i}\in (\mathtt{w}_1, \mathtt{w}_2)$ for $i = 1, 2, 3, \cdots, n$ and $j = 1, 2, \cdots, m$. Existence and uniqueness results are proved by applying classical Banach and Krasnosel'skii fixed-point theorems, and the Leray-Schauder alternative.

Very recently in ^[26] the authors established existence and uniqueness results for a class of coupled systems of nonlinear Hilfer-type fractional $\overline{\psi}_{*}$-proportional differential equations equipped with nonlocal multi-point and integro-multi-strip coupled boundary conditions of the form:

where $\mathbb{D}^{\rho_{\kappa}, \varphi_a, \vartheta_*, \overline{\psi}_{*}}_{\mathtt{w}_1+}$ and $\kappa = 1, 2$ denote the Hilfer fractional $\overline{\psi}_{*}$-proportional derivative operator of the order $\rho_\kappa \in (1, 2]$ and type $\varphi_i \in [0, 1]$, $\vartheta_*\in (0, 1],$ $\mathtt{w}_1 < \zeta_j < \xi_i < \eta_i < \mathtt{w}_2,$ $\mathtt{w}_1 < \delta_j < z_i < \epsilon_i < \mathtt{w}_2,$ $\kappa_i, \theta_j, \phi_i, \vartheta_j\in \mathbb{R},$ $i = 1, 2, \cdots, n$, $j = 1, 2, \cdots, m$, $\overline{\psi}_{*}$: $[\mathtt{w}_1, \mathtt{w}_2]\rightarrow \mathbb{R}$ is an increasing function with $\overline{\psi}_{*}'(z)\neq 0$ for all $z\in [\mathtt{w}_1, \mathtt{w}_2]$ and $\Psi_1, \Psi_2$: $[\mathtt{w}_1, \mathtt{w}_2]\times \mathbb{R\times \mathbb{R}} \to \mathbb{R}$ are continuous functions.

In this work, motivated by the above mentioned papers, we study a coupled system of $\psi$-Hilfer generalized proportional sequential fractional differential equations with mixed nonlocal and integro-multi-point boundary conditions, of the form

where ${^{H}D_{\mathtt{w}_1}^{\mathcal{X}, \beta_{\iota}, \rho, \psi} }$ denotes the $\psi$-Hilfer generalized proportional fractional derivative operator of order $\mathcal{X} \in \{\alpha_{1}, \alpha_{2} \}$ with the parameters $\beta_{\iota}, \iota \in \{1, 2\}$, $1 < \mathcal{X} \leq 2,$ $0\leq \beta_{\iota} \leq 1,$ ${^{p}I_{\mathtt{w}_1}^{\mathcal{Y}, \rho, \psi}}$ is a generalized proportional fractional integral operator of order $\mathcal{Y} > 0$, $\mathcal{Y} \in \{\Phi_{j}, \upsilon_{l} \}$, $\lambda_{1}, \lambda_{2}, \eta_{i}, \zeta_{j}, \aleph_{k}, \Theta_{l} \in \mathbb{R}\backslash\{0\}$, $\xi_{i}, \theta_{j}, \varrho_{k}, \vartheta_{l}\in (a, b)$, $i = 1, 2, \cdots, n$, $j = 1, 2, \cdots, m$, $k = 1, 2, \cdots, r$, $l = 1, 2, \cdots, q$ and $\Upsilon_1, \Upsilon_2$: $[\mathtt{w}_1, \mathtt{w}_2]\times\mathbb{R}\times\mathbb{R} \rightarrow \mathbb{R}$ are nonlinear continuous functions.

Here, we emphasize that problem (1.4) is novel, and its investigation will enhance the scope of the literature on nonlocal Hilfer-type fractional $\psi$-proportional systems. Note that, when $\psi(t) = t$, problem (1.4) reduces to a coupled system of Hilfer generalized proportional fractional differential equations with mixed nonlocal multi-point and integro-multi-point boundary conditions; while if $\rho = 1$, reduces to a coupled system of $\psi$-Hilfer fractional differential equations with mixed nonlocal multi-point and integro-multi-point boundary conditions. If $\psi(t) = t$, $\zeta_j = 0$, $\Theta_l = 0$, problem (1.4) is reduced to problem (1.2).

In solving (1.4), we first convert it into an equivalent fixed point problem, with the help of an auxiliary result based on a linear variant (1.4). Afterward, under different assumptions, we apply different fixed point theorems to establish our results on existence and uniqueness of solutions. For the first result (Theorem 3.1), we apply the Leray-Schauder's alternative to show that there exists at least one solution for the problem (1.4). The second result (Theorem 3.2), relying on Krasnosel'skii's fixed point theorem, shows that the problem (1.4) has at least one solution under different assumptions, and the last result (Theorem 3.3), shows the existence of a unique solution to the problem (1.4) by means of Banach's contraction mapping principle. In Section 4, we illustrate all the obtained theoretical results with the aid of constructed numerical examples. We emphasize that the problem (1.4) is novel and its investigation will enhance the scope of the literature on coupled systems of $\psi$-Hilfer generalized proportional fractional differential equations with mixed nonlocal and integro-multi-point boundary conditions. The used method is standard, but its configuration in the problem (1.4) is new.

The structure of the rest of the paper is organized as follows: In Section 2, some necessary definitions and preliminary results related to our problem are presented. Section 3 contains the main results for the problem (1.4), while numerical examples illustrating these results are constructed in Section 4. A brief conclusion closes the paper.

2.   Preliminaries

In this section, we introduce some necessary definitions and preliminary results needed in main results later.

Definition 2.1. ^[11,12] Let the functions $\vartheta_{0}, \vartheta_{1}$: $[0, 1]\times \mathbb{R} \rightarrow [0, \infty)$ be continuous such that for all $t\in \mathbb{R}$ and for $\rho \in [0, 1]$, we get

and

Let also $\psi(t)$ be a strictly positive increasing continuous function. So, the proportional differential operator of order $\rho$ of function $\Upsilon(t)$ with respect to function $\psi(t)$ is defined by

Moreover, if $\vartheta_{0}(\rho, t) = \rho$ and $\vartheta_{1}(\rho, t) = 1- \rho$, then operator ${^{p}D^{\rho, \psi}}$ becomes

The integral corresponding to the above proportional derivative is defined as

where

The generalized proportional integral of order $n$ corresponding to proportional derivative ${^{p}D^{n, \rho, \psi}} \Upsilon_1(t),$ is given by

Based on the generalized proportional integral of order $n$, we can obtain the following general proportional fractional integral and derivative.

Definition 2.2. ^[11,12] Let $\rho \in (0, 1]$ and $\alpha > 0.$ The fractional proportional integral of order $\alpha$ of the function $f$ with respect to function $\psi$ is defined by

Definition 2.3. ^[11,12] Let $\rho \in (0, 1]$ $\alpha > 0$ and $\psi(t)$ is a continuous function on $[\mathtt{w}_1, \mathtt{w}_2],$ $\psi'(t) > 0.$ The generalized proportional fractional derivative of order $\alpha$ of the function $\Upsilon$ with respect to function $\psi$ is defined by

where

Below we present the generalized proportional fractional derivatives of order $\alpha$ of function $\Upsilon$ with respect to another function $\psi$ in Hilfer sense.

Definition 2.4. ^[27] For $\rho\in(0, 1].$ Let functions $\Upsilon, \psi \in C^{m}([\mathtt{w}_1, \mathtt{w}_2], \mathbb{R})$ and $\psi$ be positive and strictly increasing with $\psi'(t) \not = 0,$ for all $t\in[\mathtt{w}_1, \mathtt{w}_2]$. The $\psi$-Hilfer generalized propotional fractional derivative of order $\alpha$ and type $\beta$ for $\Upsilon$ with respect to another function $\psi$ is defined by

where order $n-1 < \alpha < n$ and $0\leq\beta\leq 1.$

If $\gamma = \alpha+\beta(n-\alpha),$ then the $\psi$-Hilfer generalized proportional derivative $^{H}D^{\alpha, \beta, \rho, \psi}_{\mathtt{w}_1}$ is equivalent to

Lemma 2.1. ^[27] Let $n-1 < \alpha < n\in \mathbb{N},$ $0 < \rho\leq 1,$ $0\leq \beta\leq 1$ and $n-1 < \gamma < n$ such that $\gamma = \alpha +n\beta -\alpha\beta.$ If $\Upsilon\in C([\mathtt{w}_1, \mathtt{w}_2], \mathbb{R})$ and $I^{(n-\gamma, \rho, \psi)}_{\mathtt{w}_1} \in C ^{n}([\mathtt{w}_1, \mathtt{w}_2], \mathbb{R}),$ then

Lemma 2.2. Let $1 < \alpha_{1}, \alpha_{2} < 2,$ $0\leq \beta_{1}, \beta_{2}\leq 1,$ $\gamma_{i} = \alpha_{i} +\beta_{i}(2 - \alpha_{i}),$ $i = 1, 2$, $\ddot{\Lambda} \ne 0$ and $z, w \in C([\mathtt{w}_1, \mathtt{w}_2], \mathbb{R})$. Then the pair $(k_1, k_2)$ is the solution of the coupled system

if and only if

and

where $\ddot{\Lambda} = X_1Y_1 -X_2Y_2,$

Proof. Let the pair $(k_1, k_2)$ be the solution of the system (2.1). We take the Riemann-Liouville integrals to Eq (2.1),

Then, applying Lemma 2.1 with $n = 2$ to Eq (2.5), we get

and

where

From the conditions $k_1(\mathtt{w}_1) = 0$ and $k_2(\mathtt{w}_1) = 0$ we get $c_1 = 0$ and $c_3 = 0,$ since $\gamma_1\in [\alpha_1, 2]$ and $\gamma_2\in [\alpha_2, 2]$ (see ^[27]), and Eqs (2.6) and (2.7) are reduced to

From the boundary conditions

and

we get

and

From Eqs (2.10) and (2.11), by using the notations (2.4) we get the system

where

By solving the above system, we obtain the constants $c_{0}$ and $c_{2}$ as

Now substitute the values of $c_0$ and $c_2$ into Eqs (2.8) and (2.9) and yield Eqs (2.2) and (2.3), as desired. We can prove the converse of the lemma by direct computation. □

3.   Existence results

In this section, we prove the existence and uniqueness results for the problem (1.4) by using three fixed ponit theorems.

First, we defined the spaces

with the norm

and

with the norm

Then it is well known that $(X, \| \cdot \|)$ and $(Y, \| \cdot \|)$ are Banach apaces. Obviously, the product space of $X \times Y$ endowed with norm $\| (k_1, k_2) \| = \|k_1\| +\|k_2\|$ for $(k_1, k_2) \in X\times Y$ is a Banach space.

In view of the Lemma 2.2, we define an operator $T$: $X\times Y \rightarrow X\times Y$ by

where

and

Then, we introduce the following notation for computational convenience.

Notation 3.1. Let $\mathcal{A}_{i}, \mathcal{B}_{i}, \mathcal{C}_{i}$ for $i = 1, 2$ be the constants:

Now we prove our first existence result via Leray-Schauder alternative ^[28].

Theorem 3.1. Let $\Upsilon_1, \Upsilon_2$: $[\mathtt{w}_1, \mathtt{w}_2] \times \mathbb{R}^{2}\to \mathbb{R}$ be continuous functions. Suppose that:

$(H_1)$ There exist $u_i, v_i \geq 0$ for $i = 1, 2$ and $u_0, v_0 > 0$ such that for each $k_1, k_2\in \mathbb{R},$ $t\in [\mathtt{w}_1, \mathtt{w}_2],$

If

and

where the constants $\mathcal{A}_{i}, \mathcal{B}_{i}, \mathcal{C}_{i}$ for $i = 1, 2$ are defined in the Notation 3.1, then the problem (1.4) has at lest one solution on $[\mathtt{w}_1, \mathtt{w}_2].$

Proof. Since $f$ and $g$ are continuous functions, then $T$ is a continuous operator. We prove that $TB_r$ is uniformly bounded, where $B_r$ is the closed ball

For all $(k_1, k_2) \in B_r,$ by $(H_1)$ we have

and similarly

So, for any $k_1, k_2 \in B_r,$ we have

and hence

In the same way, we can obtain that

Consequently,

Therefore, the set $TB_r$ is uniformly bounded.

Next, it is proven that $TB_r$ is equicontinuous. Let $(k_1, k_2) \in B_r$ and $t_1, t_2 \in [\mathtt{w}_1, \mathtt{w}_2]$ with $t_1 < t_2.$ Then, we have

Then, we obtain that

when $t_2 \rightarrow t_1,$ independently of $k_1$ and $k_2.$ Similarly,

as $t_2 \rightarrow t_1.$ Therefore $TB_r$ is equicontinuous on $[\mathtt{w}_1, \mathtt{w}_2].$ From the above three steps and Arzel$\acute a$-Ascoli theorem, we conclude that $T$ is completely continuous.

Let

We prove that $U$ is bounded. Let

be any solution of $(k_1, k_2) = \mu T(k_1, k_2).$ For each $t\in [\mathtt{w}_1, \mathtt{w}_2]$, we have

Then

and thus

Similarly, we have

Thus we obtain

This imply that,

where

Then, the set $U$ is bounded. Therefore, by Leray-Schauder alternative the problem (1.4) has at least one solution on $[\mathtt{w}_1, \mathtt{w}_2].$ □

Now, we prove the second existence of results by applying Krasnosel'skii point theorem ^[29].

Theorem 3.2. Suppose $\Upsilon_1, \Upsilon_2$: $[\mathtt{w}_1, \mathtt{w}_2] \times \mathbb{R}^{2} \rightarrow \mathbb{R}$ are continuous functions. In addition we assume that:

$(H_2)$ There exist positive functions $\varphi_1, \varphi_2 \in C([\mathtt{w}_1, \mathtt{w}_2], \mathbb{R}^{+}),$ such that

If

then, the problem (1.4) has at least one solution on $[\mathtt{w}_1, \mathtt{w}_2].$

Proof. First, we separate the operator $T$ as

with

We claim that $TB_r \subset B_{r}$ where

We set

for $i = 1, 2$ and choose

Let $(k_1, k_2), (\bar k_1, \bar k_2)\, \in B_{r}.$ Acconding the proof of Theorem 3.1, the following inequalities are obtained

and therefore

Hence,

and

Consider the operators $T_{11}$ and $T_{21}.$ By continuity of $f$ and $g,$ $T_{11}$ and $T_{21}$ are continuous operators. For any $(k_1, k_2) \in B_r,$ we have

In a similar way, we can get

Hence, we obtain that

which yields that $(T_{11}, T_{12})B_r$ is uniformly bounded. For any $t_{1}, \, t_{2} \in [\mathtt{w}_1, \mathtt{w}_2],$ $t_2 > t_1$ and for all $(k_1, k_2) \in B_r$ the operators $(T_{11}, T_{12})B_r$ are equicontinuous by the proof of Theorem 3.1.

Lastly, it is proven that the operators $T_{12}$ and $T_{22}$ are contraction mappings. For all $(k_1, k_2),$ $(\bar k_1, \bar k_2) \in B_{r},$ we have:

Additionally, we also obtain that

Combining the above inequalities, we have

We have $(T_{11}, T_{21})$ is a contraction. Consequently, by Krasnosel'skii's fixed point theorem, the problem (1.4) has at least one solution on $[\mathtt{w}_1, \mathtt{w}_2].$ □

Banach's fixed point theorem ^[30] is applied to obtain our uniqueness and existence result.

Theorem 3.3. Let $\Upsilon_1, \Upsilon_2$: $[\mathtt{w}_1, \mathtt{w}_2] \times \mathbb{R}^{2} \rightarrow \mathbb{R}$ such that:

$(H_3)$ There exist positive constants $L_1, L_2,$ such that, for all $t\in [\mathtt{w}_1, \mathtt{w}_2]$ and $o_i, \bar{o}_i \in \mathbb{R},$ $i = 1, 2,$ we have

Then, the problem (1.4) has a unique solution, provided that

Proof. Let

and

with

For all $(k_1, k_2) \in B_r$ and $t\in [\mathtt{w}_1, \mathtt{w}_2].$ By applying $(H_3),$ we obtain the following inequalities

We have

Thus,

We also have

Therefore

Hence, $T(B_r) \subset B_r.$ Now, it is shown that $(T_1, T_2)$ is a contraction mapping. For all $(k_1, k_2), (\bar k_1, \bar k_2)\in B_r,$ we have

By a similar argument, we have

Hence, we obtain that

By assumption (3.4), the operator $T$ is a contraction mapping. By Banach's fixed point theorem the problem (1.4) has a unique solution. □

4.   Examples

Example 4.1. Consider the following coupled system of $\psi$-Hilfer genneralized proportional fractional differential equation,

Here, we take

From the above data, we obtain

${\rm{(i)}}$ In order to illustrate Theorem 3.1, consider the functions $f$ and $g,$ defined by

Then, we have

Thus, $(H_1)$ is satisfied with

Then

and

Thus, all assumptions of Theorem 3.1 are satisfied. Hence, the proplem (4.1), with $f$ and $g,$ given by (4.2), has at least one solution on $[1/10, 3].$

${\rm{(ii)}}$ Consider now the following functions $f$ and $g$,

It is obvious to check that the above functions satisfy

Then we find that $\mathcal{C}_1+\mathcal{C}_2 \approx0.4610 < 1.$ Hence, by Theorem 3.2, the coupled system (4.1), with $f$ and $g,$ given by (4.3), has at lest one solution on an interval $[1/10, 3]$.

$\text{(iii)}$ To illustrate Theorem 3.3, we consider the functions $f$ and $g$ as

We have

and therefore the Lipschitz condition for $f$ and $g$ is satisfied with $L_1 = 1/10$ and $L_2 = 1/9.$ In addition, we find that

Thus, by Theorem 3.3, problem (4.1), with $f$ and $g,$ given by (4.4), has a unique solution on $[1/10, 3].$

Example 4.2. We investigate the behavior of solutions by replacing the values of proportional constant $\rho$ by $0.1, 0.2, \cdots, 0.9,$ in the following coupled linear system of $\psi$-Hilfer generalized proportional fractional differential equations of the form:

Here, we set

By using integrating factor technique, we can obtain

where $\gamma_{1} = 7/4$, $\gamma_{2} = 31/20$ and

with the canstants $M, N, O, P, Q, R, S$ and $T$ are defined by

For finding the analytic solutions, we use two constants $c_1$ and $c_3$ from Table 1 and substitute them in Eqs (4.6) and (4.7), respectively.

Next, using the Matlab program, we can find the approximate analytical solutions of $k_1(t)$ and $k_2(t)$ with different values of $\rho$ as $0.1, 0.2, 0.3, \cdots, 0.9$. Two graphs of $k_1(t)$ and $k_2(t)$ can be drawn.

From Figure 1, if the value $\rho$ is increasing then the value of $k_1(t)$ is decreasing for each point $t\in [0, 1]$. From the Figure 2, we see that if the value of $\rho$ increases, then the value of $k_2(t)$ decreases for each $t\in [0, 1]$. The lower and upper bounds for the above two curves correspond to $\rho = 0.1$ and $\rho = 0.9$, respectively, when the value of $t$ increases.

5.   Conclusions

In this paper, we investigated a coupled system of $\psi$-Hilfer fractional proportional differential equations supplemented with nonlocal integro-multipoint boundary conditions. We rely on standard fixed point theorems, Banach, Krasnosel'skii and Leray-Schauder alternative to establish the desired existence and uniqueness results. The obtained theoretical results are well illustrated by numerical examples. Our results are new and contribute significialy to the existing results in the literature concerning $\psi$-Hilfer fractional proportional nonlocal integro-multi-point coupled systems.

Our results are novel and contribute to the existing literature on nonlocal systems of nonlinear $\psi$-Hilfer generalized fractional proportional differential equations. Note that the results presented in this paper are wider in scope and produced a variety of new results as special cases. For instance, fixing the parameters in the nonlocal integro-multi-point $\psi$ Hilfer generalized proportional fractional system in (1.4), we obtained some new results as special cases associated with the following:

● Nonlocal $\psi$-Hilfer generalized proportional fractional systems of order in $(1, 2]$ if $\zeta_j = 0, \; \Theta_{l} = 0,$ $j = 1, 2, \cdots, m,$ $l = 1, 2, \cdots, q.$

● Integro-multi-point nonlocal $\psi$-Hilfer generalized proportional fractional systems of order in $(1, 2]$ if $\eta_i = 0, \; \aleph_{k} = 0,$ $i = 1, 2, \cdots, n,$ $k = 1, 2, \cdots, r.$

● Nonlocal Integro-multi-point Hilfer generalized proportional fractional systems of order in $(1, 2]$ if $\psi (t) = t.$

● Integro-multi-point nonlocal Hilfer generalized fractional systems of order in $(1, 2]$ if $\rho = 1.$

Furthermore, additional new results can be recorded as special cases for different combinations of the parameters $\zeta_j, \Theta_{l}$, $j = 1, 2, \cdots, m$, $l = 1, 2, \cdots, q$, $\eta_i = 0$, $\aleph_{k} = 0$, $i = 1, 2, \cdots, n$, $k = 1, 2, \cdots, r$ involved in the system (1.4). For example, by taking all values where $\eta_j = 0$, $j = 1, 2, \cdots, m,$ we obtain the results for a coupled system of nonlinear $\psi$-Hilfer generalized proportional fractional differential equations supplemented by the following nonlocal boundary conditions:

Use of AI tools declaration

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

Acknowledgments

This research was funded by National Science, Research and Innovation Fund (NSRF) and King Mongkut's University of Technology North Bangkok with Contract No. KMUTNB-FF-66-11.

Conflict of interest

The authors declare that there are no conflicts of interest.