From the lab to the people: major challenges in the biological treatment of Down syndrome

1.   Introduction

Fractional calculus has proven in recent years to be a helpful method of tackling the complexity of complex systems from various scientific and engineering branches. It involves the generalization of the integer order differentiation and integration of a function to non-integer order, see ^[1]. In recent years, there has been considerable interest in fractional differential equations, with numerous works dedicated to the topic. Notable examples include the books by Benchohra et al. ^[2,3]. The authors of ^[4,5] investigated the qualitative theorems of solutions to diverse fractional differential equations and inclusions about memory effects and predictive behavior arguments.

In a recent publication ^[6], Diaz introduced novel definitions for the special functions ${{\mathsf{k}}}$-gamma and ${{\mathsf{k}}}$-beta. Interested readers can refer to additional sources such as ^[7,8] to delve deeper into this topic. Furthermore, in another work ^[9], Sousa et al. presented the ${\rm{\mathsf{φ}}}$-Hilfer derivative of fractional order and elucidated some crucial properties related to this type of fractional operator. Drawing inspiration from the various papers cited earlier, we have introduced a new extension of the renowned Hilfer fractional derivative ^[10,11,12].

On the other hand, delay differential equations are a type of functional differential equation that arise in various biological and physical applications and often require consideration of variable or state-dependent delays. The study of functional differential equations with delay has garnered significant attention in recent years due to their crucial applications in mathematical models of real-world phenomena. For examples, see ^[4,5] and the references therein.

In ^[13], Krim et al. studied the problem

where ${ }^{\rho} D_{0^{+}}^{{\vartheta_2}}$ is the Katugampola derivative of fractional order ${\vartheta_2} \in(0, 1]$, and

is a continuous function.

Using the Picard operator method, Krasnoselskii fixed point approach, and Gronwall's inequality lemma, Almalahi et al. ^[14] established the existence and stability theories for the problem:

where ${ }^{H} D_{0^{+}}^{{\vartheta_1}, {{\vartheta_2}}; {\rm{\mathsf{φ}}}}(\cdot)$ is the ${\rm{\mathsf{φ}}}$-Hilfer derivative of fractional order ${\vartheta_1} \in(0, 1)$ and type ${{\vartheta_2}} \in[0, 1], I_{0^{+}}^{1-{\vartheta_3}, {\rm{\mathsf{φ}}}}(\cdot)$ is the ${\rm{\mathsf{φ}}}$-Riemann-Liouville integral of fractional order $(1-{\vartheta_3})$,

are prefixed points satisfying $0 < {k}_{1} \leq {k}_{2} \leq \ldots \leq {k}_{{\jmath}} < {\mathfrak{a}_2}$, and $c_{{\jmath}} \in \mathbb{R}, \delta \in C[-{\vartheta_2}, 0]$, the function

is continuous, and

with ${\widehat{\mathfrak{g}}}({\varrho}) \leq {\varrho}, {\vartheta_2} > 0$.

In light of the above studies, we focus on a terminal-valued hybrid problem governed by a nonlinear implicit $({\mathsf{k}}, {\rm{\mathsf{φ}}})$-Hilfer fractional differential equation with mixed-type arguments (retarded and advanced):

where $^{H}_{\mathsf{k}}{\mathcal{D}}_{{\mathfrak{a}_1}+}^{{\vartheta_1}, {\vartheta_2};{\rm{\mathsf{φ}}} }$ is the $({\mathsf{k}}, {\rm{\mathsf{φ}}})$-Hilfer derivative of fractional order ${\vartheta_1} \in(0, {\mathsf{k}})$ and type ${\vartheta_2} \in [0, 1]$ defined in Section 2. Furthermore,

are pre-fixed points satisfying ${\mathfrak{a}_1} < \epsilon_1\leq\ldots\leq\epsilon_{\tilde{n}} < {\mathfrak{a}_2}$, and $\alpha_{\jmath}, {\jmath} = 1, \ldots, {\tilde{n}}$ are real numbers. For each function ${\mathsf{y}}$ defined on $\left[{\mathfrak{a}_1}-\mathsf{d}, {\mathfrak{a}_2}+\tilde{\mathsf{d}}\right]$ and for any ${\varrho}\in ({\mathfrak{a}_1}, {\mathfrak{a}_2}]$, we denote by ${\mathsf{y}}_{\varrho}$ the element defined by

The following are the primary novelties of the current paper:

● Given the diverse conditions imposed on problems (1.1)–(1.4), our study can be seen as both a continuation and a generalization of the studies mentioned above, such as the papers ^[13,14].

● The introduced $({\mathsf{k}}, {\rm{\mathsf{φ}}})$-Hilfer operator serves as an extension, encompassing previously established fractional derivatives such as the Caputo, Hadamard, and Hilfer fractional derivatives already present in the existing literature.

● The number of papers addressing a nonlocal condition combined with retarded and advanced arguments is very limited. Therefore, our work aims to fill this gap in the literature.

● The introduced $({\mathsf{k}}, {\rm{\mathsf{φ}}})$-Hilfer operator serves as an extension, encompassing previously established fractional derivatives such as the Caputo, Hadamard, and Hilfer fractional derivatives already present in the existing literature.

The structure of this paper is as follows: Section 2 presents certain notations and preliminaries about the ${\rm{\mathsf{φ}}}$-Hilfer fractional derivative, the functions ${\mathsf{k}}$-gamma and ${\mathsf{k}}$-beta, and some auxiliary results. Further, we give the definition of the $({\mathsf{k}}, {\rm{\mathsf{φ}}})$-Hilfer type fractional derivative and some essential theorems and lemmas. In Section 3, we present three existence and uniqueness results for the problems (1.1)–(1.4) that are founded on the Banach contraction principle, the Schauder and Krasnoselskii fixed point theorems. In the last section, illustrative examples are provided in support of the results obtained.

2.   Preliminaries

First, we present the weighted spaces, notations, definitions, and preliminary facts that are used in this article. Please refer to ^[3] for all details on these spaces and notations.

Let

and

The Banach space of continuous functions is denoted by $C({ \mathbb{T}}, {{\mathbb R}})$ with the norm

Let $AC^{n}({ \mathbb{T}}, {{\mathbb R}})$, $C^{n}({ \mathbb{T}}, {{\mathbb R}})$ be the spaces of continuous functions, $n$-times absolutely continuous, and $n$-times continuously differentiable functions on ${ \mathbb{T}}$, respectively.

Let

and

be the spaces gifted, respectively, with the norms

Let ${\rm{\mathsf{φ}}}\in C([{\mathfrak{a}_1}, {\mathfrak{a}_2}], \mathbb{R})$ be an increasing function such that ${\rm{\mathsf{φ}}}'({\varrho})\neq 0$, for all ${\varrho}\in { \mathbb{T}}$.

Now, let the weighted Banach space be defined as

where

with the norm

and

with the norm

Next, let us define the Banach space

with the norm

Denote $X_{{\rm{\mathsf{φ}}}}^{p}({\mathfrak{a}_1}, {\mathfrak{a}_2}), \ (1\leq p \leq \infty)$ to the space of each real-valued Lebesgue measurable functions ${\widehat{\mathfrak{g}}}$ on $[{\mathfrak{a}_1}, {\mathfrak{a}_2}]$ such that $\|{\widehat{\mathfrak{g}}}\|_{X_{{\rm{\mathsf{φ}}}}^{p}} < \infty,$ with the norm given as

where ${\rm{\mathsf{φ}}}$ is a non-deceasing and non-negative function on $[{\mathfrak{a}_1}, {\mathfrak{a}_2}]$, such that ${\rm{\mathsf{φ}}}'$ is continuous on $[{\mathfrak{a}_1}, {\mathfrak{a}_2}]$ with ${\rm{\mathsf{φ}}}(0) = 0$.

Definition 2.1. ^[6] The ${\mathsf{k}}$-gamma function is given as

where

and for ${\mathsf{k}} \rightarrow 1$, then

Furthermore the ${\mathsf{k}}$-beta function is defined as follows:

so that

and

Definition 2.2. ^[15] Let

be a non-decreasing function on $({\mathfrak{a}_1}, {\mathfrak{a}_2}]$ and ${\rm{\mathsf{φ}}}^{\prime}({\varrho}) > 0$ be continuous on $({\mathfrak{a}_1}, {\mathfrak{a}_2})$ and ${\vartheta_1} > 0$. The generalized ${\mathsf{k}}$-fractional integral operators of a function $\widehat{\mathfrak{g}}$ of order ${\vartheta_1}$ are defined by

with ${\mathsf{k}} > 0$ and

Additionally, the authors of the work ^[16] extended these operators and defined the generalized fractional integrals by

where $G(z, {\vartheta_1})\in AC[{\mathfrak{a}_1}, {\mathfrak{a}_2}]$.

Theorem 2.3. ^[16] Let ${\vartheta_1} > 0$, ${\mathsf{k}} > 0$, and consider the integrable function ${\widehat{\mathfrak{g}}}$: $\left[{\mathfrak{a}_1}, {\mathfrak{a}_2}\right] \rightarrow \mathbb{R}$. Then ${\mathcal{J}}_{G, {\mathfrak{a}_1}+}^{{\vartheta_1}, {\mathsf{k}}; {\rm{\mathsf{φ}}}}{\widehat{\mathfrak{g}}}$ exists for all ${\varrho} \in\left[{\mathfrak{a}_1}, {\mathfrak{a}_2}\right]$.

Theorem 2.4. ^[16] Let ${\widehat{\mathfrak{g}}}\in X_{{\rm{\mathsf{φ}}}}^{p}({\mathfrak{a}_1}, {\mathfrak{a}_2})$ and take ${\vartheta_1} > 0$ and ${\mathsf{k}} > 0$. Then ${\mathcal{J}}_{G, {\mathfrak{a}_1}+}^{{\vartheta_1}, {\mathsf{k}}; {\rm{\mathsf{φ}}}}{\widehat{\mathfrak{g}}}\in C([{\mathfrak{a}_1}, {\mathfrak{a}_2}], {{\mathbb R}})$.

Lemma 2.5. ^[10,11] Consider ${\vartheta_1} > 0$, ${\vartheta_2} > 0$, and ${\mathsf{k}} > 0$. Then, one has

and

Lemma 2.6. ^[10,11] Let ${\vartheta_1}, {\vartheta_2} > 0$ and ${\mathsf{k}} > 0$. Then, we have

and

Theorem 2.7. ^[10,11] Let $0 < {\mathfrak{a}_1} < {\mathfrak{a}_2} < \infty, {\vartheta_1} > 0, 0\leq {\vartheta_3} < 1$, ${\mathsf{k}} > 0$, and ${\mathsf{y}} \in C_{{\vartheta_3}, {\mathsf{k}}; {\rm{\mathsf{φ}}}}({ \mathbb{T}})$. If

then

Definition 2.8. ($({\mathsf{k}}, {\rm{\mathsf{φ}}})$-Hilfer derivative ^[10,11]) Let

with $n\in\mathbb{N}$, ${ \mathbb{T}} = [{\mathfrak{a}_1}, {\mathfrak{a}_2}]$ an interval such that

and

are two functions such that ${\rm{\mathsf{φ}}}$ is increasing and ${\rm{\mathsf{φ}}}'({\varrho})\neq 0$, for all ${\varrho}\in { \mathbb{T}}$. The $({\mathsf{k}}, {\rm{\mathsf{φ}}})$-Hilfer fractional derivative $^{H}_{\mathsf{k}}{\mathcal{D}}^{{\vartheta_1}, {\vartheta_2};{\rm{\mathsf{φ}}}}_{{\mathfrak{a}_1}+}(\cdot)$ and $^{H}_{\mathsf{k}}{\mathcal{D}}^{{\vartheta_1}, {\vartheta_2};{\rm{\mathsf{φ}}}}_{{\mathfrak{a}_2}-}(\cdot)$ of a function ${\widehat{\mathfrak{g}}}$ of order ${\vartheta_1}$ and type $0\leq {\vartheta_2} \leq 1$, with ${\mathsf{k}} > 0$ is defined by

where

Lemma 2.9. ^[10,11] Let ${\varrho} > {\mathfrak{a}_1},$ ${\vartheta_1} > 0, 0\leq {\vartheta_2} \leq1,$ and ${\mathsf{k}} > 0.$ Thus, for

and one has

Theorem 2.10. ^[10,11] If

where $n\in{{\mathbb N}}$ and ${\mathsf{k}} > 0$, then

where

Particularly, for $n = 1$, one gets

Lemma 2.11. ^[10,11] Let ${\vartheta_1} > 0, 0\leq {\vartheta_2} \leq1,$ and ${\mathsf{y}}\in C^{1}_{{\vartheta_3}, {\mathsf{k}}; {\rm{\mathsf{φ}}}}({ \mathbb{T}})$, where ${\mathsf{k}} > 0$, then for ${\varrho}\in ({\mathfrak{a}_1}, {\mathfrak{a}_2}],$ we have

3.   Main results

We start this section by taking the next fractional differential problem:

such that $0 < {\vartheta_1} < {\mathsf{k}}, 0\leq {\vartheta_2} \leq1,$ subjected to the conditions

where

${\mathsf{k}} > 0$, $\alpha_{\jmath}, {\jmath} = 1, \ldots, {\tilde{n}},$ belong to ${{\mathbb R}}$, $\alpha_{{\tilde{n}}+1} = -1$ and $\epsilon_{\jmath}, {\jmath} = 1, \ldots, {\tilde{n}}+1,$ are pre-fixed points verifying

such that

and where $\delta(\cdot) \in C({ \mathbb{T}}, \mathbb{R})$, ${\chi}(\cdot)\in\mathcal{C}$, ${\psi}\in C\left([{\mathfrak{a}_1}, {\mathfrak{a}_2}], {{\mathbb R}}\backslash\{0\}\right)$, and $\tilde{{\chi}}(\cdot)\in\tilde{\mathcal{C}}$.

Theorem 3.1. The function ${\mathsf{y}}$ verifies (3.1)–(3.4) if and only if

Proof. Assume that ${\mathsf{y}}$ satisfies Eqs (3.1)–(3.4), and by implementing the integral operator ${\mathcal{J}}_{{\mathfrak{a}_1}+}^{{\vartheta_1}, {\mathsf{k}}; {\rm{\mathsf{φ}}}}(\cdot)$ of fractional order $\vartheta_1$ on both sides of (3.1), we have

Using Theorem 2.10, we get

In what follows, by putting ${\varrho} = \epsilon_{\jmath}$ into (3.6), and applying $\alpha_{\jmath}$ to both sides, one gets

By (3.2) and (3.6) with ${\varrho} = {\mathfrak{a}_2}$, we have

which implies

Substituting (3.7) into (3.6), we obtain (3.5).

Now, we show that ${\mathsf{y}}$ verifies Eq (3.5), it follows that it also verifies (3.1)–(3.4). Applying $^{H}_{\mathsf{k}}{\mathcal{D}}_{{\mathfrak{a}_1}+}^{{\vartheta_1}, {\vartheta_2};{\rm{\mathsf{φ}}} }(\cdot)$ on both sides of (3.5), we get

In view of Lemmas 2.9 and 2.11, we find Eq (3.1). Now, taking ${\varrho} = {\mathfrak{a}_2}$ in Eq (3.5), we have

Substituting ${\varrho} = \epsilon_{\jmath}$ into (3.5), we get

Then, we have

and thus,

Then,

From (3.8) and (3.9), we find that

which implies that argument (3.2) holds. □

In sequel, we present the following finding as a consequence of Theorem 3.1.

Lemma 3.2. Let

such that $0 < {\vartheta_1} < {\mathsf{k}}$ and $0\leq {\vartheta_2}\leq1$, and suppose that ${\chi}(\cdot)\in\mathcal{C}$, $\tilde{{\chi}}(\cdot)\in\tilde{\mathcal{C}}$, and

is a continuous function. Then, ${\mathsf{y}}\in\mathbb{F}$ is a solution of problems (1.1)–(1.4) iff ${\mathsf{y}}$ is a fixed point of the mapping ${\Bbbk}$: $\mathbb{F}\rightarrow \mathbb{F}$ defined by

where $\delta$ is a function verifying

and

Next, we present the following hypotheses for using in the sequel analysis:

(Ax1)

is a continuous function.

(Ax2) There exist real numbers $\zeta_1 > 0$ and $0 < \zeta_2 < 1$, where

for any

and ${\varrho}\in ({\mathfrak{a}_1}, {\mathfrak{a}_2}]$.

(Ax3) There exist functions ${\mathfrak{m}}_1, {\mathfrak{m}}_2, {\mathfrak{m}}_3\in C({ \mathbb{T}}, {{\mathbb R}}_+)$ with

such that

for any

and ${\varrho}\in ({\mathfrak{a}_1}, {\mathfrak{a}_2}]$.

(Ax4) The function $\psi$ is continuous on ${\mathbb{T}}$ and there exists ${\mathfrak G} > 0$ such that

Now, we will study the uniqueness theorem for problems (1.1)–(1.4) by utilizing the Banach fixed point technique ^[17].

Theorem 3.3. Suppose that $(\text{Ax1})$, $(\text{Ax2})$, and $(\text{Ax4})$ are satisfied. If

then, problems (1.1)–(1.4) have a unique solution in $\mathbb{F}$.

Proof. In order to prove that the mapping ${\Bbbk}$ given in (3.10) possesses one fixed point in $\mathbb{F}.$ Let us take ${\mathsf{y}}, {\widehat{\mathsf{y}}} \in \mathbb{F}$, thus for any

we have

Thus

Further, for ${\varrho}\in ({\mathfrak{a}_1}, {\mathfrak{a}_2}]$, we have

where $\delta_1$ and $\delta_2$ are functions satisfying the functional equations

By (Ax2), we have

Then,

Therefore, for each ${\varrho}\in ({\mathfrak{a}_1}, {\mathfrak{a}_2}]$,

By Lemma $2.6,$ we have

Hence,

which implies that

Thus,

By (3.12) and (3.13), we obtain

Based on (3.11), the mapping ${\Bbbk}$ is a contraction on $\mathbb{F}$. Therefore, by the Banach fixed point technique, ${\Bbbk}$ owns one fixed point ${\mathsf{y}} \in \mathbb{F}$, which is a unique solution for problems (1.1)–(1.4). □

Our subsequent existence theorem for problems (1.1)–(1.4) will be proved by the Schauder fixed point technique ^[17].

Theorem 3.4. Suppose that $(Ax1)$, $(Ax3)$, and $(Ax4)$ are verified. If

then, problems (1.1)–(1.4) have at least one solution in $\mathbb{F}$.

Proof.

We will split the proof into several steps.

Step 1. The mapping ${\Bbbk}$ is continuous.

Consider $\{{\mathsf{y}}_n\}$ to be a convergent sequence to ${\mathsf{y}}$ in $\mathbb{F}$. For each

we have

For ${\varrho}\in ({\mathfrak{a}_1}, {\mathfrak{a}_2}]$, we have

where $\delta$ and $\delta_n$ are functions satisfying the functional equations

Since ${\mathsf{y}}_n\rightarrow {\mathsf{y}}$, then we get $\delta_n({\varrho})\rightarrow \delta({\varrho})$ as $n\rightarrow \infty$ for each ${\varrho}\in ({\mathfrak{a}_1}, {\mathfrak{a}_2}]$, and since ${\mathfrak{g}}$ is continuous, then we have

Step 2. We show ${\Bbbk}(B_M)\subset B_M$.

Consider $M$ to be a positive real number, where

Now, we present the next closed bounded ball

Then, for each ${\varrho}\in [{\mathfrak{a}_1}-\mathsf{d}, {\mathfrak{a}_1}]$, we have

and for each ${\varrho}\in \left[{\mathfrak{a}_2}, {\mathfrak{a}_2}+\tilde{\mathsf{d}}\right]$, we have

Further, for each ${\varrho}\in ({\mathfrak{a}_1}, {\mathfrak{a}_2}]$, $(3.10)$ implies that

By hypothesis (Ax3), for ${\varrho}\in({\mathfrak{a}_1}, {\mathfrak{a}_2}]$, we have

which implies that

then

Thus for ${\varrho}\in({\mathfrak{a}_1}, {\mathfrak{a}_2}]$, from $(3.15)$ we get

By Lemma $2.6,$ we have

Thus

Then, for each

we obtain

Step 3. We prove that the set ${\Bbbk}({B}_{M})$ is relatively compact.

Let

and let ${{\mathsf{y}}}\in {B}_{M}$. Then,

By Lemma $2.6,$ we get

As $k_1 \rightarrow k_2$, the right side of the above inequality tends to zero. From Steps 1–3, using the Arzela-Ascoli theorem, we infer that $\Bbbk$: $\mathbb{F} \rightarrow \mathbb{F}$ is a continuous and compact mapping. Consequently, we deduce that ${\Bbbk}$ owns at least one fixed point, which is a solution for problems $(1.1)$–$(1.4)$. □

Our third outcome depends on the Krasnoselskii fixed point technique ^[17].

Theorem 3.5. Suppose that $(Ax1)$–$(Ax4)$ are verified. If

then, problems (1.1)–(1.4) have a solution in $\mathbb{F}$.

Proof. Let us assume that the ball

with

{Next, we introduce the mappings} ${\nabla_1}$ and ${\nabla_2}$ on $B_{\omega}$ as follows:

and

where $\delta$ is a function verifying

Then $(3.10)$ can be written as

Step 1. We prove that

for any ${\mathsf{y}}, {\widehat{\mathsf{y}}} \in B_{\omega}.$

By (Ax3) and from $(3.10)$, for ${\varrho}\in({\mathfrak{a}_1}, {\mathfrak{a}_2}]$, we have

which implies that

and then

Thus, for ${\varrho}\in ({\mathfrak{a}_1}, {\mathfrak{a}_2}]$ and by $(3.17)$, we have

Then, for each ${\varrho}\in\left[{\mathfrak{a}_1}-\mathsf{d}, {\mathfrak{a}_2}+\tilde{\mathsf{d}}\right],$ we obtain

For ${\varrho}\in ({\mathfrak{a}_1}, {\mathfrak{a}_2}]$ and by $(3.18)$, we have

For each ${\varrho}\in [{\mathfrak{a}_1}-\mathsf{d}, {\mathfrak{a}_1}]$, we have

and for each ${\varrho}\in \left[{\mathfrak{a}_2}, {\mathfrak{a}_2}+\tilde{\mathsf{d}}\right]$, we have

Then, for each ${\varrho}\in\left[{\mathfrak{a}_1}-\mathsf{d}, {\mathfrak{a}_2}+\tilde{\mathsf{d}}\right]$, we get

From $(3.19)$ and $(3.20)$, for each ${\varrho}\in \left[{\mathfrak{a}_1}-\mathsf{d}, {\mathfrak{a}_2}+\tilde{\mathsf{d}}\right]$, we have

which infers that

Step 2. The mapping ${\nabla_1}$ is a contraction.

In view of the condition $(3.16)$ and Theorem $3.3$, the mapping ${\nabla_1}$ is a contraction on $\mathbb{F}$ with the norm $\|\cdot\|_{\mathbb{F}}$.

Step 3. ${\nabla_2}$ is continuous and compact.

Let $\{{\mathsf{y}}_n\}$ be a sequence such that ${\mathsf{y}}_n\longrightarrow {\mathsf{y}}$ in $\mathbb{F}$. For each

we have

For ${\varrho}\in ({\mathfrak{a}_1}, {\mathfrak{a}_2}]$, we have

such that $\delta$ and $\delta_n$ are functions verifying the functional equations

Since ${\mathsf{y}}_n\rightarrow {\mathsf{y}}$, then we get $\delta_n({\varrho})\rightarrow \delta({\varrho})$ as $n\rightarrow \infty$ for each ${\varrho}\in ({\mathfrak{a}_1}, {\mathfrak{a}_2}]$, and since ${\mathfrak{g}}$ is continuous, then we have

Then ${\nabla_2}$ is continuous. Next we prove that ${\nabla_2}$ is uniformly bounded on $B_{\omega}.$ For each

and any ${\widehat{\mathsf{y}}}\in B_{\omega},$ we get

This implies that the mapping ${\nabla_2}$ is uniformly bounded on $B_{\omega}.$ In order to show the compactness of ${\nabla_2}$, we take ${k}_1, {k}_2\in({\mathfrak{a}_1}, {\mathfrak{a}_2}]$ such that ${k}_1 < {k}_2$, and ${\mathsf{y}}\in B_{\omega}$. Then

By Lemma $2.6,$ we get

Note that

This proves that ${\nabla_2}B_{\omega}$ is equicontinuous on $({\mathfrak{a}_1}, {\mathfrak{a}_2}].$ Therefore, ${\nabla_2}$ is compact. Thus, based on the Krasnoselskii fixed point technique, we conclude that ${\Bbbk}$ possesses a fixed point, which satisfies problems (1.1)–(1.4). □

4.   Applications

We give various examples of (1.1)–(1.4), with

where ${\varrho}\in { \mathbb{T}},$ ${\mathsf{y}}\in C\left(\left[-\mathsf{d}, \tilde{\mathsf{d}}\right], {{\mathbb R}}\right)$, and ${\widehat{\mathsf{y}}}\in{{\mathbb R}}$.

Example 4.1. Taking ${\vartheta_2}\rightarrow \frac{1}{2}$, ${\vartheta_1} = \frac{1}{2}$, ${\mathsf{k}} = 1$, ${\rm{\mathsf{φ}}}({\varrho}) = {\pi}^{\varrho}$, $\alpha_1 = 1$, $\alpha_2 = 2$, $\alpha_3 = 3$, $\epsilon_1 = \frac{5}{4}$, $\epsilon_2 = \frac{4}{3}$, $\epsilon_3 = \frac{3}{2}$, ${\tilde{n}} = 3$, $\mathsf{d} = \tilde{\mathsf{d}} = \frac{1}{3}$, and ${\vartheta_3} = \frac{3}{4}$, we have the system below:

We have

and then

By continuity of the function ${\mathfrak{g}}$, the hypothesis $(Ax1)$ holds. For every

one has

Then, the condition $(Ax3)$ is satisfied with

and

The condition $(Ax4)$ is verified since we have that

We have

Hence, in view of Theorem 3.4, we infer that problems $(4.1)$–$(4.4)$ possess a solution in $\mathbb{F}$.

Example 4.2. Considering ${\vartheta_2}\rightarrow0$, ${\vartheta_1} = \frac{1}{2}$, ${\mathsf{k}} = 1$, ${\rm{\mathsf{φ}}}({\varrho}) = {\varrho}^{\rho}$, $\alpha_1 = 1$, $\alpha_2 = 1$, $\alpha_3 = 5$, $\epsilon_1 = \frac{3}{2}$, $\epsilon_2 = 2$, $\epsilon_3 = \frac{5}{2}$, ${\tilde{n}} = 3$, $\mathsf{d} = \tilde{\mathsf{d}} = \frac{1}{2}$, $\rho = \frac{1}{2}$, and ${\vartheta_3} = \frac{1}{2}$, we have the next system:

We have

and then

Additionally, for every

one has

Then, the condition $(Ax2)$ holds with

Since

Hence, all of the hypotheses of Theorem 3.3 are verified. It follows that problems $(4.5)$–$(4.8)$ possess one solution in $\mathbb{F}$.

5.   Conclusions

Our research considered a class of problems involving nonlinear implicit $({\mathsf{k}}, {\rm{\mathsf{φ}}})$-Hilfer hybrid fractional differential equations with nonlocal terminal conditions. We achieved this by proving the existence and uniqueness of solutions for these equations. Our strategy hinged on powerful mathematical tools: the Banach contraction principle, Schauder's fixed point theorem, and Krasnoselskii's fixed point techniques. To showcase the practical applications of our findings and the ease of using our theorems, we presented some illustrative examples. These illustrations effectively highlight the flexibility and wide-reaching impact of the studied operator across various cases. It is noteworthy that the introduced $({\mathsf{k}}, {\rm{\mathsf{φ}}})$-Hilfer operator operates as an extension, encompassing previously established fractional derivatives such as the Caputo, Hadamard, and Hilfer fractional derivative already present in the existing literature. This broader conceptual framework substantially contributes to the ongoing advancement of fractional calculus, thus laying the groundwork for promising directions of future exploration within this ever-evolving and dynamic domain.

Author contributions

A. Salim: conceptualization, data curation, formal analysis, investigation, methodology, writing-original draft; S. T. M. Thabet: conceptualization, data curation, formal analysis, methodology, writing-original draft; I. Kedim: data curation, formal analysis, investigation, methodology, writing-review and editing; M. Vivas-Cortez: investigation, writing-review and editing. All authors have read and agreed to the published version of the article.

Use of AI tools declaration

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

Acknowledgments

Pontificia Universidad Católica del Ecuador, Proyecto Título: "Algunos resultados Cualitativos sobre Ecuaciones diferenciales fraccionales y desigualdades integrales" Cod UIO2022. This study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2024/R/1446).

Conflict of interest

The authors declare that they have no conflicts of interest.