Theoretical analysis of a class of $\varphi$-Caputo fractional differential equations in Banach space

1.   Introduction

The present paper is devoted to analyzing the following problem with a constant coefficient $\varpi > 0$ of the form:

where $1 < \alpha < 2$, ${}^{c} \mathcal{D}^{\theta; \varphi}_{a^+}$ is the Caputo fractional derivatives concerning $\varphi$ of order $\theta \in \{\alpha, \alpha -1\}$, $g :I \times \mathbb{X} \to \mathbb{X}$ is a function satisfying some hypotheses that will be precise later and $(\mathbb{X}, \Vert \cdot \Vert)$ is a real Banach space.

The study of differential equations involving integer or non-integer order derivatives has emerged as a pivotal tool for modeling complex phenomena across diverse scientific and engineering domains. Extensive exploration by various authors, as reflected in references such as ^[1,2,3,4], underscores the multifaceted nature of this theoretical framework. In contemporary research, the incorporation of non-integer order derivatives, particularly through the $\varphi$-Caputo type introduced in ^[5], has gained prominence in concrete modeling. Applications range from anomalous diffusions, including ultra-slow processes ^[6], to financial models such as the Heston model ^[7], random walks ^[8], financial crises ^[9], and the Verhulst model ^[10]. The focus has intensified on both quantitative and qualitative properties of solutions for differential problems governed by $\varphi$-Caputo type derivatives, as evident in works like ^[11,12,13]. In this context, the present research seeks to advance the findings from ^[14] to a more general setting. Specifically, the investigation addresses scenarios where nonlinear forcing terms operate within infinite-dimensional Banach spaces. Employing a Weissinger type fixed point theorem, the study strategically sidesteps certain additional hypotheses identified in [14, Theorem 7]. Furthermore, through the integration of fixed point techniques and the measure of noncompactness (MNC) under specific growth and compactness assumptions, the research extends the existence result initially established in [14, Theorem 6]. This comprehensive approach aims to deepen the understanding of differential equations with non-integer order derivatives, offering insights into their behavior in complex and infinite-dimensional settings ^[15].

The current paper is divided into four sections: We collect in Section 2 the basic background needed in the remainder of the paper. In Section 3, Weissinger's and Meir-Keeler's fixed point theorems are used to obtain a new existence criterion. Finally, two illustrative examples are presented.

2.   Preliminaries

Throughout this paper, we endow the space $C(\mathfrak{J}, \mathbb{X})$ of continuous functions $z:\mathfrak{J}\to \mathbb{X}$ by the norm

$L^1(\mathfrak{J}, \mathbb{X})$ denotes the Banach space of Bochner integrable functions $z:\mathfrak{J}\to \mathbb{X}$ normed by

Set

Letting $\varphi \in \mathbb{S}_{a^+}^{1, +}(\mathfrak{J}, \mathbb{R})$ for $s, t \in \mathfrak{J}$, $(s < t)$, we define

Definition 2.1. ^[16] The Mittag-Leffler function $\mathbb{M}_\alpha(\cdot)$ is given by

where $\Gamma(\cdot)$ is the gamma function.

Definition 2.2. ^[5,17] The $\varphi$-fractional integral of a function $f$ of order $\alpha > 0$ is given by

with $\varphi\in \mathbb{S}_{a^+}^{1, +}(\mathfrak{J}, \mathbb{R})$.

Lemma 2.1. ^[5,17] Let $\alpha, \gamma > 0$, then

Lemma 2.2. ^[18] Let $\alpha > 1$ and $\varsigma > 0$. Then, for all $t\in \mathfrak{J}$ we have

Definition 2.3. ^[5] Let $n-1 < \alpha \leq n$ with $n\in\mathbb{N}$, $\varphi\in \mathbb{S}_{a^+}^{1, +}(\mathfrak{J}, \mathbb{R})$. The left-sided $\varphi$-Caputo FDs of a function $f$ of order $\alpha$ is defined as

Definition 2.4. ^[19] Let $\mathbb{O}\subset\mathbb{X}$ be a bounded set. The Hausdorff MNC of $\mathbb{O}$ is defined by

Lemma 2.3. ^[19] Let $\mathbb{O}, \mathbb{V}\subset \mathbb{X}$ be bounded. Then, $\Lambda$ satisfies:

(1) $\Lambda(\mathbb{O}) = 0 \Longleftrightarrow \mathbb{O}$ is relatively compact.

(2) $\mathbb{O} \subset \mathbb{V} \Longrightarrow \Lambda(\mathbb{O}) \leq \Lambda(\mathbb{V}).$

(3) $\Lambda(\mathbb{O} \cup \mathbb{V}) = \max \{\Lambda(\mathbb{O}), \Lambda(\mathbb{V})\}.$

(4) $\Lambda(\mathbb{O}) = \Lambda(\overline{\mathbb{O}}) = \Lambda(\operatorname{conv}(\mathbb{O}))$, where $\operatorname{conv} \mathbb{O}$ and $\overline{\mathbb{O}}$ denote the convex hull and the closure of $\mathbb{O}$, respectively.

(5) $\Lambda(\mathbb{O} +\mathbb{V}) \leq \Lambda(\mathbb{O})+\Lambda(\mathbb{V}).$

(6) $\Lambda(\lambda \mathbb{O}) \leq |\lambda| \Lambda(\mathbb{O})$, for any $\lambda \in \mathbb{R}.$

Lemma 2.4. ^[20] Let $\mathbb{O} \subset \mathbb{X}$ be bounded. Then, for all $\epsilon$, there is a sequence $\left\{x_n\right\}_{n = 1}^{\infty} \subset \mathbb{O}$ such that

The set $\mathbb{O} \subset L^1(\mathfrak{J}, \mathbb{X})$ is called uniformly integrable if for all $\zeta \in \mathbb{O}$ we have

with $\delta \in L^1\left(J, \mathbb{R}^{+}\right)$.

Lemma 2.5. ^[21] Assume that $\left\{\zeta_n\right\}_{n = 1}^{\infty}$ $\subset$ $L^1(\mathfrak{J}, \mathbb{X})$ is uniformly integrable, the map $t \longmapsto \Lambda\left(\left\{\zeta_n(t)\right\}_{n = 1}^{\infty}\right)$ is measurable, and furthermore

We now recall respectively the theorems of Weissinger and Meir-Keeler that we will use in the following.

Theorem 2.1. ^[3] Let $(\mathbb{E}, \|\cdot\|)$ be a Banach space and $\Theta_n \geq 0$ for every $n \in \mathbb{N}$ where $\sum_{n = 0}^{\infty} \Theta_n$ converges. If the operator $\mathcal{N} : \mathbb{E} \longrightarrow \mathbb{E}$ satisfies

for every $n \in \mathbb{N}$, then $\mathcal{N}$ has a uniquely defined fixed point $u^*$. Additionally, for any $v_0 \in \mathbb{E}$, the sequence $\left\{\mathcal{N}^n v_0\right\}_{n = 1}^{\infty}$ converges to $u^*$.

Definition 2.5. ^[22] Let $\mathbb{B}\subset \mathbb{X}$ be a nonempty set. We say that $\mathcal{N}: \mathbb{B} \longrightarrow \mathbb{B}$ is a Meir-Keeler condensing operator, if for any $\epsilon > 0$, there exists $\eta > 0$ such that

for any bounded subset $\mathbb{A}$ of $\mathbb{B}$.

Theorem 2.2. ^[22] Let $\mathbb{B}$ be a nonempty, bounded, closed, and convex subset of a Banach space $\mathbb{X}$. If $\mathcal{N}: \mathbb{B} \longrightarrow \mathbb{B}$ is a continuous and Meir-Keeler condensing operator, then $\mathcal{N}$ has at least one fixed point and the set of all fixed points of $\mathcal{N}$ in $\mathbb{B}$ is compact.

3.   Main results

Theorem 3.1. We impose the assumptions:

$(A1)$ $g: \mathfrak{J} \times \mathbb{X} \to \mathbb{X}$ is a continuous function. $(A2)$ There exists a constant $L_{g} > 0$ such that

for any $u_1, u_2 \in \mathbb{X}$ and $t\in \mathfrak{J}$.

Then, problem (1.1) admits a unique solution on $\mathfrak{J}$.

Proof. According to [14, Theorem 1], let us introduce $\mathcal{L} : C(\mathfrak{J}, \mathbb{X}) \to C(\mathfrak{J}, \mathbb{X})$ given by

Evidently, the solutions of problem (1.1) can be regarded as the fixed point of $\mathcal{L}$. We will show, with the aid of Theorem 2.1 and a suitably selected equivalent norm, that $\mathcal{L}$ admits a unique fixed point.

In this respect, let $y, z \in C(\mathfrak{J}, \mathbb{X})$. Then, for every $t \in\mathfrak{J}$ and $n \in \mathbb{N}$, since $0 < e^{-\varpi \varphi(t, s)} < 1$ for $a < s < t < b$, we have

Using (A2) and Lemma 2.1, one gets

Again, by (A2), we obtain

Lemma 2.1 entails

Repeating the process for $n = 3, 4, \cdots$, for each $t \in \mathfrak{J}$, it remains to show that

By induction, assume that (3.3) holds for some $n$ and let us prove it for $n + 1$.

One has

Lemma 2.1 yields

Hence, inequality (3.3) holds.

Therefore, we conclude that for all $n \in\mathbb{N}$, one has

By putting

we observe that

Finally, Theorem 2.1 entails that $\mathcal{L}$ admits a unique fixed point which is the unique global solution of problem (1.1). □

Now, we prove another existence result, which is based on Theorem 2.2.

Theorem 3.2. Assume that assumption (A1) holds. Furthermore, we suppose:

(A3) There exist continuous functions $\xi, \kappa : \mathfrak{J} \to \mathbb{R}_{+}$ such that

for all $t\in \mathfrak{J}$.

(A4) There exists a continuous function $\sigma : \mathfrak{J} \to \mathbb{R}_{+}$ such that for each bounded set $\mathbb{U} \subset \mathbb{X}$, and each $t\in \mathfrak{J}$, we have

(A5) The following inequality holds:

with

Then Eq (1.1) admits at least one solution.

Proof. Introduce again the operator $\mathcal{L}$ represented by (3.2) and define the ball

Step 1. $\mathcal{L}$ is a self-mapping from $\mathbf{B}_{R}$ to $\mathbf{B}_{R}$. By (A3), we have

Combining Lemma 2.1 and (A5), one gets

Thus,

This shows that $\mathcal{L}$ is a self-mapping from $\mathbf{B}_{R}$ to $\mathbf{B}_{R}$.

Step 2. $\mathcal{L}$ is continuous. Let the sequence $\{y_n\}$ such that $y_n \to y$ in $\mathbf{B}_{R}$. For all $t\in \mathfrak{J}$, we obtain

Since $g$ is continuous, we have

Step 3. $\mathcal{L} (\mathbf{B}_{R})$ is equicontinuous. Letting $y \in \mathbf{B}_{R}$ and $a < t_1 < t_2 < b$, we get

where

and

Since $e^{-\varpi \varphi(t_2, s)} < 1$, making use of (A3), one obtains

Thus,

On the other side,

Thus,

From (3.6) and (3.7), the equicontinuity of $\mathcal{L} (\mathbf{B}_{R})$ results immediately.

Step 4. Now, we prove that $\mathcal{L} : \mathbf{B}_{R} \to \mathbf{B}_{R}$ satisfies Definition 2.5.

To do this, for every bounded subset $\mathbb{J} \subset C(\mathfrak{J}, \mathbb{X})$ we define the MNC as

Next, fixing $\epsilon > 0$, we show the existence of $\eta > 0$ such that

Now, let $\mathbb{U} \subset \mathbf{B}_{R}$ and, using Lemma 2.4, it follows that for a given $\epsilon' > 0$. Then there exists a sequence $\{y_n\}_{n = 1}^{\infty} \subset \mathbb{U}$ such that, for all $t\in \mathfrak{J}$,

Then, since $\varphi'(\cdot)\varphi(\cdot, a)^{\alpha -1} \in L^{1}(\mathfrak{J}, \mathbb{R})$, it is possible to choose $\aleph$ such that

where $\sigma^{*} = \sup_{t\in \mathfrak{J}}\sigma(t)$. After that, from

we obtain

Next, using (A4), for all $\tau \in [a, s]$, we have

Thus, using Lemma 2.5, for all $t \in \mathfrak{J}$, $s \in [a, t]$, and $\tau \leq s$, one obtains

Multiplying both sides by $e^{-\aleph t}$, one obtains

So, by (3.11), (3.13), and (3.14), we have

Next, by (3.10) and (3.15), fixing $\epsilon' > 0$, we have

Then,

Observe that, from the last estimates,

Letting

one gets

which means that $\mathcal{L} : \mathbf{B}_{R} \to \mathbf{B}_{R}$ satisfies Definition 2.5. Therefore, Theorem 2.2 entails that $\mathcal{L}$ admits at least one unique fixed point in $\mathbf{B}_{R}$ which is the solution of problem (1.1). □

4.   Examples

Example 4.1. Let

be the Banach space of real sequences converging to zero, equipped by

Consider the following problem posed in $\mathbb{X}_1$:

Note that, problem (4.1) is a particular case of (1.1), where:

and $g : [0, 1]\times \mathbb{X}_1 \to \mathbb{X}_1$, given by

for $t\in [0, 1]$, $y = \{y_n\}_{n\geq 1} \in \mathbb{X}_1$.

Condition (A1) is satisfied. Moreover, for any $u_1, u_2 \in \mathbb{X}_1$ and $t \in [0, 1]$, we have

So, condition (A2) is satisfied with

Thus, with the assistance of Theorem 3.1, problem (4.1) has a unique solution $y \in C([0, 1], \mathbb{X}_1)$.

Example 4.2. Let

be the Banach space with the norm

We recall that the Hausdorff MNC in $\left(\mathbb{X}_2, \Vert \cdot\Vert \right)$ is defined as follows (see ^[19]):

Consider the following problem posed in $\mathbb{X}_2$:

where $\zeta = \min_{\alpha \in (1, 2)} \alpha \Gamma(\alpha -1)$, and we take

and $g : [0, b]\times \mathbb{X}_2 \to \mathbb{X}_2$, given by

for $t\in [0, b]$, $y = \{y_n\}_{n\geq 1} \in \mathbb{X}_2$.

Evidently, condition (A1) holds and

Thus, condition (A3) holds with $\xi(t) = \kappa(t) = \frac{1}{(e^{5t} + 13)}$, and one gets $\xi^* = \kappa^* = 0.2$. On the other side, for any bounded set $\mathbb{U} \subset \mathbb{X}_2$, we obtain

Hence, (A4) is verified. Now, we can choose $R$ such that

This function satisfies condition (A5), and from $b < \left(\frac{\zeta}{0.2}\right)^{1/\alpha}$ we get

Finally, all the assumptions of Theorem 3.2 are verified, and thus problem (4.3) has at least one solution $y \in C([0, b], \mathbb{X}_2)$.

5.   Conclusions

We concluded that the quantitative study for a class of nonlinear fractional differential equations involving $\varphi$-Caputo type of order $\alpha\in(1, 2)$ in an infinite-dimensional Banach space framework is achieved. In this context, the results proved in ^[14,23] can be regarded as a special case. Our proof combines results from MNC, Weissinger's, and Meir-Keeler's fixed point theorems. In the future, new work may explore some qualitative aspects of solutions to problem (1.1). Also, for more about fractional functions, we recommend ^{[23,24,25,26,27]}.

Use of AI tools declaration

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

Acknowledgements

The authors would like to express their gratitude to the referees for their corrections and remarks, which have significantly contributed to the improvement of the paper. The valuable feedback provided by both reviewers has been instrumental in enhancing the overall quality of the manuscript.

Conflict of interest

The authors declare no conflicts of interest.