Existence and stability analysis of a problem of the Caputo fractional derivative with mixed conditions

1.   Introduction and problem statement

The analytical apparatus of fractional integro-differentiation and the theory of fractional differential equations demonstrate high efficiency in the description and mathematical modeling of various physical and geophysical processes occurring in fractal environments. Using the concept of the effective rate of the change in the number of parameters of the modeled systems leads to differential equations that contain a composition of fractional differentiation operators of different origins; see ^[1,2,3]. More than 300 years ago, scientists used incorrect random orders to generalize ordinary differential equations and integrals using fractional differential equations. The origin of fractional calculus goes to Newton and Leibniz; see ^[4,5,6]. Fractional differential equations are useful for many models: physical, biological, genetic, and even economic phenomena. Several recent studies have been carried out by researchers to prove the existence and uniqueness of the solution for fractional differential equations with different conditions (boundary, initial, nonlocal, and integral conditions, etc.). For more details, the reader is referred to ^[7,8,9] and the references therein. For a more comprehensive and informative presentation and to illustrate the broad applicability of fractional calculus, we mention the work in ^[10] for fluid dynamics, mathematical biology ^[11,12], image denoising ^[13,14], and image super-resolution^[15,16]. In ancient times, the study of the stability of solutions of fractional differential equations was slow, but recently many researchers have done it in different articles in several ways (asymptotically stable, Ulam stable, generalized Ulam stable, ...), see ^[17,18,19].

In ^[20], the system is governed by a Caputo fractional with a nonlocal initial condition

is considered. The authors investigated the existence and uniqueness of the solution using the Banach and Krasnoselskii fixed-point theorems.

In ^[21], using the Banach contraction mapping together with the Burton-Kirk fixed-point theorem, the authors obtained certain results on the existence and uniqueness of a solution for the problem with the Caputo fractional delay

with appropriate conditions on the parameters of nonlocal and integral boundary value. In ^[22], the problem with the Riemann-Liouville fractional integral in the RHS of

is considered. Using a combination of the Krasnoselskii and Banach fixed-point theorems, results on the existence and uniqueness are proved. In addition, some suitable conditions are provided that ensure the generalized Ulam stability of the system.

In light of these studies, we shall prove that the unique solution exists with the Ulam stability of the following system:

where $\varpi_0, \varpi_1$ are real constants, $1 < \nu+\kappa\leq2,$ the functional ${^C_{\vphantom{a}}D^{\kappa}_{0^{+}}}$ is the Caputo fractional derivative of order $\kappa,$ and $\omega, \psi$, and $N$ are appropriate functions given by the following:

where $E$ is a Banach space. The existence and stability analysis for the problem (1.4) involving the Caputo fractional derivative with mixed conditions typically refers to studying the behavior and solutions of fractional differential equations (FDEs) under a specific boundary. The Caputo fractional derivative is one of the most commonly used definitions of fractional derivatives, particularly because of its useful properties in physical and engineering applications. Equation (1.4)$_1$ is studied under mixed conditions (1.4)$_2$ to extend the work in ^[22], where only the integral boundary condition was taken and the problem was with the Riemann-Liouville fractional integral in the RHS of the differential equation and the works ^[23,24], in which the authors used fractional integration in the equation with the initial condition, but here we used the fractional derivative instead with mixed conditions.

Our study is based on the three-fixed-point theorem, and the goal is to prove the existence and uniqueness results for the solution. In Section 2, some preliminary and integral equations are given. The generalized stability is shown in Section 3. It should be noted that this representation constitutes a generalization of recent results obtained in this research area. To illustrate our results, we conclude the article with examples.

2.   Some preliminary and integral equation

In this section, we present the definitions of fractional integral, fractional Caputo derivative, and some auxiliary lemmas. We refer to ^{[25,26,27,28]} for essential preliminary concepts regarding fractional calculus and fixed-point theory.

Definition 2.1. ^[25] Let $\kappa > 0$ and $\hbar:\mathbb{R}_{+}\longrightarrow \mathbb{R}.$ The Riemann-Liouville fractional integral of order $\kappa$ of a function $\hbar$ is given by

Definition 2.2. ^[28] Let $\kappa > 0$; the order $\kappa$ Caputo fractional derivative of a function $\hbar:\mathbb{R}_{+}\longrightarrow\mathbb{R}$ is defined by

where $n = [\kappa]+1$, and the right-hand side is point-wise defined on $\mathbb{R}_{+}$.

Lemma 2.1. ^[25] For $\kappa > 0$, $i \in \mathbb{R}$, and the appropriate function $\hbar(z)\in C^{n-1}[0, \infty)$, where $\hbar(z)$ exists almost everywhere in any bounded interval of $\mathbb{R}_{+},$ we have

Lemma 2.2. Let $1 < \nu+\kappa < 2$ and $\varpi_1\neq\frac{2}{ \gamma ^2}.$ Assume that $\omega, \psi$ and $N$ are three continuous functions. If $\varpi\in C(J, E)$, then $\varpi$ is a solution of (1.4) if and only if $\varpi$ satisfies the integral equation

Proof. Assume that $\varpi\in C(J, E)$ is a solution of (1.4), and the goal is to prove that $\varpi$ satisfies (2.1). By Lemma $2.1$, we obtain

Then, by (1.4), Lemma $2.1$, and Definition 2.1, we obtain

Substituting (2.3) in (2.2) with the first condition in (1.4) yields:

But we have

Then, we find

Therefore, we obtain

Finally, the integral equivalent Eq (2.1) is obtained. Conversely, using the integral Eq (2.4), which is equivalent to (2.1), we have

Applying the inverse operator of the integral operator on both sides of (2.4) and using the properties of the Caputo derivative (linearity), then

By the properties

and

will lead to

because

which means that $\varpi$ verify (1.4).

For the other side of the proof, substituting $z$ by $0$ in (2.1) will make clear that the non-local condition in (1.4) matches. Thus, $\varpi$ is a solution of (1.4). This concludes the proof. □

3.   Existence of solution with different approaches

We discuss the existence of a globally unique solution by using three fixed-point theorems in different directions. System (1.4) is transformed into a related fixed-point system as

here the operator

is given by:

Noting by

where $\mu_1$, $\mu_2$, and $\mu_3$ will be defined in (H2).

3.1.   The existence results using Leray-Schauder nonlinear alternative

Theorem 3.1. Assume $\omega, \psi\in C(J\times \mathbb{R}, \mathbb{R})$ and $N\in C(J\times J\times \mathbb{R}, \mathbb{R})$ are continuous functions. Assume that

(H0) There exist functions $f_1, f_2\in C(J, \mathbb{R}^+), \ f_3\in C(J\times J, \mathbb{R}^+),$ with $f = \max\{f_1, f_2, f_3\}$ and nondecreasing functions

with $g = \max\{g_1, g_2, g_3\}$ such that

and

for all $z\in[0, 1], s\in[0, 1], \varpi\in\mathbb{R}.$ Also, assume that there exists a constant $M > 0$ such that

Then problem (1.4) has at least one solution in $J$.

Proof. For $r > 0$, let

where $r > 0$, be a bounded set in $C([0, 1], \mathbb{R})$. We shall prove that $F$ maps bounded sets into bounded sets in $C([0, 1], \mathbb{R})$. By using the assumption (H1) and some computations (The beta function and its properties together with the gamma function)

The well-known Beta function and Gamma function are related as

and

we obtain

Let $z_1, z_2\in[0, 1]$, with $z_1 < z_2$, and $\varpi\in B_r$, in which $B_r$ is a bounded set of $C([0, 1], \mathbb{R})$. Then

If $(z_2-z_1)\longrightarrow 0$, then the RHS of (3.7) tends to zero independently of $\varpi\in B_r$. Therefore,

so $F$ maps bounded sets into equi-continuous sets of $C$.

Owing to the Arzela-Ascoli theorem, we find

is completely continuous.

Applying the Leray-Schauder nonlinear alternative when we show the boundedness of the set of all solutions to

Let $\varpi$ be a solution of (1.4). Thus, by (3.6) we obtain

which implies

Then by (H2), $\exists M > 0$ where $M\neq\|\varpi\|$. Define the set

so

is completely continuous. By the choice of $Y$, there is no $z\in\partial Y$ where

then owing to the nonlinear Leray-Schauder type, we find that $\mathcal{F}$ has a fixed point $\varpi\in\bar{Y}$ which is exactly the solution of the IVP (1.4). □

Example 3.1. Let the next fractional integro-differential system

where

Clearly

with

Since all the conditions of Theorem3.1 are satisfied, the problem (3.8) has at least one solution on $[0, 1]$.

3.2.   Owing to the Krasnoselskii's fixed point

Theorem 3.2. Let

and

be continuous functions satisfying

(H1)

hold where $L_1, L_2, L_3\geq0$ with

and

(H2) $\exists \mu^*_1, \mu_2, \mu_3\in L^{\infty}([0, 1], \mathbb{R}^{+})$ such that

If $\varLambda\leq1$ and $L\varLambda_{1}\leq 1,$ then problem (1.4) has at least one solution on $[0, 1].$

Proof. Let $\varpi\in C(J, E)$, we define

and consider the closed ball

Then, the operators $\mathcal{F}_1, \mathcal{F}_2$ are defined on $B_{r}$ as

and

For $\varpi, v^*\in B_{r},$ $z\in[0, 1]$, by fixed $r\geq\frac{\delta_1}{1-\varLambda}$ and by the assumption (H2), we find:

Therefore

This implies that $(\mathcal{F}_1u+\mathcal{F}_2v)\in B_{r}$. We use the estimations:

Now, we have to show that $\mathcal{F}_2$ is a contraction mapping. For $\varpi, v^*\in E$ and $0\leq z \leq 1,$ we have

Thus

As long as $L\varLambda_1\leq1,$ then $\mathcal{F}_2$ is a contraction. Since the functions $h, f$, and $K$ are continuous, we have the operator $\mathcal{F}_1$ is continuous. Also, $\mathcal{F}_1B_{r} \subset B_{r},$ for $\varpi\in B_{r},$ i.e., $\mathcal{F}_1$ is uniformly bounded on $B_{r}$ as

Therefore

We will now show that $\overline{(\mathcal{F}_{1}B_{r})}$ is equicontinuous. So, let us define

For any $\varpi\in B_{r}$ and for each $z_1, z_{2}\in[0, 1]$ with $z_1\leq z_{2},$ we have

The right hand side of (3.11) is independent of $\varpi$ and tends to $0$ when $|z_{2}-z_1|\longrightarrow 0,$ which means that

this implies that $\overline {\mathcal{F}_{1}B_{r}}$ is equicontinuous, then $\mathcal{F}_1$ is relatively compact on $B_{r}.$ So, owing to the Arzela-Ascoli theorem, $\mathcal{F}_1$ is compact on $B_{r},$ and then, all conditions of Theorem 3.2 are fulfilled. Then, the operator $\mathcal{F}_1+\mathcal{F}_{2}$ has a fixed point on $B_{r}.$ Therefore, problem (1.4) has at least one solution on $[0, 1].$ □

Example 3.2. Consider the following fractional integro-differential problem.

Where

For $\varpi, v^*\in E = \mathbb{R}^{+}$ and $z\in [0, 1]$, we obtain

yields

and

Using the above data, we obtain

fixing $r\geq\frac{\delta_1}{1-\varLambda} = 17.0673$ then, problem (3.12) has at least one solution on $[0, 1]$ by Theorem3.2

3.3.   The existence and uniqueness

Theorem 3.3. Let (H1) hold. If $L\varLambda < 1$, then (1.4) has a unique solution in $[0, 1].$

Proof. Let

where $M_1, M_2, M_3 > 0$ and

For

fixed, let

By (H1), we have

and

and

First step: We shall prove that $TD_{r}\subset D_{r}.$ Let $t\in [0, 1]$ and $\varpi\in D_{r},$ we have

Hence, $\mathcal{F}D_{r_1}\subset D_{r_1}.$

Second step: We prove now that $\mathcal{F}:D_{r_1}\longrightarrow D_{r}$ is a contraction. By $(H1),$ let $z\in [0, 1]$ and $\forall \varpi, v^*\in D_{r_1}$, we have

Since $L\delta_2 < 1,$ we have $\mathcal{F}$ is a contraction. All conditions of Banach's fixed point theorem are satisfied, then $\exists \varpi\in C(J, E)$ such that $\mathcal{F}\varpi = \varpi$, which is considered as a unique solution of (1.4) in $C(J, E)$. □

Example 3.3. Consider the following fractional integro-differential problem:

where

Taking $\varpi, v^*\in E = \mathbb{R}^{+}$ and $z\in [0, 1]$, and

Thus,

Then, there exists a unique solution for the problem (3.18) on $[0, 1]$ by application of Theorem 3.3

Example 3.4. Consider the following fractional integro-differential problem:

where

To illustrate our result in Theorem 3.3, let $\varpi, v^*\in E = \mathbb{R}^{+}$, $0\leq z\leq 1$, and

Thus,

then, there exists a unique solution for the problem (3.23) on $[0, 1]$ by application of Theorem 3.3.

4.   Generalized Ulam stability criterion

We show the Ulam stability of (1.4), using integration $v^*(z) = Fv^*(z)$.

Let us define the non-linear continuous operator

as follows

Definition 4.1. Let $\epsilon > 0$ and $v$ be a solution of (1.4), where

We say that problem (1.4) is Ulam-Hyers stable if there exists $\beta > 0$ and solution $\varpi \in C([0, 1], E)$ of (1.4) satisfying

here $\epsilon^* > 0$ depending on $\epsilon$.

Definition 4.2. Let $g \in C(\mathbb{R}^+, \mathbb{R}^+)$ and solution $v$ of (1.4), there exists a solution $\varpi \in C([0, 1], E)$ of (1.4) where

Then (1.4) is said to be stable in the generalized Ulam-Hyers sense.

Definition 4.3. Let $\epsilon > 0$ and $\theta \in C([0, 1], \mathbb{R}^+)$. Let $v$ be a solution of (1.4); we say that the system (1.4) is stable in the Ulam-Hyers-Rassias sense with respect to $\theta$ if

and $\exists \beta > 0$ and a solution $v \in C([0, 1], E)$ of (1.4) where

here $\epsilon_* > 0$ depending on $\epsilon$.

Theorem 4.1. Let (H1) in Theorem 3.1 hold with $L\delta < 1$. The (1.4) is both Ulam-Hyers and generalized Ulam-Hyers stable.

Proof. Let $\varpi \in C([0, 1], E)$ be a solution of (1.4), satisfying (2.1) in the sense of Theorem 3.3, and any solution $v^*$ satisfying (4.1). Then, we obtain:

because $L\delta < 1$ and $\epsilon > 0$, we obtain

Let $\epsilon_* = \frac{\epsilon}{1-L\delta}$ be fixed and $\beta = 1$, we obtain the Ulam-Hyers stability requirements. Besides, the generalized Ulam-Hyers stability is followed by taking

□

Theorem 4.2. Let (H1) holds with $L < \delta^{-1}$, and $\exists \theta\in C([0, 1], \mathbb{R}^+)$ satisfy (4.4). Then (1.4) is Ulam-Hyers-Rassias stable with respect to $\theta$.

Proof. As in the proof of Theorem 4.1, we have

where $\epsilon_* = \frac{\epsilon}{1-L\delta}$. □

5.   Conclusions

This article delves into the study of a fractional-order non-linear integro-differential equation, featuring a composition of fractional derivatives with distinct origins and mixed conditions. This equation serves as a model for the motion of a system within a fractal medium. The paper utilizes three fixed-point theorems to establish the existence and uniqueness of a solution. Subsequently, it presents the Ulam stability criterion for the obtained solution. Finally, a proposed example is provided to illustrate the main results. Complete studies are made in which our contributions extend the work in ^[22]. The problem of the fractional Caputo derivative (in the RHS and LHS of the differential equation) with mixed conditions is considered in (1.4) to improve the model (1.3), where only the integral boundary condition was taken, and the problem was with the Riemann-Liouville fractional integral in the RHS of the differential equation. This problem can be studied by a generalized Caputo derivative with changing conditions, especially when the derivative comes in the psi Caputo derivative; then study it with different origins and suitable mixed conditions.

Author contributions

All authors contributed equally to this work. All authors have read and approved the final version of the manuscript for publication.

Use of Generative-AI tools declaration

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

Acknowledgments

The Researchers would like to thank the Deanship of Graduate Studies and Scientific Research at Qassim University for financial support (QU-APC-2025).

Conflict of interest

The authors declare that they have no conflict of interest.