Stieltjes integral boundary value problem involving a nonlinear multi-term Caputo-type sequential fractional integro-differential equation

1.   Introduction

In the previous few decades, fractional differential equations have become of great interest for researchers due to its high accuracy and usability in numerous subjects of science and technology. A lot of physical and natural phenomena can be modeled through fractional differential equations, which provides better results compared to integer order differential equations. Due to this, fractional differential equations are counted as a special tool for modeling. Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, electrochemistry, aerodynamics, viscoelasticity, polymer rheology, economics, biology, electrodynamics of complex medium, etc. For details, see ^{[2,6,7,8,9,12,14,15,16,17,19,20,22,24,28,29,31,37,39]}.

Additionally, fractional differential equations serve as an excellent tool for the description of the hereditary properties of various materials and processes. Consequently, the subject of the aforementioned equations is gaining great importance and attention from the researchers. Additionally, researchers are attracted to the enriched material on theoretical aspects and analytic/numerical methods for solving fractional order models. Furthermore, the mathematical models involving fractional order derivatives are more realistic and practical compared to the classical models. In the most recent years, many researchers have focused on the existence of solutions for fractional differential equations, for instance, see ^[3,4,10] and the references therein.

In $1940$, while interacting with the mathematical community at the University of Wisconsin, Ulam expressed his concern regarding the stability of group homomorphisms ^[25]. The broader form of his views about the stability of functional equations is, "Impose constraints which converges the solutions of an inequality to the exact solutions of the corresponding equations". In $1941$, by considering Banach spaces, Hyers gave half an answer to Ulam's question about the stability of functional equations ^[18]. Due to this contribution of Hyers, Ulam's problem was refereed to as Hyers-Ulam stability of functional equations. For the first time, Hyers-Ulam stability of linear differential equations were introduced by Obloza ^[21]. Along with generalization, the work of Obloza has been enhanced with different features by using new approaches as time progressed. For more details regarding Ulam's stability with different approaches, we recommend ^{[27,33,34,35,36]}.

In ^[32], the authors studied a new class of impulsive implicit sequential fractional differential equations of the following form:

where $^{c}D^{\beta}$ denotes the Caputo fractional derivative of order $\beta$, $D$ denotes an ordinary derivative, with the lower limit $0$, $0 = x_{0} < w_{0} < x_{1} < w_{1} < \dots < x_{m} < w_{m} = T$, and $T$ is a pre-fixed number and $\beta \in \mathrm{R}^{+}$. The function $f:[0, T]\times \mathrm{R}\times \mathrm{R} \rightarrow \mathrm{R}$ is continuous and $G_{k}:[w_{k-1}, x_{k}]\times \mathrm{R}\rightarrow \mathrm{R}$ is also continuous for all $k = 1, 2, \dots, m$,

Binlin et al. ^[38] studied the existence and uniqueness ($EU$) of the solution, as well as the stability in the form of Ulam's problem, for the following FDEs with Stieltjes integral condition:

Bashir et al. ^[5] studied the $EU$ of the solution of the nonelinear multi-term fractional integro-DE with anti-periodic conditions:

where, $\lambda_1, \ \lambda_2\, \ \in\mathcal{R}, \ \lambda_1 \ne0, \, v \in [1, 2), \ \varrho \, \in (1, v), \ \omega > 0, \, \, f, g:[0, 1]\times\mathcal{R}\rightarrow \mathcal{R}$ are appropriate functions.

In this article, we analyze the multi-term, nonlinear, sequential fractional differential equation with Stieltjes integral conditions of the following form:

where, $v\in (1, 2], {\; \; } \varrho\in(1, v), \lambda_1, \lambda_2, \in \mathcal{R}, \lambda_3 \in \mathbb{R^{+}}, \lambda_1\ne0, p\in \mathbb{N}, \beta_{i}\in\mathcal{R}$ for all $i = 0, 1, \cdots, p, 0\le\beta_1\le\beta_2\le\cdots\le\beta_{p} < v$, $\beta_{0}\in[1, v), \, f, g:[0, 1]\times\mathcal{R}\rightarrow \mathcal{R}$ are appropriate functions, and the integrals presented in Boundary Condions (BCs) are Riemann-Stieltjes integrals with $x_{i}(i = 1, 2, \cdots, p)$ functions of a bounded variation. In addition to general FDEs, the multi–point boundary conditions are more valuable than the classical initial/boundary conditions, because these conditions describe the characteristics of chemical, physical or others processes happening inside the domain.

2.   Preliminaries

Here, we present necessary preliminaries so that the paper will be self contained.

Let $\mathcal{C} = {C}([0, 1], \mathcal{R})$ be the Banach space of all continuous functions endowed with the norm denoted by $\|.\|$.

Definition 2.1. ^[1] The fractional integral of order $v$ from $0$ to $\xi$ for the function $x$ is defined by the following:

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

Definition 2.2. ^[1] The Caputo derivative of fractional order v from 0 to $\xi$ for a function x can be defined as follows:

Lemma 2.1. ^[1] The FDE $^C\mathscr{D}_{0, \xi}^{v}x(\xi) = 0$ with $v > 0$, involving Caputo differential operator $^C\mathscr{D}_{0, \xi}^{v}$ has a solution in the following form:

where $c_{k}\in\mathcal{R}, k = 0, \ 1, \cdots, m-1 = :\overline{0, m-1}$ and $m = \lfloor v \rfloor +1$.

Lemma 2.2. ^[1] For each $v > 0$, we have the following:

where $c_{k}\in\mathcal{R}, \ k = \overline{0, m-1}$ and $m = \lfloor v \rfloor +1$.

Lemma 2.3. ^[38] If $\Re(v) > 0$ and $\lambda > 0,$ then

Theorem 2.1. (Krasnoselskii's fixed point theorem ^[11]) If $K$ is a closed, convex, non-empty subset of a Banach space $M$ such that $P$ and $Q$ map $K$ into $M$ and,

(ⅰ) $Px+Qy \in K$${\; \; \; }(\forall x, y\in K),$

(ⅱ) $P$ is compact and continuous,

(ⅲ) $Q$ is a contraction mapping,

then $\exists$ $y$ in $K$ such that, $Py+Qy = y$.

Theorem 2.2. (Banach fixed point theorem) Every contraction mapping $\delta$ from $B$ to $B$ has a fixed point (unique), where $B$ is a non-empty closed set in a Banach space $X$.

Definition 2.3. Consider a Cauchy problem $\frac{d}{dt}x(t) = f(t)$ with $x(t_0) = x_0$, then, a continuous function $u$ is called its mild solution if

Lemma 2.4. Suppose that $f, g\in C([0, 1]\times \mathcal{R}, \mathcal{R})$, then, the mild solution of (1.2) has the following form:

Proof. Consider problem (1.2) and apply the fractional integral of order $v$ to obtain the following:

This implies the following:

Equivalently, we obtain te following:

Now, multiplying by the integrating factor $e^{\lambda_{3}\xi}$ and then integrating from 0 to $\xi$, we obtain the following:

Therefore,

Consequently, Eq (2.2) becomes the following:

Now, the boundary condition $x(0) = 0$ implies that $c_{2} = 0$. Thus, (2.3) implies the following:

Differentiating Eq (2.4) w.r.t $\xi$ and then applying the boundary condition, we obtain the following:

Now, let $\beta_{0}\in[1, v)$, then, from ^[38], we obtain the following:

Similarly, for $\; \; 1\le\beta_1\le\beta_2\le\cdots\le\beta_p < v$, we obtain the following:

Additionally,

From Eqs (2.5) and (2.7), we have the following:

Now, Eq (2.4) becomes the following:

which is the required proof.

Remark 2.1. According to Lemma 2, and the Counter-Example 1 in ^[13], Example 3.1 from ^[23] and Fact 2, from ^[23], the existence of continuous (even Holderian) solutions of the fractional-type integral forms is not sufficient to ensure the existence of solutions to the corresponding Caputo-type fractional differential problems. For this reason, we only obtain the mild solution.

3.   Existence and uniqueness result

Throughout the paper, in the case of the Stieltjes integral, we consider the same monotonic functions, which are functions of $t$ and also functions of bounded variation.

The given problem can be converted into a fixed point problem as follows: $\; {\delta}(x) = x$, where $\; \delta:\mathcal{C}\rightarrow \mathcal{C}$ is defined by

Theorem 3.1. $H_{1}:$ Suppose $f, g\in{C}([0, 1], \mathcal{R})\rightarrow \mathcal{R}$ and

where $N = \max\{N_1, N_2 \}$, and

with $L_1, L_2 > 0 \ \forall \ \xi\in [0, 1]$ and $x, \ y\in\mathcal{R}$.

$H_2:$ $L = \max\{L_1, L_2\}$ and $\Omega_1+L \ \Omega_2 < 1$, where

Then, Eq (1.2) has a unique mild solution in $\mathcal{C}$.

Proof. Considering a closed ball $B_r = \{x\in \mathcal{C}:\|x\|\le r\}$, we show that $\delta B_r\subset B_r$, where $\delta$ is defined by Eq (3.1) and $r\ge (N\Omega_2+\Omega_3)(1-\Omega_1-L\Omega_2)^{-1}$. For any $x\in B_r$, it follows from condition ($H_1$) that

Now, as $\|\delta(x)\| = \sup_{\xi\in[0, 1]}|\delta x(\xi)|$, the following holds true:

Now, using the mean value Theorem ^[30] for Stieltjes integral with $\chi\in[0, 1]$ and $x_{i}(1) = \Re_{i} > 0$ and from ($H_1$), we obtain the following:

This implies the following:

Next, to prove the contraction, consider the following:

Now, using the mean value theorem for the Stieltjes integral with $\chi\in[0, 1]$ and $x_{i}(1) = \Re_{i} > 0$ and from ($H_1$), we obtain the following:

Thus,

From ($H_2$), we know that $(\Omega_{1}+L\Omega_{2})$ $<$ $1$, which demonstrates that the operator $\delta$ is contractive. Therefore, according to the Banach contraction principle, there exists a fixed point, which is the mild solution of the problem $(1.2)$.

Theorem 3.2. Suppose that ($\textrm{$ H_1 $}$) and the following condition are satisfied.

$\textrm{$ (H_3) $}$: $\exists$ $\mu_{1}$, $\mu_{2}$ $\in$ $\textrm{$ C $}$${([0, 1], \mathbb{R^{+})}}$ such that $|f(\xi, x)|\le \mu_{1}(\xi)$ and $|g(\xi, x)|\le\mu_{2}$, $\Omega_{1} < 1$, then, $\exists$ at least one mild solution of problem $(1.2)$ in $\mathcal{C}$.

Proof. Suppose $K_{\xi}$ = $\{x\in\mathcal{C}:\|x\|\le\xi\}$ be a closed subset of $B_r$ and define $\delta_1$ and $\delta_2$ on $K_{\xi}\rightarrow \mathcal{C}$ as follows:

and

Clearly, ${\; \delta = \delta_1+\delta_2}$. Now, we verify the hypothesis of Krasnoselskii's fixed point theorem for $x, y\in K_{\xi}$ with $\xi\ge(\mu\Omega_2+\Omega_3)(1-\Omega)^{-1}$. Presently,

Thus, $\delta_1 x+\delta_2 y \in K_{\xi}$.

$(ⅱ)$ Next, we show that $\delta_1$ is continuous and compact. Since $f$ and $g$ are continuous thus $\delta_1$ is also continuous. Additionally, $\delta_1$ is uniformly bounded, i.e.,

For compactness, assume that

and

Then, for $\xi_1 < \xi_2$, we have the following:

where $\chi_{1}, {\; \; } \chi_{2} \in [0, 1]$. If $\xi_{1}\rightarrow \xi_{2}$, then $\chi_{1} \rightarrow \chi_{2}$ and the right side approaches to 0 independent of x $\in$ $K_\xi$. Thus, $\delta_1$ is equicontinuous. Therefore, by the Arzela-Ascoli theorem, $\delta_1$ is relatively compact.

(ⅲ) As $\Omega_1 < 1$, therefore, $\delta_2$ is contraction. Thus, there is at least one mild solution to the problem in $\mathcal{C}$.

4.   Stability analysis

Let $\epsilon > 0$, $f, {\; }g \in ([0, 1]\times \mathcal{R}, \mathcal{R})$, and assume the following inequality:

Definition 4.1. (1.2) is said to be Hyers-Ulam stable if there is a constant $k_{0} \in \mathbb{R^{+}}$ such that for each $\epsilon > 0$ and each red mild solution $u\in \mathcal{C}$ of (4.1), there is a mild solution $x \in \mathcal{C}$ of (1.2) with,

${\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; }|u(\xi)-x(\xi)|\le k_{0}\epsilon {\; \; \; } \forall{\; \; \; } \xi\in[0, 1].$

Remark 4.1. A function $u\in \mathcal{C}$ is a mild solution of (4.1) if there is $\psi \in \mathcal{C}$ depending on u, such that:

(ⅰ) ${\; \; \; }(\lambda_1 \mathscr{D}^{v}+\lambda_2 \mathscr{D}^{\varrho})(\mathscr{D}+\lambda_3)u(\xi) = f(\xi, u(\xi))+I^{\omega} g(\xi, u(\xi))+\psi (\xi)$ ${\; \; }$ $\forall$ $\xi\in [0, 1].$

(ⅱ) ${\; \; \; }|\psi (\xi)|\le\epsilon {\; \; \; } \forall{\; \; } \xi\in [0, 1].$

Lemma 4.1. If $u\in \mathcal{C}$ is a mild solution of the Inequality (4.1), then u is also a mild solution of the following inequality:

${\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; }$ $|u(\xi)-x(\xi)|\le k_{0}\epsilon$.

where,

Proof. If $u$ is the mild solution of Inequality (4.1), then $u$ will also be the mild solution of the following:

The mild solution of (4.2) is given by the following:

Let the terms free of $\psi$ be denoted by $x(\xi)$; then,

Assume that,

In view of (ⅱ) of Remark 3.2, we have the following:

which is the desired result.

5.   Example

In this section, we solve some examples using the obtained theorems.

Example 5.1.

where,

Since,

From assumptions ($H_1$) and ($H_2$), we have the following:

where, ${\; \; }L_1 = \frac{1}{25}$, $L_2 = \frac{1}{7}$ and $L = \max{\{L_1, L_2\}} = \frac{1}{7}$. Calculating $\Omega_1$ and $\Omega_2$ from the given data, we obtain $\Omega_1 = 0.93461$ and $\Omega_{2} = 0.136698$. Additionally, $\Omega_{1}+L\Omega_{2}\approx 0.95414 < 1$. Hence, the given problem (5.1) has a unique mild solution in $\mathcal{C}$ for $\xi\in[0, 1]$.

Example 5.2.

where,

Therefore,

From assumption ($H_1$) and ($H_2$), we have

where, ${\; \; }L_1 = \frac{1}{17}$, $L_2 = \frac{1}{16}$ and $L = max{\{L_1, L_2\}} = \frac{1}{16}$. Calculating $\Omega_1$ and $\Omega_2$ from the given data, we obtain $\Omega_1 = 0.2188677$, $\Omega_{2} = 0.324496$. Additionally, $\Omega_{1}+L\Omega_{2}\approx 0.239415 < 1$. Hence, the given problem (5.2) has a unique mild solution in $\mathcal{C}$ for $t\, \in[0, 1]$.

6.   Conclusions

The paper established sufficient conditions that showed the existence, uniqueness and Ulam's stability for the mild solutions of Problem (1.2). The conditions were obtained from the view of fixed point theorems. Furthermore, we demonstrated the obtained results using two examples. The multi-point boundary conditions can be used to describe the characteristics of chemical, physical or others processes occouring inside the domain. Thus, the obtained results can be fruitful in the mentioned processes. The Hyers-Ulam stability means that for any approximation in a specific region we will obtain an exact mild solution. Therefore, the obtained results of the Hyers-Ulam stability can be utilized in a numerical analysis and approximation theory of the related mentioned processes.

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 is supported by the Anhui Provincial Natural Science Foundation Project (KJ2021A1175) and Funding project for cultivating top-notch talents in universities (gxgnfx2022096).

Conflicts of interest

The authors declare no conflicts of interest.