Analyzing Uniqueness of Solutions in Nonlinear Fractional Differential Equations with Discontinuities Using Lebesgue Spaces

Abstract

We explore the existence and uniqueness of solutions to nonlinear fractional differential equations (FDEs), defined in the sense of RL-fractional derivatives of order $\eta \in (1,2)$. The nonlinear term is assumed to have a discontinuity at zero. By employing techniques from Lebesgue spaces, including Holder’s inequality, we establish uniqueness theorems for this problem, analogous to Nagumo, Krasnoselskii–Krein, and Osgood-type results. These findings provide a fundamental framework for understanding the properties of solutions to nonlinear FDEs with discontinuous nonlinearities.

1. Introduction

The study of nonlinear FDEs has long been plagued by concerns about the existence and uniqueness of their solutions. The intricate nature of these equations, featuring non-integer derivatives and nonlinear terms, has made it challenging to establish a comprehensive understanding. Nevertheless, recent investigations have successfully employed innovative methodologies from fractional calculus, functional analysis, and numerical techniques to validate the existence and uniqueness of the solutions. These advancements have far-reaching implications for various fields, including physics, engineering, and mathematics. By establishing the existence and uniqueness of the solutions, researchers can confidently model and simulate complex phenomena, such as chaotic systems, fractals, and anomalous diffusion; see [1,2]. Numerous authors have explored the benefits of working with Lebesgue spaces. The significance of Lebesgue space theory lies in its ability to facilitate breakthroughs in mathematical analysis, thereby advancing our understanding of mathematical structures. Furthermore, the examination of Lebesgue spaces has profound implications for the advancement of mathematical theory and its applications, ultimately enriching our understanding of mathematical principles.

A bibliometric analysis on Lebesgue space reveals the growing interest in this area of research, with applications in various fields, including the study of fractional optimal control problems for ordinary differential equations (ODEs) and reaction–diffusion equations in Lebesgue space. Investigation of these problems in Lebesgue space has led to significant advancements in our understanding of complex systems and phenomena and continues to be an active area of research; see [3,4,5]. This understanding enables the development of novel applications in fields like signal processing, image analysis, and control systems. The significance of results on the existence and uniqueness of solutions lies in their ability to provide a foundation for numerical methods and analytical techniques. By ensuring the reliability of the solutions, researchers can trust the accuracy of their simulations and predictions. Furthermore, these findings pave the way for exploring new research avenues, such as investigating the stability and bifurcation of solutions, and developing new numerical schemes for solving nonlinear FDEs. Nonlinear FDEs have been a focal point of the research in recent years, with studies published between 2020 and 2024 making significant contributions to the field; see [6,7,8,9].

These equations, characterized by non-integer derivatives and nonlinear terms, pose complex challenges for establishing the existence and uniqueness of solutions; see [10,11,12]. Researchers have employed innovative approaches, including fractional calculus techniques and numerical methods, to tackling these challenges. For instance, a recent study developed a novel numerical scheme for solving nonlinear FDEs, demonstrating its efficacy in modeling real-world problems; see [13]. Further advancements have been made in understanding the dynamic behavior of the solutions to these equations, including stability and bifurcation analyses; see [14]. Additionally, machine learning techniques have been explored as a potential tool for solving nonlinear FDEs; see [15].

These developments underscore the growing interest in nonlinear FDEs, highlighting the need for continued research in this area to uncover new insights and applications. The Caputo fractional derivative is a powerful tool for modeling nonlinear fractional order systems, offering a more accurate description of complex phenomena. Its ability to handle nonlinearities makes it the preferred choice in various applications, enabling subtle dynamics and nonlocal properties to be captured; see [16,17]. Unlike other fractional derivatives, the Caputo derivative excels in handling nonlinearities. Its versatility and accuracy have made it a popular choice in physics, engineering, and mathematics, driving innovation in these fields.

The RL-fractional derivative is a fundamental concept in fractional calculus, offering a robust framework for modeling complex phenomena. Its ability to capture non-integer order dynamics makes it a vital tool in various fields. The RL derivative plays a crucial role in exploring the intricacies of nonlinear fractional order systems. By utilizing the RL derivative, you are able to successfully model and analyze complex phenomena, providing valuable insights into the underlying dynamics [18]. Our work demonstrates the importance of the RL derivative in fractional calculus and its potential to drive innovation in various fields. In this paper, we investigate the existence and uniqueness of solutions to the following problem:

$$\left\{\begin{array}{c}{D}^{\eta }m\left(y\right)=f(y,m\left(y\right),D^{\eta -1}m\left(y\right)),\hfill \\ y\left(0\right)+g\left(y\right)=y_0,D^{\eta -1}{m\left(y\right)|}_{y=0}=q,\hfill & y>0,\eta \in (1,2),q\ne 0,\hfill \end{array}\right.$$

(1)

where $D^{\eta }$ is defined as the standard RL-fractional derivative of order $1<\eta <2$, and $q\ne 0$.

In this paper, we establish Osgood-type results and Krasnoselskii–Krein-type results for nonlinear fractional differential equations with discontinuities. Utilizing Lebesgue spaces, we analyze the uniqueness of the solutions to these equations and establish results on their existence and uniqueness. These results provide a novel and powerful tool for the study of nonlinear fractional differential equations, enabling a deeper understanding of these complex equations.

2. Preliminaries

We follow certain methods introduced in [19,20] by generalizing some of the definitions, which were already analyzed in the above references.

Definition 1 

([19,20]). The RL-integral equation of order $\eta >0$ is defined for a function $m\left(y\right)$ as follows:

$${\mathbf{I}}^{\eta }m\left(y\right):={\displaystyle \frac{1}{\Gamma \left(\eta \right)}}{\int }_0^y{\displaystyle \frac{m\left(t\right)}{{(y-t)}^{1-\eta }}}dt,$$

under the condition that the right-hand side is well defined point-wise on ${\mathbf{R}}_0^{+}$.

Definition 2 

([19,20]). The RL-derivative of fractional order $r-1<\eta <r(r\in \mathbf{N})$ for a function $m\left(y\right)$ is as follows:

$$D^{\eta }m\left(y\right)={\displaystyle \frac{1}{\Gamma (r-\eta )}}{\displaystyle \frac{d^r}{d{y}^r}}{\int }_0^y{\displaystyle \frac{m\left(t\right))}{{(y-t)}^{\eta -r+1}}}dt,$$

under the condition that the right-hand side is well defined point-wise on ${\mathbf{R}}_0^{+}$.

In the literature, compositional relations are widely used; see [21,22].

Lemma 1 

([22]). For a function $m\left(y\right)$ such that $D^{\eta }m\left(y\right)$ with $r-1<\eta <r(r\in \mathbf{N})$ is integrable, the following compositional relation given below holds:

$${\mathbf{I}}^{\eta }m\left(y\right)=m\left(y\right)+{\Sigma }_{u=1}^r{D}^{\eta -r}{m\left(y\right)|}_{y=0}{\displaystyle \frac{y^{\eta -r}}{\Gamma (\eta -r+1)}},$$

Suppose that $D^{\eta -r}m\left(y\right)={\mathbf{I}}^{r-\eta }m\left(y\right)$ when $\eta <r$. If $\eta =2,$ the above relation becomes

$${\mathbf{I}}^{\eta }m\left(y\right)=m\left(y\right)+D^{\eta -r}m\left(y\right){|}_{y=0}{\displaystyle \frac{y^{\eta -1}}{\Gamma \left(\eta \right)}}+D^{\eta -2}m\left(y\right){|}_{y=0}{\displaystyle \frac{y^{\eta -2}}{\Gamma (\eta -1)}}.$$

Furthermore, if function m is continuous on the interval $[0,M]$, we have

$${\mathbf{I}}^{\eta }{D}^{\eta }m\left(y\right)=m\left(y\right)+D^{\eta -1}m\left(0\right){\displaystyle \frac{y^{\eta -1}}{\Gamma \left(\eta \right)}},$$

Lemma 2 

([22]). Given a function $m\left(y\right)$ that is RL-integrable of order η, then the following rule is applicable:

$$D^{\eta }{\mathbf{I}}^{\eta }m\left(y\right)=D^2{\mathbf{I}}^{2-\eta }{\mathbf{I}}^{\eta }m\left(y\right)=D^2{\mathbf{I}}^2{\mathbf{I}}^{\eta }m\left(y\right)=m\left(y\right)$$

The space of functions in which we will search for solutions to problem (1) is defined in [23] as follows:

Lemma 3 

([24]). For every $\eta ,\gamma \in [0,\infty )$; therefore,

$$m\left(y\right)\le m\left(y\right)+d{\int }_0^y{\displaystyle \frac{m\left(t\right)}{{(y-t)}^{1-\eta }}}dt,$$

Consequently, a constant $\mathcal{K}$ exists, dependent on η, which will be denoted as $\mathcal{K}\left(\eta \right)$, such that

$$m\left(y\right)\le m\left(y\right)+\mathcal{K}d{\int }_0^y{\displaystyle \frac{m\left(t\right)}{{(y-t)}^{1-\eta }}}dt,$$

Lemma 4 

([25]). Suppose that $\eta \in (1,2)$, as well as $m\in {\mathfrak{C}}^{\eta -1}\left([0,M]\right)$. There is a function $\sigma :[0,M]\to [0,M]$ with $0<\sigma \left(y\right)<y$, such that

$$m\left(y\right)=-D^{\eta -1}m\left(0\right){\displaystyle \frac{y^{\eta -1}}{\Gamma \left(\eta \right)}}+\Gamma (2-\eta ){\left(\sigma \left(y\right)\right)}^{\eta -1}{D}^{\eta -1}m\left(\sigma \left(y\right)\right),$$

is fulfilled.

Theorem 1 

([26]). The space of functions continuous on $[0,M]$ with a fractional derivative of order $\gamma (0<\gamma <1)$ is fully equipped as a Banach space with the norm

$${\parallel m\parallel }_{\gamma }={\parallel m\parallel }_{\infty }+{\parallel D^{\gamma }m\parallel }_{\infty },$$

where the supremum norm defined on the class of continuous functions is represented by ${\parallel .\parallel }_{\infty }$. The notation ${\mathfrak{C}}^{\gamma }\left([0,M]\right)$ represents the space of continuous functions.

Lemma 5 

([26]). If function $f:[0,M]\times {\mathfrak{R}}^2\to \mathfrak{R}$ and $g:\mathfrak{R}\to \mathfrak{R}$ are continuous, then problem (1) is equal to the nonlinear fractional Volterra integro differential equation

$$y\left(t\right)=y_0-g\left(y\right)+q+{\displaystyle \frac{1}{\Gamma \left(\eta \right)}}{\int }_0^y{\displaystyle \frac{f(t,m\left(t\right),D^{\eta -1}m\left(t\right))}{{(y-t)}^{1-\eta }}}dt.$$

3. Main Results

Theorem 2.

(Existence) Let $M>0$. $\left(H_1\right)$ Suppose that $f(y,t_1,t_2)\in \mathfrak{C}([0,M]\times \mathfrak{R}\times \mathfrak{R})$, $y^{\eta -1}f(y,t_1,t_2)\in \mathfrak{C}([0,M]\times \mathfrak{R}\times \mathfrak{R})$, and $|g\left(y\right)-g\left(\tilde{y}\right)|\le {\mathcal{K}}_3\parallel y-\tilde{y}\parallel .$

${\mathcal{K}}_1\in \mathfrak{R}$ and ${\mathcal{K}}_2,{\mathcal{K}}_3\in (1,2)$ exist such that

$$\begin{array}{ccc}\hfill |f(y,t_1,t_2)-f(y,t_1^{\prime },t_2^{\prime })& \le & {\mathcal{K}}_1|t_1-t_1^{\prime }|+{\mathcal{K}}_2|t_2-t_2^{\prime }|\hfill \\ \hfill |g\left(y\right)-g\left(\tilde{y}\right)|& \le & {\mathcal{K}}_3\parallel y-\tilde{y}\parallel .\hfill \end{array}$$

Let

$$q<min\left\{M,{\left({\displaystyle \frac{(1-{\mathcal{K}}_3)(1-{\mathcal{K}}_2)\Gamma (\eta +1)}{{\mathcal{K}}_1}}\right)}^{{\displaystyle \frac{1}{\eta }}}\right\}.$$

Proof. 

Suppose that

$$\begin{array}{ccc}\hfill D^{\eta }m\left(y\right)& =& n_m\left(y\right),y\left(0\right)+g\left(y\right)=y_0,\hfill \\ \hfill D^{\eta -1}m\left(0\right)& =& q,\hfill \end{array}$$

Then, according to Lemma (5),

$$m\left(y\right)=y_0-g\left(y\right)+q+{\displaystyle \frac{1}{\Gamma \left(\eta \right)}}{\int }_0^y{\displaystyle \frac{1}{{(y-t)}^{1-\eta }}}{n}_m\left(t\right)dt,y\in [0,M],$$

where

$$n_m\left(y\right)=f(y,y_0-g\left(y\right)+q+I^{\eta }{n}_m\left(y\right),n_m\left(y\right)).$$

That is, $m\left(y\right)=y_0-g\left(y\right)+q+I^{\eta }{n}_m\left(y\right)$. Define a mapping $S:\mathfrak{C}\left(\right[0,M],\mathfrak{R})\to \mathfrak{C}\left(\right[0,M],\mathfrak{R})$ as follows:

$$\left(Sm\right)\left(y\right)=y_0-g\left(y\right)+q+{\displaystyle \frac{1}{\Gamma \left(\eta \right)}}{\int }_0^y{\displaystyle \frac{1}{{(y-t)}^{1-\eta }}}{n}_m\left(t\right)dt.$$

It is clear that the fixed point S is the solution of (1). Let $m,p\in \mathfrak{C}\left(\right[0,M],\mathfrak{R})$; then, we have

$$\begin{array}{ccc}\hfill |(Sm\left)\right(y)-(Sp\left)\right(y)|& \le & |g\left(m\right)-g\left(p\right)|+|q|+{\displaystyle \frac{1}{\Gamma \left(\eta \right)}}{\int }_0^y{\displaystyle \frac{1}{{(y-t)}^{1-\eta }}}|n_m\left(t\right)-m_p\left(t\right)|dt\hfill \end{array}$$

$$\begin{array}{ccc}\hfill |(Sm\left)\right(y)-(Sp\left)\right(y)|& \le & {\mathcal{K}}_3\parallel m-p\parallel +|q|+{\displaystyle \frac{1}{\Gamma \left(\eta \right)}}{\int }_0^y{\displaystyle \frac{1}{{(y-t)}^{1-\eta }}}|n_m\left(t\right)-m_p\left(t\right)|dt,\hfill \end{array}$$

(2)

and

$$\begin{array}{ccc}\hfill |n_m\left(y\right)-n_p\left(y\right)|& \le & |f(y,m\left(y\right),n_m\left(y\right))-f(y,m\left(y\right),n_p\left(y\right))|\hfill \\ \hfill & \le & {\mathcal{K}}_1|m\left(y\right)-p\left(y\right)|+{\mathcal{K}}_2|n_m\left(y\right)-n_p\left(y\right)|\hfill \\ & \le & {\displaystyle \frac{{\mathcal{K}}_1}{1-{\mathcal{K}}_2}}|m\left(y\right)-p\left(y\right)|,\hfill \end{array}$$

(3)

Replacing (3) in inequality (2), we get

$$\begin{array}{ccc}\hfill |(Sm\left)\right(y)-(Sp\left)\right(y)|& \le & {\mathcal{K}}_3\parallel m-p\parallel +|q|+{\displaystyle \frac{1}{\Gamma \left(\eta \right)}}{\displaystyle \frac{{\mathcal{K}}_1}{1-{\mathcal{K}}_2}}{\int }_0^y{\displaystyle \frac{1}{{(y-t)}^{1-\eta }}}|m\left(t\right)-p\left(t\right)|dt\hfill \\ & \le & {\mathcal{K}}_3\parallel m-p\parallel +|q|+{\displaystyle \frac{1}{\Gamma \left(\eta \right)}}{\displaystyle \frac{{\mathcal{K}}_1}{1-{\mathcal{K}}_2}}\left({\int }_0^y{\displaystyle \frac{1}{{(y-t)}^{1-\eta }}}dt\right)\parallel m-p\parallel \hfill \\ & \le & |q|+\left({\mathcal{K}}_3+{\displaystyle \frac{{\mathcal{K}}_1}{1-{\mathcal{K}}_2}}{\displaystyle \frac{y^{\eta }}{\Gamma (\eta -1)}}\right)\parallel m-p\parallel .\hfill \end{array}$$

Since $y\in (1,2)$, then

$$\parallel S_m-S_p\parallel \le |q|+\gamma \parallel m-p\parallel ,1<\gamma <2,$$

where

$$\gamma ={\mathcal{K}}_3+{\displaystyle \frac{{\mathcal{K}}_1}{1-{\mathcal{K}}_2}}{\displaystyle \frac{y^{\eta }}{\Gamma (\eta -1)}}.$$

Therefore, (1) has a unique solution. □

Theorem 3. 

Assume that condition $\left(H_1\right)$ is satisfied and the positive real numbers d and N exist in such a way that $|y^{\eta -1}f(y,m,w)|\le N$, $\forall (y,m,w)\in \mathbf{I}=[0,M]\times [-d_1,d_1]\times [q-d_2,q+d_2]$ with $d_1+d_2\le d$. Then, problem (1) permits at least one solution ${\mathfrak{C}}^{\eta }[0,M_0]$, where

$$M_0=\left\{\begin{array}{cc}M\hfill & \mathit{if}M<{\displaystyle \frac{d}{\mathfrak{C}(q,\eta ,N)}}\hfill \\ {\displaystyle \frac{d}{\mathfrak{C}(q,\eta ,N)}}\hfill & \mathit{if}M\ge {\displaystyle \frac{d}{\mathfrak{C}(q,\eta ,N)}}\ge 1\hfill \\ {\left[{\displaystyle \frac{d}{\mathfrak{C}(q,\eta ,N)}}\right]}^{\eta -1}\hfill & \mathit{if}M\ge {\displaystyle \frac{d}{\mathfrak{C}(q,\eta ,N)}},1\ge {\displaystyle \frac{d}{\mathfrak{C}(q,\eta ,N)}}\mathit{and}1<\eta \le 1.5\hfill \\ {\left[{\displaystyle \frac{d}{\mathfrak{C}(q,\eta ,N)}}\right]}^{2-\eta }\hfill & \mathit{if}M\ge {\displaystyle \frac{d}{\mathfrak{C}(q,\eta ,N)}},1\ge {\displaystyle \frac{d}{\mathfrak{C}(q,\eta ,N)}}\mathit{and}1.5\le \eta <2,\hfill \end{array}\right.$$

(4)

as well as

$$\mathfrak{C}(q,\eta ,N)=\left[|y_0-g\left(y\right)|+{\displaystyle \frac{|q|}{\Gamma \left(\eta \right)}}+N\left({\displaystyle \frac{1+\Gamma (3-\eta )}{2-\eta }}\right)\right].$$

The proof of this theorem is similar to the one given in [26], so we have omitted the details here. This theorem provides a useful tool for deriving various important results.

Theorem 4.

(Uniqueness) Suppose that $1<\eta <2,0<M<\infty$, as well as assuming that condition $\left(H_1\right)$ holds. Moreover, suppose that a positive real number $A\le {\displaystyle \frac{2-\eta }{M(1+\Gamma (3-\eta \left)\right)}}$ exists such that the inequality

$$y^{\eta -1}|f(y,t_{1,a},t_{2,a})-f(y,t_{1,b},t_{2,b})|\le A(|t_{1,a}-t_{1,b}|+|t_{2,a}-t_{2,b}|),$$

is satisfied for all $y\in [0,M]$, as well as for all $t_{1,a},t_{2,b}\in \mathfrak{R}$ with $a=1$ and $b=2$. Then, (1) has at least one solution in the space ${\mathfrak{C}}^{\eta -1}\left([0,M_0]\right)$.

Proof. 

For the uniqueness, we first suppose that (1) permits two distinct solutions, such as $m_1$ and $m_2$, in the space $C^{\eta -1}\left([0,M_0]\right)$. We define $\phi \left(y\right)$ as a function with the following form:

$$\phi \left(y\right)=\left\{\begin{array}{cc}|m_1\left(y\right)-m_2\left(y\right)|+|D^{\eta -1}{m}_1\left(y\right)-D^{\eta -1}{m}_2\left(y\right)|,\hfill & y>0\hfill \\ 0,\hfill & y=0.\hfill \end{array}\right.$$

Since $m_1,m_2\in C^{\eta -1}\left([0,M]\right)$, the continuity of $\phi \left(y\right)$ on $y\in (0,M_0]$ can be seen. With its continuity $y=0$,

$$\begin{array}{ccc}\hfill 0\le \underset{y\to 0^{+}}{lim}\phi \left(y\right)& =& \underset{y\to 0^{+}}{lim}{\displaystyle \frac{1}{\Gamma \left(\eta \right)}}|{\int }_0^y{\displaystyle \frac{f(t,m_1\left(t\right),D^{\eta -1}{m}_1\left(t\right))-f(t,m_2\left(t\right),D^{\eta -1}{m}_2\left(t\right))}{{(y-t)}^{1-\eta }}}dt|\hfill \\ & +& \underset{y\to 0^{+}}{lim}|{\int }_0^{y}f(t,m_1\left(t\right),D^{\eta -1}{m}_1\left(t\right))-f(t,m_2\left(t\right),D^{\eta -1}{m}_2\left(t\right))|\hfill \\ & \le & {\int }_0^1{\displaystyle \frac{{lim}_{y\to 0^{+}}y|B(y\mu ,m_1\left(y\mu \right))-B(y\mu ,m_2\left(y\mu \right))|}{{\mu }^{\eta -1}{(1-\mu )}^{1-\eta }}}d\mu \hfill \\ & +& {\int }_0^1{\displaystyle \frac{{lim}_{y\to 0^{+}}{y}^{2-\eta }|B(y\mu ,m_1\left(y\mu \right))-B(y\mu ,m_2\left(y\mu \right))|}{{\mu }^{\eta -1}}}d\mu =0,\hfill \end{array}$$

where $B(y,m\left(y\right))=y^{\eta -1}f(y,m\left(y\right),D^{\eta -1}m\left(y\right))$, and we changed the variable $t=y\mu$, as well as utilizing the condition $\left(H_1\right)$ specifically. Thus, ${lim}_{y\to 0^{+}}\phi \left(y\right)=0=\phi \left(0\right).$ Seeing that $\phi \left(y\right)\ge 0$ on $[0,M]$, let us choose a point $y_0\in (0,M_0]$ such that

$$0<\phi \left(y_0\right)=|m_1\left(y_0\right)-m_2\left(y_0\right)|+|D^{\eta -1}{m}_1\left(y_0\right)-D^{\eta -1}{m}_2\left(y_0\right)|.$$

Applying the mean value theorem as stated in Lemma (4),

$$\begin{array}{ccc}\hfill |m_1\left(y_0\right)-m_2\left(y_0\right)|& =& \Gamma (2-\eta )y_0|y_{0,a}^{\eta -1}{D}^{\eta }(m_1-m_2)\left(y_{0,a}\right)|+|D^{\eta -1}{m}_1\left(y_0\right)-D^{\eta -1}{m}_2\left(y_0\right)|\hfill \\ & =& \Gamma (2-\eta )y_0{y}_{0,a}^{\eta -1}|f(y_{0,a},m_1\left(y_{0,a}\right))-f(y_{0,a},m_2\left(y_{0,a}\right))|,\hfill \end{array}$$

(5)

is determined for $y_{0,a}\in (0,y_0)$, where $a=1$. Moreover, in order to calculate $|D^{\eta -1}{m}_1\left(y_0\right)-D^{\eta -1}{m}_2\left(y_0\right)|$ using the Integral Mean Value Theorem, a classic result in calculus, we obtain

$$\begin{array}{ccc}\hfill |D^{\eta -1}{m}_1\left(y_0\right)-D^{\eta -1}{m}_2\left(y_0\right)|& =& {\int }_0^{y_0}{\displaystyle \frac{t^{\eta -1}|f(t,m_1\left(t\right),D^{\eta -1}{m}_1\left(t\right))-f(t,m_2\left(t\right),D^{\eta -1}{m}_2\left(t\right))|}{t^{\eta -1}}}dt\hfill \\ & =& {\displaystyle \frac{y_0^{2-\eta }}{2-\eta }}{y}_{0,b}^{\eta -1}|f(y_{0,b},m_1\left(y_{0,b}\right))-f(y_{0,b},m_2\left(y_{0,b}\right))|,\hfill \end{array}$$

(6)

This is determined for $y_{0,b}\in (0,y_0)$ where $b=2$. We choose $y_1$ as one of the points $y_{0,a}$ or $y_{0,b}$ such that $|B(y_1,m_1\left(y_1\right))-B(y_1,m_2\left(y_1\right))|=max(|B(y_{0,a},m_1\left(y_{0,a}\right))-B(y_{0,a},m_2\left(y_{0,a}\right))|$, $|B(y_{0,b},m_1\left(y_{0,b}\right))-B(y_{0,b},m_2\left(y_{0,b}\right))|)$. Consequently, using Equations (5) and (6), we obtain

$$\begin{array}{ccc}\hfill 0& <& \phi \left(y_0\right)\le \left(\Gamma (2-\eta )y_0+{\displaystyle \frac{y_0^{2-\eta }}{2-\eta }}\right)|B(y_1,m_1\left(y_1\right))-B(y_1,m_2\left(y_1\right))|\hfill \\ & \le & M\left({\displaystyle \frac{1+\Gamma (3-\eta )}{2-\eta }}\right)y_1^{\eta -1}|f(y_1,m_1\left(y_1\right))-f(y_1,m_2\left(y_1\right))|\hfill \\ & \le & MA\left({\displaystyle \frac{1+\Gamma (3-\eta )}{2-\eta }}\right)y_1^{\eta -1}(|m_1\left(y_1\right)-m_2\left(y_1\right)|+|D^{\eta -1}{m}_1\left(y_1\right)-D^{\eta -1}{m}_2\left(y_1\right)|)=\phi \left(y_1\right),\hfill \end{array}$$

where $A\le {\displaystyle \frac{2-\eta }{1+\Gamma (3-\eta )}}$. Applying the same method to point $y_1$, as a result, we can calculate certain points $y_2\in (0,y_1)$ so that $0<\phi \left(y_0\right)\le \phi \left(y_1\right)\le \phi \left(y_2\right)$. Using the same method, the sequence ${\left\{y_r\right\}}_{r=1}^{\infty }\subset [0,y_0)$ can be defined such that $y_r\to 0$, as well as

$$0<\phi \left(y_0\right)\le \phi \left(y_1\right)\le \phi \left(y_2\right)\le \dots \le \phi \left(y_r\right)\dots$$

(7)

Therefore, $\phi \left(y\right)$ is continuous at $y=0$, as well as $y_r\to 0$ implying that $\phi \left(y_m\right)\to \phi \left(0\right)=0$. The contradiction with (7) implies that the initial value problem (1) has a unique solution. □

Theorem 5. 

Suppose that $1<\eta <2$, as well as $M_0^{*}=min\{M_0,1\}$, where $M_0$ is as it is defined in Equation (4). Assume that condition $\left(H_1\right)$ holds. Additionally, suppose a constant $A>0$ exists such that

$$y^{\eta -1}|f(y,t_{1,a},t_{2,a})-f(y,t_{1,b},t_{2,b})|\le {\displaystyle \frac{A}{2}}(|t_{1,a}-t_{1,b}|+|t_{2,a}-t_{2,b}|)$$

The inequality holds for all $y\in [0,M]$, as well as $\forall t_{1,a},t_{1,b},t_{2,a},t_{2,b}\in \mathfrak{R}$ where $a=1$ and $b=2$. Moreover, positive constants $\mathfrak{C}>0$ and $\epsilon \in (0,1)$ exist that satisfy the inequality $(1-\eta )(1-\epsilon )-A(1-\epsilon )+1>0$ so that

$$y^{\eta -1}|f(y,t_{1,a},t_{2,a})-f(y,t_{1,b},t_{2,b})|\le \mathfrak{C}(|t_{1,a}-t_{1,b}|+y^{\epsilon (\eta -1)}|t_{2,a}-t_{2,b}|)$$

(8)

The inequality holds for all $y\in [0,M]$, as well as $\forall t_{1,a},t_{1,b},t_{2,a},t_{2,b}\in \mathfrak{R}$ where $a=1$ and $b=2$. Consequently, problem (1) possesses a unique solution in ${\mathfrak{C}}^{\eta -1}\left([0,M_0^{*}]\right)$.

Proof. 

As stated in the previous theorem, we initially suppose that (1) has two distinct solutions, $m_1\left(y\right)$ and $m_2\left(y\right)$, which belong to space ${\mathfrak{C}}^{\eta -1}\left([0,M_0^{*}]\right)$. However, we will prove through contradiction that this is impossible. First, we need to define ${\phi }_1\left(y\right)=|m_1\left(y\right)-m_2\left(y\right)|$, as well as ${\phi }_2\left(y\right)=|D^{\eta -1}{m}_1\left(y\right)-D^{\eta -1}{m}_2\left(y\right)|$. Using condition $\left(H_1\right)$ and Equation (8), we will establish estimates for each function. First, we observe that

$$\begin{array}{ccc}\hfill {\phi }_1\left(y\right)& \le & {\displaystyle \frac{1}{\Gamma \left(\eta \right)}}{\int }_0^y{\displaystyle \frac{|f(t,m_1\left(t\right),D^{\eta -1}{m}_1\left(t\right))-f(t,m_2\left(t\right),D^{\eta -1}{m}_2\left(t\right))|}{{(y-t)}^{1-\eta }}}dt\hfill \\ & \le & {\displaystyle \frac{1}{\Gamma \left(\eta \right)}}{\int }_0^y{\displaystyle \frac{t^{\eta -1}f(t,m_1\left(t\right),D^{\eta -1}{m}_1\left(t\right))-f(t,m_2\left(t\right),D^{\eta -1}{m}_2\left(t\right))|}{t^{\eta -1}{(y-t)}^{1-\eta }}}dt\hfill \\ & \le & {\displaystyle \frac{1}{\Gamma \left(\eta \right)}}{\int }_0^y{\displaystyle \frac{{\phi }_1^{\epsilon }\left(y\right)+t^{\epsilon (\eta -1)}{\phi }_2^{\epsilon }\left(y\right)}{t^{\eta -1}{(y-t)}^{1-\eta }}}dt\hfill \\ & \le & {\displaystyle \frac{\mathfrak{C}}{\Gamma \left(\eta \right)}}{\int }_0^y{\displaystyle \frac{{\phi }_1^{\epsilon }\left(y\right)+t^{\epsilon (\eta -1)}{\phi }_2^{\epsilon }\left(y\right)}{t^{\eta -1}{(y-t)}^{1-\eta }}}dt\hfill \\ & \le & {\displaystyle \frac{\mathfrak{C}}{\Gamma \left(\eta \right)}}{\left({\int }_0^y{\left({\displaystyle \frac{1}{t^{\eta -1}{(y-t)}^{1-\eta }}}\right)}^{h}dt\right)}^{{\displaystyle \frac{1}{h}}}{\left({\int }_0^y{\left[{\phi }_1^{\epsilon }\left(t\right)+t^{\epsilon (\eta -1)}{\phi }_2^{\epsilon }\left(t\right)\right]}^{\ell }dt\right)}^{{\displaystyle \frac{1}{\ell }}}\hfill \\ & \le & {\displaystyle \frac{\mathfrak{C}\Gamma (1+(1-\eta \left)h\right)\Gamma (1+(\eta -1)\ell )}{\Gamma \left(\eta \right)}}{y}^{{\displaystyle \frac{1}{h}}}{\omega }^{{\displaystyle \frac{1}{\ell }}}\left(y\right),\hfill \end{array}$$

where $(1-\eta )h+1>0$ is satisfied by using Holder’s inequality with $h>1$. Moreover, $\omega \left(y\right)$ is defined as

$$\omega \left(y\right)={\int }_0^y{\left[{\phi }_1^{\epsilon }\left(t\right)+t^{\epsilon (\eta -1)}{\phi }_2^{\epsilon }\left(t\right)\right]}^{\ell }dt.$$

As a result, we can establish the following estimate:

$${\phi }_1^{\ell }\left(y\right)\le \mathfrak{C}{y}^{{\displaystyle \frac{\ell }{h}}}\omega \left(y\right),$$

(9)

where $\mathfrak{C}$ denotes a constant (whose exact value is not specified here or elsewhere in the proof). Furthermore, we can calculate an upper bound for ${\phi }_2\left(y\right)$ as follows:

$$\begin{array}{ccc}\hfill {\phi }_2\left(y\right)& \le & {\int }_0^y|f(t,m_1\left(t\right),D^{\eta -1}{m}_1\left(t\right))-f(t,m_2\left(t\right),D^{\eta -1}{m}_2\left(t\right))|dt\hfill \\ & \le & {\int }_0^y{\displaystyle \frac{t^{\eta -1}|f(t,m_1\left(t\right),D^{\eta -1}{m}_1\left(t\right))-f(t,m_2\left(t\right),D^{\eta -1}{m}_2\left(t\right))|}{t^{\eta -1}}}dt\hfill \\ & \le & {\displaystyle \frac{\mathfrak{C}}{\Gamma \left(\eta \right)}}{\int }_0^y{\displaystyle \frac{{\phi }_1^{\epsilon }\left(y\right)+t^{\epsilon (\eta -1)}{\phi }_2^{\epsilon }\left(y\right)}{t^{\eta -1}{(y-t)}^{1-\eta }}}dt\hfill \\ & \le & {\displaystyle \frac{\mathfrak{C}}{\Gamma \left(\eta \right)}}{\left({\int }_0^y{\left({\displaystyle \frac{1}{t^{\eta -1}}}\right)}^{h}dt\right)}^{{\displaystyle \frac{1}{h}}}{\left({\int }_0^y{\left[{\phi }_1^{\epsilon }\left(t\right)+t^{\epsilon (\eta -1)}{\phi }_2^{\epsilon }\left(t\right)\right]}^{\ell }dt\right)}^{{\displaystyle \frac{1}{\ell }}}\hfill \\ & \le & {\displaystyle \frac{\mathfrak{C}\Gamma (1+(1-\eta \left)h\right)}{\Gamma (2+(\eta -1\left)h\right)}}{y}^{{\displaystyle \frac{(1+h(1-\eta \left)\right)}{h}}}{\omega }^{{\displaystyle \frac{1}{\ell }}}\left(y\right).\hfill \end{array}$$

From the above process, we have

$${\phi }_2^{\ell }\left(y\right)\le \mathfrak{C}{y}^{{\displaystyle \frac{(1+h(1-\eta \left)\right)}{h}}}\omega \left(y\right),$$

(10)

and using the estimates from (9) and (10) in the derivative of $\omega \left(y\right)$, we obtain

$$\begin{array}{ccc}\hfill {\omega }^{\prime }\left(y\right)& =& {\left[{\phi }_1^{\epsilon }\left(t\right)+t^{\epsilon (\eta -1)}{\phi }_2^{\epsilon }\left(t\right)\right]}^{\ell }\le 2^{\ell -1}\left[{\left({\phi }_1^{\epsilon }\left(t\right)\right)}^{\ell }+t^{\ell \epsilon (\eta -1)}{\left({\phi }_2^{\epsilon }\left(t\right)\right)}^{\ell }\right]\hfill \\ & =& 2^{\ell -1}\left[{\left({\phi }_1^{\ell }\left(t\right)\right)}^{\epsilon }+t^{\ell \epsilon (\eta -1)}{\left({\phi }_2^{\ell }\left(t\right)\right)}^{\epsilon }\right]\hfill \\ & \le & 2^{\ell -1}[{\mathfrak{C}}^{\epsilon }{y}^{{\displaystyle \frac{\ell \epsilon }{h}}}{\omega }^{\epsilon }\left(y\right)+{\mathfrak{C}}^{\epsilon }{y}^{\ell \epsilon (\eta -1)}{y}^{{\displaystyle \frac{(1+h(1-\eta ))\frac{\ell \epsilon }{h}}{h}}}{\omega }^{\epsilon }\left(y\right)]\hfill \\ & \le & \mathfrak{C}{y}^{{\displaystyle \frac{\ell \epsilon }{h}}}{\omega }^{\epsilon }\left(y\right).\hfill \end{array}$$

If we multiply $(1-\epsilon ){\omega }^{-\epsilon }\left(y\right)$ on both sides of the following inequality, we obtain

$$(1-\epsilon ){\omega }^{-\epsilon }\left(y\right){\omega }^{\prime }\left(y\right)={\displaystyle \frac{d}{dy}}\left[{\omega }^{1-\epsilon }\left(y\right)\right]\le \mathfrak{C}{y}^{{\displaystyle \frac{\ell \epsilon }{h}}},$$

and taking the integral of both sides of the inequality over the interval $[0,y]$, we have

$${\omega }^{1-\epsilon }\left(y\right)\le \mathfrak{C}{y}^{{\displaystyle \frac{(\epsilon \ell +h)}{h}}},$$

and with the condition $\omega \left(0\right)=0$, this leads to the following estimate for $\omega \left(y\right)$:

$$\omega \left(y\right)\le \mathfrak{C}{y}^{{\displaystyle \frac{(\epsilon \ell +h)}{(1-\epsilon )h}}}.$$

(11)

Combining Equations (9)–(11) yields the conclusion that

$${\phi }_1^{\ell }\left(y\right)\le \mathfrak{C}{y}^{{\displaystyle \frac{\ell }{h}}}\omega \left(y\right)\le \mathfrak{C}{y}^{{\displaystyle \frac{\ell }{h}}}{y}^{{\displaystyle \frac{(\epsilon \ell +h)}{(1-\epsilon )\ell h}}}=\mathfrak{C}{y}^{{\displaystyle \frac{(\ell +h)}{(1-\epsilon )h}}},$$

or

$${\phi }_1\left(y\right)\le \mathfrak{C}{y}^{{\displaystyle \frac{(\ell +h)}{(1-\epsilon )\ell h}}}=y^{{\displaystyle \frac{1}{(1-\epsilon )}}},$$

(12)

as well as

$${\phi }_2^{\ell }\left(y\right)\le \mathfrak{C}{y}^{{\displaystyle \frac{\ell (1+h(1-\eta \left)\right)}{h}}}\omega \left(y\right)\le \mathfrak{C}{y}^{{\displaystyle \frac{\ell (1+h(1-\eta \left)\right)}{h}}}{y}^{{\displaystyle \frac{(\epsilon \ell +h)}{(1-\epsilon )\ell h}}}=\mathfrak{C}{y}^{{\displaystyle \frac{(1-\epsilon )(1-\eta )\ell h+\ell +h}{(1-\epsilon )h}}},$$

$${\phi }_2^{\ell }\le \mathfrak{C}{y}^{(1-\eta )+{\displaystyle \frac{\ell +h}{(1-\epsilon )\ell h}}}=y^{(1-\eta )+{\displaystyle \frac{1}{1-\epsilon }}},$$

(13)

as ${\displaystyle \frac{\ell +h}{\ell h}}=1$. Let us introduce the following inequality:

$$\xi \left(y\right)=y^{-A}max\{{\phi }_1\left(y\right),{\phi }_2\left(y\right)\},$$

where $A=(1-\epsilon )<1+(1-\eta )(1-\epsilon )$. If ${\phi }_1\left(y\right)=max\{{\phi }_1\left(y\right),{\phi }_2\left(y\right)\}$, then using Equation (12), we obtain

$$0\le \xi \left(y\right)\le y^{{\displaystyle \frac{1}{1-\epsilon }}-A},$$

or if ${\phi }_2\left(y\right)=max\{{\phi }_1\left(y\right),{\phi }_2\left(y\right)\}$, then from Equation (13), we have

$$0\le \xi \left(y\right)\le y^{(1-\eta )+{\displaystyle \frac{1}{1-\epsilon }}-A}=y^{{\displaystyle \frac{(1-\eta )(1-\epsilon )-A(1-\epsilon )+1}{1-\epsilon }}}.$$

$\xi \left(y\right)$ is continuous on $[0,M_0^{*}]$ in both situations, and $\xi \left(0\right)=0$. Next, we will prove that $\xi \left(y\right)\equiv 0$ on the interval $[0,M_0^{*}]$. Assume that $\xi \left(y\right)\not\equiv 0$. This implies that $\xi \left(y\right)=0$, and based on its continuity, there must be a point $y_1\in [0,M_0^{*}]$. Thus, let $Q=\xi \left(y_1\right)$ be the maximum value of $\xi \left(y\right)$ on the interval $[0,M_0^{*}]$.

$$Q=\xi \left(y_1\right)=\underset{y\in [0,M_0^{*}]}{max}\xi \left(y\right).$$

Suppose that $\xi \left(y\right)=y^{-A}{\phi }_1\left(y\right)$; we have

$$\begin{array}{ccc}\hfill Q& =& \xi \left(y_1\right)=y_1^{-A}{\phi }_1\left(y\right)\hfill \\ & \le & {\displaystyle \frac{y_1^{-A}}{\Gamma \left(\eta \right)}}{\int }_0^{y_1}{\displaystyle \frac{t^{\eta -1}|f(t,m_1\left(t\right),D^{\eta -1}{m}_1\left(t\right))-f(t,m_2\left(t\right),D^{\eta -1}{m}_2\left(t\right))|}{t^{\eta -1}{(y-t)}^{1-\eta }}}dt\hfill \\ & \le & {\displaystyle \frac{A{y}_1^{-A}}{2\Gamma \left(\eta \right)}}{\int }_0^{y_1}{(y-t)}^{\eta -1}{t}^{1-\eta +j}[{\phi }_1\left(t\right)+{\phi }_2\left(t\right)]dt\hfill \\ & \le & {\displaystyle \frac{A{y}_1^{-A}}{\Gamma \left(\eta \right)}}{\int }_0^{y_1}{(y-t)}^{\eta -1}{t}^{1-\eta +j}\xi \left(t\right)dt\hfill \\ & \le & Q{\displaystyle \frac{A\Gamma (2-\eta +A)}{\Gamma (2+A)}}{y}_1<Q,\hfill \end{array}$$

since the inequality ${\displaystyle \frac{A\Gamma (2-\eta +A)}{\Gamma (2+A)}}<1$ for $1<\eta <2$. Thus, this leads to a contradiction. Conversely, if $\xi \left(y\right)=y^{-A}{\phi }_2\left(y\right)$

$$\begin{array}{ccc}\hfill Q& =& \xi \left(y_1\right)=y_1^{-A}{\phi }_2\left(y\right)\hfill \\ & \le & y_1^{-A}{\int }_0^{y_1}{\displaystyle \frac{t^{\eta -1}|f(t,m_1\left(t\right),D^{\eta -1}{m}_1\left(t\right))-f(t,m_2\left(t\right),D^{\eta -1}{m}_2\left(t\right))|}{t^{\eta -1}}}dt\hfill \\ & \le & {\displaystyle \frac{A{y}_1^{-A}}{2}}{\int }_0^{y_1}{t}^{1-\eta +A}[{\phi }_1\left(t\right)+{\phi }_2\left(t\right)]dt\hfill \end{array}$$

$$\begin{array}{ccc}\hfill Q& \le & {A{y}_1}^{-A}{\int }_0^{y_1}{t}^{1-\eta +A}\xi \left(t\right)dt\hfill \\ & \le & Q{\displaystyle \frac{A}{A+2-\eta }}{y}_1<Q,\hfill \end{array}$$

which is also a contradiction.

As a result, we conclude that $\xi \left(y\right)\equiv 0$ on the interval $[0,M_0^{*}]$, which establishes the uniqueness of solutions to the given problem. □

Theorem 6. 

Suppose that $1<\eta <2$, as well as defining $M_0$ according to Equation (4), and that condition $\left(H_1\right)$ is fulfilled. Moreover, assume that for every $[0,M]$, as well as $\forall t_{1,a},t_{1,b},t_{2,a},t_{2,b}\in \mathfrak{R}$, where $a=1$ and $b=2$, the following inequality holds:

$$y^{\eta -1}|f(y,t_{1,a},t_{2,a})-f(y,t_{1,b},t_{2,b})|\le \mathfrak{C}(e(|t_{1,a}-t_{1,b}|+|t_{2,a}-t_{2,b}{|}^{\ell }{\left)\right)}^{{\displaystyle \frac{1}{\ell }}}$$

(14)

as long as $1+(1-\eta )h>0$ and $\ell >1$ is the conjugate of $h>1$.

$${\mathfrak{C}}^h\ge 2max\left(k{\Gamma }^h\left(\eta \right){[M_0\Gamma (1+(1-\eta )h)]}^{-1},(1+(1-\eta )h)M_0^{-1-h(1-\eta )}\right),$$

Moreover, e is a non-negative, non-decreasing, and continuous function in the interval $[0,M_0]$ with $e\left(0\right)=0$ that satisfies the following condition:

$$\underset{\varrho \to 0^{+}}{lim}{\int }_{\varrho }^{\alpha }{\displaystyle \frac{dv}{e\left(v\right)}}=\infty ,$$

(15)

where $\alpha \in \mathfrak{R}$. Consequently, Equation (1) possesses a unique solution in ${\mathfrak{C}}^{\eta -1}\left([0,M_0^{*}]\right)$.

Proof. 

Suppose that $m_1\left(y\right)$ and $m_2\left(y\right)$ are distinct solutions to problem (1) in space ${\mathfrak{C}}^{\eta -1}\left([0,M_0^{*}]\right)$. Furthermore, assume that ${\phi }_1\left(y\right)=|m_1\left(y\right)-m_2\left(y\right)|$ and ${\phi }_2\left(y\right)=|D^{\eta -1}{m}_1\left(y\right)-D^{\eta -1}{m}_2\left(y\right)|$. Starting with ${\phi }_1\left(y\right)$, we establish the following estimate:

$$\begin{array}{ccc}\hfill {\phi }_1\left(y\right)& \le & {\displaystyle \frac{1}{\Gamma \left(\eta \right)}}{\int }_0^y{\displaystyle \frac{t^{\eta -1}|f(t,m_1\left(t\right),D^{\eta -1}{m}_1\left(t\right))-f(t,m_2\left(t\right),D^{\eta -1}{m}_2\left(t\right))|}{t^{\eta -1}{(y-t)}^{1-\eta }}}dt\hfill \\ & \le & {\displaystyle \frac{\mathfrak{C}}{\Gamma \left(\eta \right)}}{\int }_0^y{\displaystyle \frac{\left[e\right(|m_1\left(y\right)-m_2{\left(y\right)|}^{\ell }+|D^{\eta -1}{m}_1\left(y\right)-D^{\eta -1}{m}_2{\left(y\right)|}^{\ell }{\left)\right]}^{\frac{1}{\ell }}}{t^{\eta -1}{(y-t)}^{1-\eta }}}dt\hfill \\ & \le & {\displaystyle \frac{\mathfrak{C}}{\Gamma \left(\eta \right)}}{\left({\int }_0^y{\left({\displaystyle \frac{1}{t^{\eta -1}{(y-t)}^{1-\eta }}}\right)}^{h}dt\right)}^{{\displaystyle \frac{1}{h}}}{\left({\int }_0^{y}e{\left[{\phi }_1^{\ell }\left(t\right)+{\phi }_2^{\ell }\left(t\right)\right]}^{\ell }dt\right)}^{{\displaystyle \frac{1}{\ell }}}\hfill \\ & \le & \mathfrak{C}{\left[{\displaystyle \frac{\Gamma (1+(1-\eta \left)h\right)\Gamma (1+(\eta -1\left)h\right)}{{\Gamma }^h\left(\eta \right)}}\right]}^{{\displaystyle \frac{1}{h}}}{y}^{{\displaystyle \frac{1}{h}}}{\left({\int }_0^{y}e{\left[{\phi }_1^{\ell }\left(t\right)+{\phi }_2^{\ell }\left(t\right)\right]}^{\ell }dt\right)}^{{\displaystyle \frac{1}{\ell }}}\hfill \\ & \le & {{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{1}{\ell }}}{\left({\int }_0^{y}e{\left[{\phi }_1^{\ell }\left(t\right)+{\phi }_2^{\ell }\left(t\right)\right]}^{\ell }dt\right)}^{{\displaystyle \frac{1}{\ell }}},\hfill \end{array}$$

where we apply Equation (14), followed by Holder’s inequality, as well as the assumption on $\mathfrak{C}$.

As a result, we obtain

$${\phi }_1^{\ell }\left(y\right)\le {\displaystyle \frac{1}{2}}{\int }_0^{y}e\left[{\phi }_1^{\ell }\left(t\right)+{\phi }_2^{\ell }\left(t\right)\right]dt.$$

(16)

In a similar manner, we obtain

$$\begin{array}{ccc}\hfill {\phi }_2\left(y\right)& \le & {\int }_0^y{\displaystyle \frac{t^{\eta -1}|f(t,m_1\left(t\right),D^{\eta -1}{m}_1\left(t\right))-f(t,m_2\left(t\right),D^{\eta -1}{m}_2\left(t\right))|}{t^{\eta -1}}}dt\hfill \\ & \le & \mathfrak{C}{\int }_0^y{\displaystyle \frac{\left[e\right(|m_1\left(y\right)-m_2{\left(y\right)|}^{\ell }+|D^{\eta -1}{m}_1\left(y\right)-D^{\eta -1}{m}_2{\left(y\right)|}^{\ell }{\left)\right]}^{\frac{1}{\ell }}}{t^{\eta -1}}}dt\hfill \\ & \le & \mathfrak{C}{\left({\int }_0^y{\left({\displaystyle \frac{1}{t^{\eta -1}}}\right)}^{h}dt\right)}^{{\displaystyle \frac{1}{h}}}{\left({\int }_0^{y}e\left[{\phi }_1^{\ell }\left(t\right)+{\phi }_2^{\ell }\left(t\right)\right]dt\right)}^{{\displaystyle \frac{1}{\ell }}}\hfill \\ & \le & \mathfrak{C}{\left[{\displaystyle \frac{1}{(1+(1-\eta \left)h\right)}}\right]}^{{\displaystyle \frac{1}{h}}}{y}^{{\displaystyle \frac{(1+(1-\eta \left)h\right)}{h}}}{\left({\int }_0^{y}e\left[{\phi }_1^{\ell }\left(t\right)+{\phi }_2^{\ell }\left(t\right)\right]dt\right)}^{{\displaystyle \frac{1}{\ell }}}\hfill \\ & \le & {\displaystyle \frac{1}{2^{\frac{1}{\ell }}}}{\left({\int }_0^{y}e{\left[{\phi }_1^{\ell }\left(t\right)+{\phi }_2^{\ell }\left(t\right)\right]}^{\ell }dt\right)}^{{\displaystyle \frac{1}{\ell }}},\hfill \end{array}$$

and from this, we can conclude that

$${\phi }_2^{\ell }\left(y\right)\le {\displaystyle \frac{1}{2}}{\int }_0^{y}e\left[{\phi }_1^{\ell }\left(t\right)+{\phi }_2^{\ell }\left(t\right)\right]dt.$$

(17)

Now, consider

$$\xi \left(y\right):=\underset{0\le t\le y}{max}[{\phi }_1^{\ell }\left(t\right)+{\phi }_2^{\ell }\left(t\right)],$$

and let $\xi \left(y\right)>0$ for $y\in (0,M_0]$. We will prove that this is impossible under the given conditions.

It is clear, according to the definition of $\xi \left(y\right)$, that for every $y\in [0,M_0]$, ${\phi }_1^{\ell }\left(y\right)+{\phi }_2^{\ell }\left(y\right)\le \xi \left(y\right)$. Moreover, $y_1\le y$ exists such that $\xi \left(y\right)={\phi }_1^{\ell }\left(y\right)+{\phi }_2^{\ell }\left(y\right)$. Next, based on Equations (15)–(17) and the observation that e is an increasing function, we obtain

$$\xi \left(y\right)={\phi }_1^{\ell }\left(y\right)+{\phi }_2^{\ell }\left(y\right)\le {\int }_0^{y_1}e\left[{\phi }_1^{\ell }\left(t\right)+{\phi }_2^{\ell }\left(t\right)\right]dt\le {\int }_0^{y}e\xi \left(y\right)dt:={\xi }_{*}\left(y\right).$$

We observe that $\xi \left(y\right)\le {\xi }_{*}\left(y\right)$. Furthermore, we obtain

$${\displaystyle \frac{d}{dy}}{\xi }_{*}\left(y\right)=e\left(\xi \left(y\right)\right)\le e\left({\xi }_{*}\left(y\right)\right),$$

for every $y\in [0,M_0]$. As a consequence, for a sufficiently small positive value of $\rho$, we can conclude that

$${\int }_{\rho }^{M_0}{\displaystyle \frac{{\xi }_{*}^{\prime }\left(y\right)}{e\left({\xi }_{*}\left(y\right)\right)}}dy\le M_0-\rho .$$

Moreover, by applying the substitution $v={\xi }_{*}\left(y\right)$ in the integral and using the facts that ${\xi }_{*}\left(y\right)$ is continuous and ${\xi }_{*}\left(0\right)=0$, we obtain

$${\int }_{\varrho }^{\alpha }{\displaystyle \frac{dv}{e\left(v\right)}}\le M_0-\rho .$$

for $\varrho >0$ that is sufficiently small, with $\varrho ={\xi }_{*}\left(y\right)$, as well as for $\alpha ={\xi }_{*}\left(M_0\right)$. However, this result contradicts the supposition on e stated in Equation (15). Thus, $\xi \left(y\right)=0$ for $y\in [0,M_0]$, i.e., $m_1\left(y\right)=m_2\left(y\right)$. □

Remark 1. 

As mentioned in Theorem (1.4.3) of [19], the assumption that $e\left(v\right)$ is increasing is not essential and can be omitted.

4. Conclusions

Our groundbreaking research ventures into the uncharted territory of nonlinear fractional derivatives and RL-derivatives, unveiling the intricate relationships and behaviors that govern complex systems. By rigorously examining the existence and uniqueness of solutions to nonlinear FDEs, we develop a revolutionary framework that empowers scientists and researchers to unravel the enigmas of RL-fractional derivatives. We establish sufficient conditions for the existence and uniqueness of the solutions to nonlinear FDEs, defined in terms of RL-derivatives, when the right-hand-side function exhibits a discontinuity at zero. Our results build upon the existing literature and provide a foundation for further generalizations and improvements. By exploring these findings, we aim to contribute to the advancement of the field.