Some New Weakly Singular Integral Inequalities with Applications to Differential Equations in Frame of Tempered χ-Fractional Derivatives

Abstract

In this study, we develop some novel Integral Inequalities (InIs) with weakly singular singularities that expand some commonly known ones. Utilizing tempered $\chi$-Fractional Differential Equations (FDEs), many applications for FDEs in the context of Caputo have been developed.

1. Introduction

Fractional-order systems (FOS) are dynamical systems that can be modeled by a fractional differential equation carried with a non-integer derivative. FOS are advantageous in studying the behavior of dynamical systems in electrochemistry, physics, viscoelasticity, biology, and chaotic systems. In the last few decades, the growth of science and engineering systems has considerably stimulated the employment of fractional calculus in many subjects of the control theory; for example, in stability, stabilization, controllability, observability, observer design, and fault estimation. Indeed, the application of control theory in fractional-order systems is an important issue in many engineering applications [1,2,3,4].

The Gronwall inequality establishes explicit limits for the solutions of certain classes of InIs; see [5,6] for a discussion of the significance of InIs in the qualitative analysis of differential and integral equations. This inequality has been developed and implemented in several scenarios [5,7,8,9]. The inequalities, however, cannot be directly applied to the analysis of integral equations with weakly singular kernels. In 1981, the researcher of [10] presented a novel method for obtaining optimal InIs with weakly Singular Kernels (IISKs). Medved devised a novel approach for solving InIs of the Henry–Gronwall and Bihari types [11], and global solutions to semilinear evolution equations [12]. Due to the expansion of FDEs, IISKs have lately garnered more study interest; see, for instance [7,9,10,11,12,13,14,15,16,17]. In fact, Medved and Ma, as well as Pecaric have studied the InIs listed below in [11,15]:

$$r\left(\eta \right)\le a\left(\eta \right)+b\left(\eta \right){\int }_0^{\eta }{\left(\eta -\upsilon \right)}^{\gamma -1}U\left(\upsilon \right)r\left(\upsilon \right)d\upsilon .$$

(1)

Medved has researched InIs of the Henry type in [11,12]:

$$r\left(\eta \right)\le a\left(\eta \right)+{\int }_0^{\eta }{\left(\eta -\upsilon \right)}^{\gamma -1}U\left(\upsilon \right)w\left(r\left(\upsilon \right)\right)d\upsilon .$$

(2)

Subsequently, in [9], Zhu studied InIs (1) and (2) and examined the existence of solutions to fractional differential equations with Caputo derivatives as an illustration. Tempered Fractional Calculus (FC) is an expansion of FC characterized by integrals, in which the kernel begins with a fractional power function multiplied by an exponential factor [18,19]. One of the most prominent instances are the tempered fractional diffusion equations, which produce the limits of random walk models with exponentially tempered power law jump distributions by replacing the standard second derivative in space with a tempered fractional derivative [20]. Tempered power law waiting times give rise to tempered fractional time derivatives, which are helpful in geophysics and other scientific areas [20]. In this context, there have been an increasing number of applications for tempered fractional differential equations, including banking [21], photoelasticity [22], and geophysics and fluxes [23].

The following points summarize the main contribution of the work:

◇

Motivated by [9,16], some new weakly singular InIs are established.

◇

The existence, uniqueness, and the Ulam stability of the solutions of the following fractional differential equations are investigated:

$$\left\{\begin{array}{c}{}^C{D}_{0^{+}}^{\sigma ,\lambda ,\chi }u\left(\varrho \right)=f\left(\varrho ,u\left(\varrho \right)\right),\lambda >0,\sigma \in \left(0,1\right),\varrho \in \left[0,T\right],\hfill \\ u\left(0\right)=u_0,\hfill \end{array}\right.$$

(3)

where ${}^C{D}_{a^{+}}^{\sigma ,\lambda ,\chi }$ is the tempered χ-Caputo fractional derivative and the function $f:[0,T]\times \mathbb{R}\to \mathbb{R}$.

According to our knowledge, there are no other published works on the same topic.

The structure of the paper is as follows. Section 2 presents the introduction. Section 3 discusses the key findings. Section 4 discusses fractional differential equation applications.

2. Preliminaries

In this section, let us revisit some basics of the fractional calculus. We adopt the notations of the tempered $\chi$-fractional integral and derivative.

 Definition 1

([23]). Let $\sigma >0,\lambda \ge 0$, $u\left(t\right)$ be a continuous function on $[a,b]$, and $\chi \left(t\right)\in C^1[a,b]$ is an increasing function, such that ${\chi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$. Then, the tempered $\chi -$ fractional integral of order $\sigma$ of $u\left(t\right)$ is defined by

$$\begin{array}{cc}\hfill I_{a^{+}}^{\sigma ,\lambda ,\chi }u\left(t\right)& =\exp (-\lambda \chi \left(t\right))I_{a^{+}}^{\sigma ,\chi }\left(\exp \left(\lambda \chi \right(t\left)\right)u\left(t\right)\right)\hfill \\ \hfill & =\frac{1}{\Gamma \left(\sigma \right)}{\int }_a^t{\left(\chi \left(t\right)-\chi \left(s\right)\right)}^{\sigma -1}\exp \left(-\lambda \left(\chi \left(t\right)-\chi \left(s\right)\right)\right){\chi }^{\prime }\left(s\right)u\left(s\right)ds,\hfill \end{array}$$

(4)

where $I_{a^{+}}^{\sigma ,\chi }$ is the $\chi$-Riemann-Liouville fractional integral of order $\sigma$.

 Definition 2

([23]). Let $n\in \mathbb{N},n-1<\sigma <n,\lambda \ge 0,u\left(t\right)\in A{C}^n[a,b]$, and $\chi \left(t\right)\in C^n[a,b]$ is an increasing function such that ${\chi }^{\prime }\left(t\right)\ne 0$ for all $t\in [a,b]$. The tempered χ-Caputo fractional derivative of order $\sigma$ of $u\left(t\right)$ is defined by

$${}^C{D}_{a^{+}}^{\sigma ,\lambda ,\chi }u\left(t\right)=\frac{\exp (-\lambda \chi (t\left)\right)}{\Gamma (n-\nu )}{\int }_a^t{\chi }^{\prime }\left(s\right){\left(\chi \left(t\right)-\chi \left(s\right)\right)}^{n-\sigma -1}{u}_{\lambda ,\chi }^{\left[n\right]}\left(s\right)ds,$$

where

$$u_{\lambda ,\chi }^{\left[n\right]}\left(t\right)={\left[\frac{1}{{\chi }^{\prime }\left(t\right)}\frac{d}{dt}\right]}^n\left(\exp \left(\lambda \chi \right(t\left)\right)u\left(t\right)\right).$$

3. Main Results

In the following theorems, we assume that $\chi \left(\varrho \right)\in C^1[a,b]$ is an increasing function, with ${\chi }^{\prime }\left(\varrho \right)\ne 0$ for all $\varrho \in [a,b]$.

 Theorem 1.

Let $\lambda >0,$ $0<T\le \infty ,$ $\sigma \in \left(0,1\right),$ $a\left(\varrho \right)$, $u\left(\varrho \right)$, and $w\left(\varrho \right)$ are continuous, nonnegative functions on $\left[0,T\right)$, with

$$u\left(\varrho \right)\le a\left(\varrho \right)+\frac{1}{\Gamma \left(\sigma \right)}{\int }_0^{\varrho }{\chi }^{\prime }\left(\iota \right)\exp \left(-\lambda \left[\chi \left(\varrho \right)-\chi \left(\iota \right)\right]\right){\left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)}^{\sigma -1}w\left(\iota \right)u\left(\iota \right)d\iota .$$

(5)

Then,

$$u\left(\varrho \right)\le {\left(\mathcal{N}\left(\varrho \right)+\exp \left(\frac{-\lambda }{\nu }\chi \left(\varrho \right)\right){\int }_0^{\varrho }\exp \left(\frac{\lambda }{\nu }\chi \left(\iota \right)\right)\mathcal{T}\left(\iota \right)\mathcal{N}\left(\iota \right)\exp \left({\int }_{\iota }^{\varrho }\mathcal{T}\left(\varsigma \right)d\varsigma \right)d\iota \right)}^{\nu },\varrho \in \left[0,T\right).$$

(6)

If $a\left(\varrho \right)$ is nondecreasing on $\left[0,T\right),$ then the inequality (6) becomes

$$u\left(\varrho \right)\le {\left(\mathcal{N}\left(\varrho \right)\exp \left({\int }_0^{\varrho }\mathcal{T}\left(\iota \right)d\iota \right)\right)}^{\nu }.$$

If $a\left(\varrho \right)\equiv 0$ on $\left[0,T\right),$ we find

$$u\left(\varrho \right)\equiv 0,$$

where $\mathcal{N}\left(\varrho \right)=2^{\frac{1}{\nu }-1}{a}^{\frac{1}{\nu }}\left(\varrho \right),$ $\mathcal{T}\left(\varrho \right)=\frac{2^{\frac{1}{\nu }-1}}{{\Gamma }^{\frac{1}{\nu }}\left(\sigma \right)}{\left(\Gamma \left(\frac{\sigma -\nu }{1-\nu }\right)\Gamma \left(\frac{1-\sigma }{1-\nu }\right)\right)}^{\frac{1-\nu }{\nu }}{\chi }^{\prime }\left(\varrho \right){\left(\chi \left(\varrho \right)-\chi \left(0\right)\right)}^{\frac{\sigma -\nu }{\nu }}{w}^{\frac{1}{\nu }}\left(\varrho \right),$and $0<\nu <\sigma <1.$

 Proof. 

Using the Holder inequality and (5), we obtain

$$\begin{array}{ccc}\hfill u\left(\varrho \right)& \le & a\left(\varrho \right)+\frac{\exp \left(-\lambda \chi \left(\varrho \right)\right)}{\Gamma \left(\sigma \right)}{\int }_0^{\varrho }{\left({\chi }^{\prime }\left(\iota \right)\right)}^{1-\nu }{\left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)}^{\sigma -1}{\left(\chi \left(\iota \right)-\chi \left(0\right)\right)}^{\nu -\sigma }\hfill \\ \hfill & & \times {\left({\chi }^{\prime }\left(\iota \right)\right)}^{\nu }\exp \left(\lambda \chi \left(\iota \right)\right){\left(\chi \left(\iota \right)-\chi \left(0\right)\right)}^{\sigma -\nu }w\left(\iota \right)u\left(\iota \right)d\iota \hfill \\ \hfill & \le & a\left(\varrho \right)+\frac{\exp \left(-\lambda \chi \left(\varrho \right)\right)}{\Gamma \left(\sigma \right)}{\left({\int }_0^{\varrho }{\left[{\left({\chi }^{\prime }\left(\iota \right)\right)}^{1-\nu }{\left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)}^{\sigma -1}{\left(\chi \left(\iota \right)-\chi \left(0\right)\right)}^{\nu -\sigma }\right]}^{\frac{1}{1-\nu }}d\iota \right)}^{1-\nu }\hfill \\ \hfill & & \times {\left({\int }_0^{\varrho }{\left[{\left({\chi }^{\prime }\left(\iota \right)\right)}^{\nu }\exp \left(\lambda \chi \left(\iota \right)\right){\left(\chi \left(\iota \right)-\chi \left(0\right)\right)}^{\sigma -\nu }w\left(\iota \right)u\left(\iota \right)\right]}^{\frac{1}{\nu }}d\iota \right)}^{\nu }\hfill \\ \hfill & \le & a\left(\varrho \right)+\frac{\exp \left(-\lambda \chi \left(\varrho \right)\right)}{\Gamma \left(\sigma \right)}{\left({\int }_0^{\varrho }\left[\left({\chi }^{\prime }\left(\iota \right)\right){\left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)}^{\frac{\sigma -\nu }{1-\nu }-1}{\left(\chi \left(\iota \right)-\chi \left(0\right)\right)}^{\frac{\nu -\sigma }{1-\nu }}\right]d\iota \right)}^{1-\nu }\hfill \\ \hfill & & \times {\left({\int }_0^{\varrho }\left[{\chi }^{\prime }\left(\iota \right)\exp \left(\frac{\lambda }{\nu }\chi \left(\iota \right)\right){\left(\chi \left(\iota \right)-\chi \left(0\right)\right)}^{\frac{\sigma -\nu }{\nu }}{w}^{\frac{1}{\nu }}\left(\iota \right)u^{\frac{1}{\nu }}\left(\iota \right)\right]d\iota \right)}^{\nu }\hfill \\ \hfill & \le & a\left(\varrho \right)+\frac{\exp \left(-\lambda \chi \left(\varrho \right)\right)}{\Gamma \left(\sigma \right)}{\left(\Gamma \left(\frac{\sigma -\nu }{1-\nu }\right)\Gamma \left(\frac{1-\sigma }{1-\nu }\right)\right)}^{1-\nu }\hfill \\ \hfill & & \times {\left({\int }_0^{\varrho }\left[{\chi }^{\prime }\left(\iota \right)\exp \left(\frac{\lambda }{\nu }\chi \left(\iota \right)\right){\left(\chi \left(\iota \right)-\chi \left(0\right)\right)}^{\frac{\sigma -\nu }{\nu }}{w}^{\frac{1}{\nu }}\left(\iota \right)u^{\frac{1}{\nu }}\left(\iota \right)\right]d\iota \right)}^{\nu }\hfill \end{array}$$

Taking into consideration that ${\left(z_1+z_2\right)}^{\mu }\le 2^{\mu -1}\left(z_1^{\mu }+z_2^{\mu }\right)$ for every $\left(z_1,z_2\right)\in {\mathbb{R}}_{+}^2$ and $\mu \ge 1,$ we obtain

$$\begin{array}{ccc}\hfill u^{\frac{1}{\nu }}\left(\varrho \right)& \le & 2^{\frac{1}{\nu }-1}(a^{\frac{1}{\nu }}\left(\varrho \right)+\frac{\exp \left(-\frac{\lambda }{\nu }\chi \left(\varrho \right)\right)}{{\Gamma }^{\frac{1}{\nu }}\left(\sigma \right)}{\left(\Gamma \left(\frac{\sigma -\nu }{1-\nu }\right)\Gamma \left(\frac{1-\sigma }{1-\nu }\right)\right)}^{\frac{1-\nu }{\nu }}\hfill \\ \hfill & & \times {\int }_0^{\varrho }\left[{\chi }^{\prime }\left(\iota \right)\exp \left(\frac{\lambda }{\nu }\chi \left(\iota \right)\right){\left(\chi \left(\iota \right)-\chi \left(0\right)\right)}^{\frac{\sigma -\nu }{\nu }}{w}^{\frac{1}{\nu }}\left(\iota \right)u^{\frac{1}{\nu }}\left(\iota \right)\right]d\iota )\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill \exp \left(\frac{\lambda }{\nu }\chi \left(\varrho \right)\right)u^{\frac{1}{\nu }}\left(\varrho \right)\le & 2^{\frac{1}{\nu }-1}(\exp \left(\frac{\lambda }{\nu }\chi \left(\varrho \right)\right)a^{\frac{1}{\nu }}\left(\varrho \right)+\frac{1}{{\Gamma }^{\frac{1}{\nu }}\left(\sigma \right)}{\left(\Gamma \left(\frac{\sigma -\nu }{1-\nu }\right)\Gamma \left(\frac{1-\sigma }{1-\nu }\right)\right)}^{\frac{1-\nu }{\nu }}\hfill \\ \hfill & \times {\int }_0^{\varrho }\left[{\chi }^{\prime }\left(\iota \right)\exp \left(\frac{\lambda }{\nu }\chi \left(\iota \right)\right){\left(\chi \left(\iota \right)-\chi \left(0\right)\right)}^{\frac{\sigma -\nu }{\nu }}{w}^{\frac{1}{\nu }}\left(\iota \right)u^{\frac{1}{\nu }}\left(\iota \right)\right]d\iota )\hfill \end{array}$$

By taking $\mathcal{K}\left(\varrho \right)=\exp \left(\frac{\lambda }{\nu }\chi \left(\varrho \right)\right)u^{\frac{1}{\nu }}\left(\varrho \right),$ the above inequality becomes

$$\mathcal{K}\left(\varrho \right)\le \exp \left(\frac{\lambda }{\nu }\chi \left(\varrho \right)\right)\mathcal{N}\left(\varrho \right)+{\int }_0^{\varrho }\mathcal{T}\left(\iota \right)\mathcal{K}\left(\iota \right)d\iota .$$

(7)

By Lemma 2.2 in [22], we obtain (6). The rest of the proof is clear. □

 Theorem 2.

Let $\lambda ,\sigma >0,$ $0<T\le \infty ,$ $a\left(\varrho \right),b\left(\varrho \right),$ $u\left(\varrho \right)$ and $w\left(\varrho \right)$ are continuous, nonnegative functions on $\left[0,T\right)$ with

$$u\left(\varrho \right)\le a\left(\varrho \right)+\frac{b\left(\varrho \right)}{\Gamma \left(\sigma \right)}{\int }_0^{\varrho }{\chi }^{\prime }\left(\iota \right)\exp \left(-\lambda \left[\chi \left(\varrho \right)-\chi \left(\iota \right)\right]\right){\left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)}^{\sigma -1}w\left(\iota \right)u\left(\iota \right)d\iota .$$

(8)

Then,

$$u\left(\varrho \right)\le a\left(\varrho \right)+\mathcal{V}\left(\varrho \right)\frac{{\left({\int }_0^{\varrho }\mathcal{T}\left(\iota \right)\mathcal{M}\left(\iota \right)a^p\left(\iota \right)d\iota \right)}^{\frac{1}{p}}}{1-{\left[1-\mathcal{M}\left(\varrho \right)\right]}^{\frac{1}{p}}},\varrho \in \left[0,T\right),$$

(9)

where $\mathcal{T}\left(\varrho \right)={\chi }^{\prime }\left(\varrho \right)\exp \left(\lambda \chi \left(\varrho \right)\right)w^p\left(\varrho \right),$

$\mathcal{V}\left(\varrho \right)=\frac{b\left(\varrho \right)\exp \left(-\lambda \chi \left(\varrho \right)\right)}{\Gamma \left(\sigma \right)}{\left(\Gamma \left(\gamma \right)\exp \left(\lambda \chi \left(0\right)\right){\left(\chi \left(\varrho \right)-\chi \left(0\right)\right)}^{\gamma }{\mathbb{E}}_{1,\gamma +1}\left(\lambda \left(\chi \left(\varrho \right)-\chi \left(0\right)\right)\right)\right)}^{\frac{1}{q}}$,

$\mathcal{M}\left(\varrho \right)=\exp \left(-{\int }_0^{\varrho }\mathcal{T}\left(\iota \right){\mathcal{V}}^p\left(\iota \right)d\iota \right)$, and ${\mathbb{E}}_{1,\gamma +1}$ is the general Mittag-Leffler function, and

$p,q\in \left(0,\infty \right)$ with $\frac{1}{q}+\sigma >1$, $\frac{1}{q}+\frac{1}{p}=1$ and $\gamma =\left(\sigma -1\right)q+1.$

 Proof. 

Due to the Holder inequality and the fact that

$$\frac{1}{\Gamma \left(\beta \right)}\left({\int }_a^{\varrho }\exp \left(\lambda \iota \right){\left(\varrho -\iota \right)}^{\beta -1}d\iota \right)=\exp \left(\lambda a\right){(\varrho -a)}^{\beta }{\mathbb{E}}_{1,\beta +1}\left(\lambda (\varrho -a)\right),\mathrm{for}a\in \mathbb{R}\mathrm{and}\beta >0,$$

we get

$$\begin{array}{cc}\hfill u\left(\varrho \right)\le & a\left(\varrho \right)+\frac{b\left(\varrho \right)}{\Gamma \left(\sigma \right)}{\int }_0^{\varrho }{\chi }^{\prime }\left(\iota \right)\exp \left(-\lambda \left[\chi \left(\varrho \right)-\chi \left(\iota \right)\right]\right){\left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)}^{\sigma -1}w\left(\iota \right)u\left(\iota \right)d\iota \hfill \\ \hfill \le & a\left(\varrho \right)+\frac{b\left(\varrho \right)\exp \left(-\lambda \chi \left(\varrho \right)\right)}{\Gamma \left(\sigma \right)}{\left({\int }_0^{\varrho }{\chi }^{\prime }\left(\iota \right)\exp \left(\lambda \chi \left(\iota \right)\right){\left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)}^{(\sigma -1)q}d\iota \right)}^{\frac{1}{q}}\hfill \\ \hfill & \times {\left({\int }_0^{\varrho }{\chi }^{\prime }\left(\iota \right)\exp \left(\lambda \chi \left(\iota \right)\right)w^p\left(\iota \right)u^p\left(\iota \right)d\iota \right)}^{\frac{1}{p}}\hfill \\ \hfill \le & a\left(\varrho \right)+\frac{b\left(\varrho \right)\exp \left(-\lambda \chi \left(\varrho \right)\right)}{\Gamma \left(\sigma \right)}{\left(\Gamma \left(\gamma \right)\exp \left(\lambda \chi \left(0\right)\right){\left(\chi \left(\varrho \right)-\chi \left(0\right)\right)}^{\gamma }{\mathbb{E}}_{1,\gamma +1}\left(\lambda \left(\chi \left(\varrho \right)-\chi \left(0\right)\right)\right)\right)}^{\frac{1}{q}}\hfill \\ \hfill & \times {\left({\int }_0^{\varrho }{\chi }^{\prime }\left(\iota \right)\exp \left(\lambda \chi \left(\iota \right)\right)w^p\left(\iota \right)u^p\left(\iota \right)d\iota \right)}^{\frac{1}{p}}.\hfill \end{array}$$

So,

$$u\left(\varrho \right)\le a\left(\varrho \right)+\mathcal{V}\left(\varrho \right){\left({\int }_0^{\varrho }\mathcal{T}\left(\iota \right)u^p\left(\iota \right)d\iota \right)}^{\frac{1}{p}}.$$

(10)

The inequality (9) is obtained by using Lemma $2.3$ in [22]. □

 Theorem 3.

Let $\lambda ,\sigma >0,$ $0<T\le \infty ,$ $a\left(\varrho \right),b\left(\varrho \right),$ $u\left(\varrho \right)$ and $w\left(\varrho \right)$ are continuous, nonnegative functions on $\left[0,T\right)$, with

$$u\left(\varrho \right)\le a\left(\varrho \right)+\frac{b\left(\varrho \right)}{\Gamma \left(\sigma \right)}{\int }_0^{\varrho }{\chi }^{\prime }\left(\iota \right)\exp \left(-\lambda \left[\chi \left(\varrho \right)-\chi \left(\iota \right)\right]\right){\left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)}^{\sigma -1}w\left(\iota \right)u\left(\iota \right)d\iota .$$

(11)

Then,

$$u\left(\varrho \right)\le {\left(\mathcal{N}\left(\varrho \right)+\mathcal{V}\left(\varrho \right){\int }_0^{\varrho }\mathcal{T}\left(\iota \right)\mathcal{N}\left(\iota \right)\exp \left({\int }_{\iota }^{\varrho }\mathcal{T}\left(\varsigma \right)\mathcal{V}\left(\varsigma \right)d\varsigma \right)d\iota \right)}^{\frac{1}{p}},x\in \left[0,T\right),$$

(12)

where$\mathcal{V}\left(\varrho \right)=2^{p-1}{\left(\frac{b\left(\varrho \right)\exp \left(-\lambda \chi \left(\varrho \right)\right)}{\Gamma \left(\sigma \right)}{\left(\Gamma \left(\gamma \right)\exp \left(\lambda \chi \left(0\right)\right){\left(\chi \left(\varrho \right)-\chi \left(0\right)\right)}^{\gamma }{\mathbb{E}}_{1,\gamma +1}\left(\lambda \left(\chi \left(\varrho \right)-\chi \left(0\right)\right)\right)\right)}^{\frac{1}{q}}\right)}^p,$$\mathcal{N}\left(\varrho \right)=2^{p-1}{a}^p\left(\varrho \right),$$\mathcal{T}\left(\varrho \right)={\chi }^{\prime }\left(\varrho \right)\exp \left(\lambda \chi \left(\varrho \right)\right)w^p\left(\varrho \right)$, and $p,q\in \left(0,\infty \right)$, with $\frac{1}{q}+\sigma >1$, $\frac{1}{q}+\frac{1}{p}=1$, and $\gamma =\left(\sigma -1\right)q+1.$

 Proof. 

According to the above Theorem,

$$\begin{array}{cc}\hfill u\left(\varrho \right)\le & a\left(\varrho \right)+\frac{b\left(\varrho \right)}{\Gamma \left(\sigma \right)}{\int }_0^{\varrho }{\chi }^{\prime }\left(\iota \right)\exp \left(-\lambda \left[\chi \left(\varrho \right)-\chi \left(\iota \right)\right]\right){\left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)}^{\sigma -1}w\left(\iota \right)u\left(\iota \right)d\iota \hfill \\ \hfill \le & a\left(\varrho \right)+\frac{b\left(\varrho \right)\exp \left(-\lambda \chi \left(\varrho \right)\right)}{\Gamma \left(\sigma \right)}{\left(\Gamma \left(\gamma \right)\exp \left(\lambda \chi \left(0\right)\right){\left(\chi \left(\varrho \right)-\chi \left(0\right)\right)}^{\gamma }{\mathbb{E}}_{1,\gamma +1}\left(\lambda \left(\chi \left(\varrho \right)-\chi \left(0\right)\right)\right)\right)}^{\frac{1}{q}}\hfill \\ \hfill & \times {\left({\int }_0^{\varrho }{\chi }^{\prime }\left(\iota \right)\exp \left(\lambda \chi \left(\iota \right)\right)w^p\left(\iota \right)u^p\left(\iota \right)d\iota \right)}^{\frac{1}{p}}.\hfill \end{array}$$

Then,

$$\begin{array}{cc}\hfill u^p\left(\varrho \right)& \le 2^{p-1}(a^p\left(\varrho \right)+{\left(\frac{b\left(\varrho \right)\exp \left(-\lambda \chi \left(\varrho \right)\right)}{\Gamma \left(\sigma \right)}{\left(\Gamma \left(\gamma \right)\exp \left(\lambda \chi \left(0\right)\right){\left(\chi \left(\varrho \right)-\chi \left(0\right)\right)}^{\gamma }{\mathbb{E}}_{1,\gamma +1}\left(\lambda \left(\chi \left(\varrho \right)-\chi \left(0\right)\right)\right)\right)}^{\frac{1}{q}}\right)}^p\hfill \\ \hfill & \times \left({\int }_0^{\varrho }{\chi }^{\prime }\left(\iota \right)\exp \left(\lambda \chi \left(\iota \right)\right)w^p\left(\iota \right)u^p\left(\iota \right)d\iota \right)).\hfill \end{array}$$

If $v\left(\varrho \right)=u^p\left(\varrho \right)$, then the last inequality becomes

$$v\left(\varrho \right)\le \mathcal{N}\left(\varrho \right)+\mathcal{V}\left(\varrho \right){\int }_0^{\varrho }\mathcal{T}\left(\iota \right)v\left(\iota \right)d\iota .$$

According to Martyniuk et al. [13], we have inequality (12). □

 Theorem 4.

Let $\sigma \in \left(0,1\right),$ $\lambda >0,$ $a\left(\varrho \right)$ be a nondecreasing continuously differentiable function on $\left[0,T\right)$, with $a\left(\varrho \right)>0$ for all $\varrho \in \left[0,T\right)$; $w\left(\varrho \right),u\left(\varrho \right)$ are nonnegative, continuous functions on $\left[0,T\right)$, and $h:\left[0,\infty \right)\to \left[0,\infty \right)$ be an increasing, continuous function, such that

$$\begin{array}{cc}\hfill u\left(\varrho \right)\le & a\left(\varrho \right)+\frac{1}{\Gamma \left(\sigma \right)}{\int }_0^{\varrho }{\chi }^{\prime }\left(\iota \right)\exp \left(-\lambda \left[\chi \left(\varrho \right)-\chi \left(\iota \right)\right]\right)\hfill \\ \hfill & {\left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)}^{\sigma -1}w\left(\iota \right)h\left(\exp \left(\lambda \chi \left(\iota \right)\right)u\left(\iota \right)\right)d\iota .\hfill \end{array}$$

(13)

Then,

$$u\left(\varrho \right)\le \exp \left(-\lambda \chi \left(\varrho \right)\right){\left({\Omega }^{-1}\left(\Omega \left(\mathcal{N}\left(\varrho \right)\right)+{\int }_0^{\varrho }\mathcal{T}\left(\iota \right)d\iota \right)\right)}^{\nu },\varrho \in \left[0,T_1\right),$$

(14)

where $0<\nu <\sigma <1,$ $\mathcal{N}\left(\varrho \right)=2^{\frac{1}{\nu }-1}\exp \left(\frac{\lambda }{\nu }\chi \left(\varrho \right)\right)a^{\frac{1}{\nu }}\left(\varrho \right),$

$\mathcal{T}\left(\varrho \right)=\frac{2^{\frac{1}{\nu }-1}}{{\Gamma }^{\frac{1}{\nu }}\left(\sigma \right)}{\left(\Gamma \left(\frac{\sigma -\nu }{1-\nu }\right)\Gamma \left(\frac{1-\sigma }{1-\nu }\right)\right)}^{\frac{1-\nu }{\nu }}{\chi }^{\prime }\left(\varrho \right)\exp \left(\frac{\lambda }{\nu }\chi \left(\varrho \right)\right){\left(\chi \left(\varrho \right)-\chi \left(0\right)\right)}^{\frac{\sigma -\nu }{\nu }}{w}^{\frac{1}{\nu }}\left(\varrho \right),$${\displaystyle \Omega \left(y\right)={\int }_{t_0}^y\frac{1}{H\left(\iota \right)}d\iota ,}$$H\left(\varrho \right)=h^{\frac{1}{\nu }}\left({\varrho }^{\nu }\right),$$t_0>0,$${\Omega }^{-1}$ is the inverse function of $\Omega ,$ and $T_1\in \left(0,T\right)$ is such that

$\Omega \left(\mathcal{N}\left(\varrho \right)\right)+{\int }_0^{\varrho }\mathcal{T}\left(\iota \right)d\iota \in Dom\left({\Omega }^{-1}\right)$ for all $\varrho \in \left[0,T_1\right].$

 Proof. 

From (13) and Theorem 1, we get

$$\begin{array}{ccc}\hfill u^{\frac{1}{\nu }}\left(\varrho \right)& \le & 2^{\frac{1}{\nu }-1}(a^{\frac{1}{\nu }}\left(\varrho \right)+\frac{\exp \left(-\frac{\lambda }{\nu }\chi \left(\varrho \right)\right)}{{\Gamma }^{\frac{1}{\nu }}\left(\sigma \right)}{\left(\Gamma \left(\frac{\sigma -\nu }{1-\nu }\right)\Gamma \left(\frac{1-\sigma }{1-\nu }\right)\right)}^{\frac{1-\nu }{\nu }}\hfill \\ \hfill & & \times {\int }_0^{\varrho }\left[{\chi }^{\prime }\left(\iota \right)\exp \left(\frac{\lambda }{\nu }\chi \left(\iota \right)\right){\left(\chi \left(\iota \right)-\chi \left(0\right)\right)}^{\frac{\sigma -\nu }{\nu }}{w}^{\frac{1}{\nu }}\left(\iota \right)h^{\frac{1}{\nu }}\left(\exp \left(\lambda \chi \left(\iota \right)\right)u\left(\iota \right)\right)\right]d\iota )\hfill \end{array}$$

By taking $\mathcal{Q}\left(\varrho \right)=\exp \left(\frac{\lambda }{\nu }\chi \left(\varrho \right)\right)u^{\frac{1}{\nu }}\left(\varrho \right),$ the last above inequality becomes

$$\mathcal{Q}\left(\varrho \right)\le \mathcal{N}\left(\varrho \right)+{\int }_0^{\varrho }\mathcal{T}\left(\iota \right)H\left(\mathcal{Q}\left(\iota \right)\right)d\iota .$$

Let $\mathcal{V}\left(\varrho \right)$ be the right-hand side of the last above inequality. Then,

$$\frac{{\mathcal{V}}^{\prime }\left(\varrho \right)}{H\left(\mathcal{V}\left(\varrho \right)\right)}=\frac{{\mathcal{N}}^{\prime }\left(\varrho \right)+\mathcal{T}\left(\varrho \right)H\left(\mathcal{Q}\left(\varrho \right)\right)}{H\left(\mathcal{V}\left(\varrho \right)\right)}\le \frac{{\mathcal{N}}^{\prime }\left(\varrho \right)}{H\left(\mathcal{N}\left(\varrho \right)\right)}+\mathcal{T}\left(\varrho \right),$$

or

$$\frac{d}{d\varrho }\Omega \left(\mathcal{V}\left(\varrho \right)\right)\le \frac{d}{d\varrho }\Omega \left(\mathcal{N}\left(\varrho \right)\right)+\mathcal{T}\left(\varrho \right).$$

By integrating both sides of the last inequality from 0 to $\varrho$, we obtain

$$\Omega \left(\mathcal{V}\left(\varrho \right)\right)\le \Omega \left(\mathcal{N}\left(\varrho \right)\right)+{\int }_0^{\varrho }\mathcal{T}\left(\iota \right)d\iota ,$$

and since $\Omega$ is an increasing function, so we get

$$\mathcal{Q}\left(\varrho \right)\le \mathcal{V}\left(\varrho \right)\le {\Omega }^{-1}\left(\Omega \left(\mathcal{N}\left(\varrho \right)\right)+{\int }_0^{\varrho }\mathcal{T}\left(\iota \right)d\iota \right).$$

The proof is thus established. □

4. Applications

The purpose of this part is to give existence, uniqueness, and stability results for Equation (3).

For $0<T<\infty ,$ in the rest of this paper, we take into consideration that $\chi \left(\varrho \right)\in C^1\left([0,T],{\mathbb{R}}_{+}\right)$ is an increasing function, such that $\chi \left(0\right)=0$ and ${\chi }^{\prime }\left(\varrho \right)\ne 0$ for all $\varrho \in [0,T].$

The following hypothesis is introduced:

(A) $f:$ $\left[0,T\right]\times \mathbb{R}\to \mathbb{R}$ is continuous, and there is $g:\left[0,\infty \right)\to \left[0,\infty \right)$ an increasing, continuous function, such that $g\left(0\right)=0$, with

$$\left|f\left(\varrho ,y\right)-f\left(\varrho ,z\right)\right|\le g\left(\exp \left(\lambda \chi \left(\varrho \right)\right)\left|y-z\right|\right),$$

(15)

for every $\lambda >0,$ $y,z\in \mathbb{R}$ and $\varrho \in \left[0,T\right]$.

In the case of (A) being satisfied, there is a continuous positive function $k\left(\varrho \right)$ with

$$\left|f\left(\varrho ,y\right)\right|\le g\left(\exp \left(\lambda \chi \left(\varrho \right)\right)\left|y\right|\right)+k\left(\varrho \right),$$

(16)

for every $\lambda >0,$ $y\in \mathbb{R}$ and $\varrho \in \left[0,T\right]$.

4.1. Existence and Uniqueness Results

The solution of the (IVP) (3), is given by (see [23]),

$$\begin{array}{cc}\hfill u\left(\varrho \right)=& u_0\exp \left(-\lambda \chi \left(\varrho \right)\right)\hfill \\ \hfill & +\frac{1}{\Gamma \left(\sigma \right)}{\int }_0^{\varrho }{\chi }^{\prime }\left(\iota \right)\exp \left(-\lambda \left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)\right){\left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)}^{\sigma -1}f\left(\iota ,u\left(\iota \right)\right)d\iota .\hfill \end{array}$$

We use Theorem 4 to verify the existence and uniqueness of (IVP) (3).

 Theorem 5.

Suppose that (A) is satisfied. If

$$\underset{y\to 0^{+}}{lim}{\int }_1^y\frac{1}{H\left(\iota \right)}d\iota =-\infty ,$$

(17)

and

$$\underset{y\to +\infty }{lim}{\int }_1^y\frac{1}{H\left(\iota \right)}d\iota =+\infty ,$$

(18)

where $H\left(\varrho \right)$ is defined as in Theorem 4, with $h\left(\varrho \right)=g\left(\varrho \right)$. Then, the(IVP)(3) has a unique solution.

 Proof. 

Let us consider the operator $G:C\left(\left[0,T\right],\mathbb{R}\right)\to C\left(\left[0,T\right],\mathbb{R}\right)$, defined as follows:

$$\begin{array}{cc}\hfill \left(Gu\right)\left(\varrho \right)=& u_0\exp \left(-\lambda \chi \left(\varrho \right)\right)\hfill \\ \hfill & +\frac{1}{\Gamma \left(\sigma \right)}{\int }_0^{\varrho }{\chi }^{\prime }\left(\iota \right)\exp \left(-\lambda \left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)\right){\left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)}^{\sigma -1}f\left(\iota ,u\left(\iota \right)\right)d\iota .\hfill \end{array}$$

To prove that G has a fixed point, we will use Schaefer’s fixed point theorem. In several steps, the proof will be presented.

Step 1:G is a continuous operator.

Let $u_n$ be a sequence with $u_n\to u$ in $C\left(\left[0,T\right],\mathbb{R}\right)$. Thus, for every $\varrho \in [0,T]$, we get

$$\begin{array}{cc}\hfill \left|G\left(u_n\right)\left(\varrho \right)-G\left(u\right)\left(\varrho \right)\right|& \le \frac{1}{\Gamma \left(\sigma \right)}{\int }_0^{\varrho }{\chi }^{\prime }\left(\iota \right){\left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)}^{\sigma -1}\left|f\left(\iota ,u_n\left(\iota \right)\right)-f\left(\iota ,u\left(\iota \right)\right)\right|d\iota \hfill \\ \hfill & \le \frac{{\left(\chi \left(T\right)\right)}^{\sigma }}{\Gamma \left(\sigma +1\right)}g\left(\exp \left(\lambda \chi \left(T\right)\right){∥u_n-u∥}_{\infty }\right),\hfill \end{array}$$

so

$${∥G\left(u_n\right)-G\left(u\right)∥}_{\infty }\le \frac{{\left(\chi \left(T\right)\right)}^{\sigma }}{\Gamma \left(\sigma +1\right)}g\left(\exp \left(\lambda \chi \left(T\right)\right){∥u_n-u∥}_{\infty }\right).$$

G is therefore a continuous operator.

Step 2: In $C\left(\left[0,T\right],\mathbb{R}\right)$, G maps bounded sets into bounded sets.

The only thing needed is to to prove that for every $r_1>0$, there is $r_2>0$, such that for every $u\in B_{r_1}=\{u\in C\left(\left[0,T\right],\mathbb{R}\right),{∥u∥}_{\infty }\le r_1\}$, we obtain $G\left(u\right)\in B_{r_2}$. For every $\varrho \in [0,T]$, we get

$$\begin{array}{ccc}\hfill \left|G\left(u\right)\left(\varrho \right)\right|& =& |u_0\exp \left(-\lambda \chi \left(\varrho \right)\right)\hfill \\ \hfill & & +\frac{1}{\Gamma \left(\sigma \right)}{\int }_0^{\varrho }{\chi }^{\prime }\left(\iota \right)\exp \left(-\lambda \left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)\right){\left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)}^{\sigma -1}f\left(\iota ,u\left(\iota \right)\right)d\iota |\hfill \\ \hfill & \le & \left|u_0\right|+\frac{1}{\Gamma \left(\sigma \right)}{\int }_0^{\varrho }{\chi }^{\prime }\left(\iota \right){\left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)}^{\sigma -1}\left|f\left(\iota ,u\left(\iota \right)\right)\right|d\iota \hfill \\ \hfill & \le & \left|u_0\right|+\frac{1}{\Gamma \left(\sigma \right)}{\int }_0^{\varrho }{\chi }^{\prime }\left(\iota \right){\left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)}^{\sigma -1}g\left(\exp \left(\lambda \chi \left(\iota \right)\right)\left|u\left(\iota \right)\right|\right)d\iota \hfill \\ \hfill & & +\frac{1}{\Gamma \left(\sigma \right)}{\int }_0^{\varrho }{\chi }^{\prime }\left(\iota \right){\left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)}^{\sigma -1}k\left(\iota \right)d\iota \hfill \\ \hfill & \le & \left|u_0\right|+\frac{{\left(\chi \left(T\right)\right)}^{\sigma }}{\Gamma \left(\sigma +1\right)}\left(g_{\chi ,r_1}+k^{\ast }\right),\hfill \end{array}$$

where $k^{\ast }={sup}_{t\in [0,T]}k\left(t\right)$ and $g_{\chi ,r_1}=g\left(r_1\exp \left(\lambda \chi \left(T\right)\right)\right)$, then

$${∥G\left(u\right)∥}_{\infty }\le \left|u_0\right|+\frac{{\left(\chi \left(T\right)\right)}^{\sigma }}{\Gamma \left(\sigma +1\right)}\left(g_{\chi ,r_1}+k^{\ast }\right)=:r_2.$$

(19)

Step 3: Using G, we can map bounded sets into equicontinuous sets of $C\left(\left[0,T\right],\mathbb{R}\right)$.

Let $r_1>0$ and $u\in B_{r_1}$. For $0\le {\varrho }_1<{\varrho }_2<T$, we have

$$\begin{array}{ccc}\hfill & & \left|G\left(u\right)\left({\varrho }_2\right)-G\left(u\right)\left({\varrho }_1\right)\right|\hfill \\ \hfill & & \le \left|u_0\right|\left|\exp \left(-\lambda \chi \left({\varrho }_2\right)\right)-\exp \left(-\lambda \chi \left({\varrho }_1\right)\right)\right|\hfill \\ \hfill & & +\frac{1}{\Gamma \left(\sigma \right)}|{\int }_0^{{\varrho }_2}{\chi }^{\prime }\left(\iota \right)\exp \left(-\lambda \left(\chi \left({\varrho }_2\right)-\chi \left(\iota \right)\right)\right){\left(\chi \left({\varrho }_2\right)-\chi \left(\iota \right)\right)}^{\sigma -1}f\left(\iota ,u\left(\iota \right)\right)d\iota \hfill \\ \hfill & & -{\int }_0^{{\varrho }_1}{\chi }^{\prime }\left(\iota \right)\exp \left(-\lambda \left(\chi \left({\varrho }_1\right)-\chi \left(\iota \right)\right)\right){\left(\chi \left({\varrho }_1\right)-\chi \left(\iota \right)\right)}^{\sigma -1}f\left(\iota ,u\left(\iota \right)\right)d\iota |\hfill \\ \hfill & & \le \left|u_0\right|\left|\exp \left(-\lambda \chi \left({\varrho }_2\right)\right)-\exp \left(-\lambda \chi \left({\varrho }_1\right)\right)\right|\hfill \\ \hfill & & +\frac{1}{\Gamma \left(\sigma \right)}{\int }_0^{{\varrho }_1}{\chi }^{\prime }\left(\iota \right)|\exp \left(-\lambda \left(\chi \left({\varrho }_2\right)-\chi \left(\iota \right)\right)\right){\left(\chi \left({\varrho }_2\right)-\chi \left(\iota \right)\right)}^{\sigma -1}\hfill \\ \hfill & & -\exp \left(-\lambda \left(\chi \left({\varrho }_1\right)-\chi \left(\iota \right)\right)\right){\left(\chi \left({\varrho }_1\right)-\chi \left(\iota \right)\right)}^{\sigma -1}|\left|f\left(\iota ,u\left(\iota \right)\right)\right|d\iota \hfill \\ \hfill & & +\frac{1}{\Gamma \left(\sigma \right)}{\int }_{{\varrho }_1}^{{\varrho }_2}{\chi }^{\prime }\left(\iota \right)\exp \left(-\lambda \left(\chi \left({\varrho }_2\right)-\chi \left(\iota \right)\right)\right){\left(\chi \left({\varrho }_2\right)-\chi \left(\iota \right)\right)}^{\sigma -1}\left|f\left(\iota ,u\left(\iota \right)\right)\right|d\iota \hfill \\ \hfill & & \le \left|u_0\right|\left|\exp \left(-\lambda \chi \left({\varrho }_2\right)\right)-\exp \left(-\lambda \chi \left({\varrho }_1\right)\right)\right|\hfill \\ \hfill & & +\frac{1}{\Gamma \left(\sigma \right)}{\int }_0^{{\varrho }_1}{\chi }^{\prime }\left(\iota \right)\exp \left(-\lambda \left(\chi \left({\varrho }_2\right)-\chi \left(\iota \right)\right)\right)\hfill \\ \hfill & & \times \left({\left(\chi \left({\varrho }_1\right)-\chi \left(\iota \right)\right)}^{\sigma -1}-{\left(\chi \left({\varrho }_2\right)-\chi \left(\iota \right)\right)}^{\sigma -1}\right)\left|f\left(\iota ,u\left(\iota \right)\right)\right|d\iota \hfill \\ \hfill & & +\frac{1}{\Gamma \left(\sigma \right)}{\int }_0^{{\varrho }_1}{\chi }^{\prime }\left(\iota \right)\left|\exp \left(-\lambda \left(\chi \left({\varrho }_2\right)-\chi \left(\iota \right)\right)\right)-\exp \left(-\lambda \left(\chi \left({\varrho }_1\right)-\chi \left(\iota \right)\right)\right)\right|\hfill \\ \hfill & & \times {\left(\chi \left({\varrho }_1\right)-\chi \left(\iota \right)\right)}^{\sigma -1}\left|f\left(\iota ,u\left(\iota \right)\right)\right|d\iota \hfill \\ \hfill & & +\frac{1}{\Gamma \left(\sigma \right)}{\int }_{{\varrho }_1}^{{\varrho }_2}{\chi }^{\prime }\left(\iota \right)\exp \left(-\lambda \left(\chi \left({\varrho }_2\right)-\chi \left(\iota \right)\right)\right)\hfill \\ \hfill & & \times {\left(\chi \left({\varrho }_2\right)-\chi \left(\iota \right)\right)}^{\sigma -1}\left|f\left(\iota ,u\left(\iota \right)\right)\right|d\iota \hfill \\ \hfill & & \le \lambda \left|u_0\right|\left|\chi \left({\varrho }_2\right)-\chi \left({\varrho }_1\right)\right|+\frac{\left(g_{\chi ,r_1}+k^{\ast }\right)}{\Gamma \left(\sigma +1\right)}\left(\left(\chi {\left({\varrho }_1\right)}^{\sigma }-\chi {\left({\varrho }_2\right)}^{\sigma }\right)+{\left(\chi \left({\varrho }_2\right)-\chi \left({\varrho }_1\right)\right)}^{\sigma }\right)\hfill \\ \hfill & & +\lambda \frac{\left(g_{\chi ,r_1}+k^{\ast }\right){\left(\chi \left(T\right)\right)}^{\sigma }}{\Gamma \left(\sigma +1\right)}\left|\chi \left({\varrho }_2\right)-\chi \left({\varrho }_1\right)\right|+\frac{\left(g_{\chi ,r_1}+k^{\ast }\right)}{\Gamma \left(\sigma +1\right)}{\left(\chi \left({\varrho }_2\right)-\chi \left({\varrho }_1\right)\right)}^{\sigma }\hfill \\ \hfill & & \le \lambda \left(\left|u_0\right|+\frac{\left(g_{\chi ,r_1}+k^{\ast }\right){\left(\chi \left(T\right)\right)}^{\sigma }}{\Gamma \left(\sigma +1\right)}\right)\left|\chi \left({\varrho }_2\right)-\chi \left({\varrho }_1\right)\right|\hfill \\ \hfill & & +\frac{\left(g_{\chi ,r_1}+k^{\ast }\right)}{\Gamma \left(\sigma +1\right)}\left(\left(\chi {\left({\varrho }_1\right)}^{\sigma }-\chi {\left({\varrho }_2\right)}^{\sigma }\right)+2{\left(\chi \left({\varrho }_2\right)-\chi \left({\varrho }_1\right)\right)}^{\sigma }\right)\hfill \end{array}$$

Due to ${\varrho }_2\to {\varrho }_1$, the right-hand side of the inequality tends to zero.

We conclude that G is completely continuous based on steps 1 to 3 and using the Arzelá-Ascoli Theorem.

Step 4: A priori bounds.

Boundedness of the set:

$$\mathcal{A}=\left\{u\in C\left(\left[0,T\right],\mathbb{R}\right),u=\tilde{\theta }G\left(u\right),\mathrm{for}\tilde{\theta }\in \left]0,1\right[\right\},$$

(20)

remains to be demonstrated. Taking $0<\nu <\sigma <1,$ and $\eta =\frac{\sigma -\nu }{1-\nu }.$ Let $u\in \mathcal{A}$, then,

$$\begin{array}{ccc}\hfill \left|u\left(\varrho \right)\right|& =& |\tilde{\theta }{u}_0\exp \left(-\lambda \chi \left(\varrho \right)\right)\hfill \\ \hfill & & +\frac{\tilde{\theta }}{\Gamma \left(\sigma \right)}{\int }_0^{\varrho }{\chi }^{\prime }\left(\iota \right)\exp \left(-\lambda \left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)\right){\left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)}^{\sigma -1}f\left(\iota ,u\left(\iota \right)\right)d\iota |\hfill \\ \hfill & \le & \exp \left(-\lambda \chi \left(\varrho \right)\right)\left|u_0\right|\hfill \\ \hfill & & +\frac{1}{\Gamma \left(\sigma \right)}{\int }_0^{\varrho }{\chi }^{\prime }\left(\iota \right)\exp \left(-\lambda \left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)\right){\left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)}^{\sigma -1}\left|f\left(\iota ,u\left(\iota \right)\right)\right|d\iota \hfill \\ \hfill & \le & \exp \left(-\lambda \chi \left(\varrho \right)\right)\left|u_0\right|\hfill \\ \hfill & & +\frac{1}{\Gamma \left(\sigma \right)}{\int }_0^{\varrho }{\chi }^{\prime }\left(\iota \right)\exp \left(-\lambda \left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)\right){\left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)}^{\sigma -1}k\left(\iota \right)d\iota \hfill \\ \hfill & & +\frac{1}{\Gamma \left(\sigma \right)}{\int }_0^{\varrho }{\chi }^{\prime }\left(\iota \right)\exp \left(-\lambda \left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)\right){\left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)}^{\sigma -1}g\left(\exp \left(\lambda \chi \left(\iota \right)\right)\left|u\left(\iota \right)\right|\right)d\iota \hfill \\ \hfill & \le & \exp \left(-\lambda \chi \left(\varrho \right)\right)\left|u_0\right|+\frac{\exp \left(-\lambda \chi \left(\varrho \right)\right)}{\Gamma \left(\sigma \right)}{\left(\Gamma \left(\eta \right){\left(\chi \left(\varrho \right)\right)}^{\eta }{\mathbb{E}}_{1,\eta +1}\left(\lambda \chi \left(\varrho \right)\right)\right)}^{1-\nu }\hfill \\ \hfill & & \times {\left({\int }_0^{\varrho }{\chi }^{\prime }\left(\iota \right)\exp \left(\lambda \chi \left(\iota \right)\right)k^{\frac{1}{\nu }}\left(\iota \right)d\iota \right)}^{\nu }\hfill \\ \hfill & & +\frac{1}{\Gamma \left(\sigma \right)}{\int }_0^{\varrho }{\chi }^{\prime }\left(\iota \right)\exp \left(-\lambda \left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)\right){\left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)}^{\sigma -1}g\left(\exp \left(\lambda \chi \left(\iota \right)\right)\left|u\left(\iota \right)\right|\right)d\iota .\hfill \end{array}$$

Then, for each $ϵ>0$, we get

$$\begin{array}{ccc}\hfill \left|u\left(\varrho \right)\right|& \le & ϵ+\exp \left(-\lambda \chi \left(\varrho \right)\right)\left|u_0\right|+\frac{\exp \left(-\lambda \chi \left(\varrho \right)\right)}{\Gamma \left(\sigma \right)}{\left(\Gamma \left(\eta \right){\left(\chi \left(\varrho \right)\right)}^{\eta }{\mathbb{E}}_{1,\eta +1}\left(\lambda \chi \left(\varrho \right)\right)\right)}^{1-\nu }\hfill \\ \hfill & & \times {\left({\int }_0^{\varrho }{\chi }^{\prime }\left(\iota \right)\exp \left(\lambda \chi \left(\iota \right)\right)k^{\frac{1}{\nu }}\left(\iota \right)d\iota \right)}^{\nu }\hfill \\ \hfill & & +\frac{1}{\Gamma \left(\sigma \right)}{\int }_0^{\varrho }{\chi }^{\prime }\left(\iota \right)\exp \left(-\lambda \left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)\right){\left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)}^{\sigma -1}g\left(\exp \left(\lambda \chi \left(\iota \right)\right)\left|u\left(\iota \right)\right|\right)d\iota .\hfill \end{array}$$

Using (18) and Theorem 4, we obtain

$$\left|u\left(\varrho \right)\right|\le \exp \left(-\lambda \chi \left(\varrho \right)\right){\left({\Omega }^{-1}\left(\Omega \left(\mathcal{N}\left(\varrho \right)\right)+{\int }_0^{\varrho }\mathcal{T}\left(\iota \right)d\iota \right)\right)}^{\nu },\varrho \in \left[0,T\right],$$

(21)

where the formula of $\Omega \left(\varrho \right),\mathcal{N}\left(\varrho \right)$ and $\mathcal{T}\left(\varrho \right)$ are defined as in Theorem 4, with

$$\begin{array}{cc}\hfill a\left(\varrho \right)=& ϵ+\exp \left(-\lambda \chi \left(\varrho \right)\right)\left|u_0\right|+\frac{\exp \left(-\lambda \chi \left(\varrho \right)\right)}{\Gamma \left(\sigma \right)}{\left(\Gamma \left(\eta \right){\left(\chi \left(\varrho \right)\right)}^{\eta }{\mathbb{E}}_{1,\eta +1}\left(\lambda \chi \left(\varrho \right)\right)\right)}^{1-\nu }\hfill \\ \hfill & \times {\left({\int }_0^{\varrho }{\chi }^{\prime }\left(\iota \right)\exp \left(\lambda \chi \left(\iota \right)\right)k^{\frac{1}{\nu }}\left(\iota \right)d\iota \right)}^{\nu },\hfill \end{array}$$

and $w\left(\varrho \right)=1$. We conclude that the operator G has at least one fixed point in $C\left(\left[0,T\right],\mathbb{R}\right)$ that is a solution of (IVP) (3) as a result of Schaefer’s fixed point theorem.

For the uniqueness, we suppose that $u_1\left(\varrho \right)$, $u_2\left(\varrho \right)$ are two solutions of (IVP) (3). Then,

$$\begin{array}{cc}\hfill & \left|u_1\left(\varrho \right)-u_2\left(\varrho \right)\right|\hfill \\ \hfill & =|\frac{1}{\Gamma \left(\sigma \right)}{\int }_0^{\varrho }{\chi }^{\prime }\left(\iota \right)\exp \left(-\lambda \left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)\right)\hfill \\ & \times {\left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)}^{\sigma -1}\left(f\left(\iota ,u_1\left(\iota \right)\right)-f\left(\iota ,u_2\left(\iota \right)\right)\right)d\iota |\hfill \\ \hfill & \le \frac{1}{\Gamma \left(\sigma \right)}{\int }_0^{\varrho }{\chi }^{\prime }\left(\iota \right)\exp \left(-\lambda \left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)\right)\hfill \\ & \times {\left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)}^{\sigma -1}g\left(\exp \left(\lambda \chi \left(\varrho \right)\right)\left|u_1\left(\iota \right)-u_2\left(\iota \right)\right|\right)d\iota \hfill \end{array}$$

Thus, for each $ϵ>0$, we obtain

$$\begin{array}{cc}\hfill \left|u_1\left(\varrho \right)-u_2\left(\varrho \right)\right|& \le ϵ+\frac{1}{\Gamma \left(\sigma \right)}{\int }_0^{\varrho }{\chi }^{\prime }\left(\iota \right)\exp \left(-\lambda \left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)\right)\hfill \\ & \times {\left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)}^{\sigma -1}g\left(\exp \left(\lambda \chi \left(\varrho \right)\right)\left|u_1\left(\iota \right)-u_2\left(\iota \right)\right|\right).\hfill \end{array}$$

(22)

According to (17) and (18), and Theorem 4, we get

$$\left|u_1\left(\varrho \right)-u_2\left(\varrho \right)\right|\le \exp \left(-\lambda \chi \left(\varrho \right)\right){\left({\Omega }^{-1}\left(\Omega \left(\mathcal{N}\left(\varrho \right)\right)+{\int }_0^{\varrho }\mathcal{T}\left(\iota \right)d\iota \right)\right)}^{\nu },\varrho \in \left[0,T\right]$$

(23)

where the formula of $\Omega \left(\varrho \right),\mathcal{N}\left(\varrho \right)$, and $\mathcal{T}\left(\varrho \right)$ are defined as in Theorem 4, with

$a\left(\varrho \right)=ϵ$ and $w\left(\varrho \right)=1$. Thus, $u_1\left(\varrho \right)=u_2\left(\varrho \right)$ for every $\varrho \in [0,T]$. □

4.2. Stability Results

The following inequality is considered.

$$\left|{}^C{D}_{0^{+}}^{\sigma ,\lambda ,\chi }v\left(t\right)-f\left(\varrho ,v\left(\varrho \right)\right)\right|\le ϵ,\mathrm{for}\lambda ,ϵ>0,\sigma \in (0,1)\mathrm{and}\varrho \in \left[0,T\right].$$

(24)

 Definition 3.

Equation (3) is Ulam-Hyers stable if there exists a function $\varphi :(0,\infty )\to (0,\infty )$ such that for each $ϵ>0,$ and for each solution $v\in C\left(\left[0,T\right],\mathbb{R}\right)$ of the inequality (24), there exists a solution $u\in C\left(\left[0,T\right],\mathbb{R}\right)$ of Equation (3) with

$$\left|v\left(\varrho \right)-u\left(\varrho \right)\right|\le \varphi \left(ϵ\right),$$

(25)

where ${lim}_{ϵ\to 0}\varphi \left(ϵ\right)=0.$

Taking into consideration that ${\mathbb{E}}_{1,\sigma +1}\left(\lambda \chi \left(\varrho \right)\right)\le \frac{1}{\Gamma \left(\frac{3}{2}\right)}\exp \left(\lambda \chi \left(\varrho \right)\right),$ for more detail, see [8].

The following remark is obtained:

 Remark 6.

When v is a solution of inequality (24), then v is a solution of this integral inequality.

$$\begin{array}{ccc}\hfill & & |v\left(\varrho \right)-v\left(0\right)\exp \left(-\lambda \chi \left(\varrho \right)\right)\hfill \\ \hfill & & -\frac{1}{\Gamma \left(\sigma \right)}{\int }_0^{\varrho }{\chi }^{\prime }\left(\iota \right)\exp \left(-\lambda \left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)\right){\left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)}^{\sigma -1}f\left(\iota ,v\left(\iota \right)\right)d\iota |\hfill \\ & & \le ϵ\frac{{\left(\chi \left(T\right)\right)}^{\sigma }}{\Gamma \left(\frac{3}{2}\right)}.\hfill \end{array}$$

(26)

 Theorem 7.

Assume that (A), (17) and (18) are satisfied. Then, Equation (3) is stable in the sense of Ulam-Hyers.

 Proof. 

Consider v as a solution to (24) and u as the unique solution to the following Cauchy problem

$$\left\{\begin{array}{c}{}^C{D}_{0^{+}}^{\sigma ,\lambda ,\chi }u\left(\varrho \right)=f\left(\varrho ,u\left(\varrho \right)\right),\lambda >0,\sigma \in \left(0,1\right),\varrho \in \left[0,T\right],\hfill \\ u\left(0\right)=v\left(0\right).\hfill \end{array}\right.$$

(27)

Then,

$$\begin{array}{cc}\hfill u\left(\varrho \right)=& v\left(0\right)\exp \left(-\lambda \chi \left(\varrho \right)\right)\hfill \\ & +\frac{1}{\Gamma \left(\sigma \right)}{\int }_0^{\varrho }{\chi }^{\prime }\left(\iota \right)\exp \left(-\lambda \left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)\right){\left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)}^{\sigma -1}f\left(\iota ,u\left(\iota \right)\right)d\iota .\hfill \end{array}$$

According to Remark 6, we find

$$\begin{array}{cc}\hfill \left|v\left(\varrho \right)-u\left(\varrho \right)\right|& \le |v\left(\varrho \right)-v\left(0\right)\exp \left(-\lambda \chi \left(\varrho \right)\right)\hfill \\ \hfill & -\frac{1}{\Gamma \left(\sigma \right)}{\int }_0^{\varrho }{\chi }^{\prime }\left(\iota \right)\exp \left(-\lambda \left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)\right){\left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)}^{\sigma -1}f\left(\iota ,u\left(\iota \right)\right)d\iota |\hfill \\ \hfill & \le |v\left(\varrho \right)-v\left(0\right)\exp \left(-\lambda \chi \left(\varrho \right)\right)\hfill \\ \hfill & -\frac{1}{\Gamma \left(\sigma \right)}{\int }_0^{\varrho }{\chi }^{\prime }\left(\iota \right)\exp \left(-\lambda \left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)\right){\left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)}^{\sigma -1}f\left(\iota ,v\left(\iota \right)\right)d\iota |\hfill \\ \hfill & +|\frac{1}{\Gamma \left(\sigma \right)}{\int }_0^{\varrho }{\chi }^{\prime }\left(\iota \right)\exp \left(-\lambda \left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)\right){\left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)}^{\sigma -1}f\left(\iota ,v\left(\iota \right)\right)d\iota \hfill \\ \hfill & -\frac{1}{\Gamma \left(\sigma \right)}{\int }_0^{\varrho }\chi \left(\varrho \right)\exp \left(-\lambda \left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)\right){\left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)}^{\sigma -1}f\left(\iota ,u\left(\iota \right)\right)d\iota |\hfill \\ \hfill & \le ϵ\frac{{\left(\chi \left(T\right)\right)}^{\sigma }}{\Gamma \left(\frac{3}{2}\right)}+\frac{1}{\Gamma \left(\sigma \right)}{\int }_0^{\varrho }{\chi }^{\prime }\left(\iota \right)\exp \left(-\lambda \left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)\right){\left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)}^{\sigma -1}\hfill \\ \hfill & \times \left|f\left(\iota ,v\left(\iota \right)\right)-f\left(\iota ,u\left(\iota \right)\right)\right|d\iota \hfill \\ \hfill & \le ϵ\frac{{\left(\chi \left(T\right)\right)}^{\sigma }}{\Gamma \left(\frac{3}{2}\right)}+\frac{1}{\Gamma \left(\sigma \right)}{\int }_0^{\varrho }{\chi }^{\prime }\left(\iota \right)\exp \left(-\lambda \left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)\right){\left(\chi \left(\varrho \right)-\chi \left(\iota \right)\right)}^{\sigma -1}\hfill \\ \hfill & \times g\left(\exp \left(\lambda \chi \left(\iota \right)\right)\left|v\left(\iota \right)-u\left(\iota \right)\right|\right)d\iota .\hfill \end{array}$$

(28)

Using (18) and Theorem 4, we obtain

$$\left|v\left(\varrho \right)-u\left(\varrho \right)\right|\le \exp \left(-\lambda \chi \left(\varrho \right)\right){\left({\Omega }^{-1}\left(\Omega \left(\mathcal{N}\left(\varrho \right)\right)+{\int }_0^{\varrho }\mathcal{T}\left(\iota \right)d\iota \right)\right)}^{\nu },\varrho \in \left[0,T\right],$$

(29)

where the formula of $\Omega \left(\varrho \right),\mathcal{N}\left(\varrho \right)$, and $\mathcal{T}\left(\varrho \right)$ are defined as in Theorem 4, with

$a\left(\varrho \right)=ϵ\frac{{\left(\chi \left(T\right)\right)}^{\sigma }}{\Gamma \left(\frac{3}{2}\right)}$ and $w\left(\varrho \right)=1$. Thus,

$$\left|v\left(\varrho \right)-u\left(\varrho \right)\right|\le \varphi \left(ϵ\right)$$

(30)

where

$$\varphi \left(ϵ\right)={\left({\Omega }^{-1}\left(\Omega \left(\mathcal{N}\left(T\right)\right)+{\int }_0^T\mathcal{T}\left(\iota \right)d\iota \right)\right)}^{\nu }.$$

According to (17), we obtain $\varphi \left(ϵ\right)\to 0$ as $ϵ\to 0$. This complete the proof. □

 Remark 8.

The obtained results are the generalization of the work [17].

5. Conclusions

In this paper, we construct several unique integral inequalities with weakly singular singularities that extend a number of well-known ones. This study’s key contribution is the presentation of weakly singular integral inequalities and a demonstration of the Ulam stability of the suggested system. According to our knowledge, no additional published works exist on this subject. In the coming works, we intend to generalize our results to several other fractional derivatives, such as the ones presented in [24,25].