Analysis of Solutions of the Integro-Differential Equations with Generalized Liouville–Caputo Fractional Derivative by $\rho$-Laplace Transform

Abstract

The aim of this manuscript is two-fold. First, we analyze a linear fractional differential equation for the existence, uniqueness and Ulam–Hyers stabilities and then we consider a nonlinear fractional differential equation for the said analysis. For the linear problem, an important tool that is, modified Laplace transform is considered. By employing the $\rho$-Laplace transform method, we verify that the linear fractional matrix-valued differential equation is Ulam–Hyers stable. For the nonlinear fractional integro-differential equation the arguments for the existence and uniqueness of solutions are on account of Schaefer’s fixed point approach and Banach contraction principle. Through extended Gronwall’s inequality, we achieve the stability results. At the end, with the help of an example, we demonstrate our main theoretical work.

Introduction

In the last few decades, much attention has been paid to Fractional Calculus (FC) due to its wide range applications into real world phenomena. Different attempts have been made in order to present these phenomena in a superior way and to explore new fractional operators with different kernels such as Riemann–Liouville, Caputo, Hadamard, and generalized Liouville–Caputo [3, 7, 22, 29]. In [8], a new and practical fractional derivative operator has been introduced called the Atangana–Baleanu fractional derivative.

Integral inequalities has a major contribution in flourishing the analysis of differential and integral equations. Among the well known inequalities, Gronwall inequality [10] is a most popular one, because of its applications in many areas of mathematics. A generalized Gronwall inequality on a Riemann–Liouville fractional differential equations was considered in [30]. While in [1], the Gronwall inequality was obtained for q-fractional operators. Liu et al. [22] presented the extended Gronwall inequality for the analysis of Liouville–Caputo fractional differential equations.

In the literature of FC, there exist many stability perceptions, for instance: Mittag–Leffler stability, asymptotic stability, global asymptotic uniform stability, synchronization problems, stabilization problems and fractional input stability [20, 26]. But the most simple and easy to use when dealing with models which describe real world phenomena is the Ulam stability, mentioned by the Ulam [27] in 1940, then splendidly solved by Hyers [15] in 1941 and called it Ulam–Hyers (UH) stability. Subsequently, many renown mathematicians explored the above said stability brilliantly; for details, one can see [4,5,6, 9, 11,12,13, 24, 31,32,33]. Razaei et al. [25] studied the UH stability of linear differential equations. Recently in [28], the authors presented the UH stability of two types of linear differential equations.

In most of our analysis, when dealing with differential equations including physical models, there are some strong and interesting tools found in the literature like, Riemann–Liouville, Caputo and Riez fractional derivatives, integral transforms including Laplace, Mellin and Fourier for finding the exact solution. Recently, a new generalized fractional derivatives and integrals presented by Katugampola [18, 19], which consolidates the Riemann–Liouville and the Hadamard derivatives and integrals respectively.

Liu et al. [22] investigated the UH stability of linear differential equations with the help of $\rho$-Laplace transform. After that, they studied the following Cauchy problem of nonlinear equations

$$\begin{aligned} \left\{ \begin{array}{l} {^{c}D_{0^{+}}^{\gamma ,\rho }}f(t)=g(t,f(t)),\\ f(0)=f_{0}, \end{array} \right. \end{aligned}$$

where $\rho >0$, $0<\gamma <1$.

Inspired by the work mentioned above [22], our first task is to investigate the following fractional differential equations for the said analysis

$$\begin{aligned} \left\{ \begin{array}{l} {^{c}D_{t_0}^{\gamma ,\rho }}{\mathcal {W}}(t)=A{\mathcal {W}}(t)+q(t),\quad t\in [t_0,T],\;\gamma \in (0,1),\\ {\mathcal {W}}(0)=P, \end{array} \right. \end{aligned}$$

(1)

where ${^{c}D_{t_0}^{\gamma ,\rho }}$ denotes the left generalized $\gamma$ order Liouville–Caputo fractional derivative defined componentwise. A is n-th order matrix over the real field ${\mathbb {R}}$, q(t) is n-dimensional locally integrable column vector function on the closed interval $[t_0,T]$ and its entries admits the $\rho$-Laplace transformation, ${\mathcal {W}}(t)=(w_{1}(t),w(t),\dots ,w_{n}(t))^{T}$ is an unknown vector function, while P is a specified vector.

Then, we consider the following integro-differential equation with generalized Liouville–Caputo fractional derivative of the form

$$\begin{aligned} \left\{ \begin{array}{l} {^{c}D_{t_{0}}^{\gamma ,\rho }}\alpha (t)=\aleph (t,\alpha (t),\int _{t_0}^{t}l(s,\alpha (s))ds), \quad t \in J:=[t_0,T], \\ {\mathcal {J}}_{t_{0}}^{1-\gamma ,\rho }\alpha (t)|_{t=t_{0}}=\vartheta , \end{array} \right. \end{aligned}$$

(2)

where $\rho >0$, $\gamma \in (0,1]$, $^{c}D_{t_{0+}}^{\gamma ,\rho }\alpha (t)$ denotes the left generalized $\gamma$ order Liouville–Caputo fractional derivative for $\alpha$ with the parameter $\rho$, $l:\Delta \times {\mathbb {R}} \rightarrow {\mathbb {R}}$. For the sake of simplicity, we take

$$\begin{aligned} \mathrm {L}\alpha (t)=\int _{t_0}^{t}l(s,\alpha (s))ds. \end{aligned}$$

The manuscript is organized as follows: In section “Preliminaries”, we recall some important definitions, lemmas and basic properties related to $\rho$-Laplace transform. In section “Stability Analysis of (1)”, we concentrate on the stability of problem (1). Section “Existence Result of Nonlinear Model (2)” is devoted to the existence and uniqueness of problem (2). Section “Stability Analysis for Nonlinear Model” is managed to discuss the stability analysis of problem (2). In section “Example”, we mention the applications of our obtained results. Last Section of this article is dedicated to the conclusion.

Preliminaries

In this section, we discuss some imperative definitions of generalized fractional derivatives and integrals. To prove our main results, we state some basic definitions, lemmas and properties of the $\rho$-Laplace transform.

Assume that the space of all continuous functions from J into ${\mathbb {R}}$ is denoted by $C(J,{\mathbb {R}})$ with norm:

$$\begin{aligned} \Vert v\Vert _{C}=\sup \{|v(x)|: x\in J\}. \end{aligned}$$

Definition 2.1

[19] Let $\theta :[a,\infty )\rightarrow {\mathbb {R}}$, then the generalized left-sided integral is defined as

$$\begin{aligned} ({I_{a^{+}}^{\gamma ,\rho }}\theta )(t)=\frac{1}{\Gamma (\gamma )}\int _{a}^{t}\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1}\theta (\tau )\frac{d\tau }{\tau ^{1-\rho }},\quad t>a, \end{aligned}$$

where $\gamma \in (0,1)$, $\rho >0$.

Similarly, the generalized right-sided integral is defined as

$$\begin{aligned} ({I_{b^{-}}^{\gamma ,\rho }}\theta )(t)=\frac{1}{\Gamma (\gamma )}\int _{t}^{b}\left( \frac{\tau ^{\rho }-t^{\rho }}{\rho }\right) ^{\gamma -1}\theta (\tau )\frac{d\tau }{\tau ^{1-\rho }},\quad \rho >0. \end{aligned}$$

Definition 2.2

[18] Suppose that $0<\gamma <1$, $\rho >0$. The generalized Liouville–Caputo derivative of the function $\theta :[0,+\infty )\rightarrow {\mathbb {R}}$ is expressed in the form

$$\begin{aligned} ({^{c}D_{0^{+}}^{\gamma ,\rho }}\theta )(t)=\frac{1}{\Gamma (1-\gamma )}\int _{0}^{t}\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{-\gamma }\theta '(\tau )\frac{d\tau }{\tau ^{1-\rho }}, \quad t>0. \end{aligned}$$

Definition 2.3

[23] The standard Mittag–Leffler function for $a \in {\mathbb {C}}$, the set of all complex numbers, defined by the series

$$\begin{aligned} {\mathbb {E}}_{a}(z)=\sum _{j=0}^{\infty }\frac{z^{j}}{\Gamma (aj+1)}, \quad \Re (a)>0,\;z \in {\mathbb {C}}, \end{aligned}$$

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

The two parameter Mittag–Leffler function for $a,b \in {\mathbb {C}}$ is given by

$$\begin{aligned} {\mathbb {E}}_{a,b}(z)=\sum _{j=0}^{\infty }\frac{z^{j}}{\Gamma (aj+b)}, \quad \Re (a)>0,\;\Re (b)>0,\; z \in {\mathbb {C}}, \end{aligned}$$

where ${\mathbb {C}}$ denotes the set of all complex numbers. The series is convergent when a and b are strictly positive. It is abbreviated as ${\mathbb {E}}_{a}(z)={\mathbb {E}}_{a,1}(z)$, when $b=1$. We acquired the classical exponential function, when $a=b=1$.

Definition 2.4

The Mittag–Leffler matrix ${\mathbb {E}}_{a}(A)$ is defined as;

$$\begin{aligned} {\mathbb {E}}_{a}(A)=\sum _{j=0}^{\infty }\frac{A^{j}}{\Gamma (aj+1)}=I_{n}+\frac{A}{\Gamma (a+1)}+\frac{A^{2}}{\Gamma (2a+1)}+\cdots , \end{aligned}$$

where $a>0$ and $A_{n} \in M_{n}$.

Theorem 2.5

[16] Let $\rho >0$, $\gamma \in (0,1)$. The $\rho$-Laplace transform of the function z of the generalized fractional derivative in the Liouville–Caputo sense can be defined as

$$\begin{aligned} {\mathcal {L}}_{\rho }\{{^{c}D_{0^{+}}^{\gamma ,\rho }}z(t)\}(s)=\tau ^{\gamma }{\mathcal {L}}_{\rho }\{z(t)\}(s)-\tau ^{\gamma -1}z(0). \end{aligned}$$

The $\rho$-Laplace transform of the function z is defined as;

$$\begin{aligned} {\mathcal {L}}_{\rho }\{z(t)\}(s)=\int _{0}^{\infty }e^{-s\frac{t^{\rho }}{\rho }}z(t)\frac{dt}{t^{1-\rho }}. \end{aligned}$$

Then

$$\begin{aligned} {\mathcal {L}}_{\rho }\{z(t)\}(\tau )={\mathcal {L}}\{z((\rho t)^{\frac{1}{\rho }})\}(\tau ), \end{aligned}$$

where, ${\mathcal {L}}_{\rho }\{z\}$ represents the usual Laplace transform of z.

Theorem 2.6

[16] Let $\rho >0$ and z be a piecewise continuous function on interval [0, T] and of exponential order. Then the $\rho$-Laplace transform of the left generalized fractional integral starting at 0 is expressed in the following form:

$$\begin{aligned} {\mathcal {L}}_{\rho }\{({I_{0^{+}}^{\gamma ,\rho }}z)(t)\}(\tau )=\tau ^{-1}{\mathcal {L}}_{\rho }\{z(t)\}(\tau ). \end{aligned}$$

In the following theorem, we provided the linearity property.

Theorem 2.7

[16] If the $\rho$-Laplace transform of the functions $x:[0,\infty )\rightarrow {\mathbb {R}}$ and $y:[0,\infty )\rightarrow {\mathbb {R}}$ exists respectively for $\tau >a_{1}$ and $\tau >a_{2}$ and suppose ${\mathcal {L}}_{\rho }\{x(t)\}=X(\tau )$, ${\mathcal {L}}_{\rho }\{y(t)\}=Y(\tau )$. Then, the $\rho$-Laplace transform of $px+qy$ exists for any constants p and q and

$$\begin{aligned} {\mathcal {L}}_{\rho }\{px(t)+q y(t)\}(\tau )&=p{\mathcal {L}}_{\rho }\{x(t)\}(\tau )+q{\mathcal {L}}_{\rho }\{y(t)\}(\tau ) ,\quad \tau >\max \{a_{1},a_{2}\}\\&=p X(\tau )+q Y(\tau ). \end{aligned}$$

We need to state the $\rho$-convolution integral, so as to find $\rho$-Laplace transforms of the generalized fractional integrals and derivatives.

Definition 2.8

[16] Let x and y be piecewise continuous functions on interval [0, T] and of exponential order $e^{a\frac{t^{\rho }}{\rho }}$. Then the $\rho$-convolution of x and y is defined as

$$\begin{aligned} (x*_{\rho }y)(t)=\int _{0}^{t}x((t^{\rho }-\tau ^{\rho })^{\frac{1}{\rho }})y(\tau )\frac{d\tau }{\tau ^{1-\rho }}. \end{aligned}$$

In the following Lemma, we defined the commutativity of $\rho$-convolution of two functions.

Lemma 2.9

[16] Let x and y be piecewise continuous functions at each interval [0, T] and of exponential order. Then

$$\begin{aligned} x*_{\rho }y=y*_{\rho }x, \quad \rho >0 \end{aligned}$$

and

$$\begin{aligned} {\mathcal {L}}_{\rho }\{x*_{\rho }y\}={\mathcal {L}}_{\rho }\{x\}{\mathcal {L}}_{\rho }\{y\}. \end{aligned}$$

Lemma 2.10

[16] Let $s>0$, $\rho >0$ and $|\frac{\mu }{s^{\gamma }}|<1$.

• (i) ${\mathcal {L}}_{\rho }\{1\}(s)=\frac{1}{s}$.

• (ii) ${\mathcal {L}}_{\rho }\{e^{\mu \frac{t^{\rho }}{\rho }}\}(s)=\frac{1}{s-\mu }$.

• (iii) ${\mathcal {L}}_{\rho }\{t^{n}\}(s)=\rho ^{\frac{n}{\rho }}\frac{\Gamma (1+\frac{n}{\rho })}{s^{1+\frac{n}{\rho }}}, \; n \in {\mathbb {R}}$.

Theorem 2.11

[14] Let (W, d) be a complete metric space and S be a nonempty and closed subset of W. If a map $\Delta :S\rightarrow S$ be a contraction, then T has specifically a unique fixed point $w^{*} \in S$.

Theorem 2.12

[14] Let $\mathrm {Y}$ be a Banach space and $\Delta :\mathrm {Y}\rightarrow \mathrm {Y}$ be a continuous and compact mapping. If the set

$$\begin{aligned} \Omega =\{y \in \mathrm {Y}: y=\mu \Delta (y)\;\text {for some}\; 0<\mu <1\}, \end{aligned}$$

is bounded. Then $\Delta$ has a fixed point in $\mathrm {Y}$.

Next, we recall the generalized Gronwall’s inequality [2].

Lemma 2.13

Let $\gamma ,\rho >0$, $z(t),\alpha (t)$ be nonnegative functions and $\beta (t)$ be nondecreasing and nonnegative function, $\beta (t)\le A$, where A is a constant. If

$$\begin{aligned} z(t)\le \alpha (t)+\beta (t)\int _{t_{0}}^{t}\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1}z(\tau )\frac{d\tau }{\tau ^{1-\rho }}, \quad t \in [t_{0},T), \; T>0, \end{aligned}$$

then we have the following inequality

$$\begin{aligned} z(t)\le \alpha (t){\mathbb {E}}_{\gamma }\left( \beta (t)\Gamma (\gamma )\left( \frac{t^{\rho }-t_{0}^{\rho }}{\rho }\right) ^{\gamma }\right) . \end{aligned}$$

In the following lemma, we present the semigroup properties.

Lemma 2.14

[17]

• (i) The fractional integral ${I_{t_0}^{\rho ,\gamma }}$ satisfy the semigroup property for $\gamma ,\beta >0$

  $$\begin{aligned} ({I_{t_0}^{\rho ,\gamma }}{I_{t_0}^{\rho ,\beta }}h)(t)=({I_{t_0}^{\rho ,\gamma +\beta }}h)(t). \end{aligned}$$

• (ii) Let $\gamma ,\beta >0$. The operator ${^{c}D_{t_0}^{\rho ,\gamma }}$ satisfies the semigroup property

  $$\begin{aligned} ({^{c}D_{t_0}^{\rho ,\gamma }} {I_{t_0}^{\rho ,\gamma }}h)(t)=h(t), ~ ({^{c}D_{t_0}^{\rho ,\gamma }} {I_{t_0}^{\rho ,\beta }}h)(t)=({I_{t_0}^{\rho ,\beta -\gamma }}h)(t). \end{aligned}$$

• (iii) Let $0<\gamma <1$, if $h(t)\in C[t_{0},b]$ and ${I_{t_0}^{1-\gamma ,\rho }}h \in C^{1}[t_{0},b]$. Then, we have

  $$\begin{aligned} ({I_{t_0}^{\rho ,\gamma }} ({^{c}D_{t_0}^{\rho ,\gamma }}h))(t)=h(t)-\frac{({I_{t_0}^{\rho ,1-\gamma }}h)(t_0)}{\Gamma (\gamma )}\left( \frac{t^{\rho }-t_{0}^{\rho }}{\rho }\right) ^{\gamma -1}, \quad \text {for all} \quad t \in (t_{0},b]. \end{aligned}$$

We present the following definitions from [21, 29].

Definition 2.15

Problem (1) is said to be UH stable if there exists a real number $Z>0$, such that for every $\varepsilon >0$ and for each solution ${\mathcal {W}}\in M_{n,m}$ (the class of $n\times m$ matrices) of the inequality

$$\begin{aligned} \Phi (q,{\mathcal {W}},({^{c}D_{t_0}^{\gamma _{1},\rho }}{\mathcal {W}}),({^{c}D_{t_0}^{\gamma _{2},\rho }}{\mathcal {W}}),\dots ,({^{c}D_{t_0}^{\gamma _{n},\rho }}{\mathcal {W}}))\le \varepsilon , \end{aligned}$$

there exists a solution ${\mathcal {W}}_{a}\in M_{n,m}$ of (1) satisfying

$$\begin{aligned} |{\mathcal {W}}(t)-{\mathcal {W}}_{a}(t)|\le Z\varepsilon ,\quad t \in J. \end{aligned}$$

Definition 2.16

Problem (1) is said to be generalized UH–Rassias stable with respect to $Q \in C([0,T],{\mathbb {R}})$ if there exists a constant $u_{Q}>0$, such that for every solution ${\mathcal {W}}\in M_{n,m}$ of the inequality

$$\begin{aligned} \Phi (q,{\mathcal {W}},({^{c}D_{t_0}^{\gamma _{1},\rho }}{\mathcal {W}}),({^{c}D_{t_0}^{\gamma _{2},\rho }}{\mathcal {W}}),\dots ,({^{c}D_{t_0}^{\gamma _{n},\rho }}{\mathcal {W}}))\le Q(t), \end{aligned}$$

there exists a solution ${\mathcal {W}}_{a}\in M_{n,m}$ of (1) satisfying

$$\begin{aligned} |{\mathcal {W}}(t)-{\mathcal {W}}_{a}(t)|\le u_{Q} Q(t), ~ t \in J. \end{aligned}$$

Lemma 2.17

If the matrix $(\tau ^{\gamma }I_{n}-A)$ is invertible, then the linear fractional differential equation

$$\begin{aligned} \left\{ \begin{array}{l} {^{c}D_{t_0}^{\gamma ,\rho }}{\mathcal {W}}(t)=A{\mathcal {W}}(t)+q(t),\\ {\mathcal {W}}(t_0)=P, \end{array} \right. \end{aligned}$$

(3)

has solution given by

$$\begin{aligned} {\mathcal {W}}(t)=P{\mathbb {E}}_{\gamma }\left( A\left( \frac{t^{\rho }}{\rho }\right) ^{\gamma }\right) +\int _{t_0}^{t}\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1}{\mathbb {E}}_{\gamma ,\gamma } \left( A\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma }\right) q(\tau )\frac{d\tau }{\tau ^{1-\rho }}. \end{aligned}$$

Proof

By applying the $\rho$-Laplace transform to both sides of (3) and using Theorem 2.5, we have

$$\begin{aligned} \tau ^{\gamma }{\mathcal {L}}_{\rho }\{{\mathcal {W}}(t)\}(s)-\tau ^{\gamma -1}{\mathcal {W}}(0)=A{\mathcal {L}}_{\rho }\{{\mathcal {W}}(t)\}(s)+{\mathcal {L}}_{\rho }\{q(t)\}(s). \end{aligned}$$

It follows that

$$\begin{aligned} {\mathcal {L}}_{\rho }\{{\mathcal {W}}(t)\}(s)&=\tau ^{\gamma -1}(\tau ^{\gamma }I_{n}-A)^{-1}P+(\tau ^{\gamma }I_{n}-A)^{-1}{\mathcal {L}}_{\rho }\{q(t)\}\\&=P{\mathcal {L}}_{\rho }\left\{ {\mathbb {E}}_{\gamma }\left( A\left( \frac{t^{\rho }}{\rho }\right) ^{\gamma }\right) \right\} +{\mathcal {L}}_{\rho }\left\{ \left( \frac{t^{\rho }}{\rho }\right) ^{\gamma -1} {\mathbb {E}}_{\gamma ,\gamma }\left( A\left( \frac{t^{\rho }}{\rho }\right) ^{\gamma }\right) \right\} {\mathcal {L}}_{\rho }\{q(t)\}\\&=P{\mathcal {L}}_{\rho }\left\{ {\mathbb {E}}_{\gamma }\left( A\left( \frac{t^{\rho }}{\rho }\right) ^{\gamma }\right) \right\} \\&\quad +{\mathcal {L}}_{\rho }\left\{ \int _{t_0}^{t}\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1} {\mathbb {E}}_{\gamma ,\gamma }\left( A\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma }\right) q(\tau )\frac{d\tau }{\tau ^{1-\rho }}\right\} \\&={\mathcal {L}}_{\rho }\left\{ P\left\{ {\mathbb {E}}_{\gamma }\left( A\left( \frac{t^{\rho }}{\rho }\right) ^{\gamma }\right) \right\} \right. \\&\quad +\left. \int _{t_0}^{t}\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1} {\mathbb {E}}_{\gamma ,\gamma }\left( A\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma }\right) q(\tau )\frac{d\tau }{\tau ^{1-\rho }}\right\} . \end{aligned}$$

This implies that,

$$\begin{aligned} {\mathcal {W}}(t)=P{\mathbb {E}}_{\gamma }\left( A\left( \frac{t^{\rho }}{\rho }\right) ^{\gamma }\right) +\int _{t_0}^{t}\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1}{\mathbb {E}}_{\gamma ,\gamma } \left( A\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma }\right) q(\tau )\frac{d\tau }{\tau ^{1-\rho }}. \end{aligned}$$

$\square$

Lemma 2.18

Let $\rho >0$, $\gamma \in (0,1]$ and $\aleph :J\times {\mathbb {R}} \times {\mathbb {R}}\rightarrow {\mathbb {R}}$, $l:\Delta \times {\mathbb {R}} \rightarrow {\mathbb {R}}$ be continuous. Then, solution of the Cauchy problem

$$\begin{aligned} \left\{ \begin{array}{l} {^{c}D_{t_0}^{\gamma ,\rho }}\alpha (t)=\aleph (t,\alpha (t),\mathrm {L}\alpha (t)), \quad t \in J:=[t_0,T], \\ {{\mathcal {J}}_{t_0}^{1-\gamma ,\rho }}\alpha (t)|_{t=t_{0}}=\vartheta , \end{array} \right. \end{aligned}$$

(4)

is given by

$$\begin{aligned} \alpha (t)=\frac{\vartheta }{\Gamma (\gamma )}\left( \frac{t^{\rho }-t_{0}^{\rho }}{\rho }\right) ^{\gamma -1}+\frac{1}{\Gamma (\gamma )}\int _{t_{0}}^{t}\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1} \aleph (\tau ,\alpha (\tau ),\mathrm {L}\alpha (\tau ))\frac{d\tau }{\tau ^{1-\rho }}. \end{aligned}$$

Proof

Let $\alpha (t)$ satisfies (4). If $t \in [t_0,T]$, then

$$\begin{aligned} ^{c}D^{\gamma ,\rho }\alpha (t)=\aleph (t,\alpha (t),\mathrm {L}\alpha (t)). \end{aligned}$$

Applying ${^{c}{\mathcal {J}}_{t_0}^{\gamma ,\rho }}$ to both sides we get,

$$\begin{aligned}&\alpha (t)- \frac{({{\mathcal {J}}_{t_0}^{1-\gamma ,\rho }}\alpha )(t_{0})}{\Gamma (\gamma )}\left( \frac{t^{\rho }-t_{0}^{\rho }}{\rho }\right) ^{\gamma -1}\\&\quad =\frac{1}{\Gamma (\gamma )}\int _{t_{0}}^{t}\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1} \aleph (\tau ,\alpha (\tau ),\mathrm {L}\alpha (\tau ))\frac{d\tau }{\tau ^{1-\rho }}. \end{aligned}$$

By using initial condition, we get

$$\begin{aligned} \alpha (t)= \frac{\vartheta }{\Gamma (\gamma )}\left( \frac{t^{\rho }-t_{0}^{\rho }}{\rho }\right) ^{\gamma -1} +\frac{1}{\Gamma (\gamma )}\int _{t_{0}}^{t}\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1} \aleph (\tau ,\alpha (\tau ),\mathrm {L}\alpha (\tau ))\frac{d\tau }{\tau ^{1-\rho }}. \end{aligned}$$

(5)

Next, we prove the sufficient part.

Let $\alpha \in C_{1-\gamma ,\rho }^{\gamma }[t_0,T]$ satisfies (5), then $^{c}D_{t_{0+}}^{\gamma ,\rho } \in C_{1-\gamma ,\rho }[t_0,T]$.

By applying the operator $^{c}D_{t_{0+}}^{\gamma ,\rho }$ to both sides of (5), we obtain

$$\begin{aligned}&{^{c}D_{t_0}^{\gamma ,\rho }}\left( \alpha (t)-\frac{\vartheta }{\Gamma (\gamma )}\left( \frac{t^{\rho }-t_{0}^{\rho }}{\rho }\right) ^{\gamma -1}\right) (t)\\&\quad ={^{c}D_{t_0}^{\gamma ,\rho }}\left( \frac{1}{\Gamma (\gamma )}\int _{t_{0}}^{t}\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1} \aleph (\tau ,\alpha (\tau ),\mathrm {L}\alpha (\tau ))\frac{d\tau }{\tau ^{1-\rho }}\right) . \end{aligned}$$

Since,

$$\begin{aligned} {^{c}D_{t_0}^{\gamma ,\rho }}\bigg ((t^{\rho }-t_{0}^{\rho })\bigg )(t)=0. \end{aligned}$$

Thus, we obtain

$$\begin{aligned} {^{c}D_{t_0}^{\gamma ,\rho }}\alpha (t)=\aleph (t,\alpha (t),\mathrm {L}\alpha (t)). \end{aligned}$$

To show that $\alpha (t)$ satisfies the initial condition, we apply the operator ${\mathcal {J}}_{t_0}^{1-\gamma ,\rho }$ on both sides of (5)

$$\begin{aligned} {\mathcal {J}}_{t_0}^{1-\gamma ,\rho }\alpha (t)&={\mathcal {J}}_{t_0}^{1-\gamma ,\rho }\left( \frac{\vartheta }{\Gamma (\gamma )}\bigg (\frac{t^{\rho }-t_{0}^{\rho }}{\rho }\bigg )^{\gamma -1}\right. \\&\quad +\left. \frac{1}{\Gamma (\gamma )}\int _{t_{0}}^{t}\bigg (\frac{t^{\rho }-\tau ^{\rho }}{\rho }\bigg )^{\gamma -1}\aleph (\tau ,\alpha (\tau ),L\alpha (\tau ))\frac{d\tau }{\tau ^{1-\rho }}\right) \\&={\mathcal {J}}_{t_0}^{1-\gamma ,\rho }\frac{\vartheta }{\Gamma (\gamma )}\bigg (\frac{t^{\rho }-t_{0}^{\rho }}{\rho }\bigg )^{\gamma -1}(t)\\&\quad +{\mathcal {J}}_{t_0}^{1-\gamma ,\rho } \frac{1}{\Gamma (\gamma )}\int _{t_{0}}^{t}\bigg (\frac{t^{\rho }-\tau ^{\rho }}{\rho }\bigg )^{\gamma -1}\aleph (\tau ,\alpha (\tau ),\mathrm {L}\alpha (\tau ))\frac{d\tau }{\tau ^{1-\rho }}\\&={\mathcal {J}}_{t_0}^{1-\gamma ,\rho }\frac{\vartheta }{\Gamma (\gamma )}\bigg (\frac{t^{\rho }-t_{0}^{\rho }}{\rho }\bigg )^{\gamma -1}(t)+({\mathcal {J}}_{t_0}^{1,\rho }\aleph (\tau ,\alpha (\tau ),\mathrm {L}\alpha (\tau ))). \end{aligned}$$

Now, taking the limit $t\rightarrow t_{0}$ to both sides, we get

$$\begin{aligned} {\mathcal {J}}_{t_0}^{1-\gamma ,\rho }\alpha (t_{0})=\vartheta . \end{aligned}$$

$\square$

Stability Analysis of (1)

Theorem 3.1

Let $\gamma \in (0,1)$, $\rho >0$, A is n-th order matrix and q(t) is specified vector. If ${\mathcal {W}}(t)\in M_{n,m}$ satisfies the following inequality

$$\begin{aligned} |{^{c}D_{t_0}^{\gamma ,\rho }}{\mathcal {W}}(t)-A{\mathcal {W}}(t)-q(t)|\le \varepsilon , \end{aligned}$$

for each $\varepsilon >0$, then there exists a solution ${\mathcal {W}}_{a}$ of problem (1) such that

$$\begin{aligned} |{\mathcal {W}}(t)-{\mathcal {W}}_{a}(t)|\le \left( \frac{T^{\rho }}{\rho }\right) ^{\gamma }{\mathbb {E}}_{\gamma ,\gamma +1}\left( A\left( \frac{T^{\rho }}{\rho }\right) ^{\gamma }\right) \varepsilon . \end{aligned}$$

(6)

Proof

Suppose

$$\begin{aligned} {\mathcal {W}}_{1}(t)=({^{c}D_{t_0}^{\gamma ,\rho }}{\mathcal {W}})(t)-A{\mathcal {W}}(t)-q(t). \end{aligned}$$

(7)

By taking the $\rho$-Laplace transform of (7) and using Theorem 2.5, we get

$$\begin{aligned} {\mathcal {L}}_{\rho }\{{\mathcal {W}}_{1}(t)\}(\tau )&={\mathcal {L}}_{\rho }\{({^{c}D_{t_0}^{\gamma ,\rho }}{\mathcal {W}})(t)-A{\mathcal {W}}(t)-q(t)\}(\tau )\nonumber \\&={\mathcal {L}}_{\rho }\{({^{c}D_{t_0}^{\gamma ,\rho }}{\mathcal {W}})(t)\}(\tau )-A{\mathcal {L}}_{\rho }\{{\mathcal {W}}(t)\}(\tau )-{\mathcal {L}}_{\rho }\{q(t)\}(\tau )\nonumber \\&=\tau ^{\gamma }{\mathcal {L}}_{\rho }\{{\mathcal {W}}(t)\}(\tau )-\tau ^{\gamma -1}P-A{\mathcal {L}}_{\rho }\{{\mathcal {W}}(t)\}(\tau )-{\mathcal {L}}_{\rho }\{q(t)\}(\tau ), \end{aligned}$$

(8)

where, ${\mathcal {L}}_{\rho }\{{\mathcal {W}}_{1}(t)\}$ is the $\rho$-Laplace transform of the function ${\mathcal {W}}_{1}$.

Thus, from (8), we have

$$\begin{aligned} {\mathcal {L}}_{\rho }\{{\mathcal {W}}(t)\}(\tau )&=\tau ^{\gamma -1}(\tau ^{\gamma }I_{n}-A)^{-1}P+(\tau ^{\gamma }I_{n}-A)^{-1}{\mathcal {L}}_{\rho }\{{\mathcal {W}}_{1}(t)\}(\tau )\nonumber \\&\quad +(\tau ^{\gamma }I_{n}-A)^{-1}{\mathcal {L}}_{\rho }\{q(t)\}(\tau )\nonumber \\&=P{\mathcal {L}}_{\rho }\left\{ {\mathbb {E}}_{\gamma }\left( A\left( \frac{t^{\rho }}{\rho }\right) ^{\gamma }\right) \right\} (\tau )\nonumber \\&\quad +{\mathcal {L}}_{\rho }\left\{ \left( \frac{t^{\rho }}{\rho }\right) ^{\gamma -1}{\mathbb {E}}_{\gamma ,\gamma }\left( A\left( \frac{t^{\rho }}{\rho }\right) ^{\gamma }\right) \right\} (s) \cdot {\mathcal {L}}_{\rho }\{{\mathcal {W}}_{1}(t)\}(\tau )\nonumber \\&\quad +{\mathcal {L}}_{\rho }\left\{ \left( \frac{t^{\rho }}{\rho }\right) ^{\gamma -1}{\mathbb {E}}_{\gamma ,\gamma }\left( A\left( \frac{t^{\rho }}{\rho }\right) ^{\gamma }\right) \right\} (\tau ) \cdot {\mathcal {L}}_{\rho }\{q(t)\}(\tau )\nonumber \\&=P{\mathcal {L}}_{\rho }\left\{ {\mathbb {E}}_{\gamma }\left( A\left( \frac{t^{\rho }}{\rho }\right) ^{\gamma }\right) \right\} (\tau )\nonumber \\&\quad +{\mathcal {L}}_{\rho }\left\{ \int _{t_0}^{t}\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1} {\mathbb {E}}_{\gamma ,\gamma }\left( A\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma }\right) \cdot {\mathcal {W}}_{1}(t)\frac{d\tau }{\tau ^{1-\rho }}\right\} \nonumber \\&\quad +{\mathcal {L}}_{\rho }\left\{ \int _{t_0}^{t}\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1}{\mathbb {E}}_{\gamma ,\gamma }\left( A\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma }\right) \cdot q(t) \frac{d\tau }{\tau ^{1-\rho }}\right\} . \end{aligned}$$

(9)

Set

$$\begin{aligned} {\mathcal {W}}_{a}(t)=P{\mathbb {E}}_{\gamma }\left( A\left( \frac{t^{\rho }}{\rho }\right) ^{\gamma }\right) +\int _{t_0}^{t}(\frac{t^{\rho }-\tau ^{\rho }}{\rho })^{\gamma -1}{\mathbb {E}}_{\gamma ,\gamma }\left( A\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma }\right) \cdot q(t) \frac{d\tau }{\tau ^{1-\rho }}. \end{aligned}$$

(10)

Now, taking the $\rho$-Laplace transform of (10), we obtain

$$\begin{aligned} {\mathcal {L}}_{\rho }\{{\mathcal {W}}_{a}(t)\}(\tau )&=P{\mathcal {L}}_{\rho }\left\{ {\mathbb {E}}_{\gamma }\left( A\left( \frac{t^{\rho }}{\rho }\right) ^{\gamma }\right) \right\} (\tau )\nonumber \\&\quad +{\mathcal {L}}_{\rho }\left\{ \int _{t_0}^{t}\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1}{\mathbb {E}}_{\gamma ,\gamma }\left( A\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma }\right) \cdot q(t) \frac{d\tau }{\tau ^{1-\rho }}\right\} (\tau )\nonumber \\&=P{\mathcal {L}}_{\rho }\left\{ {\mathbb {E}}_{\gamma }\left( A\left( \frac{t^{\rho }}{\rho }\right) ^{\gamma }\right) \right\} (\tau )\nonumber \\&\quad +{\mathcal {L}}_{\rho }\left\{ \left( \frac{t^{\rho }}{\rho }\right) ^{\gamma -1}{\mathbb {E}}_{\gamma ,\gamma }\left( A\left( \frac{t^{\rho }}{\rho }\right) ^{\gamma }\right) *_{\rho } q(t)\right\} (\tau )\nonumber \\&=P{\mathcal {L}}_{\rho }\left\{ {\mathbb {E}}_{\gamma }\left( A\left( \frac{t^{\rho }}{\rho }\right) ^{\gamma }\right) \right\} (\tau )\nonumber \\&\quad +{\mathcal {L}}_{\rho }\left\{ \left( \frac{t^{\rho }}{\rho }\right) ^{\gamma -1}{\mathbb {E}}_{\gamma ,\gamma }\left( A\left( \frac{t^{\rho }}{\rho }\right) ^{\gamma }\right) \right\} (\tau ). {\mathcal {L}}_{\rho }\{q(t)\}(\tau )\nonumber \\&=\tau ^{\gamma -1}(\tau ^{\gamma }I_{n}-A)^{-1}P+(\tau ^{\gamma }I_{n}-A)^{-1}{\mathcal {L}}_{\rho }\{q(t)\}(\tau ). \end{aligned}$$

(11)

By Theorem 2.5 and (11), we get

$$\begin{aligned} {\mathcal {L}}_{\rho }\{({^{c}D_{t_0}^{\gamma ,\rho }}{\mathcal {W}}_{a})(t)-A{\mathcal {W}}_{a}(t)\}(\tau )&=\tau ^{\gamma }{\mathcal {L}}_{\rho }\{{\mathcal {W}}_{a}(t)\}(\tau )\\&\quad -\tau ^{\gamma -1}P-A{\mathcal {L}}_{\rho }\{{\mathcal {W}}_{a}(t)\}(\tau )\\&=(\tau ^{\gamma }I_{n}-A)(\tau ^{\gamma }I_{n}-A)^{-1}P-\tau ^{\gamma -1}P\\&\quad +(\tau ^{\gamma }I_{n}-A)(\tau ^{\gamma }I_{n}-A)^{-1}{\mathcal {L}}_{\rho }\{q(t)\}(\tau )\\&={\mathcal {L}}_{\rho }\{q(t)\}(\tau ). \end{aligned}$$

Due to the fact that ${\mathcal {L}}_{\rho }$ is one-to-one infers that ${\mathcal {W}}_{a}$ is a solution of (1).

Equations (9) and (11) implies that,

$$\begin{aligned}&{\mathcal {L}}_{\rho }\{{\mathcal {W}}(t)-{\mathcal {W}}_{a}(t)\}(\tau )\\&\quad =P{\mathcal {L}}_{\rho }\left\{ {\mathbb {E}}_{\gamma }\left( A\left( \frac{t^{\rho }}{\rho }\right) ^{\gamma }\right) \right\} (\tau )\\&\qquad +{\mathcal {L}}_{\rho }\left\{ \int _{t_0}^{t}\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1} {\mathbb {E}}_{\gamma ,\gamma }\left( A\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma }\right) {\mathcal {W}}_{1}(t)\frac{d\tau }{\tau ^{1-\rho }}\right\} \\&\qquad +{\mathcal {L}}_{\rho }\left\{ \int _{t_0}^{t}\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1}{\mathbb {E}}_{\gamma ,\gamma }\left( A\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma }\right) q(t) \frac{d\tau }{\tau ^{1-\rho }}\right\} \\&\qquad -P{\mathcal {L}}_{\rho }\left\{ {\mathbb {E}}_{\gamma }\left( A\left( \frac{t^{\rho }}{\rho }\right) ^{\gamma }\right) \right\} (\tau )\\&\qquad -{\mathcal {L}}_{\rho }\left\{ \int _{t_0}^{t}\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1}{\mathbb {E}}_{\gamma ,\gamma }\left( A\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma }\right) q(t) \frac{d\tau }{\tau ^{1-\rho }}\right\} (\tau )\\&\quad ={\mathcal {L}}_{\rho }\left\{ \int _{t_0}^{t}\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1}{\mathbb {E}}_{\gamma ,\gamma }\left( A\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma }\right) {\mathcal {W}}_{1}(t)\frac{d\tau }{\tau ^{1-\rho }}\right\} . \end{aligned}$$

This implies that,

$$\begin{aligned} |{\mathcal {W}}(t)-{\mathcal {W}}_{a}(t)|&=\left| \int _{t_0}^{t}\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1}{\mathbb {E}}_{\gamma ,\gamma }\left( A\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma }\right) {\mathcal {W}}_{1}(t) \frac{d\tau }{\tau ^{1-\rho }}\right| \\&\le \int _{t_0}^{t}\left| \left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1}{\mathbb {E}}_{\gamma ,\gamma }\left( A\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma }\right) \right| |{\mathcal {W}}_{1}(t)| \frac{d\tau }{\tau ^{1-\rho }}\\&\le \varepsilon \int _{t_0}^{t}\left| \left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1}{\mathbb {E}}_{\gamma ,\gamma }\left( A\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma }\right) \right| \frac{d\tau }{\tau ^{1-\rho }}\\&=\varepsilon \sum _{j={t_0}}^{\infty }\frac{A^{j}}{\Gamma (\gamma j+\gamma )}\int _{t_0}^{t}\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma j+\gamma -1}d\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) \\&=\varepsilon \sum _{j=0}^{\infty }\frac{A^{j}}{\Gamma (\gamma j+\gamma +1)}\left( \frac{t^{\rho }}{\rho }\right) ^{\gamma j+\gamma }\\&=\varepsilon \left( \frac{t^{\rho }}{\rho }\right) ^{\gamma }{\mathbb {E}}_{\gamma ,\gamma +1}\left( A\left( \frac{t^{\rho }}{\rho }\right) ^{\gamma }\right) \\&\le Z \varepsilon . \end{aligned}$$

Hence, we can say that (1) is UH stable with constant $Z=(\frac{T^{\rho }}{\rho })^{\gamma }{\mathbb {E}}_{\gamma ,\gamma +1}(A\frac{T^{\rho }}{\rho })^{\gamma }$ for $t\in [t_0,T]$. Note that, (1) is not UH stable, at $t=\infty$. $\square$

Theorem 3.2

Suppose that $\rho >0$, $\gamma \in (0,1)$, $A \in M_{n,n}$, $P\in M_{n,m}$ and q(t) be a locally column vector containing real continuous function on $[t_0,\infty )$. If the function ${\mathcal {W}}\in {M}_{n,m}$ satisfies the following inequality

$$\begin{aligned} |{^{c}D_{t_0}^{\gamma ,\rho }}{\mathcal {W}}(t)-A{\mathcal {W}}(t)-q(t)|\le Q(t), \end{aligned}$$

then there exists a solution ${\mathcal {W}}_{a}:[t_0,\infty )\rightarrow {\mathbb {R}}$ of problem (1) satisfying

$$\begin{aligned} |{\mathcal {W}}(t)-{\mathcal {W}}_{a}(t)|\le u_{Q}Q(t). \end{aligned}$$

Proof

Let

$$\begin{aligned} |{\mathcal {W}}_{1}(t)|\le Q(t), \end{aligned}$$

where, ${\mathcal {W}}_{1}$ is defined in (7).

Following the same procedure like in Theorem 3.1, we obtain

$$\begin{aligned} |{\mathcal {W}}(t)-{\mathcal {W}}_{a}(t)|&=\left| \int _{t_0}^{t}\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1}{\mathbb {E}}_{\gamma ,\gamma }\left( A\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma }\right) {\mathcal {W}}_{1}(t) \frac{d\tau }{\tau ^{1-\rho }}\right| \nonumber \\&\le Q\int _{t_0}^{t}\left| \left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1}{\mathbb {E}}_{\gamma ,\gamma }\left( A\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma }\right) \right| \frac{d\tau }{\tau ^{1-\rho }}\nonumber \\&\le Q\int _{t_0}^{t}\left| \left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1}{\mathbb {E}}_{\gamma ,\gamma }\left( A\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma }\right) \right| d\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) \nonumber \\&\le Q \int _{t_0}^{\frac{t^{\rho }}{\rho }} x^{\gamma -1}{\mathbb {E}}_{\gamma ,\gamma }(Ax^{\gamma })dx\nonumber \\&\le Q \int _{t_0}^{\frac{t^{\rho }}{\rho }}x^{\gamma -1}\sum _{j=0}^{\infty }\frac{A^{j}}{\Gamma (\gamma j+\gamma )}x^{\gamma j}dx\nonumber \\&\le Q\sum _{j=0}^{\infty }\frac{|A^{j}|}{\Gamma (\gamma j+\gamma +1)}(\frac{t^{\rho }}{\rho })^{\gamma j+\gamma }\nonumber \\&\le Q\left( \frac{t^{\rho }}{\rho }\right) ^{\gamma }{\mathbb {E}}_{\gamma ,\gamma +1}\left( A\left( \frac{t^{\rho }}{\rho }\right) ^{\gamma j}\right) \nonumber \\&\le u_{q}Q. \end{aligned}$$

(12)

From (12), we conclude that (1) is generalized UH–Rassias stable on $[t_0,T]$. $\square$

Existence Result of Nonlinear Model (2)

We start this section by considering the following assumptions.

Suppose that

$(\mathrm {A}_{1})$:

$\aleph :J\times {\mathbb {R}}\times {\mathbb {R}}\rightarrow {\mathbb {R}}$ is continuous.

$(\mathrm {A}_{2})$:

For any $a,{\tilde{a}},b,{\tilde{b}}\in {\mathbb {R}}$, there exists a constant ${\mathcal {P}}>0$, such that

$$\begin{aligned} |\aleph (t,a,b)-\aleph (t,{\tilde{a}},{\tilde{b}})|\le {\mathcal {P}}(|a-{\tilde{a}}|+|b-{\tilde{b}}|). \end{aligned}$$

$(\mathrm {A}_{3})$:

There exists a constant $\Re >0$ such that for each $t \in J$ and for every $a,b \in {\mathbb {R}}$ satisfying

$$\begin{aligned} |\aleph (t,a,b)|\le \Re . \end{aligned}$$

$(\mathrm {A}_{4})$:

The function $l:\Delta \times {\mathbb {R}}\rightarrow {\mathbb {R}}$ is continuous and there exists a constant $\mathrm {L}>0$, such that

$$\begin{aligned} |l(t,s,a)-l(t,s,{\tilde{a}})|\le \mathrm {L}_{1}|a-{\tilde{a}}|. \end{aligned}$$

We transform problem (2) into a fixed point problem.

Consider an operator $\Psi :C(J,{\mathbb {R}})\rightarrow C(J,{\mathbb {R}})$ defined by

$$\begin{aligned} \Psi (\alpha )(t)&=\frac{\vartheta }{\Gamma (\gamma )}\left( \frac{t^{\rho }-t_{0}^{\rho }}{\rho }\right) ^{\gamma -1}\nonumber \\&\quad +\frac{1}{\Gamma (\gamma )}\int _{t_{0}}^{t} \left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1}\aleph (\tau ,\alpha (\tau ),\mathrm {L}\alpha (\tau ))\frac{d\tau }{\tau ^{1-\rho }},\quad \forall \;t \in J. \end{aligned}$$

(13)

Our first result for the existence of solutions is based on the Schaefer’s fixed point theorem.

Theorem 4.1

Let assumptions $(\mathrm {A}_{1})$ and $(\mathrm {A}_{3})$ holds. Then, (2) has atleast one solution.

Proof

The proof is accomplish in several steps.

• Step 1. The operator (13) is continuous.

  $$\begin{aligned} |\Psi (\alpha _{n})(t)-\Psi (\alpha )(t)|&=\frac{1}{\Gamma (\gamma )}\left| \int _{t_{0}}^{t} \left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1}\aleph (\tau ,\alpha _{n}(\tau ),\mathrm {L}\alpha _{n}(\tau ))\frac{d\tau }{\tau ^{1-\rho }}\right. \\&\quad -\left. \int _{t_{0}}^{t}\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1}\aleph (\tau ,\alpha (\tau ),\mathrm {L}\alpha (\tau ))\frac{d\tau }{\tau ^{1-\rho }}\right| \\&\le \frac{1}{\Gamma (\gamma )}\int _{t_{0}}^{t}\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1}\\&\quad \times |\aleph (\tau ,\alpha _{n}(\tau ),\mathrm {L}\alpha _{n}(\tau ))-\aleph (\tau ,\alpha (\tau ),\mathrm {L}\alpha (\tau ))|\frac{d\tau }{\tau ^{1-\rho }}\\&\le \frac{1}{\Gamma (\gamma +1)}\left( \frac{T^{\rho }}{\rho }\right) ^{\gamma }\Vert \aleph (\cdot ,\alpha _{n},\mathrm {L}\alpha _{n})-\aleph (\cdot ,\alpha ,\mathrm {L}\alpha )\Vert _{C}. \end{aligned}$$

  As, $\aleph$ is continuous function, we have

  $$\begin{aligned} \Vert \aleph (\cdot ,\alpha _{n},\mathrm {L}\alpha _{n})-\aleph (\cdot ,\alpha ,\mathrm {L}\alpha )\Vert _{C}\rightarrow 0 \quad as \quad n\rightarrow \infty . \end{aligned}$$

  Hence, we can say that $\Psi$ is continuous.

• Step 2. $\Psi$ maps bounded sets into bounded sets of $C([t_0,T],{\mathbb {R}})$. In fact, we need to prove that for each $\alpha \in \mathrm {M}_{\xi }=\{\alpha \in C(J,{\mathbb {R}}):\Vert \alpha \Vert \le \xi \}$ and for every $\xi >0$ there exists a constant $\eta >0$, such that $\Vert \Psi (\alpha )\Vert _{c}\le \eta$. From $(\mathrm {A}_{3})$ and for any $t \in [t_0,T]$, we have

  $$\begin{aligned} |\Psi (\alpha )(t)|&\le \frac{1}{\Gamma (\gamma )}|\vartheta \rho ^{1-\gamma }(t^{\rho })^{\gamma -1}|\\&\quad +\frac{1}{\Gamma (\gamma )}\left| \int _{t_{0}}^{t}\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1} \aleph (\tau ,\alpha (\tau ),\mathrm {L}\alpha (\tau ))\frac{d\tau }{\tau ^{1-\rho }}\right| \\&\le \frac{1}{\Gamma (\gamma )}|\vartheta \rho ^{1-\gamma }(t^{\rho })^{\gamma -1}|\\&\quad +\frac{1}{\Gamma (\gamma )}\int _{t_{0}}^{t}\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1} |\aleph (\tau ,\alpha (\tau ),\mathrm {L}\alpha (\tau ))|\frac{d\tau }{\tau ^{1-\rho }}\\&\le \frac{1}{\Gamma (\gamma )}|\vartheta \rho ^{1-\gamma }(T^{\rho })^{\gamma -1}|+\frac{\Re T^{\rho \gamma }}{\Gamma (\gamma +1)\rho ^{\gamma }}:=\eta , \end{aligned}$$

  which implies that

  $$\begin{aligned} \Vert \Psi (\alpha )(t)\Vert _{C}\le \eta . \end{aligned}$$

  Hence, $\Psi$ is bounded.

• Step 3. The operator $\Psi$ maps bounded sets into equicontinuous sets in $C([t_0,T],{\mathbb {R}})$. Let $\aleph \in \mathrm {M}_{\xi }$ with $t_{1},t_{2} \in J$ and $t_0\le t_{1}<t_{2}\le T$. Then

  $$\begin{aligned} |\Psi (\alpha )(t_{1})-\Psi (\alpha )(t_{2})|&\le \frac{\vartheta \rho ^{1-\gamma }}{\Gamma (\gamma )}\big [(t_{1}^{\rho })^{\gamma -1}-(t_{2}^{\rho })^{\gamma -1}\big ]\\&\quad +\frac{1}{\Gamma (\gamma )}\int _{t_0}^{t_{1}}\left( \frac{t_{1}^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1}|\aleph (\tau ,\alpha (\tau ),\mathrm {L}\alpha (\tau ))|\frac{d\tau }{\tau ^{1-\rho }}\\&\quad -\frac{1}{\Gamma (\gamma )}\int _{t_0}^{t_{2}}\left( \frac{t_{2}^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1} |\aleph (\tau ,\alpha (\tau ),\mathrm {L}\alpha (\tau ))|\frac{d\tau }{\tau ^{1-\rho }}\\&\le \frac{\vartheta \rho ^{1-\gamma }}{\Gamma (\gamma )}\big [(t_{1}^{\rho })^{\gamma -1}-(t_{2}^{\rho })^{\gamma -1}\big ]\\&\quad +\frac{\Re }{\Gamma (\gamma )}\left( \int _{t_0}^{t_{1}}\left[ \left( \frac{t_{1}^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1} -\left( \frac{t_{2}^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1}\right] \frac{d\tau }{\tau ^{1-\rho }}\right. \\&\quad +\left. \int _{t_{1}}^{t_{2}}\left( \frac{t_{2}^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1}\frac{d\tau }{\tau ^{1-\rho }}\right) \\&\le \frac{\vartheta \rho ^{1-\gamma }}{\Gamma (\gamma )}\big [(t_{1}^{\rho })^{\gamma -1}-(t_{2}^{\rho })^{\gamma -1}\big ]\\&\quad +\frac{\Re }{\Gamma (\gamma +1)\rho ^{\gamma }}\big (t_{2}^{\gamma \rho }-t_{1}^{\gamma \rho }-2(t_{2}^{\rho }-t_{1}^{\rho })^{\gamma }\big ). \end{aligned}$$

  As, $t_{1}\rightarrow t_{2}$, then the right hand side of the above inequality tends to zero. Consequently, $\Psi$ is equicontinuous. As a consequence of Steps 1–3, together with the Arzela–Ascoli theorem, we can conclude that $\Psi$ is completely continuous.

• Step 4. Next, we show that the set $\Lambda =\{\alpha \in C(J,{\mathbb {R}}):\alpha =\kappa \Omega (\alpha )\;\text {for some}\; 0<\kappa <1\}$ is bounded. Let $\alpha \in \Lambda$ and for each $t \in J$, we have:

  $$\begin{aligned} |\alpha (t)|&\le \kappa \left[ \frac{\vartheta }{\Gamma (\gamma )}\left( \frac{t^{\rho }}{\rho }\right) ^{\gamma -1} +\frac{1}{\Gamma (\gamma )}\int _{t_0}^{t}\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1} |\aleph (\tau ,\alpha (\tau ),\mathrm {L}\alpha (\tau ))|\frac{d\tau }{\tau ^{1-\rho }}\right] \\&\le \frac{\vartheta }{\Gamma (\gamma )}\left( \frac{t^{\rho }}{\rho }\right) ^{\gamma -1}+\frac{\Re T^{\rho \gamma }}{\Gamma (\gamma +1)\rho ^{\gamma }}:={\mathcal {Q}}, \end{aligned}$$

which implies that,

$$\begin{aligned} \Vert \alpha (t)\Vert _{C}\le {\mathcal {Q}}. \end{aligned}$$

Thus, the set $\Lambda$ is bounded. $\square$

Our next analysis is to show that the operator $\Psi$ has a unique fixed point.

Theorem 4.2

Assume that $(A_{1}),(A_{2})$ and $(A_{4})$ holds. Then, the problem (2) has a unique solution on $[t_0,T]$, provided that

$$\begin{aligned} \frac{{\mathcal {P}}(1+\mathrm {L}_{1})T^{\rho \gamma }}{\Gamma (\gamma +1)\rho ^{\gamma }}<1. \end{aligned}$$

(14)

Proof

By the use of Banach fixed point theorem, we can prove that the operator defined by (13) has a fixed point.

For $\alpha _{1},\alpha _{2} \in C(J,{\mathbb {R}})$ and $t \in [t_0,T]$, we have

$$\begin{aligned} |\Psi (\alpha _{1})(t)-\Psi (\alpha _{2})(t)|&\le \frac{1}{\Gamma (\gamma )}\int _{t_0}^{t}\bigg (\frac{t^{\rho }-\tau ^{\rho }}{\rho }\bigg )^{\gamma -1} |\aleph (\tau ,\alpha _{1}(\tau ),\mathrm {L}\alpha _{1}(\tau ))\\&\quad -\aleph (\tau ,\alpha _{2}(\tau ),\mathrm {L}\alpha _{2}(\tau ))|\frac{d\tau }{\tau ^{1-\rho }}\\&\le \frac{1}{\Gamma (\gamma )}\int _{t_0}^{t}\bigg (\frac{t^{\rho }-\tau ^{\rho }}{\rho }\bigg )^{\gamma -1}{\mathcal {P}}\big [|\alpha _{1}-\alpha _{2}|+\mathrm {L}_{1} |\alpha _{1}-\alpha _{2}|\big ]\frac{d\tau }{\tau ^{1-\rho }}. \end{aligned}$$

On further simplifications, we get

$$\begin{aligned} \Vert \Psi \alpha _{1}-\Psi \alpha _{2}\Vert _{C}\le \frac{{\mathcal {P}}(1+\mathrm {L}_{1})T^{\rho \gamma }}{\Gamma (\gamma +1)\rho ^{\gamma }}\Vert \alpha _{1}-\alpha _{2}\Vert _{C}. \end{aligned}$$

Hence, $\Psi$ is a contraction. As an application of Banach fixed point theorem, we conclude that $\Psi$ has a fixed point which is the solution of (2). $\square$

Stability Analysis for Nonlinear Model

In this section, we are interested in the stability concepts from the Ulam’s point of view for the problem (2).

We consider the following definitions.

Definition 5.1

Problem (2) is said to be UH stable, if there exists a constant $M>0$ such that for every $\xi >0$ and for each solution $\alpha \in C(J,{\mathbb {R}})$ of the inequality

$$\begin{aligned} |{^{c}D_{t_0}^{\gamma ,\rho }}\alpha (t)-\aleph (t,\alpha (t),\mathrm {L}\alpha (t))|\le \xi , \quad t \in J, \end{aligned}$$

(15)

there exists a solution $\alpha ^{*}\in C(J,{\mathbb {R}})$ of the problem (2) satisfying

$$\begin{aligned} |\alpha (t)-\alpha ^{*}(t)|\le M\xi , \quad t \in J. \end{aligned}$$

Definition 5.2

Problem (2) is said to be generalized UH stable, if there is $\phi \in C({\mathbb {R}}_{+},{\mathbb {R}}_{+})$, $\phi (0)=0$ such that for every $\xi >0$ and for each solution $\alpha \in C(J,{\mathbb {R}})$ satisfying

$$\begin{aligned} |{^{c}D_{t_0}^{\gamma ,\rho }}\alpha (t)-\aleph (t,\alpha (t),\mathrm {L}\alpha (t))|\le \xi , \quad t \in J, \end{aligned}$$

(16)

there exists a solution $\alpha ^{*}\in C(J,{\mathbb {R}})$ of the problem (2) with

$$\begin{aligned} |\alpha (t)-\alpha ^{*}(t)|\le \phi (\xi ),\quad t \in J. \end{aligned}$$

Definition 5.3

Problem (2) is said to be UH–Rassias stable with respect to $\Theta \in C(J,{\mathbb {R}})$, if there exists a constant $M_{\Theta }>0$ such that for every $\xi >0$ and for each solution $\alpha \in C(J,{\mathbb {R}})$ of the inequality

$$\begin{aligned} |{^{c}D_{t_0}^{\gamma ,\rho }}\alpha (t)-\aleph (t,\alpha (t),\mathrm {L}\alpha (t))|\le \Theta (t)\xi , \quad t \in J, \end{aligned}$$

(17)

there exists a solution $\alpha ^{*}\in C(J,{\mathbb {R}})$ of the problem (2) satisfying

$$\begin{aligned} |\alpha (t)-\alpha ^{*}(t)|\le M_{\Theta }\Theta (t)\xi , \quad t \in J. \end{aligned}$$

Definition 5.4

Problem (2) is said to be generalized UH–Rassias stable with respect to $\Theta \in C(J,{\mathbb {R}})$, if there exists a constant $M_{\Theta }>0$ and for each solution $\alpha \in C(J,{\mathbb {R}})$ of the inequality

$$\begin{aligned} |{^{c}D_{t_0}^{\gamma ,\rho }}\alpha (t)-\aleph (t,\alpha (t),\mathrm {L}\alpha (t))|\le \Theta (t), \quad t \in J, \end{aligned}$$

(18)

there exists a solution $\alpha ^{*}\in C(J,{\mathbb {R}})$ of the problem (2) satisfying

$$\begin{aligned} |\alpha (t)-\alpha ^{*}(t)|\le M_{\Theta }\Theta (t), \quad t \in J. \end{aligned}$$

Remark 5.5

It is clear that

• (i) Definition (15) $\Rightarrow$ Definition (16);

• (ii) Definition (17) $\Rightarrow$ Definition (18);

• (iii) Definition (17) $\Rightarrow$ Definition (15), for $\Phi (t)=1$.

Remark 5.6

We say that $\alpha \in C(J,{\mathbb {R}})$ is a solution of the inequality (15), if there exists a function $\Theta \in C(J,{\mathbb {R}})$ which depends on $\alpha$ such that

• (I) $|\Theta (t)|\le \delta , \; t \in J;$

• (II) $^{c}D_{t_{0+}}^{\gamma ,\rho }\alpha (t)=\aleph (t,\alpha (t),\mathrm {L}\alpha (t))+\Theta (t), \; t\in J$.

Similar remarks can be stated for the inequalities (17) and (18).

Remark 5.7

Let $\rho >0$, $\alpha \in (0,1]$, if $\alpha \in C(J,{\mathbb {R}})$ is a solution of the inequality (15), then we have

$$\begin{aligned}&\left| \alpha (t)-\frac{\vartheta }{\Gamma (\gamma )}\left( \frac{t^{\rho }-t_{0}^{\rho }}{\rho }\right) ^{\gamma -1}-\frac{1}{\Gamma (\gamma )}\int _{t_{0}}^{t} \left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1}\aleph (\tau ,\alpha (\tau ),\mathrm {L}\alpha (\tau ))\frac{d\tau }{\tau ^{1-\rho }}\right| \\&\quad \le \frac{T^{\rho \gamma }\delta }{\rho ^{\gamma }\Gamma (\gamma +1)},\quad t \in J. \end{aligned}$$

Proof

In fact, by (II) of Remark 5.6, we have

$$\begin{aligned} {^{c}D_{t_0}^{\gamma ,\rho }}\alpha (t)=\aleph (t,\alpha (t),\mathrm {L}\alpha (t))+\Theta (t), \quad t \in J. \end{aligned}$$

Then,

$$\begin{aligned} \alpha (t)&=\frac{\vartheta }{\Gamma (\gamma )}\left( \frac{t^{\rho }-t_{0}^{\rho }}{\rho }\right) ^{\gamma -1}+\frac{1}{\Gamma (\gamma )}\int _{t_{0}}^{t} \left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1}\aleph (\tau ,\alpha (\tau ),\mathrm {L}\alpha (\tau ))\frac{d\tau }{\tau ^{1-\rho }}\\&\quad +\frac{1}{\Gamma (\gamma )}\int _{t_{0}}^{t}\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1}\Theta (\tau )\frac{d\tau }{\tau ^{1-\rho }} ,\quad \forall \, t \in J. \end{aligned}$$

It follows that

$$\begin{aligned}&\left| \alpha (t)-\frac{\vartheta }{\Gamma (\gamma )}\left( \frac{t^{\rho }-t_{0}^{\rho }}{\rho }\right) ^{\gamma -1}-\frac{1}{\Gamma (\gamma )}\int _{t_{0}}^{t} \left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1}\aleph (\tau ,\alpha (\tau ),\mathrm {L}\alpha (\tau ))\frac{d\tau }{\tau ^{1-\rho }}\right| \\&\quad =\frac{1}{\Gamma (\gamma )}\left| \int _{t_{0}}^{t}\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1}\Theta (\tau )\frac{d\tau }{\tau ^{1-\rho }}\right| \\&\quad \le \frac{1}{\Gamma (\gamma )}\int _{t_{0}}^{t}\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1}|\Theta (\tau )|\frac{d\tau }{\tau ^{1-\rho }}\\&\quad \le \frac{T^{\rho \gamma }\delta }{\rho ^{\gamma }\Gamma (\gamma +1)}. \end{aligned}$$

One can state similar remarks for the solutions of the inequalities (17) and (18). $\square$

Before stating our main result, we need the following assumption.

$(\mathrm {A}_{5})$:

Suppose that $\Theta$ be nondecreasing function and there exists a constant $\zeta _{\Theta }>0$ such that

$$\begin{aligned} \int _{t_0}^{t}(t^{\rho }-s^{\rho })^{\gamma -1}(\tau ^{\rho })' \Theta (\tau )d\tau \le \zeta _{\Theta } \Theta , \end{aligned}$$

(19)

for all $t \in [t_0,T]$.

Theorem 5.8

Let the assumptions $(A_{1}),(A_{2}),(A_{4})$, and $(A_5)$ hold. Then, (2) is said to be generalized Ulam–Hyers–Rassias stable on J.

Proof

Suppose that $\alpha \in C(J,{\mathbb {R}})$ be the solution of the inequality (18). From Theorem 4.2, $\alpha ^{*} \in C(J,{\mathbb {R}})$ be the unique solution of the following

$$\begin{aligned} \left\{ \begin{array}{l} {^{c}D_{t_0}^{\gamma ,\rho }}\alpha ^{*}(t)=\aleph (t,\alpha ^{*}(t),\int _{t_0}^{t}l(t,s,\alpha ^{*}(s)))ds, \quad t \in J:=[t_0,T], \\ {{\mathcal {J}}_{t_0}^{1-\gamma ,\rho }}\alpha (t)|_{t=t_{0}}=\vartheta . \end{array} \right. \end{aligned}$$

(20)

We write the solution of (20) as

$$\begin{aligned} \alpha ^{*}(t)&=\frac{\vartheta }{\Gamma (\gamma )}\left( \frac{t^{\rho }-t_{0}^{\rho }}{\rho }\right) ^{\gamma -1}\\&\quad +\frac{1}{\Gamma (\gamma )}\int _{t_{0}}^{t} \left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1}\aleph (\tau ,\alpha ^{*}(\tau ),\mathrm {L}\alpha ^{*}(\tau ))\frac{d\tau }{\tau ^{1-\rho }} ,\quad \forall \, t \in J. \end{aligned}$$

Now, integrating the inequality (18) and by condition $(A_{5})$, we have

$$\begin{aligned}&\left| \alpha (t)-\frac{\vartheta }{\Gamma (\gamma )}\left( \frac{t^{\rho }-t_{0}^{\rho }}{\rho }\right) ^{\gamma -1}-\frac{1}{\Gamma (\gamma )}\int _{t_{0}}^{t} \left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1}\aleph (\tau ,\alpha (\tau ),\mathrm {L}\alpha (\tau ))\frac{d\tau }{\tau ^{1-\rho }}\right| \\&\quad \le \frac{1}{\Gamma (\gamma )}\int _{t_{0}}^{t}\bigg (\frac{t^{\rho }-\tau ^{\rho }}{\rho }\bigg )^{\gamma -1}\Theta (\tau )\frac{d\tau }{\tau ^{1-\rho }}\\&\quad =\frac{1}{\Gamma (\gamma )\rho ^{\gamma }}\int _{t_{0}}^{t}(s^{\rho })'(t^{\rho }-\tau ^{\rho })^{\gamma -1}\Theta (\tau )d\tau \\&\quad \le \frac{1}{\Gamma (\gamma )\rho ^{\gamma }}\zeta _{\Theta }\Theta (t). \end{aligned}$$

Accordingly,

$$\begin{aligned}&|\alpha (t)-\alpha ^{*}(t)|\\&\quad \le \left| \alpha (t)-\frac{\vartheta }{\Gamma (\gamma )}\left( \frac{t^{\rho }-t_{0}^{\rho }}{\rho }\right) ^{\gamma -1}-\frac{1}{\Gamma (\gamma )}\int _{t_{0}}^{t} \left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1}\aleph (\tau ,\alpha ^{*}(\tau ),\mathrm {L}\alpha ^{*}(\tau ))\frac{d\tau }{\tau ^{1-\rho }}\right| \\&\quad \le \left| \alpha (t)-\frac{\vartheta }{\Gamma (\gamma )}\left( \frac{t^{\rho }-t_{0}^{\rho }}{\rho }\right) ^{\gamma -1}-\frac{1}{\Gamma (\gamma )}\int _{t_{0}}^{t} \left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1}\aleph (\tau ,\alpha (\tau ),\mathrm {L}\alpha (\tau ))\frac{d\tau }{\tau ^{1-\rho }}\right. \\&\qquad +\frac{1}{\Gamma (\gamma )}\int _{t_{0}}^{t} \left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1}\aleph (\tau ,\alpha (\tau ),\mathrm {L}\alpha (\tau ))\frac{d\tau }{\tau ^{1-\rho }}\\&\qquad -\left. \frac{1}{\Gamma (\gamma )}\int _{t_{0}}^{t} \left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1}\aleph (\tau ,\alpha ^{*}(\tau ),\mathrm {L}\alpha ^{*}(\tau ))\frac{d\tau }{\tau ^{1-\rho }}\right| \\&\quad \le \left| \alpha (t)-\frac{\vartheta }{\Gamma (\gamma )}\left( \frac{t^{\rho }-t_{0}^{\rho }}{\rho }\right) ^{\gamma -1}-\frac{1}{\Gamma (\gamma )}\int _{t_{0}}^{t} \left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1}\aleph (\tau ,\alpha (\tau ),\mathrm {L}\alpha (\tau ))\frac{d\tau }{\tau ^{1-\rho }}\right| \\&\qquad +\frac{1}{\Gamma (\gamma )}\int _{t_{0}}^{t}\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1}|\aleph (\tau ,\alpha (\tau ),\mathrm {L}\alpha (\tau ))-\aleph (\tau ,\alpha ^{*}(\tau ),\mathrm {L}\alpha ^{*}(\tau ))| \frac{d\tau }{\tau ^{1-\rho }}\\&\quad \le \frac{1}{\Gamma (\gamma )\rho ^{\gamma }}\zeta _{\Theta }\Theta (t)+\frac{{\mathcal {P}}(1+\mathrm {L}_{1})}{\Gamma (\gamma )}\int _{t_{0}}^{t}\left( \frac{t^{\rho }-\tau ^{\rho }}{\rho }\right) ^{\gamma -1}|\alpha (\tau )-\alpha ^{*}(\tau )|. \end{aligned}$$

Hence, by utilizing Lemma 2.13, we acquire

$$\begin{aligned} |\alpha (t)-\alpha ^{*}(t)|&\le \frac{1}{\Gamma (\gamma )\rho ^{\gamma }}\zeta _{\Theta }\Theta (t){\mathbb {E}}_{\gamma }\left( \frac{{\mathcal {P}}(1+\mathrm {L}_{1})}{\Gamma (\gamma )}\Gamma (\gamma )\left( \frac{t^{\rho }-t_{0}^{\rho }}{\rho }\right) ^{\gamma }\right) \\&\le \frac{\zeta _{\Theta }}{\Gamma (\gamma )\rho ^{\gamma }}{\mathbb {E}}_{\gamma }\left( {\mathcal {P}}(1+\mathrm {L}_{1})\left( \frac{T^{\rho }}{\rho }\right) ^{\gamma }\right) \Theta (t)\\&\le M_{\Theta }\Theta (t). \end{aligned}$$

Here we set,

$$\begin{aligned} M_{\Theta }=\frac{\zeta _{\Theta }}{\Gamma (\gamma )\rho ^{\gamma }}{\mathbb {E}}_{\gamma }\left( {\mathcal {P}}(1+\mathrm {L}_{1})\left( \frac{T^{\rho }}{\rho }\right) ^{\gamma }\right) . \end{aligned}$$

Hence, the proposed problem (2) is generalized UH–Rassias stable with respect to $\Theta (t)$ on $[t_0,T]$. $\square$

Example

This section is devoted to present the application of our main results.

Example 6.1

Consider the following nonlinear fractional integro-differential equation with Liouville–Caputo derivative

$$\begin{aligned} \left\{ \begin{array}{l} {^{c}D_{0^{+}}^{\frac{1}{3},3}}\alpha (t)=\frac{1}{2\sqrt{4+t^{2}}}\left[ \frac{|\alpha (t)|}{1+|\alpha (t)|}+\int _{0}^{t}e^{3s}\frac{|\alpha (s)|}{1+|\alpha (s)|}ds\right] , \quad t \in [0,2], \\ {{\mathcal {J}}_{0^{+}}^{\frac{3}{2},3}}\alpha (t)|_{t=0}=2. \end{array} \right. \end{aligned}$$

(21)

Note that, $\gamma =\frac{1}{3}$, $\rho =3$, $\aleph (t,\alpha (t),\int _{0}^{t}l(s,\alpha (s))ds)=\frac{1}{2\sqrt{4+t^{2}}}\bigg [\frac{|\alpha (t)|}{1+|\alpha (t)|}+\int _{0}^{t}e^{3s}\frac{|\alpha (s)|}{1+|\alpha (s)|}ds\bigg ]$ and $(t,\alpha ) \in [0,2] \times {\mathbb {R}}$.

$$\begin{aligned}&|\aleph (t,\alpha _{1}(t),\mathrm {L}\alpha _{1}(t))-\aleph (t,\alpha _{2}(t),\mathrm {L}\alpha _{2}(t))|\\&\quad \le \frac{1}{2\sqrt{4+t^{2}}}\left[ \left| \frac{|\alpha _{1}(t)|}{1+|\alpha _{1}(t)|}-\frac{|\alpha _{1}(t)|}{1+|\alpha _{1}(t)|}\right| \right. \\&\quad \quad +\left. \int _{0}^{t}e^{3s}\left| \frac{|\alpha _{1}(s)|}{1+|\alpha _{1}(s)|}-\frac{|\alpha _{2}(s)|}{1+|\alpha _{2}(s)|}\right| ds\right] \\&\quad \le \frac{1}{2\sqrt{4+t^{2}}}\bigg [||\alpha _{1}(t)|-|\alpha _{2}(t)||\\&\quad \quad +\int _{0}^{t}e^{3s}||\alpha _{1}(s)|-|\alpha _{2}(s)||ds\bigg ]\\&\quad \le \frac{1}{2\sqrt{4+t^{2}}}\bigg [|\alpha _{1}(t)-\alpha _{2}(t)|+\frac{1}{3}e^{3t}|\alpha _{1}(t)-\alpha _{2}(t)|\bigg ]. \end{aligned}$$

Here, ${\mathcal {P}}=\frac{1}{4}$ and $\mathrm {L}_{1}=\frac{1}{3}$. Then $\frac{{\mathcal {P}}(1+\mathrm {L}_{1})T^{\rho \gamma }}{\Gamma (\gamma +1)\rho ^{\rho }}=0.5175<1$.

By Theorem 4.2, we can say that problem (21) has a unique solution. Next, we will check the assumption $(A_{5})$.

Let $\Theta (t)=e^{t}$, $\zeta _{\Theta }=\frac{2}{3}>0$ for $t \in [0,2]$.

$$\begin{aligned} \int _{0}^{t}(t^{\rho }-s^{\rho })^{\gamma -1}(\tau ^{\rho })' \Theta (\tau )d\tau&=\int _{0}^{t}(t^{3}-s^{3})^{\frac{1}{3}-1}(\tau ^{3})'e^{\tau }d\tau \\&\le \frac{1}{3}te^{t} \le \zeta _{\Theta } \Theta (t), \quad \text {for all}\quad t \in [0,2]. \end{aligned}$$

As a consequence, we can say that problem (21) is generalized UH–Rassias stable.

Conclusion

The $\rho$-Laplace transform is sufficient to deal with linear problems. In fact, we utilize the modified Laplace transform method to investigate the UH and generalized UH–Rassias stability of Liouville–Caputo fractional differential equations. The best tools to work with nonlinear models are the fixed point approaches and Gronwall’s inequality. Hence, in view of Schaefer’s and Banach fixed point theorem, we procured our qualitative analysis i.e., existence and uniqueness of solutions of problem (2). To achieve the stability results for nonlinear fractional integro-differential equations, we provided some hypothesis along with the extended Gronwall’s inequality. To justify our work, an application is provided.