A Fixed-Point Approach to the Hyers–Ulam Stability of Caputo–Fabrizio Fractional Differential Equations

Abstract

In this paper, we study Hyers–Ulam and Hyers–Ulam–Rassias stability of nonlinear Caputo–Fabrizio fractional differential equations on a noncompact interval. We extend the corresponding uniqueness and stability results on a compact interval. Two examples are given to illustrate our main results.

1. Introduction

In 1940, Ulam posed a question concerning the stability of homomorphisms into metric groups, a question which is regarded as the origin of the problem of stability in the theory of functional equations. In 1941, Hyers [1] answered the problem for a linear functional equation on the Banach space and established a new concept on the stability of functional equation, now called Hyers–Ulam stability. In 1978, Rassias [2] introduced a new definition of generalized Hyers–Ulam stability by the constant $\epsilon$ by a variable, and obtained the stability of Hyers–Ulam–Rassias for functional equation. There is a rich literature on this topic for standard integer-order equations (see [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]). In addition, the same stability concepts are introduced to find approximate solutions to fractional differential equations, see [18,19] and the references therein.

In 2015, Caputo and Fabrizio [20] gave a new definition of fractional derivative with a smooth kernel. Losada and Nieto [21] introduced Caputo–Fabrizio fractional differential equation the newly developed Caputo–Fabrizio fractional derivative and obtained the existence and uniqueness results under some strong restriction. Baleanu et al. [22] obtained the approximate solution for some infinite coefficient-symmetric Caputo–Fabrizio fractional integro-differential equations. Goufo [23] used the fractional derivative of the newly developed Caputo–Fabrizio without singular kernel to establish the Korteweg–de Vries–Burgers equation with two perturbation levels. Atangana and Nieto [24] studied the numerical approximation of this new fractional derivative and established an improved RLC circuit model. Moore et al. [25] developed and analyzed a Caputo–Fabrizio fractional derivative model for the HIV epidemic which includes an antiretroviral treatment compartment. Dokuyucu et al. [26] applied the fractional derivative of Caputo–Fabrizio to model the cancer treatment by radiotherapy.

Recently, Başcı et al. [27] applied the Laplace transform method to study the Hyers–Ulam stability of the following linear differential equations with Caputo–Fabrizio fractional derivative (see Definition 1):

$$\begin{array}{c}\hfill {(}^{CF}{\mathbb{D}}^{\alpha }y)(t)=f(t),0<\alpha <1,\end{array}$$

and

$$\begin{array}{c}\hfill {(}^{CF}{\mathbb{D}}^{\alpha }y)(t)-\lambda y(t)=f(t),0<\alpha <1.\end{array}$$

Meanwhile, Liu et al. [4] presented the Hyers–Ulam stability of linear differential equations with two term Caputo–Fabrizio derivatives as follows

$$\begin{array}{c}\hfill {(}^{CF}{\mathbb{D}}^{\alpha }y)(t)-\lambda {(}^{CF}{\mathbb{D}}^{\beta }y)(t)=u(t),0<\alpha ,\beta <1,\end{array}$$

and applied fixed-point theorems to derive the existence and uniqueness of solution to nonlinear equations as follows

$$\begin{array}{c}\hfill {(}^{CF}{\mathbb{D}}^{\alpha }f)(t)=g(t,f(t)),0<\alpha <1,\end{array}$$

(1)

and obtained the generalized Hyers–Ulam–Rassias stability via the Gronwall’s inequality.

Observing that ([4], Theorem 3) adopted the generalized Banach fixed-point theorem instead of the standard Banach contraction mapping and weakened the condition $a_{\alpha }L+b_{\alpha }TL<1$ in ([21], Theorem 1) to $a_{\alpha }L<1$ where $k>0$ denoted by the Lipschtiz constant of g, T denoted by the step of the interval and

$$a_{·}=\frac{2(1-·)}{(2-·)M(·)},b_{·}=\frac{2·}{(2-·)M(·)}.$$

(2)

and $M(·)$ denotes a normalization constant depending on ·.

Based on the above observation, we apply a new fixed-point approach to show the existence and uniqueness and stability for (1) on a compact interval to a noncompact interval $J=[{\tau }_0,{\tau }_0+k),k>0$.

2. Preliminaries

Definition 1

(see [20]). Let $0<\gamma <1$, the Caputo–Fabrizio fractional derivative of order γ for a function f can be written as

$$\begin{array}{c}\hfill {}^{CF}{\mathbb{D}}^{\gamma }f(\tau )=\frac{(2-\gamma )M(\gamma )}{2(1-\gamma )}{\int }_a^{\tau }exp(-\frac{\gamma }{1-\gamma }(\tau -s))f^{\prime }(s)ds,\tau >a,\end{array}$$

where $M(\gamma )$ is a normalization constant depending on γ. Please note that ${(}^{CF}{\mathbb{D}}^{\gamma })(f)=0$ if and only if f is a constant function.

Definition 2

(see [21] or ([4], Definition 2)). Let $0<\gamma <1$. The Caputo–Fabrizio fractional integral of order γ for a function f is defined as

$$\begin{array}{c}\hfill {}^{CF}{I}^{\gamma }f(\tau )=\frac{2(1-\gamma )}{(2-\gamma )M(\gamma )}f(\tau )+\frac{2\gamma }{(2-\gamma )M(\gamma )}{\int }_a^{\tau }f(s)ds,\tau >a.\end{array}$$

Let $\mathsf{\Omega }$ be a nonempty set, we present the following definition of generalized metric on $\mathsf{\Omega }$.

Definition 3

(see [3]). A function $\rho :$ $\mathsf{\Omega }\times \mathsf{\Omega }\to [0,\infty ]$ is called a generalized metric on Ω if and only if ρ satisfies

(i) 

$\rho ({\tau }_1,{\tau }_2)=0$ if and only if ${\tau }_1={\tau }_2$;

(ii) 

$\rho ({\tau }_1,{\tau }_2)=\rho ({\tau }_2,{\tau }_1)$ for all ${\tau }_1,{\tau }_2\in \mathsf{\Omega }$;

(iii) 

$\rho ({\tau }_1,{\tau }_3)\le \rho ({\tau }_1,{\tau }_2)+\rho ({\tau }_2,{\tau }_3)$ for all ${\tau }_1,{\tau }_2,{\tau }_3\in \mathsf{\Omega }$;

Theorem 1

(see [28]). Let $(\mathsf{\Omega },\rho )$ is a generalized complete metric space. Suppose $P:\mathsf{\Omega }\to \mathsf{\Omega }$ is a strictly contractive operator with the Lipschitz constant $K<1$. If there exists a nonnegative integer l such that $\rho (P^{l+1}\tau ,P^l\tau )<\infty$ for some $\tau \in \mathsf{\Omega }$, then the followings are true:

(i) 

The sequence $\{P^n\tau \}$ converges to a fixed point ${\tau }^{*}$ of P;

(ii) 

${\tau }^{*}$ is the unique fixed point of P in

$${\mathsf{\Omega }}^{*}=\{\tilde{\tau }\in \mathsf{\Omega }|\rho (P^l\tau ,\tilde{\tau })<\infty \};$$

(iii) 

If $\tilde{\tau }\in {\mathsf{\Omega }}^{*}$, then

$$\rho (\tilde{\tau },{\tau }^{*})\le \frac{1}{1-K}\rho (P\tilde{\tau },\tilde{\tau }).$$

Definition 4

(see [4]). Let $g:J\times \mathbb{R}\to \mathbb{R}$ be a continuous function. Equation (7) is Hyers–Ulam stable if there exists a real number $N>0$, such that for each $ϵ>0$ and for any solution $f\in C(J,\mathbb{R})$ of

$${|}^{CF}{\mathbb{D}}^{\gamma }f(\tau )-g(\tau ,f(\tau ))|\le ϵ,\forall \tau \in J,$$

(3)

there exists a solution $h\in C(J,\mathbb{R})$ of (1) with

$$|f(\tau )-h(\tau )|\le Nϵ,\forall \tau \in J.$$

Definition 5

(see [4]). Let $\varphi :J\to {\mathbb{R}}_{+}$ and $g:J\times \mathbb{R}\to \mathbb{R}$ be continuous functions. Equation (7) is generalized Hyers–Ulam–Rassias stable with respect to $\varphi \in C(J,{\mathbb{R}}_{+})$, if there exists a constant $c_{f,\varphi }>0$ such that for any solution $f\in C(J,\mathbb{R})$ of

$${|}^{CF}{\mathbb{D}}^{\gamma }f(\tau )-g(\tau ,f(\tau ))|\le \varphi (\tau ),\forall \tau \in J,$$

(4)

there exists a solution $h\in C(J,\mathbb{R})$ of (1) with

$$|f(\tau )-h(\tau )|\le c_{f,\varphi }\varphi (\tau ),\forall \tau \in J.$$

3. Main Results

Throughout this section, we denote the set Y of all continuous functions on J by

$$Y:=\{g:J\to \mathbb{R}|g\mathrm{is}\mathrm{continuous}\}=C(J,\mathbb{R})$$

(5)

Lemma 1

(see ([3], Theorem 3.1)). Define the function $d:$ $Y\times Y\to [0,\infty ]$ with

$$\begin{array}{c}\hfill d(f,g):=inf\{M\in [0,\infty ]||f(\tau )-g(\tau )|\le M\psi (\tau ),\forall \tau \in J\}\end{array}$$

where $\psi :J\to [0,\infty )$ is a given continuous function. Then $(Y,d)$ is a generalized complete metric space.

We give the following conditions:

$\left[A_1\right]$

The function $g:J\times \mathbb{R}\to \mathbb{R}$ is continuous and locally Lipschitz in $\tau$.

$\left[A_2\right]$

There exists a constant $L>0$ such that

$$|g(\tau ,y_1)-g(\tau ,y_2)|\le L|y_1-y_2|,\forall y_1,y_2\in \mathbb{R},\tau \in J.$$

Now, we prove the Hyers–Ulam stability of (7).

Theorem 2.

Assume that $\left[A_1\right]$ and $\left[A_2\right]$ and $|a_{\gamma }|<1/(L+1)$ hold. If the function $h:$ $J\to \mathbb{R}$ is continuously differentiable and satisfies

$$|{(}^{CF}{\mathbb{D}}^{\gamma }h)(\tau )-g(\tau ,h(\tau ))|\le ϵ$$

(6)

for all $\tau \in J$ and for some $ϵ>0$, then there exists a unique solution $f(\tau )$ of

$$\begin{array}{c}\hfill {(}^{CF}{\mathbb{D}}^{\gamma }f)(\tau )=g(\tau ,f(\tau )),0<\gamma <1,\end{array}$$

(7)

satisfying

$$|h(\tau )-f(\tau )|\le (L+1)(|a_{\gamma }|+|b_{\gamma }|k)ϵ$$

(8)

for all $\tau \in J$, where $a_{\gamma }$ and $b_{\gamma }$ are defined in (2).

Proof. 

We introduce a function $d_1:$ $Y\times Y\to [0,\infty ]$, where Y defined by (5) with

$$\begin{array}{c}\hfill d_1(f,g):=inf\{M\in [0,\infty ]||f(\tau )-g(\tau )|e^{-K(\tau -{\tau }_0)}\le M,\forall \tau \in J\},\end{array}$$

(9)

where $K=\frac{(L+1)|b_{\gamma }|}{1-(L+1)|a_{\gamma }|}>0$ and $a_{\gamma },b_{\gamma }$ are given in (2)

Let $\psi (·)=e^{K(·-{\tau }_0)}$ in Lemma 1, we obtain $(Y,d_1)$ is a generalized complete metric space.

Next, we consider the operator $P:Y\to Y$ as follows:

$$\begin{array}{c}\hfill (Pf)(\tau ):=f_0+a_{\gamma }g(\tau ,f(\tau ))+b_{\gamma }{\int }_{{\tau }_0}^{\tau }g(s,f(s))ds,\tau \in J.\end{array}$$

(10)

for any $f,g\in Y$, where $f_0=f({\tau }_0)$. Please note that any fixed point of P solves (7). Indeed, the function $u-a_{\gamma }g(\tau ,u)=v$ in (10) is invertible, it is increasing. We denote its inverse $u=G(\tau ,v)$, and G is globally Lipschitz in v and locally Lipschitz in $\tau$ by our assumptions. So, any fixed point of (10) satisfies

$$f(\tau )=G(\tau ,b_{\gamma }{\int }_{{\tau }_0}^{\tau }g(s,f(s))ds+f_0).$$

(11)

Now clearly the function $\tau \to b_{\gamma }{\int }_{{\tau }_0}^{\tau }g(s,f(s))ds+f_0$ is locally Lipschitz in $\tau$, we see that the composition function $\tau \to G(\tau ,b_{\gamma }{\int }_{{\tau }_0}^{\tau }g(s,f(s))ds+f_0)$ is also locally Lipschitz in $\tau$. So, any fixed point $f(\tau )$ of (10) is a locally Lipschitz function, and thus it is locally absolute continuous on J. So really (10) gives solutions of (7). As a matter of fact, we need just that $u-a_{\gamma }g(\tau ,u)=v$ is invertible, i.e., $u-a_{\gamma }g(\tau ,u)$ is strictly monotonic in u, and we can extend our results for more general case. We shall consider (11) instead of (10).

We prove that $Pf$ is continuous. Let ${\tau }_1,{\tau }_2\in J$, and ${\tau }_1<{\tau }_2$, we have

$$\begin{array}{cc}& |Pf({\tau }_1)-Pf({\tau }_2)|\hfill \\ =& |a_{\gamma }g({\tau }_1,f({\tau }_1))+b_{\gamma }{\int }_{{\tau }_0}^{{\tau }_1}g(s,f(s))ds-a_{\gamma }g({\tau }_2,f({\tau }_2))-b_{\gamma }{\int }_{{\tau }_0}^{{\tau }_2}g(s,f(s))ds|\hfill \\ \le & |a_{\gamma }||g({\tau }_1,f({\tau }_1))-g({\tau }_2,f({\tau }_2))|+|b_{\gamma }||{\int }_{{\tau }_0}^{{\tau }_1}g(s,f(s))ds-{\int }_{{\tau }_0}^{{\tau }_2}g(s,f(s))ds|\hfill \\ \le & |a_{\gamma }||g({\tau }_1,f({\tau }_1))-g({\tau }_1,f({\tau }_2))|+|a_{\gamma }||g({\tau }_1,f({\tau }_2))-g({\tau }_2,f({\tau }_2))|+|b_{\gamma }||{\int }_{{\tau }_1}^{{\tau }_2}g(s,f(s))ds|\hfill \\ \le & |a_{\gamma }||g({\tau }_1,f({\tau }_1))-g({\tau }_1,f({\tau }_2))|+|a_{\gamma }||g({\tau }_1,f({\tau }_2))-g({\tau }_2,f({\tau }_2))|\hfill \\ & +|b_{\gamma }|({\int }_{{\tau }_1}^{{\tau }_2}|g(s,f(s))-g(s,0)|ds+{\int }_{{\tau }_1}^{{\tau }_2}|g(s,0)|ds)\hfill \\ \le & |a_{\gamma }||g({\tau }_1,f({\tau }_1))-g({\tau }_1,f({\tau }_2))|+|a_{\gamma }||g({\tau }_1,f({\tau }_2))-g({\tau }_2,f({\tau }_2))|\hfill \\ & +|b_{\gamma }{|(L\parallel f\parallel }_{C(J,\mathbb{R})}({\tau }_2-{\tau }_1)+{\parallel g\parallel }_{C(J,\mathbb{R})}({\tau }_2-{\tau }_1)).\hfill \end{array}$$

Then, for all $f\in Y$, as ${\tau }_1\to {\tau }_2$, the right-hand side of the above inequality tends to zero (due to $\left[A_1\right]$ and $f\in Y$). Thus, $Pf$ is continuous, i.e., $Pf\in Y$ for all $f\in Y$.

Then, we have

$$|(P{f}_0)(\tau )-f_0(\tau )|e^{-K(\tau -{\tau }_0)}\le {\parallel P{f}_0-f_0\parallel }_{C(J,\mathbb{R})}max\{1,e^{-Kk}\}<\infty ,$$

for all $f_0\in Y,$ and $\tau \in J$. Therefore, by (9), we obtain $d_1(P{f}_0,f_0)<\infty ,f_0\in Y.$

Similarly, we have

$$|(f_0)(\tau )-f(\tau )|e^{-K(\tau -{\tau }_0)}\le {\parallel f_0-f\parallel }_{C(J,\mathbb{R})}max\{1,e^{-Kk}\}<\infty ,$$

for all $f\in Y,$ and $\tau \in J$, which implies that

$$d_1(f_0,f)<\infty ,\forall f\in Y,$$

that is $\{f\in Y|d_1(f_0,f)<\infty \}=Y.$

Next, we show that P is strictly contractive on Y. For any $l,n\in Y$, we get

$$\begin{array}{cc}& |(Pl)(\tau )-(Pn)(\tau )|\hfill \\ \le & |a_{\gamma }||g(\tau ,l(\tau ))-g(\tau ,n(\tau ))|+|b_{\gamma }|{\int }_{{\tau }_0}^{\tau }|g(s,l(s))-g(s,n(s))|ds\hfill \\ \le & L|a_{\gamma }||l(\tau )-n(\tau )|+L|b_{\gamma }|{\int }_{{\tau }_0}^{\tau }|l(s)-n(s)|ds\hfill \\ \le & L|a_{\gamma }||l(\tau )-n(\tau )|+L|b_{\gamma }|{\int }_{{\tau }_0}^{\tau }|l(s)-n(s)|e^{-K(s-{\tau }_0)}{e}^{K(s-{\tau }_0)}ds\hfill \\ \le & L|a_{\gamma }||l(\tau )-n(\tau )|+L|b_{\gamma }|d_1(l,n){\int }_{{\tau }_0}^{\tau }{e}^{K(s-{\tau }_0)}ds\hfill \\ \le & L|a_{\gamma }||l(\tau )-n(\tau )|+\frac{L|b_{\gamma }|}{K}{d}_1(l,n)(e^{K(\tau -{\tau }_0)}-1)\hfill \\ \le & L|a_{\gamma }||l(\tau )-n(\tau )|+\frac{L|b_{\gamma }|}{K}{d}_1(l,n)e^{K(\tau -{\tau }_0)}\hfill \end{array}$$

for all $\tau \in J$. Thus, for any $l,n\in Y$ and all $\tau \in J$, we have

$$\begin{array}{ccc}\hfill |(Pl)(\tau )-(Pn)(\tau )|e^{-K(\tau -{\tau }_0)}& \le & L|a_{\gamma }||l(\tau )-n(\tau )|e^{-K(\tau -{\tau }_0)}+\frac{L|b_{\gamma }|}{K}{d}_1(l,n)\hfill \\ & \le & L|a_{\gamma }|d_1(l,n)+\frac{L|b_{\gamma }|}{K}{d}_1(l,n)\hfill \\ & =& L(|a_{\gamma }|+\frac{|b_{\gamma }|}{K})d_1(l,n)\hfill \\ & =& \frac{L}{L+1}{d}_1(l,n).\hfill \end{array}$$

Hence, we obtain

$$d_1(Pl,Pn)\le \frac{L}{L+1}{d}_1(l,n).$$

Therefore, P is strictly contractive on Y.

When $k=1$ and $Y={\mathsf{\Omega }}^{*}$, the operator P satisfies all the conditions of Theorem 1.

On the other hand, by (6), we have

$$-ϵ\le {(}^{CF}{\mathbb{D}}^{\gamma }h)(\tau )-g(\tau ,h(\tau ))\le ϵ\forall \tau \in J.$$

Similar to the approach in ([4], Theorem 2), we can obtain

$$|h(\tau )-h_0-a_{\gamma }g(\tau ,h(\tau ))-b_{\gamma }{\int }_{{\tau }_0}^{\tau }g(s,f(s))ds|\le ϵ(|a_{\gamma }|+|b_{\gamma }|k)$$

(12)

for all $\tau \in J$. From (10), (12) is equivalent to

$$|h(\tau )-(Ph)(\tau )|\le ϵ(|a_{\gamma }|+|b_{\gamma }|k).$$

(13)

Multiply both sides of (13) by $e^{-K(\tau -{\tau }_0)}$,

$$\begin{array}{c}\hfill |h(\tau )-(Ph)(\tau )|e^{-K(\tau -{\tau }_0)}\le ϵ(|a_{\gamma }|+|b_{\gamma }|k)e^{-K(\tau -{\tau }_0)}\le M:=ϵ(|a_{\gamma }|+|b_{\gamma }|k)max\{1,e^{-Kk}\}\end{array}$$

for all $\tau \in J$. Then

$$\begin{array}{c}\hfill d_1(Ph,h)\le ϵ(|a_{\gamma }|+|b_{\gamma }|k)e^{-K(\tau -{\tau }_0)}.\end{array}$$

By Theorem 1, there exists a unique solution $f:$ $J\to \mathbb{R}$ of (7) satisfying

$$\begin{array}{c}\hfill d_1(h,f)\le \frac{1}{1-L/(L+1)}{d}_1(Ph,h)\le (L+1)ϵ(|a_{\gamma }|+|b_{\gamma }|k)e^{-K(\tau -{\tau }_0)},\tau \in J,\end{array}$$

by (9), we have

$$\begin{array}{c}\hfill |h(\tau )-f(\tau )|e^{-K(\tau -{\tau }_0)}\le (L+1)ϵ(|a_{\gamma }|+|b_{\gamma }|k)e^{-K(\tau -{\tau }_0)},\tau \in J,\end{array}$$

which implies that (8) holds. □

Remark 1.

From Definition 4, (8) shows (7) is Hyers–Ulam stable with the constant $N=(L+1)(|a_{\gamma }|+|b_{\gamma }|k)$ provided that $0<k<+\infty$. Of course, (7) is not Hyers–Ulam stable if $k=+\infty$. Theorem 2 covers the result in ([27], Theorem 2.6) and shows that the condition $0<\lambda <\frac{(2-\alpha )M(\alpha )}{2(1-\alpha )}$ can be removed.

Now we will prove the Hyers–Ulam–Rassias stability of (7).

Theorem 3.

Assume that $\left[A_1\right]$ and $\left[A_2\right]$ and $|a_{\gamma }|<1/(L+1)$ hold. If a continuously differentiable function $h:$ $J\to \mathbb{R}$ satisfies

$$|{(}^{CF}{\mathbb{D}}^{\gamma }h)(\tau )-g(\tau ,h(\tau ))|\le G(\tau )$$

(14)

for all $\tau \in J$ and for some $G:J\to (0,\infty )$ is a nondecreasing continuous function satisfying

$$\left|{\int }_{{\tau }_0}^{\tau }G(s)ds\right|\le F_{G}G(\tau ),F_G>0,$$

(15)

for all $\tau \in J$, then there exists a unique solution $f(\tau )$ of (7) satisfying

$$|h(\tau )-f(\tau )|\le (L+1)(a_{\gamma }+b_{\gamma }{F}_G)G(\tau )$$

(16)

for all $\tau \in J$.

Proof. 

We introduce a function $d_2:$ $Y\times Y\to [0,\infty ]$, where Y defined by (5) with

$$\begin{array}{c}\hfill d_2(f,g):=inf\{M\in [0,\infty ]||f(\tau )-g(\tau )|e^{-K(\tau -{\tau }_0)}\le MG(\tau ),\forall \tau \in J,K\in \mathbb{R}\}\end{array}$$

(17)

Let $\psi (·)=e^{K(·-{\tau }_0)}G(·)$ in the Lemma 1, $(Y,d_2)$ is a generalized complete metric space.

Consider $P:Y\to Y$ defined in (10). Similar to the method of Theorem 2, we can conclude that $d_2(P{f}_0,f)<\infty$ for each $f_0\in X$ and $\{f\in Y|d_2(f_0,f)<\infty \}=Y$.

Next, we prove that P is strictly contractive on Y. Note

$$\begin{array}{ccc}\hfill {\int }_{{\tau }_0}^{\tau }G(s)e^{K(s-{\tau }_0)}ds& \le & G(\tau ){\int }_{{\tau }_0}^{\tau }{e}^{K(s-{\tau }_0)}ds\hfill \\ & =& \frac{1}{K}G(\tau ){\int }_{{\tau }_0}^{\tau }d{e}^{K(s-{\tau }_0)}\hfill \\ & \le & \frac{1}{K}G(\tau )(e^{K(\tau -{\tau }_0)}-1)\hfill \\ & \le & \frac{1}{K}G(\tau )e^{K(\tau -{\tau }_0)}\hfill \end{array}$$

for all $\tau \in J.$

For any $l,n\in Y$, let $M_{l,n}\in [0,\infty ]$ be an arbitrary constant with $d_2(l,n)\le M_{l,n}$, by (17), we obtain

$$|l(\tau )-n(\tau )|e^{-K(\tau -{\tau }_0)}\le M_{l,n}G(\tau ),forall\tau \in J.$$

Then, for each $l,n\in Y$, we have

$$\begin{array}{cc}& |(Pl)(\tau )-(Pn)(\tau )|\hfill \\ \le & |a_{\gamma }||g(\tau ,l(\tau ))-g(\tau ,n(\tau ))|+|b_{\alpha }|{\int }_{{\tau }_0}^{\tau }|g(s,l(s))-g(s,n(s))|ds\hfill \\ \le & L|a_{\gamma }||l(\tau )-n(\tau )|+L|b_{\gamma }|{\int }_{{\tau }_0}^{\tau }|l(s)-n(s)|ds\hfill \\ \le & L|a_{\gamma }||l(\tau )-n(\tau )|+L|b_{\gamma }|{\int }_{{\tau }_0}^{\tau }|l(s)-n(s)|e^{-K(s-{\tau }_0)}{e}^{K(s-{\tau }_0)}ds\hfill \\ \le & L|a_{\gamma }||l(\tau )-n(\tau )|+L|b_{\gamma }|M_{l,n}{\int }_{{\tau }_0}^{\tau }G(s)e^{K(s-{\tau }_0)}ds\hfill \\ \le & L|a_{\gamma }||l(\tau )-n(\tau )|+L|b_{\gamma }|M_{l,n}\frac{1}{K}G(\tau )e^{K(\tau -{\tau }_0)}\hfill \end{array}$$

for all $\tau \in J$. Thus, for any $l,n\in Y$ and all $\tau \in J$, we have

$$\begin{array}{ccc}\hfill |(Pl)(\tau )-(Pn)(\tau )|e^{-K(\tau -{\tau }_0)}& \le & L|a_{\gamma }||l(\tau )-n(\tau )|e^{-K(\tau -{\tau }_0)}+\frac{L|b_{\gamma }|}{K}{M}_{l,n}G(\tau )\hfill \\ & \le & L|a_{\gamma }|M_{l,n}G(\tau )+\frac{L|b_{\gamma }|}{K}{M}_{l,n}G(\tau )\hfill \\ & =& L(|a_{\gamma }|+\frac{|b_{\gamma }|}{K})M_{l,n}G(\gamma )\hfill \\ & =& \frac{L}{L+1}{M}_{l,n}G(\tau ),\hfill \end{array}$$

that is, $d_2(Pl,Pn)\le \frac{L}{L+1}{M}_{l,n},\forall \tau \in J.$ Hence, we obtain

$$d_2(Pl,Pn)\le \frac{L}{L+1}{d}_2(l,n),\forall \tau \in J.$$

Therefore, P is strictly contractive on Y. When $k=1$ and $Y={\mathsf{\Omega }}^{*}$, the operator P satisfies all the conditions of Theorem 1.

On the other hand, by (14), we have

$$-G(\tau )\le {(}^{CF}{\mathbb{D}}^{\gamma }h)(\tau )-g(\tau ,h(\tau ))\le G(\tau ),\forall \tau \in J.$$

By simple computation, we can obtain

$$\begin{array}{cc}& |h(\tau )-h_0-a_{\gamma }g(\tau ,h(\tau ))-b_{\gamma }{\int }_{{\tau }_0}^{\tau }g(s,f(s))ds|\hfill \\ \le & |a_{\gamma }|G(\tau )+|b_{\gamma }|{\int }_{{\tau }_0}^{\tau }G(s)ds\hfill \\ \le & (|a_{\gamma }|+|b_{\gamma }|F_G)G(\tau ),\forall \tau \in J.\hfill \end{array}$$

This yields that

$$|h(\tau )-(Ph)(\tau )|\le (|a_{\gamma }|+|b_{\gamma }|F_G)G(\tau ),\forall \tau \in J.$$

(18)

Multiply both sides of (18) by $e^{-K(\tau -{\tau }_0)}$, then,

$$\begin{array}{c}\hfill |h(\tau )-(Ph)(\tau )|e^{-K(\tau -{\tau }_0)}\le (|a_{\gamma }|+|b_{\gamma }|F_G)G(\tau )e^{-K(\tau -{\tau }_0)},\forall \tau \in J.\end{array}$$

Then

$$\begin{array}{c}\hfill d_2(Ph,h)\le (|a_{\gamma }|+|b_{\gamma }|F_G)G(\tau )e^{-K(\tau -{\tau }_0)},\forall \tau \in J.\end{array}$$

By Theorem 1, there exists a unique solution $f:$ $J\to \mathbb{R}$ of (7) satisfying

$$\begin{array}{c}\hfill d_2(h,f)\le \frac{1}{1-L/(L+1)}{d}_2(Ph,h)\le (L+1)(|a_{\gamma }|+|b_{\gamma }|F_G)G(\tau )e^{-K(\tau -{\tau }_0)},\forall \tau \in J.\end{array}$$

By (17), we have

$$\begin{array}{c}\hfill |h(\tau )-f(\tau )|e^{-K(\tau -{\tau }_0)}\le (L+1)(|a_{\gamma }|+|b_{\gamma }|F_G)G(\tau )e^{-K(\tau -{\tau }_0)},\forall \tau \in J,\end{array}$$

which implies (16) holds. The proof is complete. □

Remark 2.

By the Definition 5, (16) shows (7) is generalized Hyers–Ulam–Rassias stable with the constant $c_{f,G}=(L+1)(|a_{\gamma }|+|b_{\gamma }|F_G)$. Theorem 3 extend the result in ([27], Corollary 2.8) and also shows that the condition $0<\lambda <\frac{(2-\alpha )M(\alpha )}{2(1-\alpha )}$ can be removed.

Remark 3.

Compared to ([4], Theorems 3 and 5), we extend the existence and uniqueness result and the generalized Hyers–Ulam–Rassias stability result for (1) on the noncompact interval and also remove the condition $L|a_{\alpha }|<1$ from the assumptions.

4. Examples

Assume that $M(·)$ in Definition 1 is the solution of the following equation:

$$\frac{2(1-·)}{(2-·)M(·)}+\frac{2·}{(2-·)M(·)}=1.$$

Then one can derive an explicit formula $M(·)=\frac{2}{2-·}$ (see ([21], p. 89)).

Example 1.

We consider the following equation:

$${(}^{CF}{\mathbb{D}}^{\gamma }f)(\tau )-\lambda f(\tau )=g(\tau ),\tau \in [0,k),k>0,$$

(19)

and let $g(\tau ,f(\tau ))=g(\tau )+\lambda f(\tau )$. Obviously, $|g(\tau ,f_1(\tau ))-g(\tau ,f_2(\tau ))|=|\lambda ||f_1(\tau )-f_2(\tau )|,\tau \in [0,k)$ and the Lipschitz condition holds with the Lipschitz constant $L=|\lambda |$. Then, (19) is Hyers–Ulam stable on J, for all $\lambda \in \mathbb{R}$ and $\alpha \in (0,1)$.

Now, let $\gamma =\frac{1}{2},\lambda =-2,f(0)=0$, and $g(\tau )=4\tau -4+4e^{-\tau }-\frac{1}{2}{e}^{-2\tau }+2{\tau }^2$. We consider the following equation:

$${(}^{CF}{\mathbb{D}}^{\frac{1}{2}}f)(\tau )+2f(\tau )=g(\tau ),\tau \in [0,k),k>0.$$

(20)

Let $h(\tau )={\tau }^2$, for $ϵ=\frac{1}{2}$ by simple calculation, we have

$${(}^{CF}{\mathbb{D}}^{\frac{1}{2}}h)(\tau )=4\tau -4+4e^{-\tau }$$

then

$$|{(}^{CF}{\mathbb{D}}^{\frac{1}{2}}h)(\tau )+2f(\tau )-g(\tau )|=\frac{1}{2}{e}^{-2\tau }\le \frac{1}{2}=ϵ,\tau \in [0,k),k>0.$$

Integrating (20) from 0 to τ, we get

$$f(\tau )={\tau }^2-\frac{1}{12}{e}^{-2\tau }+\frac{1}{12}{e}^{-\frac{1}{2}\tau }$$

then

$$\begin{array}{ccc}\hfill |h(\tau )-f(\tau )|& =& |\frac{1}{12}{e}^{-2\tau }-\frac{1}{12}{e}^{-\frac{1}{2}\tau }|=\frac{1}{12}{e}^{-\frac{1}{2}\tau }|1-e^{-\frac{3}{2}\tau }|\hfill \\ & \le & \frac{1}{6}\times \frac{1}{2}=\frac{1}{6}ϵ.\hfill \end{array}$$

(21)

So (20) is Hyers–Ulam stable (see Figure 1). Please note that the condition $\lambda >0$ in ([27], Theorem 2.6) is not required here, and moreover, (20) is Hyers–Ulam stable, too.

On the other hand, (21) implies that (20) is also Hyers–Ulam stable even for $\tau =+\infty$, which shows that ([27], Remark 2.7) is not suitable.

Example 2.

We consider the following fractional problem

$${(}^{CF}{\mathbb{D}}^{\frac{1}{3}}f)(\tau )=\frac{5}{1+e^{\tau }}\frac{|f|}{1+|f|},\tau \in [0,+\infty ),$$

(22)

and the inequality

$$|{(}^{CF}{\mathbb{D}}^{\frac{1}{3}}f)(\tau )-\frac{5}{1+e^{\tau }}\frac{|f|}{1+|f|}|\le G(\tau ),\tau \in [0,+\infty ).$$

Let $g(\tau ,f(\tau ))=\frac{5}{1+e^{\tau }}\frac{|f|}{1+|f|},(\tau ,f)\in [0,+\infty )\times \mathbb{R}$. Obviously $\left[A_1\right]$ holds. For any $\tau \in [0,+\infty )$ and $f_1,f_2\in \mathbb{R}$, we have

$$\begin{array}{ccc}\hfill |g(\tau ,f_1)-g(\tau ,f_2)|& =& \frac{5}{1+e^{\tau }}|\frac{|f_1|}{1+|f_1|}-\frac{|f_2|}{1+|f_2|}|\le \frac{5|f_1-f_2|}{(1+|f_1|)(1+|f_2|)}\hfill \\ & \le & 5|f_1-f_2|.\hfill \end{array}$$

Then the condition $\left[A_2\right]$ hold and $L=5$ and $k_{\alpha }=5$ in ([4], Theorem 5).

Let $G(\tau )=e^{\tau }\in C([0,+\infty ),(0,+\infty ))$ and ${\int }_0^{\tau }G(s)ds={\int }_0^{\tau }{e}^{s}ds=e^{\tau }-1\le e^{\tau }$. (15) holds for $F_G=1>0$. Therefore, in view of Theorem 3, (22) is generalized Hyers–Ulam–Rassias stable.

Here $\gamma =\frac{1}{3}$, by calculation, we have $M(\frac{1}{3})=\frac{6}{5}$, $a_{\frac{1}{3}}=\frac{24}{25}$. Then $a_{\gamma }{k}_f=\frac{24}{25}\times 5=\frac{24}{5}>1$. Thus $a_{\alpha }{k}_f<1$ condition of Theorem 5 in [4] does not hold in this problem. Thus, ([4], Theorem 5) does not work even on $[0,2]$.