Stability of Nonlinear Fractional Delay Differential Equations

Abstract

This article discusses several forms of Ulam stability of nonlinear fractional delay differential equations. Our investigation is based on a generalised Gronwall’s inequality and Picard operator theory. Implementations are provided to demonstrate the stability results obtained for finite intervals.

1. Introduction

The stability theory of functional equations has advanced significantly in the last thirty years. This topic has benefited greatly from the contributions of others (see [1,2,3,4,5,6,7]). Our findings are related to recent works [3,8] (where integral and differential equations are considered), as well as papers of [9,10] (where the Ulam–Hyers stability for operatorial equations and inclusions is examined). For more details on Ulam–Hyers stability (see [11,12,13,14,15,16,17,18,19,20,21,22,23]).

In 2013, Rabha [24] discussed different types of the generalized Ulam–Hyers stability for a univalent solution and studied the existence and uniqueness of a solution.

Since then, there have been many contributions in the form of generalization, refinement, and modification on this subject. In particular, in [25], the authors studied Ulam-type stabilities for Volterra delay integrodifferential equations on a finite integral.

This paper aims to discuss different types of Ulam stability of the form

$$\begin{array}{ccc}\hfill D_{\tau }^{\alpha }{\eta }_1\left(\tau \right)& =& f(\tau ,{\eta }_1\left(\tau \right),{\eta }_1\left(h\left(\tau \right)\right)),\tau \in I=[0,d],d>0,\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill {\eta }_1\left(\tau \right)& =& \phi \left(\tau \right),\tau \in [-\lambda ,0]\mathrm{and}\phi \in \left(\right[-\lambda ,0],\mathbb{R}),\hfill \end{array}$$

(2)

where $D_{\tau }^{\alpha }{\eta }_1\left(\tau \right)$ is the fractional derivative of ${\eta }_1$ of order $\alpha ,$ $0<\alpha <1,$ $f\in C(I\times {\mathbb{R}}^2,\mathbb{R}),$ $h\in C(I,[-\lambda ,d\left]\right),$ $0<\lambda <\infty$ and $h\left(\tau \right)\le \tau .$

2. Preliminaries

In this section, we outline a list of important notations, definitions, and lemmas that will be used in our main results.

Definition 1.

Following this [26,27], we realize the Riemann–Liouville derivatives and integral of fractional order α as follows

$$\begin{array}{ccc}\hfill D_{\tau }^{\alpha }f\left(\tau \right)& =& \frac{1}{\Gamma (1-\alpha )}\frac{d}{d\tau }\underset{0}{\overset{\tau }{\int }}\frac{f\left(\zeta \right)}{{(\tau -\zeta )}^{\alpha }}d\zeta ,\hfill \\ \hfill I_{\tau }^{\alpha }f\left(\tau \right)& =& \frac{1}{\Gamma \left(\alpha \right)}\underset{0}{\overset{\tau }{\int }}f\left(\zeta \right){(\tau -\zeta )}^{\alpha -1}d\zeta .\hfill \end{array}$$

Definition 2.

Equation (1) is Hyers–Ulam stable if $\exists c>0$ s.t. $\forall ϵ>0$ and ${\eta }_0\in C^1([-\lambda ,d],\mathbb{R})$, with

$$\left|D_{\tau }^{\alpha }{\eta }_0\left(\tau \right)-f(\tau ,{\eta }_0\left(\tau \right),{\eta }_0\left(h\left(\tau \right)\right))\right|\le ϵ,\tau \in I,$$

(3)

$\exists {\eta }_1\in C^1([-\lambda ,d],\mathbb{R})$ of Equation (1) s.t.

$$\left|{\eta }_0\left(\tau \right)-{\eta }_1\left(\tau \right)\right|\le cϵ,\tau \in [-\lambda ,d].$$

Definition 3.

Equation (1) is a generalized Hyers–Ulam stable if $\exists \psi \in C({\mathbb{R}}_{+},{\mathbb{R}}_{+})$ s.t. for ${\eta }_0\in C^1([-\lambda ,d],\mathbb{R})$ of (3), $\exists {\eta }_1\in C^1([-\lambda ,d],\mathbb{R})$ of Equation (1), with

$$\left|{\eta }_0\left(\tau \right)-{\eta }_1\left(\tau \right)\right|\le \psi \left(ϵ\right),\tau \in [-\lambda ,d].$$

Definition 4.

Equation (1) is Ulam–Hyers–Rassias stable if $\exists c>0$ s.t. $\forall ϵ>0$ and ${\eta }_0\in C^1([-\lambda ,d],\mathbb{R})$, with

$$\left|D_{\tau }^{\alpha }{\eta }_0\left(\tau \right)-f(\tau ,{\eta }_0\left(\tau \right),{\eta }_0\left(h\left(\tau \right)\right))\right|\le ϵ\phi \left(\tau \right),\tau \in I,$$

(4)

$\exists {\eta }_1\in C^1([-\lambda ,d],\mathbb{R})$ of Equation (1) s.t.

$$\left|{\eta }_0\left(\tau \right)-{\eta }_1\left(\tau \right)\right|\le cϵ\phi \left(\tau \right),\tau \in [-\lambda ,d],$$

where $\phi :[-\lambda ,d]\to {\mathbb{R}}_{+}.$

Remark 1.

A function ${\eta }_0\in C^1(I,\mathbb{R})$ is a solution of (3) if $\exists g\in C(I,\mathbb{R})$ (depends on ${\eta }_0$) s.t.

(a) 

$\left|g\left(\tau \right)\right|\le ϵ,$$\tau \in I,$

(b) 

$D_{\tau }^{\alpha }{\eta }_0\left(\tau \right)=f(\tau ,{\eta }_0\left(\tau \right),{\eta }_0\left(h\left(\tau \right)\right))+g\left(\tau \right),$$\tau \in I.$

Similarly, similar arguments apply to inequality (4).

Remark 2.

If ${\eta }_0\in C^1(I,\mathbb{R})$ satisfies (3), then ${\eta }_0$ is a solution to the following inequality

$$\left|{\eta }_0\left(\tau \right)-{\eta }_0\left(0\right)-\frac{1}{\Gamma \left(\alpha \right)}\underset{0}{\overset{\tau }{\int }}{(\tau -\zeta )}^{\alpha -1}f(\zeta ,{\eta }_0\left(\zeta \right),{\eta }_0\left(h\left(\zeta \right)\right))d\zeta \right|\le ϵ\tau ,\tau \in I.$$

Indeed, if ${\eta }_0\in C^1(I,\mathbb{R})$ satisfies (3), then by Remark 1, we get

$$D_{\tau }^{\alpha }{\eta }_0\left(\tau \right)=f(\tau ,{\eta }_0\left(\tau \right),{\eta }_0\left(h\left(\tau \right)\right))+g\left(\tau \right),\tau \in I.$$

This gives

$$\left|{\eta }_0\left(\tau \right)-{\eta }_0\left(0\right)-\frac{1}{\Gamma \left(\alpha \right)}\underset{0}{\overset{\tau }{\int }}{(\tau -\zeta )}^{\alpha -1}f(\zeta ,{\eta }_0\left(\zeta \right),{\eta }_0\left(h\left(\zeta \right)\right))d\zeta \right|\le \underset{0}{\overset{\tau }{\int }}g\left(\zeta \right)d\zeta \le ϵ\tau ,\tau \in I.$$

Similar estimates can also be obtained for the inequality (4).

The following inequality is the key to obtaining our main results.

Lemma 1

(A generalized Gronwall lemma [28]). Let $\lambda >0,$ $a\left(\tau \right)\ge 0$ is a locally integrable function on $0\le \tau <T\le +\infty$, and $g\left(\tau \right)\ge 0$ is a nondecreasing continuous function on $0\le \tau <T,$ $g\left(\tau \right)\le M$ (constant), and suppose that $\omega \left(\tau \right)\ge 0$ is locally integrable on $0\le \tau <T$ with

$$\omega \left(\tau \right)\le a\left(\tau \right)+g\left(\tau \right)\underset{0}{\overset{\tau }{\int }}{(\tau -\zeta )}^{\lambda -1}\omega \left(\zeta \right)d\zeta .$$

Then

$$\omega \left(\tau \right)\le a\left(\tau \right)+\underset{0}{\overset{\tau }{\int }}\left(\sum _{n=1}^{\infty }{\displaystyle \frac{{(g\left(\tau \right)\Gamma \left(\lambda \right))}^n}{\Gamma \left(n\lambda \right)}}{\tau -\zeta )}^{\lambda -1}a\left(\zeta \right)\right)d\zeta ,0\le \tau <T.$$

Definition 5

([28]). Assume that $(V,d)$ is a metric space. An operator $G:V\to V$ is a Picard operator if $\exists u^{*}\in V$ s.t.

(a) 

$F_G=\left\{u^{*}\right\},$ where $F_G=\left\{u\in V:G\left(u\right)=u\right\}$ is the fixed point set of $G;$

(b) 

${\left\{G^n\left(u_0\right)\right\}}_{n\in \mathbb{N}}$ converges to $u^{*}$$\forall u_0\in V.$

Lemma 2

([28]). Assume that $(V,$$d,$$\le )$ is an ordered metric space and $G:V\to V$ is an increasing Picard operator $(F_G=u_G^{*}).$ Then for $u\in V,$ $u\le G\left(u\right)$$\Rightarrow u\le u_G^{*},$ while $u\ge G\left(u\right)$$\Rightarrow u\ge u_G^{*}.$

Definition 6

(Contraction principle). Every contraction in a complete metric space admits a unique fixed point.

3. Ulam Stabilities for Nonlinear Fractional Delay Differential Equations

In this part, we are going to provide our results of Hyers–Ulam’s stability for Equation (1).

Theorem 1.

Equation (1) has a unique solution in $C([-\lambda ,d],\mathbb{R})\cap C^1([0,d],\mathbb{R})$ and Ulam–Hyers–Rassias stable w.s.t. the function φ if

(A1) 

$f\in C(I\times {\mathbb{R}}^2,\mathbb{R})$ and $h\in C(I,[-\lambda ,d\left]\right)$ are continuous with the Lipschitz condition:

$$\left|f(\tau ,{\eta }_1,{\eta }_2)-f(\tau ,{\varrho }_1,{\varrho }_2)\right|\le \sum _{k=1}^2\Omega \left|{\eta }_k-{\varrho }_k\right|,$$

$\Omega >0,$$\Omega \ne \Gamma (\alpha +1)$$\forall \tau \in I$ and $x_k,y_k\in \mathbb{R}.$

(A2) 

$\frac{2\Omega }{\Gamma (\alpha +1)}{d}^{\alpha }<1.$

(A3) 

$\phi :[-\lambda ,d]\to {\mathbb{R}}_{+}$ is a positive continuous nondecreasing function and $\exists \rho >0$ s.t.

$$\underset{0}{\overset{\tau }{\int }}\phi \left(\lambda \right)d\lambda \le \rho \phi \left(\tau \right),\tau \in I.$$

Proof .

(i) We first note that in view of (A1), Equation (1) is equivalent to the following integral equations:

$$\begin{array}{ccc}\hfill \eta \left(\tau \right)\hfill & =& \hfill \varphi \left(0\right)+\frac{1}{\Gamma \left(\alpha \right)}\underset{0}{\overset{\tau }{\int }}{(\tau -\zeta )}^{\alpha -1}f(\zeta ,\eta \left(\zeta \right),\eta \left(h\left(\zeta \right)\right))d\zeta ,\tau \in I.\hfill \\ \hfill \eta \left(\tau \right)\hfill & =& \varphi \left(\tau \right),\tau \in [-\lambda ,0].\hfill \end{array}$$

where $f\in C(I\times {\mathbb{R}}^2,\mathbb{R}),$ $h\in C(I,[-\lambda ,d\left]\right),$ $0<\lambda <\infty$ and $h\left(\tau \right)\le \tau .$ Consider the Banach space $V=C\left(\right[-\lambda ,d],\mathbb{R})$ with Chebyshev norm ${∥·∥}_C$ and define the operator ${\Lambda }_f:V\to V$ by

$$\begin{array}{ccc}\hfill {\Lambda }_f\left(\eta \right)\left(\tau \right)& =& \varphi \left(0\right)+\frac{1}{\Gamma \left(\alpha \right)}\underset{0}{\overset{\tau }{\int }}{(\tau -\zeta )}^{\alpha -1}f(\zeta ,\eta \left(\zeta \right),\eta \left(h\left(\zeta \right)\right))d\zeta ,\tau \in I.\hfill \\ \hfill {\Lambda }_f\left(\eta \right)\left(\tau \right)& =& \varphi \left(\tau \right),\tau \in [-\lambda ,0].\hfill \end{array}$$

Using the contraction principle, we show that $d_f$ has a fixed point. In fact, it is clear that

$$\left|{\Lambda }_f\left(\eta \right)\left(\tau \right)-{\Lambda }_f\left(\varrho \right)\left(\tau \right)\right|=0,\eta ,\varrho \in C([-r,d],\mathbb{R}),\tau \in [-\lambda ,0].$$

Next, for any $\tau \in I$, we get

$$\begin{array}{cc}\hfill \left|{\Lambda }_f\left(\eta \right)\left(\tau \right)-{\Lambda }_f\left(\varrho \right)\left(\tau \right)\right|\le & \left|\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^{\tau }{(\tau -\zeta )}^{\alpha -1}[f(\zeta ,\eta \left(\zeta \right),\eta \left(h\left(\zeta \right)\right))-f(\zeta ,\varrho \left(\zeta \right),\varrho \left(h\left(\zeta \right)\right))]d\zeta \right|\hfill \\ \hfill \le & \frac{\Omega }{\Gamma \left(\alpha \right)}{\int }_0^{\tau }{(\tau -\zeta )}^{\alpha -1}\left\{\right|\eta \left(\zeta \right)-\varrho \left(\zeta \right)|+|\eta \left(h\left(\zeta \right)\right)-\varrho \left(h\left(\zeta \right)\right)\left|\right\}d\zeta \hfill \\ \hfill \le & \frac{\Omega }{\Gamma \left(\alpha \right)}{\int }_0^{\tau }{(\tau -\zeta )}^{\alpha -1}\left(\underset{0\le \zeta \le d}{max}|\eta \left(\zeta \right)-\varrho \left(\zeta \right)|\right.\hfill \\ \hfill & +\underset{-\lambda \le \zeta \le d}{max}|\eta \left(h\left(\zeta \right)\right)-\varrho \left(h\left(\zeta \right)\right)|d\zeta \hfill \\ \hfill \le & \frac{2\Omega }{\Gamma (\alpha +1)}{d}^{\alpha }{\parallel \eta \left(\zeta \right)-\varrho \left(\zeta \right)\parallel }_c.\hfill \end{array}$$

Thus it follows that

$$∥{\Lambda }_f\left(\eta \right)\left(\tau \right)-{\Lambda }_f\left(\varrho \right)\left(\tau \right)∥\le \frac{2\Omega }{\Gamma (\alpha +1)}{d}^{\alpha }{∥\eta \left(\zeta \right)-\varrho \left(\zeta \right)∥}_c,\eta ,\varrho \in C([-\lambda ,d],\mathbb{R}),\tau \in I.$$

As $\frac{2\Omega }{\Gamma (\alpha +1)}{d}^{\alpha }<1,$ the operator $d_f$ is a contraction on the complete space $V.$ Hence, $d_f$ has a fixed point ${\eta }^{*}:[-\lambda ,0]\to \mathbb{R},$ which is a solution of Equation (1).

(ii) Assume that $\varrho \in C([-\lambda ,d],\mathbb{R})\cap C^1([0,d],\mathbb{R})$ is a solution to the inequality (4). Denote by $\eta \in C([-\lambda ,d],\mathbb{R})\cap C^1([0,d],\mathbb{R})$ the unique solution of the problem:

$$\begin{array}{ccc}\hfill D_{\tau }^{\alpha }\eta \left(\tau \right)& =& f(\tau ,\eta (\tau ),\eta (h\left(\tau \right)\left)\right),\tau \in I=[0,d],d>0,\hfill \\ \hfill \eta \left(\tau \right)& =& \varrho \left(\tau \right),\tau \in [-\lambda ,0].\hfill \end{array}$$

Then assumption (A1) allows to write the following integral equation (equivalent to the above problem):

$$\eta \left(\tau \right)=\varrho \left(0\right)+\frac{1}{\Gamma \left(\alpha \right)}\underset{0}{\overset{\tau }{\int }}{(\tau -\zeta )}^{\alpha -1}f(\zeta ,\eta \left(\zeta \right),\eta \left(h\left(\zeta \right)\right))d\zeta ,\tau \in I.$$

(5)

$$\eta \left(\tau \right)=\varrho \left(\tau \right),\tau \in [-\lambda ,0].$$

If $\varrho \in C([-\lambda ,d],\mathbb{R})\cap C^1([0,d],\mathbb{R})$ satisfies the inequality (4), then using assumption (A3) and Remarks 1 and 2, we obtain

$$\begin{array}{ccc}& & \left|\varrho \left(\tau \right)-\varrho \left(0\right)-\frac{1}{\Gamma \left(\alpha \right)}\underset{0}{\overset{\tau }{\int }}{(\tau -\zeta )}^{\alpha -1}f(\zeta ,\varrho \left(\zeta \right),\varrho \left(h\left(\zeta \right)\right))d\zeta \right|\hfill \\ & \le & \underset{0}{\overset{\tau }{\int }}\left|g\left(\zeta \right)\right|d\zeta \le \underset{0}{\overset{\tau }{\int }}ϵ\phi \left(\zeta \right)d\zeta \le \rho ϵ\phi \left(\tau \right),\tau \in I.\hfill \end{array}$$

(6)

Note that $\left|\varrho \left(\tau \right)-\eta \left(\tau \right)\right|=0$ for $\tau \in [-\lambda ,0].$ Next, by the assumption (A1), Equation (5) and the estimate in (6), for any $\tau \in I,$ we can write

$$\begin{array}{ccc}\hfill \left|\varrho \left(\tau \right)-\eta \left(\tau \right)\right|& =& \left|\varrho \left(\tau \right)-\varrho \left(0\right)-\frac{1}{\Gamma \left(\alpha \right)}\underset{0}{\overset{\tau }{\int }}{(\tau -\zeta )}^{\alpha -1}f(\zeta ,\eta \left(\zeta \right),\eta \left(h\left(\zeta \right)\right))d\zeta \right|\hfill \\ \hfill & \le & \left|\varrho \left(\tau \right)-\varrho \left(0\right)-\frac{1}{\Gamma \left(\alpha \right)}\underset{0}{\overset{\tau }{\int }}{(\tau -\zeta )}^{\alpha -1}f(\zeta ,\varrho \left(\zeta \right),\varrho \left(h\left(\zeta \right)\right))d\zeta \right|\hfill \\ & & +\frac{1}{\Gamma \left(\alpha \right)}\underset{0}{\overset{\tau }{\int }}\left|{(\tau -\zeta )}^{\alpha -1}f(\zeta ,\varrho \left(\zeta \right),\varrho \left(h\left(\zeta \right)\right))\right.\hfill \\ & & -\left.{(\tau -\zeta )}^{\alpha -1}f(\zeta ,\eta \left(\zeta \right),\eta \left(h\left(\zeta \right)\right))\right|d\zeta \hfill \\ \hfill & \le & \rho ϵ\phi \left(\tau \right)+\frac{L}{\Gamma \left(\alpha \right)}\underset{0}{\overset{\tau }{\int }}\left|{(\tau -\zeta )}^{\alpha -1}\right|\left|\varrho \left(\zeta \right)-\eta \left(\zeta \right)\right|d\zeta \hfill \\ & & +\frac{L}{\Gamma \left(\alpha \right)}\underset{0}{\overset{\tau }{\int }}\left|{(\tau -\zeta )}^{\alpha -1}\right|\left|\varrho \left(h\right(\zeta \left)\right)-\eta \left(h\right(\zeta \left)\right)\right|d\zeta .\hfill \end{array}$$

(7)

According to above inequality, we consider the operator $G:C([-\lambda ,d],{\mathbb{R}}_{+})\to C([-\lambda ,d],{\mathbb{R}}_{+})$ defined by

$$\begin{array}{ccc}\hfill G\left(m\right)\left(\tau \right)\hfill & =& \hfill \rho ϵ\phi \left(\tau \right)+\frac{\Omega }{\Gamma \left(\alpha \right)}\underset{0}{\overset{\tau }{\int }}{(\tau -\zeta )}^{\alpha -1}\left(m\left(\zeta \right)+m\left(h\right(\zeta \left)\right)\right)d\zeta ,\tau \in [0,d],\hfill \\ \hfill G\left(m\right)\left(\tau \right)\hfill & =& 0,\tau \in [-\lambda ,0].\hfill \end{array}$$

Next, we prove that G is a Picard operator. To this end, observe first that for any $m,n\in C([-\lambda ,d],{\mathbb{R}}_{+}),$ we have

$$\left|G\left(m\right)\left(\tau \right)-G\left(n\right)\left(\tau \right)\right|=0,\tau \in [-\lambda ,0].$$

Now, using $\left(\mathrm{A}1\right),$$\forall \tau \in I,$ we have

$$\begin{array}{ccc}\hfill \left|G\left(m\right)\left(\tau \right)-G\left(n\right)\left(\tau \right)\right|& \le & \frac{\Omega }{\Gamma \left(\alpha \right)}\underset{0}{\overset{\tau }{\int }}\left|{(\tau -\zeta )}^{\alpha -1}\right|\left\{\left|m\left(\zeta \right)-n\left(\zeta \right)\right|+\left|m\left(h\right(\zeta \left)\right)-n\left(h\right(\zeta \left)\right)\right|\right\}d\zeta \hfill \\ & \le & \frac{2\Omega }{\Gamma \left(\alpha \right)}\underset{0}{\overset{\tau }{\int }}{(\tau -\zeta )}^{\alpha -1}\underset{-r\le \zeta \le d}{max}\left|m\left(\zeta \right)-n\left(\zeta \right)\right|d\zeta \hfill \\ & \le & \frac{2\Omega }{\Gamma (\alpha +1)}{d}^{\alpha }{∥m-n∥}_c.\hfill \end{array}$$

Therefore,

$${∥G\left(m\right)\left(\tau \right)-G\left(n\right)\left(\tau \right)∥}_C\le \frac{2\Omega }{\Gamma (\alpha +1)}{d}^{\alpha }{∥m-n∥}_c,\mathrm{for}\mathrm{all}m,n\in C([-\lambda ,d],{\mathbb{R}}_{+}).$$

As $\frac{2\Omega }{\Gamma (\alpha +1)}{d}^{\alpha }<1,$ G is a contraction on $C([-\lambda ,d],{\mathbb{R}}_{+}),$ using Banach contraction principle, G is a Picard operator and $F_G=\left\{m^{*}\right\}.$ Then, for $\tau \in I$, we have

$$m^{*}\left(\tau \right)=\rho ϵ\phi \left(\tau \right)+\frac{\Omega }{\Gamma \left(\alpha \right)}\underset{0}{\overset{\tau }{\int }}{(\tau -\zeta )}^{\alpha -1}\left(m^{*}\left(\zeta \right)+m^{*}\left(h\left(\zeta \right)\right)\right)d\zeta .$$

As $m^{*}$ is increasing, then $m^{*}\left(h\left(\tau \right)\right)\le m^{*}\left(\tau \right)$ for $h\left(\tau \right)\le \tau ,$ $\tau \in I,$ and hence

$$m^{*}\left(\tau \right)=\rho ϵ\phi \left(\tau \right)+\frac{2\Omega }{\Gamma \left(\alpha \right)}\underset{0}{\overset{\tau }{\int }}{(\tau -\zeta )}^{\alpha -1}{m}^{*}\left(\zeta \right)d\zeta .$$

Next, applying the inequality given in Lemma 1, we obtain

$$m^{*}\left(\tau \right)\le \rho ϵ\phi \left(\tau \right)+\underset{0}{\overset{\tau }{\int }}\left(\sum _{n=1}^{\infty }{\displaystyle \frac{{(\frac{2\Omega }{\Gamma \left(\alpha \right)}\Gamma \left(\alpha \right))}^n}{\Gamma \left(n\alpha \right)}}{(\tau -\zeta )}^{\alpha -1}\rho ϵ\phi \left(\zeta \right)\right)d\zeta .$$

As $\phi$ is positive and nondecreasing, then

$$m^{*}\left(\tau \right)\le \rho ϵ\phi \left(\tau \right)+\rho ϵ\phi \left(\tau \right)\underset{0}{\overset{\tau }{\int }}\left(\sum _{n=1}^{\infty }{\displaystyle \frac{{(\frac{2\Omega }{\Gamma \left(\alpha \right)}\Gamma \left(\alpha \right))}^n}{\Gamma \left(n\alpha \right)}}{(\tau -\zeta )}^{\alpha -1}\right)d\zeta .$$

So, clearly, if we put

$$k\left(\tau \right)=max\underset{0}{\overset{\tau }{\int }}\left(\sum _{n=1}^{\infty }{\displaystyle \frac{{(\frac{2\Omega }{\Gamma \left(\alpha \right)}\Gamma \left(\alpha \right))}^n}{\Gamma \left(n\alpha \right)}}{(\tau -\zeta )}^{\alpha -1}\right)d\zeta ,$$

then

$$m^{*}\left(\tau \right)\le \rho ϵ\left(1+k\left(\tau \right)\right)\phi \left(\tau \right).$$

Taking $C_{\phi }=\rho \left(1+k\left(\tau \right)\right),$ then we get

$$m^{*}\left(\tau \right)\le C_{\phi }ϵ\phi \left(\tau \right),\tau \in [-\lambda ,d].$$

For $m\left(\tau \right)=\left|\varrho \left(\tau \right)-\eta \left(\tau \right)\right|,$ the inequality (7) leads to

$$m\left(\tau \right)\le G\left(m\right)\left(\tau \right).$$

So, we have proved that $G:C([-\lambda ,d],{\mathbb{R}}_{+})\to C([-\lambda ,d],{\mathbb{R}}_{+})$ is an increasing Picard operator for $m\in C([-\lambda ,d],{\mathbb{R}}_{+}),$$m\left(\tau \right)\le Gm\left(\tau \right)$ and $F_G=\left\{m^{*}\right\}.$ Thus, applying the abstract Gromwell lemma (Lemma 2), we get $m\left(\tau \right)\le m^{*}\left(\tau \right),$$\tau \in [-r,d],$ implying that

$$\left|\varrho \left(\tau \right)-\eta \left(\tau \right)\right|\le C_{\phi }ϵ\phi \left(\tau \right),\mathrm{for}\mathrm{all}\tau \in [-\lambda ,d].$$

Hence, Equation (1) is Ulam–Hyers–Rassias stable with respect to $\phi .$ □

Corollary 1.

Assume that F and g in (1) satisfy the hypothesis (A1), (A2) and (A3). Then the problem (1) has a unique solution and Ulam–Hyers stable.

Proof. 

By taking $\phi \left(\tau \right)=1,$ $\forall \tau \in [-\lambda ,d]$ in Theorem 1, we obtain

$$\left|\varrho \left(\tau \right)-\eta \left(\tau \right)\right|\le Cϵ,\tau \in [-\lambda ,d],$$

and the result follows. □

Remark 3.

It is easy to show that (1) has generalized Ulam–Hyers stability by taking $\psi \left(ϵ\right)=Cϵ$ in Corollary 1.

4. Applications

In this section, we consider some important particular cases of the problem (1).

Example 1.

Let $\lambda >0$ and $h_1\left(\tau \right)={\displaystyle \frac{\tau }{2}},$ $\tau \in [0,d].$ Then we get the following special case of the problem (1):

$$\begin{array}{c}{D}_{\tau }^{\alpha }\eta \left(\tau \right)=f_1\left(\tau ,\eta \left(\tau \right),\eta \left(h\left({\displaystyle \frac{\tau }{2}}\right)\right)\right),\tau \in [0,d],\\ \eta \left(0\right)=\varrho \left(0\right).\end{array}$$

(8)

Now, consider the following inequality:

$$\left|D_{\tau }^{\alpha }\eta \left(\tau \right)-f_1\left(\tau ,\eta \left(\tau \right),\eta \left(h\left({\displaystyle \frac{\tau }{2}}\right)\right)\right)\right|\le ϵ\phi \left(\tau \right),\tau \in [0,d],$$

where $f_1$, ϵ and φ are as specified in Section 3.

Using Theorem 1, we obtain the following result.

Theorem 2.

If $f_1$ and $g_1$ satisfying (A1), (A2), and (A3), then the problem (8) has a unique solution and Ulam–Hyers–Rassias stable.

Example 2.

Let $h_2\left(\tau \right)={\tau }^2,$$\tau \in [0,1].$ Then we get the specific case of the problem(1):

$$\begin{array}{c}{D}_{\tau }^{\alpha }\eta \left(\tau \right)=f_2\left(\tau ,\eta \left(\tau \right),\eta \left(h\left({\tau }^2\right)\right)\right),\tau \in [0,1],\\ \eta \left(0\right)=\varrho \left(0\right),\end{array}$$

(9)

which is an initial value problem for a nonlinear Volterra fractional delay integrodifferential equation. Consider the inequality:

$$\left|D_{\tau }^{\alpha }\eta \left(\tau \right)-f_2\left(\tau ,\eta \left(\tau \right),\eta \left(h\left({\tau }^2\right)\right)\right)\right|\le ϵ\phi \left(\tau \right),\tau \in [0,1],$$

where $f_2$ϵ and φ are as defined in Section 3.

Applying Theorem 1, we arrive at the following result for the problem (9).

Theorem 3.

if $f_2$ and $g_2$ satisfying (A1), (A2), and (A3), then the problem (8) has a unique solution and Ulam–Hyers–Rassias stable.

Other Ulam type stability results for Equation (9) can be obtained by Using the corresponding results from Section 3.

5. Examples

In this part, we offer two examples to demonstrate the key findings.

Example 3.

Consider the non-linear fractional delay differential equations

$$\begin{array}{ccc}\hfill D_{\tau }^{\frac{1}{2}}\eta \left(\tau \right)& =& {\displaystyle \frac{\sqrt{\tau }\left(sin\left(\eta \right(\tau \left)\right)+cos\left(\eta \right(\tau -1\left)\right)\right)}{200}},\tau \in [1,4],\hfill \\ \hfill \eta \left(\tau \right)& =& 1,\tau \in [0,1].\hfill \end{array}$$

(10)

As $f(\tau ,\eta \left(\tau \right),\eta \left(h\left(\tau \right)\right))=\frac{\sqrt{\tau }\left(sin\left(\eta \right(\tau \left)\right)+cos\left(\eta \right(\tau -1\left)\right)\right)}{200}$ and f is continuous

$$\begin{array}{cc}\hfill |f(\tau ,{\eta }_1,{\eta }_2)-f(\tau ,{\varrho }_1,{\varrho }_2)|& \le {\displaystyle \frac{1}{200}}\sqrt{\tau }\left(\right|sin{\eta }_1-sin{\varrho }_1|+|sin{\eta }_2-sin{\varrho }_2\left|\right)\hfill \\ \hfill & \le {\displaystyle \frac{\sqrt{\tau }}{200}}\left(\right|{\eta }_1-{\varrho }_1|+|{\eta }_2-{\varrho }_2\left|\right)\hfill \\ \hfill & \le {\displaystyle \frac{1}{100}}\left(\right|{\eta }_1-{\varrho }_1|+|{\eta }_2-{\varrho }_2\left|\right).\hfill \end{array}$$

Thus the Lipschitz constant is $\Omega =\frac{1}{100}$. Moreover, we have

$$\frac{2\Omega }{\Gamma (\alpha +1)}{d}^{\alpha }={\displaystyle \frac{1}{50\Gamma (3/2)}}{\left(4\right)}^{1/2}<1.$$

Thus, according to Corollary 1, (10) is Ulam–Hyers stable.

Example 4.

Consider the non-linear fractional delay differential equations

$$\begin{array}{ccc}\hfill D_{\tau }^{\frac{1}{3}}\eta \left(\tau \right)& =& {\displaystyle \frac{\left|\eta \left(\tau \right)\right|+|\eta (\tau -\frac{1}{2})|}{200}},\tau \in [1,3],\hfill \\ \hfill \eta \left(\tau \right)& =& \tau ,\tau \in [\frac{1}{2},1].\hfill \end{array}$$

(11)

As $f(\tau ,\eta \left(\tau \right),\eta \left(h\left(\tau \right)\right))=\frac{\left|\eta \right(\tau \left)\right|+\left|\varrho \right(\tau \left)\right|}{200}$, f is continuous and $\Omega =\frac{1}{200}$. Now

$$\frac{2\Omega }{\Gamma (\alpha +1)}{d}^{\alpha }={\displaystyle \frac{1}{100\Gamma (4/3)}}{\left(3\right)}^{1/2}<1.$$

Thus, according to Corollary 1, (11) is Ulam–Hyers stable.

6. Conclusions

In this manuscript, we discussed different types of Ulam stability for the first-order nonlinear fractional delay differential equation in the problem (1), using a generalized Gronwall’s inequality and Picard operator theory, we discussed some applications to illustrate the stability results obtained in the case of a finite interval. Our obtained results generalize those of Otrocol [29] in the case take $\alpha =1.$

7. Future Direction

It could be interesting to study different future types of Ulam stability for first-order nonlinear fractional Volterra integral equations. It is also interesting to discuss different future types of Ulam stability for the case of impulsive Volterra delay integrodifferential equations. Moreover we expect to get in these cases richer results with more attributes.