Open AccessArticle

Stability of Nonlinear Fractional Delay Differential Equations

D. A. Refaai

^1,*,

M. M. A. El-Sheikh

²,

Gamal A. F. Ismail

¹,

Mohammed Zakarya

^3,4

Ghada AlNemer

^5,*

and

Haytham M. Rezk

⁶

Department of Mathematics, Collage for Women, Ain Shams University, Cairo 11566, Egypt

Department of Mathematics and Computer Science, Faculty of Science, Menoufia University, Shebin El-Koom 32511, Egypt

Department of Mathematics, College of Science, King Khalid University, P.O. Box 9004, Abha 61413, Saudi Arabia

⁴

Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt

⁵

Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia

⁶

Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Egypt

Authors to whom correspondence should be addressed.

Symmetry 2022, 14(8), 1606; https://doi.org/10.3390/sym14081606

Submission received: 4 July 2022 / Revised: 27 July 2022 / Accepted: 28 July 2022 / Published: 4 August 2022

(This article belongs to the Special Issue Symmetry in Fractional Calculus and Inequalities)

Download Versions Notes

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.

Keywords:

Ulam–Hyers stability; delay differential equations; fractional nonlinear differential equations

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{matrix} D_{τ}^{α} η_{1} (τ) & = & f (τ, η_{1} (τ), η_{1} (h (τ))), τ \in I = [0, d], d > 0, \end{matrix}

(1)

\begin{matrix} η_{1} (τ) & = & φ (τ), τ \in [- λ, 0] and φ \in ([- λ, 0], R), \end{matrix}

(2)

where

D_{τ}^{α} η_{1} (τ)

is the fractional derivative of

η_{1}

of order

α,

0 < α < 1,

f \in C (I \times R^{2}, R),

h \in C (I, [- λ, d]),

0 < λ < \infty

and

h (τ) \leq τ .

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{matrix} D_{τ}^{α} f (τ) & = & \frac{1}{Γ (1 - α)} \frac{d}{d τ} \int_{0}^{τ} \frac{f (ζ)}{{(τ - ζ)}^{α}} d ζ, \\ I_{τ}^{α} f (τ) & = & \frac{1}{Γ (α)} \int_{0}^{τ} f (ζ) {(τ - ζ)}^{α - 1} d ζ . \end{matrix}

Definition 2.

Equation (1) is Hyers–Ulam stable if

\exists c > 0

s.t.

\forall ϵ > 0

and

η_{0} \in C^{1} ([- λ, d], R)

, with

|D_{τ}^{α} η_{0} (τ) - f (τ, η_{0} (τ), η_{0} (h (τ)))| \leq ϵ, τ \in I,

(3)

\exists η_{1} \in C^{1} ([- λ, d], R)

of Equation (1) s.t.

|η_{0} (τ) - η_{1} (τ)| \leq c ϵ, τ \in [- λ, d] .

Definition 3.

Equation (1) is a generalized Hyers–Ulam stable if

\exists ψ \in C (R_{+}, R_{+})

s.t. for

η_{0} \in C^{1} ([- λ, d], R)

of (3),

\exists η_{1} \in C^{1} ([- λ, d], R)

of Equation (1), with

|η_{0} (τ) - η_{1} (τ)| \leq ψ (ϵ), τ \in [- λ, d] .

Definition 4.

Equation (1) is Ulam–Hyers–Rassias stable if

\exists c > 0

s.t.

\forall ϵ > 0

and

η_{0} \in C^{1} ([- λ, d], R)

, with

|D_{τ}^{α} η_{0} (τ) - f (τ, η_{0} (τ), η_{0} (h (τ)))| \leq ϵ φ (τ), τ \in I,

(4)

\exists η_{1} \in C^{1} ([- λ, d], R)

of Equation (1) s.t.

|η_{0} (τ) - η_{1} (τ)| \leq c ϵ φ (τ), τ \in [- λ, d],

where

φ : [- λ, d] \to R_{+} .

Remark 1.

A function

η_{0} \in C^{1} (I, R)

is a solution of (3) if

\exists g \in C (I, R)

(depends on

η_{0}

) s.t.

(a): $|g (τ)| \leq ϵ,$ $τ \in I,$
(b): $D_{τ}^{α} η_{0} (τ) = f (τ, η_{0} (τ), η_{0} (h (τ))) + g (τ),$ $τ \in I .$

Similarly, similar arguments apply to inequality (4).

Remark 2.

η_{0} \in C^{1} (I, R)

satisfies (3), then

η_{0}

is a solution to the following inequality

|η_{0} (τ) - η_{0} (0) - \frac{1}{Γ (α)} \int_{0}^{τ} {(τ - ζ)}^{α - 1} f (ζ, η_{0} (ζ), η_{0} (h (ζ))) d ζ| \leq ϵ τ, τ \in I .

Indeed, if

η_{0} \in C^{1} (I, R)

satisfies (3), then by Remark 1, we get

D_{τ}^{α} η_{0} (τ) = f (τ, η_{0} (τ), η_{0} (h (τ))) + g (τ), τ \in I .

This gives

|η_{0} (τ) - η_{0} (0) - \frac{1}{Γ (α)} \int_{0}^{τ} {(τ - ζ)}^{α - 1} f (ζ, η_{0} (ζ), η_{0} (h (ζ))) d ζ| \leq \int_{0}^{τ} g (ζ) d ζ \leq ϵ τ, τ \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

λ > 0,

a (τ) \geq 0

is a locally integrable function on

0 \leq τ < T \leq + \infty

, and

g (τ) \geq 0

is a nondecreasing continuous function on

0 \leq τ < T,

g (τ) \leq M

(constant), and suppose that

ω (τ) \geq 0

is locally integrable on

0 \leq τ < T

with

ω (τ) \leq a (τ) + g (τ) \int_{0}^{τ} {(τ - ζ)}^{λ - 1} ω (ζ) d ζ .

Then

ω (τ) \leq a (τ) + \int_{0}^{τ} (\sum_{n = 1}^{\infty} \frac{{(g (τ) Γ (λ))}^{n}}{Γ (n λ)} {τ - ζ)}^{λ - 1} a (ζ)) d ζ, 0 \leq τ < 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): $F_{G} = {u^{*}},$ where $F_{G} = \{u \in V : G (u) = u\}$ is the fixed point set of $G;$
(b): ${G^{n} (u_{0})}_{n \in N}$ converges to $u^{*}$ $\forall u_{0} \in V .$

Lemma 2

([28]). Assume that

(V,

d,

\leq)

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 \leq G (u)

\Rightarrow u \leq u_{G}^{*},

while

u \geq G (u)

\Rightarrow u \geq 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 ([- λ, d], R) \cap C^{1} ([0, d], R)

and Ulam–Hyers–Rassias stable w.s.t. the function φ if

(A1): $f \in C (I \times R^{2}, R)$ and $h \in C (I, [- λ, d])$ are continuous with the Lipschitz condition:

$|f (τ, η_{1}, η_{2}) - f (τ, ϱ_{1}, ϱ_{2})| \leq \sum_{k = 1}^{2} Ω |η_{k} - ϱ_{k}|,$

$Ω > 0,$ $Ω \neq Γ (α + 1)$ $\forall τ \in I$ and $x_{k}, y_{k} \in R .$
(A2): $\frac{2 Ω}{Γ (α + 1)} d^{α} < 1 .$
(A3): $φ : [- λ, d] \to R_{+}$ is a positive continuous nondecreasing function and $\exists ρ > 0$ s.t.

$\int_{0}^{τ} φ (λ) d λ \leq ρ φ (τ), τ \in I .$

Proof .

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

\begin{matrix} η (τ) & = & ϕ (0) + \frac{1}{Γ (α)} \int_{0}^{τ} {(τ - ζ)}^{α - 1} f (ζ, η (ζ), η (h (ζ))) d ζ, τ \in I . \\ η (τ) & = & ϕ (τ), τ \in [- λ, 0] . \end{matrix}

where

f \in C (I \times R^{2}, R),

h \in C (I, [- λ, d]),

0 < λ < \infty

and

h (τ) \leq τ .

Consider the Banach space

V = C ([- λ, d], R)

with Chebyshev norm

{∥\cdot∥}_{C}

and define the operator

Λ_{f} : V \to V

\begin{array}{c} Λ_{f} (η) (τ) & = & ϕ (0) + \frac{1}{Γ (α)} \int_{0}^{τ} {(τ - ζ)}^{α - 1} f (ζ, η (ζ), η (h (ζ))) d ζ, τ \in I . \\ Λ_{f} (η) (τ) & = & ϕ (τ), τ \in [- λ, 0] . \end{array}

Using the contraction principle, we show that

d_{f}

has a fixed point. In fact, it is clear that

|Λ_{f} (η) (τ) - Λ_{f} (ϱ) (τ)| = 0, η, ϱ \in C ([- r, d], R), τ \in [- λ, 0] .

Next, for any

τ \in I

, we get

\begin{matrix} |Λ_{f} (η) (τ) - Λ_{f} (ϱ) (τ)| \leq & |\frac{1}{Γ (α)} \int_{0}^{τ} {(τ - ζ)}^{α - 1} [f (ζ, η (ζ), η (h (ζ))) - f (ζ, ϱ (ζ), ϱ (h (ζ)))] d ζ| \\ \leq & \frac{Ω}{Γ (α)} \int_{0}^{τ} {(τ - ζ)}^{α - 1} {| η (ζ) - ϱ (ζ) | + | η (h (ζ)) - ϱ (h (ζ)) |} d ζ \\ \leq & \frac{Ω}{Γ (α)} \int_{0}^{τ} {(τ - ζ)}^{α - 1} (max_{0 \leq ζ \leq d} | η (ζ) - ϱ (ζ) | \\ + max_{- λ \leq ζ \leq d} | η (h (ζ)) - ϱ (h (ζ)) | d ζ \\ \leq & \frac{2 Ω}{Γ (α + 1)} d^{α} {∥ η (ζ) - ϱ (ζ) ∥}_{c} . \end{matrix}

Thus it follows that

∥Λ_{f} (η) (τ) - Λ_{f} (ϱ) (τ)∥ \leq \frac{2 Ω}{Γ (α + 1)} d^{α} {∥η (ζ) - ϱ (ζ)∥}_{c}, η, ϱ \in C ([- λ, d], R), τ \in I .

\frac{2 Ω}{Γ (α + 1)} d^{α} < 1,

the operator

d_{f}

is a contraction on the complete space

V .

Hence,

d_{f}

has a fixed point

η^{*} : [- λ, 0] \to R,

which is a solution of Equation (1).

(ii) Assume that

ϱ \in C ([- λ, d], R) \cap C^{1} ([0, d], R)

is a solution to the inequality (4). Denote by

η \in C ([- λ, d], R) \cap C^{1} ([0, d], R)

the unique solution of the problem:

\begin{array}{c} D_{τ}^{α} η (τ) & = & f (τ, η (τ), η (h (τ))), τ \in I = [0, d], d > 0, \\ η (τ) & = & ϱ (τ), τ \in [- λ, 0] . \end{array}

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

η (τ) = ϱ (0) + \frac{1}{Γ (α)} \int_{0}^{τ} {(τ - ζ)}^{α - 1} f (ζ, η (ζ), η (h (ζ))) d ζ, τ \in I .

(5)

η (τ) = ϱ (τ), τ \in [- λ, 0] .

ϱ \in C ([- λ, d], R) \cap C^{1} ([0, d], R)

satisfies the inequality (4), then using assumption (A3) and Remarks 1 and 2, we obtain

\begin{matrix} |ϱ (τ) - ϱ (0) - \frac{1}{Γ (α)} \int_{0}^{τ} {(τ - ζ)}^{α - 1} f (ζ, ϱ (ζ), ϱ (h (ζ))) d ζ| \\ \leq & \int_{0}^{τ} |g (ζ)| d ζ \leq \int_{0}^{τ} ϵ φ (ζ) d ζ \leq ρ ϵ φ (τ), τ \in I . \end{matrix}

(6)

Note that

|ϱ (τ) - η (τ)| = 0

for

τ \in [- λ, 0] .

Next, by the assumption (A1), Equation (5) and the estimate in (6), for any

τ \in I,

we can write

\begin{array}{c} |ϱ (τ) - η (τ)| & = & |ϱ (τ) - ϱ (0) - \frac{1}{Γ (α)} \int_{0}^{τ} {(τ - ζ)}^{α - 1} f (ζ, η (ζ), η (h (ζ))) d ζ| \\ \leq & |ϱ (τ) - ϱ (0) - \frac{1}{Γ (α)} \int_{0}^{τ} {(τ - ζ)}^{α - 1} f (ζ, ϱ (ζ), ϱ (h (ζ))) d ζ| \\ + \frac{1}{Γ (α)} \int_{0}^{τ} |{(τ - ζ)}^{α - 1} f (ζ, ϱ (ζ), ϱ (h (ζ))) \\ - {(τ - ζ)}^{α - 1} f (ζ, η (ζ), η (h (ζ)))| d ζ \\ \leq & ρ ϵ φ (τ) + \frac{L}{Γ (α)} \int_{0}^{τ} |{(τ - ζ)}^{α - 1}| |ϱ (ζ) - η (ζ)| d ζ \\ + \frac{L}{Γ (α)} \int_{0}^{τ} |{(τ - ζ)}^{α - 1}| |ϱ (h (ζ)) - η (h (ζ))| d ζ . \end{array}

(7)

According to above inequality, we consider the operator

G : C ([- λ, d], R_{+}) \to C ([- λ, d], R_{+})

defined by

\begin{matrix} G (m) (τ) & = & ρ ϵ φ (τ) + \frac{Ω}{Γ (α)} \int_{0}^{τ} {(τ - ζ)}^{α - 1} (m (ζ) + m (h (ζ))) d ζ, τ \in [0, d], \\ G (m) (τ) & = & 0, τ \in [- λ, 0] . \end{matrix}

Next, we prove that G is a Picard operator. To this end, observe first that for any

m, n \in C ([- λ, d], R_{+}),

we have

|G (m) (τ) - G (n) (τ)| = 0, τ \in [- λ, 0] .

Now, using

(A 1),

\forall τ \in I,

we have

\begin{matrix} |G (m) (τ) - G (n) (τ)| & \leq & \frac{Ω}{Γ (α)} \int_{0}^{τ} |{(τ - ζ)}^{α - 1}| \{|m (ζ) - n (ζ)| + |m (h (ζ)) - n (h (ζ))|\} d ζ \\ \leq & \frac{2 Ω}{Γ (α)} \int_{0}^{τ} {(τ - ζ)}^{α - 1} max_{- r \leq ζ \leq d} |m (ζ) - n (ζ)| d ζ \\ \leq & \frac{2 Ω}{Γ (α + 1)} d^{α} {∥m - n∥}_{c} . \end{matrix}

Therefore,

{∥G (m) (τ) - G (n) (τ)∥}_{C} \leq \frac{2 Ω}{Γ (α + 1)} d^{α} {∥m - n∥}_{c}, for all m, n \in C ([- λ, d], R_{+}) .

\frac{2 Ω}{Γ (α + 1)} d^{α} < 1,

G is a contraction on

C ([- λ, d], R_{+}),

using Banach contraction principle, G is a Picard operator and

F_{G} = {m^{*}} .

Then, for

τ \in I

, we have

m^{*} (τ) = ρ ϵ φ (τ) + \frac{Ω}{Γ (α)} \int_{0}^{τ} {(τ - ζ)}^{α - 1} (m^{*} (ζ) + m^{*} (h (ζ))) d ζ .

m^{*}

is increasing, then

m^{*} (h (τ)) \leq m^{*} (τ)

for

h (τ) \leq τ,

τ \in I,

and hence

m^{*} (τ) = ρ ϵ φ (τ) + \frac{2 Ω}{Γ (α)} \int_{0}^{τ} {(τ - ζ)}^{α - 1} m^{*} (ζ) d ζ .

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

m^{*} (τ) \leq ρ ϵ φ (τ) + \int_{0}^{τ} (\sum_{n = 1}^{\infty} \frac{{(\frac{2 Ω}{Γ (α)} Γ (α))}^{n}}{Γ (n α)} {(τ - ζ)}^{α - 1} ρ ϵ φ (ζ)) d ζ .

φ

is positive and nondecreasing, then

m^{*} (τ) \leq ρ ϵ φ (τ) + ρ ϵ φ (τ) \int_{0}^{τ} (\sum_{n = 1}^{\infty} \frac{{(\frac{2 Ω}{Γ (α)} Γ (α))}^{n}}{Γ (n α)} {(τ - ζ)}^{α - 1}) d ζ .

So, clearly, if we put

k (τ) = m a x \int_{0}^{τ} (\sum_{n = 1}^{\infty} \frac{{(\frac{2 Ω}{Γ (α)} Γ (α))}^{n}}{Γ (n α)} {(τ - ζ)}^{α - 1}) d ζ,

then

m^{*} (τ) \leq ρ ϵ (1 + k (τ)) φ (τ) .

Taking

C_{φ} = ρ (1 + k (τ)),

then we get

m^{*} (τ) \leq C_{φ} ϵ φ (τ), τ \in [- λ, d] .

For

m (τ) = |ϱ (τ) - η (τ)|,

the inequality (7) leads to

m (τ) \leq G (m) (τ) .

So, we have proved that

G : C ([- λ, d], R_{+}) \to C ([- λ, d], R_{+})

is an increasing Picard operator for

m \in C ([- λ, d], R_{+}),

m (τ) \leq G m (τ)

and

F_{G} = {m^{*}} .

Thus, applying the abstract Gromwell lemma (Lemma 2), we get

m (τ) \leq m^{*} (τ),

τ \in [- r, d],

implying that

|ϱ (τ) - η (τ)| \leq C_{φ} ϵ φ (τ), for all τ \in [- λ, d] .

Hence, Equation (1) is Ulam–Hyers–Rassias stable with respect to

φ .

□

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

φ (τ) = 1,

\forall τ \in [- λ, d]

in Theorem 1, we obtain

|ϱ (τ) - η (τ)| \leq C ϵ, τ \in [- λ, d],

and the result follows. □

Remark 3.

It is easy to show that (1) has generalized Ulam–Hyers stability by taking

ψ (ϵ) = C ϵ

in Corollary 1.

4. Applications

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

Example 1.

Let

λ > 0

and

h_{1} (τ) = \frac{τ}{2},

τ \in [0, d] .

Then we get the following special case of the problem (1):

\begin{matrix} D_{τ}^{α} η (τ) = f_{1} (τ, η (τ), η (h (\frac{τ}{2}))), τ \in [0, d], \\ η (0) = ϱ (0) . \end{matrix}

(8)

Now, consider the following inequality:

|D_{τ}^{α} η (τ) - f_{1} (τ, η (τ), η (h (\frac{τ}{2})))| \leq ϵ φ (τ), τ \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.

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} (τ) = τ^{2},

τ \in [0, 1] .

Then we get the specific case of the problem(1):

\begin{matrix} D_{τ}^{α} η (τ) = f_{2} (τ, η (τ), η (h (τ^{2}))), τ \in [0, 1], \\ η (0) = ϱ (0), \end{matrix}

(9)

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

|D_{τ}^{α} η (τ) - f_{2} (τ, η (τ), η (h (τ^{2})))| \leq ϵ φ (τ), τ \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.

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}{c} D_{τ}^{\frac{1}{2}} η (τ) & = & \frac{\sqrt{τ} (sin (η (τ)) + cos (η (τ - 1)))}{200}, τ \in [1, 4], \\ η (τ) & = & 1, τ \in [0, 1] . \end{array}

(10)

f (τ, η (τ), η (h (τ))) = \frac{\sqrt{τ} (sin (η (τ)) + cos (η (τ - 1)))}{200}

and f is continuous

\begin{matrix} | f (τ, η_{1}, η_{2}) - f (τ, ϱ_{1}, ϱ_{2}) | & \leq \frac{1}{200} \sqrt{τ} (| sin η_{1} - sin ϱ_{1} | + | sin η_{2} - sin ϱ_{2} |) \\ \leq \frac{\sqrt{τ}}{200} (| η_{1} - ϱ_{1} | + | η_{2} - ϱ_{2} |) \\ \leq \frac{1}{100} (| η_{1} - ϱ_{1} | + | η_{2} - ϱ_{2} |) . \end{matrix}

Thus the Lipschitz constant is

Ω = \frac{1}{100}

. Moreover, we have

\frac{2 Ω}{Γ (α + 1)} d^{α} = \frac{1}{50 Γ (3 / 2)} {(4)}^{1 / 2} < 1 .

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

Example 4.

Consider the non-linear fractional delay differential equations

\begin{matrix} D_{τ}^{\frac{1}{3}} η (τ) & = & \frac{| η (τ) | + | η (τ - \frac{1}{2}) |}{200}, τ \in [1, 3], \\ η (τ) & = & τ, τ \in [\frac{1}{2}, 1] . \end{matrix}

(11)

f (τ, η (τ), η (h (τ))) = \frac{| η (τ) | + | ϱ (τ) |}{200}

, f is continuous and

Ω = \frac{1}{200}

. Now

\frac{2 Ω}{Γ (α + 1)} d^{α} = \frac{1}{100 Γ (4 / 3)} {(3)}^{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

α = 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.

Author Contributions

All authors contributed equally. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Acknowledgments

Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2022R45), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Conflicts of Interest

The authors declare no conflict of interest.

References

Guo, D.; Lakshmikantham, V.; Liu, X. Nonlinear Integral Equations in Abstract Spaces; Kuwer: Dordrecht, The Netherlands, 1996; p. 373. [Google Scholar]
Hyers, D.H.; Isac, G.; Rassias, T.M. Stability of Functional Equations in Several Variables; Progr. Nonlinear Differential Equations Appl., 34; Springer: Boston, MA, USA; Birkhäuser: Basel, Switzerland, 1998. [Google Scholar]
Jung, S.M. A fixed point approach to the stability of a Volterra integral equation. Fixed Point Theory Appl. 2007, 2007, 57064. [Google Scholar] [CrossRef] [Green Version]
Kolmanovskií, V.; Myshkis, A. Applied Theory of Functional-Differential Equations, Math. Appl. (Soviet Ser.); Kluwer: Dordrecht, The Netherlands, 1992; p. 85. [Google Scholar]
Radu, V. The fixed point alternative and the stability of functional equations. Fixed Point Theory 2003, 4, 91–96. [Google Scholar]
Rassias, T.M. On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 1978, 72, 297–300. [Google Scholar] [CrossRef]
Ulam, S.M. A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics; Interscience: New York, NY, USA; London, UK, 1960; p. 8. [Google Scholar]
Castro, L.P.; Ramos, A. Hyers–Ulam–Rassias stability for a class of nonlinear Volterra integral equations. Banach J. Math. Anal. 2009, 3, 36–43. [Google Scholar] [CrossRef]
Bota-Boriceanu, M.F.; Petrusel, A. Ulam-Hyers stability for operatorial equations. An. Stiintifice Ale Univ. 2011, 57, 65–74. [Google Scholar] [CrossRef]
Petru, T.P.; Petruşel, A.; Yao, J.C. Ulam–Hyers stability for operatorial equations and inclusions via nonself operators. Taiwan. J. Math. 2011, 15, 2195–2212. [Google Scholar] [CrossRef]
Brzdęk, J.; Eghbali, N. On approximate solutions of some delayed fractional differential equations. Appl. Math. Lett. 2016, 54, 31–35. [Google Scholar] [CrossRef]
Huang, J.; Li, Y. Hyers-Ulam stability of linear functional differential equations. J. Math. Anal. Appl. 2015, 426, 1192–1200. [Google Scholar] [CrossRef]
Jan, M.N.; Zaman, G.; Ahmad, I.; Ali, N.; Nisar, K.S.; Abdel-Aty, A.H.; Zakarya, M. Existence Theory to a Class of Fractional Order Hybrid Differential Equations. Fractals 2022, 30, 2240022. [Google Scholar] [CrossRef]
Li, Y.; Shen, Y. Hyers Ulam stability of linear differential equations of second order. Appl. Math. 2010, 23, 306–309. [Google Scholar] [CrossRef] [Green Version]
Li, T.; Zada, A.; Faisal, S. Hyers Ulam stability of nth order linear differential equations. J. Nonlinear Sci. 2016, 9, 2070–2075. [Google Scholar] [CrossRef]
Miura, T.; Miyajima, S.; Takahasi, S.E. A characterization of Hyers-Ulam stability of first order linear differential operators. J. Math. Anal. 2003, 286, 136–146. [Google Scholar] [CrossRef] [Green Version]
Rezapour, S.; Abbas, M.I.; Etemad, S.; Minh, D.N. On a multi-point p p-Laplacian fractional differential equation with generalized fractional derivatives. Math. Methods Appl. Sci. 2022; in press. [Google Scholar]
Rezapour, S.; Souid, M.S.; Bouazza, Z.; Hussain, A.; Etemad, S. On the fractional variable order thermostat model: Existence theory on cones via piece-wise constant functions. J. Funct. Spaces, 2022; in press. [Google Scholar]
Saker, S.; Kenawy, M.; AlNemer, G.; Zakarya, M. Some fractional dynamic inequalities of Hardy’s type via conformable calculus. Mathematics 2020, 8, 434. [Google Scholar] [CrossRef] [Green Version]
Tang, S.; Zada, A.; Faisal, S.; El-Sheikh, M.M.A.; Li, T. Stability of higher-order nonlinear impulsive differential equations. J. Nonlinear Sci. Appl. 2016, 9, 4713–4721. [Google Scholar] [CrossRef] [Green Version]
Turab, A.; Mitrović, Z.D.; Savić, A. Existence of solutions for a class of nonlinear boundary value problems on the hexasilinane graph. Adv. Differ. Equ. 2021, 2021, 494. [Google Scholar] [CrossRef]
Zada, A.; Ali, W.; Farina, S. Hyers Ulam stability of nonlinear differential equations with fractional integrable impulses. Math. Methods 2017, 40, 5502–5514. [Google Scholar] [CrossRef]
Zada, A.; Faisal, S.; Li, Y. Hyers Ulam Rassias stability of non linear delay differential equations. J. Nonlinear Sci. 2017, 504, 510. [Google Scholar] [CrossRef] [Green Version]
Ibrahim, R.W. Ulam Stability of Boundary Value Problem. J. Math. 2013, 37, 287–297. [Google Scholar]
Kucche, K.D.; Shikhare, P.U. Ulam Stabilities for Nonlinear Voltera Delay Integro-differential Equations. J. Contemp. Math. Anal. 2019, 54, 276–287. [Google Scholar] [CrossRef]
Podlubny, I. Fractional Differential Equations; Academic Press: San Diego, CA, USA, 1999. [Google Scholar]
Srivastava, H.M.; Owa, S. Univalent Functions, Fractional Calculus, and Their Applications; Halsted Press: Sydney, Australia; John Wiley and Sons: New York, NY, USA; Chichester, UK; Brisban, Australia; Toronto, ON, Canada, 1989. [Google Scholar]
Ye, H.; Gao, J.; Ding, Y. A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 2007, 328, 1075–1081. [Google Scholar] [CrossRef] [Green Version]
Otrocol, D.; Ilea, V. Ulam stability for a delay differential equation. Cent. Eur. J. Math. 2013, 11, 1296–1303. [Google Scholar] [CrossRef] [Green Version]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

Share and Cite

MDPI and ACS Style

Refaai, D.A.; El-Sheikh, M.M.A.; Ismail, G.A.F.; Zakarya, M.; AlNemer, G.; Rezk, H.M. Stability of Nonlinear Fractional Delay Differential Equations. Symmetry 2022, 14, 1606. https://doi.org/10.3390/sym14081606

AMA Style

Refaai DA, El-Sheikh MMA, Ismail GAF, Zakarya M, AlNemer G, Rezk HM. Stability of Nonlinear Fractional Delay Differential Equations. Symmetry. 2022; 14(8):1606. https://doi.org/10.3390/sym14081606

Chicago/Turabian Style

Refaai, D. A., M. M. A. El-Sheikh, Gamal A. F. Ismail, Mohammed Zakarya, Ghada AlNemer, and Haytham M. Rezk. 2022. "Stability of Nonlinear Fractional Delay Differential Equations" Symmetry 14, no. 8: 1606. https://doi.org/10.3390/sym14081606

APA Style

Refaai, D. A., El-Sheikh, M. M. A., Ismail, G. A. F., Zakarya, M., AlNemer, G., & Rezk, H. M. (2022). Stability of Nonlinear Fractional Delay Differential Equations. Symmetry, 14(8), 1606. https://doi.org/10.3390/sym14081606

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Menu

Stability of Nonlinear Fractional Delay Differential Equations

Abstract

1. Introduction

2. Preliminaries

3. Ulam Stabilities for Nonlinear Fractional Delay Differential Equations

4. Applications

5. Examples

6. Conclusions

7. Future Direction

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

Share and Cite

Article Metrics

Article Access Statistics

Further Information

Guidelines

MDPI Initiatives

Follow MDPI