Open AccessArticle

Some Stability Results and Existence of Solutions for a Backward Differential Equation with Time Advance via ζ—Caputo Fractional Derivative

Abdellatif Ben Makhlouf

Lassaad Mchiri

^2,* and

Mohamed Rhaima

Department of Mathematics, Faculty of Sciences of Sfax, University of Sfax, Sfax 1171, Tunisia

ENSIIE, University of Evry-Val-d’Essonne, 1 square de la Résistance, CEDEX, 91025 Evry-Courcouronnes, France

Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Author to whom correspondence should be addressed.

Axioms 2023, 12(6), 581; https://doi.org/10.3390/axioms12060581

Submission received: 6 May 2023 / Revised: 24 May 2023 / Accepted: 7 June 2023 / Published: 12 June 2023

(This article belongs to the Section Mathematical Analysis)

Download

Browse Figures

Versions Notes

Abstract

In this paper, using a fixed point method, we proved the existence and uniqueness of solutions for a backward differential equation with time advance via

ζ -

Caputo fractional derivative. Furthermore, the Ulam–Hyers–Rassias and the Ulam–Hyers stabilities of the backward differential equation with time advance via

ζ -

Caputo fractional derivative are investigated. Finally, some experiments are given to illustrate the theoretical results.

Keywords:

fixed point method; backward differential equation; time advance; ζ-Caputo fractional derivative; stability; existence

MSC:

26A33; 34A08; 35R11

1. Introduction

The study of ordinary differential equations and partial differential equations of fractional order has interested several researchers for a long time [1,2,3,4]. Problems of the stability of these equations have been studied in a very large way. For example, the authors in [5,6] studied the finite-time stability, and in [7,8,9,10], the authors studied the stability in the sense of Ulam–Hyers–Rassias (UHR) and Ulam–Hyers (UH). Additionally, some problems of controllability and optimal control of these equations have been completed by several researchers [11,12,13,14,15].

In the literature, several types of fractional derivatives have been introduced by some researchers such as [16,17,18]. These different types of fractional derivatives have greatly contributed to the development and enrichment of many basic sciences such as mathematics, physics, medicine, engineering, stochastics, etc. For example, the authors in [19] studied a fractional model for COVID-19, in [20] the authors studied a fractional order eco-epidemiological system with infected prey, and a fractional differential system in hepatitis B has been investigated in [21].

The existence of a solution and the stability in the sense of Ulam have been studied abundantly by several researchers [7,8,9,10,22]. In [2,23,24,25], and the stability with delay was investigated for different types of fractional derivatives. To the best of our knowledge, the study of stability with time advance for a backward differential equation with time advance via

ζ -

Caputo fractional derivative has never been investigated. In this paper, we considered a backward differential equation with time advance via

ζ -

Caputo fractional derivative. In the first time, we proved the existence and uniqueness of solutions by using a fixed point method. Next, we gave two stability results, in the sense of Ulam, for the backward differential equation with time advance. Finally, some numerical experiments have been presented at the end of the paper to illustrate the theoretical results. Then, we can summarize the novelties of this work as follows:

⋄: The presentation a new problem defined by a backward differential equation with time advance via $ζ -$ Caputo fractional derivative.
⋄: The study of the existence and uniqueness of solutions for the backward differential equation with time advance via $ζ -$ Caputo fractional derivative by using Banach fixed-point Theorem.
⋄: Study of the UHR and UH stabilities for the backward differential equation with time advance via $ζ -$ Caputo fractional derivative.
⋄: Numerical implementations.

In Section 2, the main results of the paper are given. Section 4 is devoted to the numerical implementations and discussion of the numerical experiments.

2. Preliminaries and Definitions

In this paper, we consider following notations:

The space $AC ([a, b]; R)$ defined by:

$AC ([a, b]; R) = \{f : [a, b] ⟶ R, f is absolutely continuous\} .$
The Banach space $C ([a, b]; R)$ of continuous functions defined from $[a, b]$ into $R$ .
The Banach space $D_{r} ([- r, 0]; R)$ of continuous functions defined from $[- r, 0]$ into $R$ , where $r > 0$ .

Let

a, b, r

be some positive real numbers such that

a < b

and

r > 0

. We define a metric space

(E, d)

, where the space

E = C ([a, b + r]; R)

and the metric

d

is given by

\begin{matrix} d (u, v) = sup_{x \in [a, b + r]} \{\frac{| u (x) - v (x) |}{σ (x) δ (x)}\}, \end{matrix}

where the function

σ

and

δ

are defined by

\begin{matrix} σ (x) & = & \{\begin{matrix} 1, & \forall & x \in [b, b + r], \\ e^{λ (ζ (b) - ζ (x))}, & \forall & x \in [a, b] . \end{matrix} \\ δ (x) & = & \{\begin{matrix} γ (b), & \forall & x \in [b, b + r], \\ γ (x), & \forall & x \in [a, b], \end{matrix} \end{matrix}

where

γ

is a non-increasing continuous positive function and

ζ

is an increasing continuous function. It is clear that the two functions

σ

and

δ

are non-increasing functions on

[a, b]

Definition 1

([26]). Let

α \in (0, 1)

and

ζ \in C^{1} ([a, b])

be functions such that ζ is increasing and

ζ^{'} (t) \neq 0

, for all

t \in [a, b]

. The

ζ -

Caputo fractional derivative of a function

v (t)

is defined by:

^{C} D_{b}^{α, ζ} v (t) = \frac{1}{Γ (1 - α)} (- \frac{1}{ζ^{'} (t)} \frac{d}{d t}) \int_{t}^{b} ζ^{'} (s) {(ζ (s) - ζ (t))}^{- α} v (s) d s .

Lemma 1.

v \in AC ([a, b]; R)

, then the

ζ -

Caputo fractional derivative of the function

v (t)

is given by

^{C} D_{b}^{α, ζ} v (t) = - \frac{1}{Γ (1 - α)} \int_{t}^{b} {(ζ (s) - ζ (t))}^{- α} v^{'} (s) d s,

where

α \in (0, 1)

and

ζ \in C^{1} ([a, b])

is a functions such that ζ is increasing and

ζ^{'} (t) \neq 0

, for all

t \in [a, b]

Theorem 1

(Theorem 2.1 in [9]). Suppose

(F, d)

is a complete metric space and

L : F ⟶ F

is a contraction (with

τ \in [0, 1)

). Suppose that

v \in F

λ > 0

and

d (v, L (v)) \leq λ

. So, there exists a unique

β \in F

that satisfies

β = L (β)

. Moreover,

d (v, β) \leq \frac{λ}{1 - τ} .

3. Main Results

Consider the backward differential equation with time advance:

\begin{matrix} ^{C} D_{T}^{α, ζ} u (t) = f (t, u^{t}), \forall t \in [t_{0}, T], \end{matrix}

(1)

\begin{matrix} u (t) = φ (T - t), \forall t \in [T, T + r], \end{matrix}

(2)

where

^{C} D_{T}^{α, ζ} (\cdot)

is the well-known

ζ -

Caputo fractional derivative with order

α \in (0, 1)

, see [26]. The variable

u^{t}

is defined by

\begin{matrix} u^{t} (x) = u (t - x), f o r a l l x \in [- r, 0], r > 0 . \end{matrix}

(3)

Let

ι = T - t

, for

t \in [T, T + r]

, then

ι \in [- r, 0]

. Consequently, the initial condition defined by Equation (2) can be rewritten as follows:

u^{T} (ι) = φ (ι), \forall ι \in [- r, 0],

where

φ

is a continuous function belonging to the space

D_{r} = C ([- r, 0]; R)

. The second member f in Equation (1) is a continuous function and is defined by

f : [t_{0}, T] \times D_{r} ⟶ R .

The correspondent integral equation of (1) is given by [26]

\begin{matrix} u (t) & = & u (T) + \frac{1}{Γ (α)} \int_{t}^{T} ζ^{'} (s) {(ζ (s) - ζ (t))}^{α - 1} f (s, u^{s}) d s . \end{matrix}

Let us now consider the following assumption:

\begin{matrix} (H) : | f (t, v_{1}) - f (t, v_{2}) | \leq L ∥ v_{1} - v_{2} ∥, \forall v_{1}, v_{2} \in D_{r} a n d \forall t \in [t_{0}, T], \end{matrix}

where

L

is a positive constant.

Definition 2.

The problem (1)–(2) is UH stable if there exists a real number

λ_{f} > 0

such that for every

ε > 0

and for each solution

\tilde{u} \in AC ([t_{0}, T + r]; R)

of the following inequality:

|^{C} D_{T}^{α, ζ} \tilde{u} (t) - f (t, {\tilde{u}}^{t} (t)) | \leq ε,

there exists a solution

u

of Equation (1) with

u (t) = \tilde{u} (t), \forall t \in [T, T + r],

such that

| \tilde{u} (t) - u (t) | \leq λ_{f} ε, t \in [t_{0}, T] .

Definition 3.

The problem (1)–(2) is UHR stable with respect to

γ \in C ([t_{0}, T]; R)

, if there exists a real number

C_{f, γ} > 0

such that for every

ε > 0

and for each solution

\tilde{u} \in AC ([t_{0}, T + r]; R)

of the following inequality:

|^{C} D_{T}^{α, ζ} \tilde{u} (t) - f (t, {\tilde{u}}^{t} (t)) | \leq ε γ (t), t \in [t_{0}, T],

there exists a solution

u

of Equation (1) with

u (t) = \tilde{u} (t), \forall t \in [T, T + r],

| \tilde{u} (t) - u (t) | \leq C_{f, γ} ε γ (t), t \in [t_{0}, T] .

Let

M_{T}^{α, ζ}

be the constant defined by

\begin{matrix} M_{T}^{α, ζ} = \frac{λ^{α} {(ζ (T) - ζ (t_{0}))}^{α}}{(λ^{α} - L) Γ (α + 1)} e^{λ (ζ (T) - ζ (t_{0}))}, \end{matrix}

(4)

where

λ

is a positive constant such that

L < λ^{α}

We present in the following the first main result, Theorem 2, which expresses the UHR stability for the system (1)–(2).

Theorem 2.

Suppose that the assumption

(H)

holds. If

y \in AC ([t_{0}, T + r]; R)

satisfies the inequality:

\begin{matrix} |^{C} D_{T}^{α, ζ} y (t) - f (t, y^{t}) | \leq ε γ (t), f o r a l l t \in [t_{0}, T], \end{matrix}

(5)

where

ε > 0

and γ is a non-increasing continuous positive function, then there exists a unique solution

u^{*}

of (1)–(2) with

u^{*} (t) = y (t), \forall t \in [T, T + r],

such that

| u^{*} (t) - y (t) | \leq ε M_{T}^{α, ζ} γ (t), \forall t \in [t_{0}, T],

where the constant

M_{T}^{α, ζ}

is given by the relation (4).

Remark 1.

Let

y \in C ([t_{0}, T]; R)

. Then, the function y is a solution to the inequality (5) if and only if there exists a function

θ \in C ([t_{0}, T]; R)

and a function

χ \in C ([t_{0}, T]; R)

^{C} D_{T}^{α, ζ} y (t) = f (t, y^{t}) + θ (t) a n d | θ (t) | \leq ε χ (t), \forall t \in [t_{0}, T] .

Let

E = C ([t_{0}, T + r]; R)

. Now, we define the operator

A : E ⟶ E

as follows:

\begin{matrix} (A u) (t) = \{\begin{matrix} y (t), \forall t \in [T, T + r], \\ y (T) + \frac{1}{Γ (α)} \int_{t}^{T} ζ^{'} (s) {(ζ (s) - ζ (t))}^{α - 1} f (s, u^{s}) d s, \forall t \in [t_{0}, T] . \end{matrix} \end{matrix}

Immediately, we have the following result.

Proposition 1.

The operator

A : E ⟶ E

is contractive.

Proof.

Let

u_{1}, u_{2} \in E

. Then, we have

(A u_{1}) (t) - (A u_{2}) (t) = 0, \forall t \in [T, T + r] .

For

t \in [t_{0}, T]

, we obtain

\begin{matrix} | (A u_{1}) (t) - (A u_{2}) (t) | & = & | \frac{1}{Γ (α)} \int_{t}^{T} ζ^{'} (s) {(ζ (s) - ζ (t))}^{α - 1} [f (s, u_{1}^{s}) - f (s, u_{2}^{s})] d s |, \\ \leq & \frac{L}{Γ (α)} \int_{t}^{T} ζ^{'} (s) {(ζ (s) - ζ (t))}^{α - 1} ∥ u_{1}^{s} - u_{2}^{s} ∥ d s, \end{matrix}

where:

\begin{matrix} ∥ u_{1}^{s} - u_{2}^{s} ∥ = sup_{ι \in [- r, 0]} (| u_{1} (s - ι) - u_{2} (s - ι) |) . \end{matrix}

For

s \in [t, T]

, there is

ι \in [- r, 0]

such that

\begin{matrix} ∥ u_{1}^{s} - u_{2}^{s} ∥ & = & | u_{1} (s - ι) - u_{2} (s - ι) |, \\ = & \frac{| u_{1} (s - ι) - u_{2} (s - ι) |}{σ (s - ι) δ (s - ι)} σ (s - ι) δ (s - ι), \\ \leq & d (u_{1}, u_{2}) σ (s) δ (s) . \end{matrix}

Therefore,

\begin{matrix} | (A u_{1}) (t) - (A u_{2}) (t) | & \leq & \frac{L d (u_{1}, u_{2})}{Γ (α)} \int_{t}^{T} ζ^{'} (s) {(ζ (s) - ζ (t))}^{α - 1} σ (s) δ (s) d s, \\ \leq & \frac{L d (u_{1}, u_{2})}{Γ (α)} δ (t) \int_{t}^{T} ζ^{'} (s) {(ζ (s) - ζ (t))}^{α - 1} e^{λ (ζ (T) - ζ (s))} d s . \end{matrix}

(6)

Let

ρ = ζ (s)

. Then,

d ρ = ζ^{'} (s) d s

. Consequently, we obtain

\begin{matrix} \int_{t}^{T} ζ^{'} (s) {(ζ (s) - ζ (t))}^{α - 1} e^{λ (ζ (T) - ζ (s))} d s & = & \int_{ζ (t)}^{ζ (T)} {(ρ - ζ (t))}^{α - 1} e^{λ (ζ (T) - ρ)} d ρ, \\ = & \int_{ζ (t)}^{ζ (T)} {(ρ - ζ (t))}^{α - 1} e^{λ (ζ (T) - ζ (t))} e^{- λ (ρ - ζ (t))} d ρ, \\ = & e^{λ (ζ (T) - ζ (t))} \int_{ζ (t)}^{ζ (T)} {(ρ - ζ (t))}^{α - 1} e^{- λ (ρ - ζ (t))} d ρ . \end{matrix}

Let

s = λ (ρ - ζ (t))

. Then,

d s = λ d ρ

. Therefore,

\begin{matrix} \int_{t}^{T} ζ^{'} (s) {(ζ (s) - ζ (t))}^{α - 1} e^{λ (ζ (T) - ζ (s))} d s & = & e^{λ (ζ (T) - ζ (t))} \int_{0}^{λ (ζ (T) - ζ (t)} \frac{s^{α - 1}}{λ^{α}} e^{- s} d s, \\ \leq & \frac{e^{λ (ζ (T) - ζ (t))}}{λ^{α}} Γ (α) . \end{matrix}

(7)

Thus, from relations (6) and (7), we deduce that

\begin{matrix} | (A u_{1}) (t) - (A u_{2}) (t) | \leq \frac{L}{λ^{α}} d (u_{1}, u_{2}) δ (t) e^{λ (ζ (T) - ζ (t))} = \frac{L}{λ^{α}} d (u_{1}, u_{2}) σ (t) δ (t), \forall t \in [t_{0}, T] . \end{matrix}

Therefore, we obtain

\begin{matrix} d (A u_{1}, A u_{2}) \leq \frac{L}{λ^{α}} d (u_{1}, u_{2}) . \end{matrix}

Recall that

L < λ^{α}

. Thus, the operator

A

is contractive. □

We can now establish the proof of Theorem 2.

Proof of Theorem 2.

We have

(A y) (t) - y (t) = 0, f o r a l l t \in [T, T + r] .

It follows from (5) that

\begin{matrix} | y (t) - (A y) (t) | & \leq & \frac{ε}{Γ (α)} \int_{t}^{T} γ (s) ζ^{'} (s) {(ζ (s) - ζ (t))}^{α - 1} d s, \\ \leq & \frac{ε γ (t)}{Γ (α)} \int_{t}^{T} ζ^{'} (s) {(ζ (s) - ζ (t))}^{α - 1} d s, \\ \leq & \frac{ε γ (t)}{Γ (α + 1)} {(ζ (T) - ζ (t))}^{α} . \end{matrix}

Therefore,

d (y, A y) \leq \frac{ε}{Γ (α + 1)} {(ζ (T) - ζ (t_{0}))}^{α} .

Using Theorem 2.1 in [9], there exists a unique solution

u^{*}

of (1)–(2), with

u^{*} (t) = y (t)

for all

t \in [T, T + r]

, such that

\begin{matrix} d (y, u^{*}) & \leq & \frac{ε}{Γ (α + 1)} {(ζ (T) - ζ (t_{0}))}^{α} \frac{1}{1 - \frac{L}{λ^{α}}}, \\ \leq & \frac{λ^{α} {(ζ (T) - ζ (t_{0}))}^{α}}{(λ^{α} - L) Γ (α + 1)} ε . \end{matrix}

Therefore,

| y (t) - u^{*} (t) | \leq \frac{λ^{α} {(ζ (T) - ζ (t_{0}))}^{α}}{(λ^{α} - L) Γ (α + 1)} e^{λ (ζ (T) - ζ (t_{0}))} γ (t) ε = ε M_{T}^{α, ζ} γ (t), \forall t \in [t_{0}, T] .

□

The second main result of this paper is given by the following Corollary 1, which expresses the UH stability for the system (1)–(2).

Corollary 1.

Suppose that the assumption

(H)

holds. If

y \in AC ([t_{0}, T + r]; R)

satisfies the identity

\begin{matrix} |^{C} D_{T}^{α, ζ} y (t) - f (t, y^{t}) | \leq ε, f o r a l l t \in [t_{0}, T], \end{matrix}

(8)

where

ε > 0

, then there exists a unique solution

u^{*}

of (1)–(2) with

u^{*} (t) = y (t), \forall t \in [T, T + r],

such that

\begin{matrix} | u^{*} (t) - y (t) | \leq M_{T}^{α, ζ} ε, \forall t \in [t_{0}, T], \end{matrix}

(9)

where the constant

M_{T}^{α, ζ}

is given by the relation (4).

Proof.

The proof of Corollary 1 can be deduced from that of Theorem 2, where the considered metric function

d

, in this case, is defined by

\begin{matrix} d (u, v) = sup_{x \in [t_{0}, T + r]} \{\frac{| u (x) - v (x) |}{β (x)}\}, \end{matrix}

where the positive function

β

is given by

\begin{matrix} β (x) = \{\begin{matrix} 1, & \forall & x \in [T, T + r], \\ e^{λ (ζ (T) - ζ (x))}, & \forall & x \in [t_{0}, T] . \end{matrix} \end{matrix}

□

4. Numerical Illustration

In this section, we consider the case when

f (s, u^{s})

is written in the form

g (s, u (s + r))

Consider the following integral equation:

\begin{matrix} u (t) & = & u (T) + \frac{1}{Γ (α)} \int_{t}^{T} ζ^{'} (s) {(ζ (s) - ζ (t))}^{α - 1} g (s, u (s + r)) d s . \end{matrix}

(10)

We divide the interval

[t_{0}, T]

into N sub-intervals

[t_{i}, t_{i + 1}]

, for

i = 0, \dots, N - 1

, of equal amplitude h, where

t_{i} = t_{0} + i h, i = 0, \dots, N a n d h = \frac{T - t_{0}}{N} .

Then, it is clear that

\begin{matrix} T & = & t_{N}, \\ h & = & t_{i + 1} - t_{i}, 0, \dots, N - 1 \end{matrix}

At the grid point

t_{i}

, for

i = N - 1, \dots, 0

, Equation (10) takes the form

\begin{matrix} u (t_{i}) & = & u (t_{N}) + \frac{1}{Γ (α)} \int_{t_{i}}^{T} ζ^{'} (s) {(ζ (s) - ζ (t_{i}))}^{α - 1} g (s, u (s + r)) d s, \\ = & u (t_{N}) + \frac{1}{Γ (α)} \sum_{k = i}^{N - 1} \int_{t_{k}}^{t_{k + 1}} ζ^{'} (s) {(ζ (s) - ζ (t_{i}))}^{α - 1} g (s, u (s + r)) d s, \\ \approx & u (t_{N}) + \frac{1}{Γ (α)} \sum_{k = i}^{N - 1} g (t_{k}, u (t_{k} + r)) \int_{t_{k}}^{t_{k + 1}} ζ^{'} (s) {(ζ (s) - ζ (t_{i}))}^{α - 1} d s . \end{matrix}

(11)

By integrating the integral in the right side of Equation (11), we obtain

\begin{matrix} u (t_{i}) = u (t_{N}) + \frac{1}{Γ (1 + α)} \sum_{k = i}^{N - 1} g (t_{k}, u (t_{k} + r)) ω_{k}^{i}, \end{matrix}

(12)

where the coefficients

ω_{k}^{i + 1}

are given by

ω_{k}^{i} = {(ζ (t_{k + 1}) - ζ (t_{i}))}^{α} - {(ζ (t_{k}) - ζ (t_{i}))}^{α}, k = i, \dots, N - 1, i = N - 1, \dots, 0 .

Now, let us consider the following notation:

u_{i} = u (t_{i}), i = 0, \dots, N .

Assume that there exists a positive integer

p > 0

such that

r = p h

. The coefficient

u (t_{k} + r)

can be rewritten as

u (t_{k} + r) = u (t_{0} + k h + p h) = u (t_{0} + (k + p) h) = u (t_{k + p}) \approx u_{k + p} .

Therefore, Equation (12) can be rewritten as

\begin{matrix} u_{i} = u_{N} + \frac{1}{Γ (1 + α)} \sum_{k = i}^{N - 1} g (t_{k}, u_{k + p}) ω_{k}^{i}, i = N - 1, \dots, 0 . \end{matrix}

(13)

Example 1.

Let

[t_{0}, T] = [0, 1]

r = 0

. Consider the system:

\begin{matrix} ^{C} D_{T}^{α, ζ} u (t) & = & 5 u (t) + F (t), \forall t \in [t_{0}, T], \end{matrix}

(14)

\begin{matrix} u (T) & = & u_{T} . \end{matrix}

(15)

whose exact solution is

u (t) = 1 + t^{2}

. The initial condition

u (T) = u (1) = 2

and

α = 0.75, ζ (t) = t

. The source term ϝ is given by

F (t) = \frac{2 t^{2 - α}}{Γ (3 - α)} - 5 (1 + t^{2}) .

The system (14)–(15) is solved using the scheme (13) by the software Matlab 7.5.0 (R2007b). Figure 1 show the convergence of the numerical solution to the exact solution.

Example 2.

This example is devoted to the UHR stability.

The data used in this experiment are chosen as follows:

[t_{0}, T] = [1, e], [- r, 0] = [- \frac{1}{10} (e - 1), 0], λ = 0.2, α = 0.75 .

Let

ζ (t) = ln (t)

. In this case, we obtain

^{C} D_{T}^{α, ζ} u (t) =^{C H} D_{e}^{0.75} u (t)

, where

^{C H} D_{e}^{0.75} u (t)

is the Caputo–Hadamard fractional derivative of

u (t)

(see [26]).

Consider the Caputo–Hadamard fractional problem:

\begin{matrix} ^{C H} D_{e}^{0.75} u (t) & = & f (t, u^{t}), \forall t \in [1, e], \end{matrix}

(16)

where

f (t, u^{t}) = \frac{1}{10} cos (t) u (t + r)

. Note that the function f satisfies the assumption

(H)

| f (t, u) - f (t, v) | \leq \frac{1}{10} ∥ u - v ∥, \forall t \in [1, e] .

Now, let us define the fractional system:

\begin{matrix} ^{C H} D_{e}^{0.75} y (t) & = & f (t, y^{t}) + θ (t), \forall t \in [1, e], \\ y (t) & = & 0.6 cos (2.5 π (e - t)) e^{- 3 (e - t)}, \forall t \in [e, e + r], \end{matrix}

where

θ (t) = \frac{1}{30 t^{2} + 1}

. Since, we have

| θ (t) | = \frac{1}{30 t^{2} + 1} \leq \frac{1}{30} \frac{1}{t^{2}}, \forall t \in [1, e],

then, from Remark 1,

y \in C ([1, e]; R)

is a solution to the inequality (5), with

ε = \frac{1}{30}

and

γ (t) = \frac{1}{t^{2}}

\begin{matrix} |^{C H} D_{e}^{0.75} y (t) - f (t, y^{t}) | \leq \frac{1}{30} \frac{1}{t^{2}}, \forall t \in [1, e] . \end{matrix}

Therefore, from Theorem 2, we deduce that Equation (16) has a unique solution

u^{*}

such that

u^{*} (t) = y (t), \forall t \in [e, e + r],

and

| u^{*} (t) - y (t) | \leq \frac{1}{30} \frac{1}{t^{2}} M_{e}^{0.75, ζ}, \forall t \in [1, e],

where the constant

M_{e}^{0.75, ζ}

is given by

\begin{matrix} M_{e}^{0.75, ζ} & = & \frac{λ^{α} {(ζ (T) - ζ (t_{0}))}^{α}}{(λ^{α} - L) Γ (α + 1)} e^{λ (ζ (T) - ζ (t_{0}))}, \\ = & \frac{0 . 2^{0.75} {(ln (e) - ln (1))}^{0.75}}{(0 . 2^{0.75} - 0.1) Γ (0.75 + 1)} e^{0.2 (ln (e) - ln (1))} \approx 1.9966 . \end{matrix}

So, we obtain

| u^{*} (t) - y (t) | \leq 0.06655 \frac{1}{t^{2}}, \forall t \in [1, e] .

In Figure 2, we plotted the solution

u^{*}

of Equation (16) for

t \in [1, e]

and the corresponding initial condition

u (t) = y (t) = φ (e - t)

, for

t \in [e, e + r]

. In Figure 3, we plotted the solution

u^{*}

of Equation (16) and the solution

y (t)

of the inequality (5), for

t \in [1, e]

, with the same initial condition

u (t) = y (t) = φ (e - t)

, for

t \in [e, e + r]

. The difference

| u^{*} (t) - y (t) |

and the curve of the function

ε γ (t) M_{e}^{0.75, ζ}

, for

t \in [1, e]

, are plotted in Figure 4. In fact, it is clear that there is consistency between the UHR stability result and the numerical experiment given in Figure 4.

Example 3.

This example is devoted to the UH stability.

The data used in this experiment are chosen as follows:

[t_{0}, T] = [1, 2 e], [- r, 0] = [- \frac{1}{5} (2 e - 1), 0], λ = 0.2, α = 0.7 .

Let

ζ (t) = \sqrt{t^{2} + 1}

. Consider the system defined by

ζ -

Caputo fractional derivative:

\begin{matrix} ^{C} D_{2 e}^{0.7, ζ} u (t) & = & f (t, u^{t}), \forall t \in [1, 2 e], \end{matrix}

(17)

where

f (t, u^{t}) = 0.3 sin (5 t) u (t + r)

. Note that the function f satisfies the assumption

(H)

| f (t, u) - f (t, v) | \leq 0.3 ∥ u - v ∥, \forall t \in [1, 2 e] .

Now, let us define the fractional system:

\begin{matrix} ^{C} D_{2 e}^{0.7, ζ} y (t) & = & f (t, y^{t}) + ε, \forall t \in [1, 2 e], \\ y (t) & = & 6 cos (12 π (2 e - t)) sin (13 π (2 e - t)), \forall t \in [2 e, 2 e + r], \end{matrix}

where

ε = 0.01

. If

y \in C ([1, 2 e]; R)

is a solution to the inequality

\begin{matrix} |^{C} D_{2 e}^{0.7, ζ} y (t) - f (t, y^{t}) | \leq ε, \forall t \in [1, 2 e], \end{matrix}

then, from Corollary 1, we deduce that Equation (17) has a unique solution

u^{*}

such that

u^{*} (t) = y (t), \forall t \in [2 e, 2 e + r],

and

| u^{*} (t) - y (t) | \leq ε M_{2 e}^{0.7, ζ}, \forall t \in [1, 2 e],

where the constant

M_{2 e}^{0.7, ζ}

is given by:

\begin{matrix} M_{2 e}^{0.7, ζ} & = & \frac{λ^{α} {(ζ (T) - ζ (t_{0}))}^{α}}{(λ^{α} - L) Γ (α + 1)} e^{λ (ζ (T) - ζ (t_{0}))}, \\ = & \frac{0 . 2^{0.7} {(ζ (2 e) - ζ (1))}^{0.7}}{(0 . 2^{0.7} - 0.3) Γ (0.7 + 1)} e^{0.2 (ζ (2 e) - ζ (1))} \approx 90.5732 . \end{matrix}

So, we obtain

| u^{*} (t) - y (t) | \leq 0.905732, \forall t \in [1, 2 e] .

In Figure 5, the solution

u^{*}

of Equation (17) for

t \in [1, 2 e]

and the corresponding initial condition

u (t) = y (t) = φ (2 e - t)

, for

t \in [2 e, 2 e + r]

are plotted. In Figure 6, we plotted the solution

u^{*}

of Equation (17) and the solution

y (t)

of the inequality (8), for

t \in [1, 2 e]

, with the same initial condition

u (t) = y (t) = φ (2 e - t)

, for

t \in [2 e, 2 e + r]

. The difference

| u^{*} (t) - y (t) |

and the horizontal line

ξ (t) = ε M_{2 e}^{0.7, ζ}

, for

t \in [1, 2 e]

, are plotted in Figure 7. Again, it is clear that there is consistency between the UH stability result and the numerical experiment given in Figure 7.

5. Conclusions

In this work, several goals are achieved. We have proved the existence and the uniqueness of solutions for the problem defined by backward differential equations with time advance via

ζ -

Caputo fractional derivative. Moreover, two stability results, in the sense of UHR and UH, have been established. Finally, we presented numerical results to confirm the theoretical results obtained.

In future work, it would be interesting to study the finite-time stability for this type of backward differential equation with time advance with other types of fractional derivatives.

Author Contributions

Conceptualization, L.M.; methodology, A.B.M.; writing—original draft, M.R. All authors have read and agreed to the published version of the manuscript.

Funding

This research is funded by ”Researchers Supporting Project number (RSPD2023R683), King Saud University, Riyadh, Saudi Arabia”.

Data Availability Statement

Not applicable.

Acknowledgments

The authors extend their appreciation to King Saud University in Riyadh, Saudi Arabia for funding this research work through Researchers Supporting Project number (RSPD2023R683).

Conflicts of Interest

The authors declare no conflict of interest.

References

Telli, B.; Souid, M.S.; Stamova, I. Boundary-Value Problem for Nonlinear Fractional Differential Equations of Variable Order with Finite Delay via Kuratowski Measure of Noncompactness. Axioms 2021, 12, 80. [Google Scholar] [CrossRef]
Arfaoui, H.; Ben Makhlouf, A. Some results for a class of two-dimensional fractional hyperbolic differential systems with time delay. J. Appl. Math. Comput. 2022, 68, 2389–2405. [Google Scholar] [CrossRef] [PubMed]
Raja, M.M.; Vijayakumar, V. Approximate controllability results for the Sobolev type fractional delay impulsive integrodifferential inclusions of order r∈(1,2) via sectorial operator. Fract. Calc. Appl. Anal. 2023, 2023. [Google Scholar] [CrossRef]
Houas, M.; Martinez, F.; Samei, M.E.; Kaabar, M.K.A. Uniqueness and Ulam-Hyers-Rassias stability results for sequential fractional pantograph q-differential equations. J. Inequalities Appl. 2022, 2022, 1–24. [Google Scholar] [CrossRef]
Du, F.; Lu, J.G. Finite-time stability of fractional-order fuzzy cellular neural networks with time delays. Fuzzy Sets Syst. 2022, 438, 107–122. [Google Scholar] [CrossRef]
Elshenhab, A.M.; Wang, X.; Cesarano, C.; Almarri, B.; Moaaz, O. Finite-Time Stability Analysis of Fractional Delay Systems. Mathematics 2022, 10, 1883. [Google Scholar] [CrossRef]
Hanan, A.; Wahasha; Satish, K.; Panchala; Mohammed, S.; Abdob. Existence and stability of a nonlinear fractional differential equation involving a ζ-Caputo operator. Adv. Theory Nonlinear Anal. Its Appl. 2020, 4, 266–278. [Google Scholar] [CrossRef]
Ahmadova, A.; Mahmudov, N.I. Ulam-Hyers stability of Caputo type fractional stochastic neutral differential equations. Stat. Probab. Lett. 2020, 168, 108949. [Google Scholar] [CrossRef]
Kahouli, O.; Ben Makhlouf, A.; Mchiri, L.; Rguigui, H. Hyers-Ulam stability for a class of Hadamard fractional Itô-Doob stochastic integral equations. Chaos Solitons Fractals 2023, 166, 112918. [Google Scholar] [CrossRef]
Ben Makhlouf, A.; Boucenna, D. Ulam-Hyers-Rassias Mittag-Leffler stability for the Darboux problem for partial fractional differential equations. Rocky Mountain J. Math. 2021, 51, 1541–1551. [Google Scholar] [CrossRef]
Selvam, A.P.; Vellappandi, M.; Govindaraj, V. Controllability of fractional dynamical systems with ζ--Caputo fractional derivative. Phys. Scr. 2023, 98, 025206. [Google Scholar] [CrossRef]
Mophou, G. Controllability of a backward fractional semilinear differential equation. Appl. Math. Comput. 2014, 242, 168–178. [Google Scholar] [CrossRef]
Vigya; Mahto, T.; Malik, H.; Mukherjee, V.; Alotaibi, M.A.; Almutairi, A. Renewable generation based hybrid power system control using fractional order-fuzzy controller. Energy Rep. 2021, 7, 641–653. [Google Scholar] [CrossRef]
Mirrezapour, S.Z.; Zare, A.; Hallaji, M. A new fractional sliding mode controller based on nonlinear fractional-order proportional integral derivative controller structure to synchronize fractional-order chaotic systems with uncertainty and disturbances. J. Vib. Control. 2021, 28, 773–785. [Google Scholar] [CrossRef]
Hassani, H.; Tenreiro Machado, J.A.; Avazzadeh, Z.; Naraghirad, E.; Dahaghin, M.S. Generalized Bernoulli Polynomials: Solving Nonlinear 2D Fractional Optimal Control Problems. J. Sci. Comput. 2020, 83, 1–21. [Google Scholar] [CrossRef]
Podlubny, I. Fractional Differential Equations; Academic Press: New York, NY, USA, 1999. [Google Scholar]
Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations; Elsevier: Amsterdam, The Netherlands, 2006. [Google Scholar]
Atangana, A.; Baleanu, D. New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model. Therm. Sci. 2016, 20, 763–769. [Google Scholar] [CrossRef] [Green Version]
Higazy, M.; Allehiany, F.M.; Mahmoud, E.E. Numerical study of fractional order COVID-19 pandemic transmission model in context of ABO blood group. Results Phys. 2021, 22, 103852. [Google Scholar] [CrossRef]
Moustafa, M.; Mohd, M.H.; Ismail, A.I.; Abdullah, F.A. Global stability of a fractional order eco-epidemiological system with infected prey. Int. J. Math. Model. Numer. Optim. 2021. [Google Scholar] [CrossRef]
Cardoso, L.C.; Camargo, R.F.; Dos Santos, F.L.P.; Dos Santos, P.C. Global stability analysis of a fractional differential system in hepatitis B. Chaos Solitons Fractals 2021, 143, 110619. [Google Scholar] [CrossRef]
Wang, X.; Luo, D.; Luo, Z.; Zada, A. Ulam-Hyers Stability of Caputo-Type Fractional Stochastic Differential Equations with Time Delays. Math. Probl. Eng. 2021, 2021, 5599206. [Google Scholar] [CrossRef]
Abbas, S.; Benchohra, M. Darboux problem for perturbed partial differential equations of fractional order with finite delay. Nonlinear Anal. Hybrid Syst. 2020, 381, 68–77. [Google Scholar] [CrossRef]
Abbas, S.; Benchohra, M.; Cabada, A. Partial neutral functional integro-differential equations of fractional order with delay. Bound. Value Probl. 2012, 2012, 1–13. [Google Scholar] [CrossRef] [Green Version]
Benchohra, M.; Hellal, M. Global uniqueness results for fractional partial hyperbolic differential equations with state-dependent delay. Ann. Pol. Math. 2014, 110, 259–281. [Google Scholar] [CrossRef]
Almeida, R. A Caputo fractional derivative of a function with respect to another function. Commun. Nonlinear Sci. Numer. Simul. 2017, 44, 460–481. [Google Scholar] [CrossRef] [Green Version]

Figure 1. The exact solution and the numerical solution for a time step

h = 10^{- 3}

Figure 1. The exact solution and the numerical solution for a time step

h = 10^{- 3}

Figure 2. The numerical solution

u^{*} (t)

for

t \in [1, e]

, with the initial condition

φ (e - t)

, for

t \in [e, e + r]

Figure 2. The numerical solution

u^{*} (t)

for

t \in [1, e]

, with the initial condition

φ (e - t)

, for

t \in [e, e + r]

Figure 3. The numerical solution

u^{*} (t)

and the numerical solution

y (t)

, for

t \in [1, e]

Figure 3. The numerical solution

u^{*} (t)

and the numerical solution

y (t)

, for

t \in [1, e]

Figure 4. The difference

| u^{*} (t) - y (t) |

and the curve of the function

ε γ (t) M_{e}^{0.75, ζ}

, for

t \in [1, e]

Figure 4. The difference

| u^{*} (t) - y (t) |

and the curve of the function

ε γ (t) M_{e}^{0.75, ζ}

, for

t \in [1, e]

Figure 5. We plotted the solution

u^{*}

of Equation (17) for

t \in [1, 2 e]

, and the corresponding initial condition

u (t) = y (t) = φ (2 e - t)

, for

t \in [2 e, 2 e + r]

Figure 5. We plotted the solution

u^{*}

of Equation (17) for

t \in [1, 2 e]

, and the corresponding initial condition

u (t) = y (t) = φ (2 e - t)

, for

t \in [2 e, 2 e + r]

Figure 6. We plotted the solution

u^{*}

of Equation (17) and the solution

y (t)

of the inequality (5), for

t \in [1, 2 e]

, with the same initial condition

u (t) = y (t) = φ (2 e - t)

, for

t \in [2 e, 2 e + r]

Figure 6. We plotted the solution

u^{*}

of Equation (17) and the solution

y (t)

of the inequality (5), for

t \in [1, 2 e]

, with the same initial condition

u (t) = y (t) = φ (2 e - t)

, for

t \in [2 e, 2 e + r]

Figure 7. We plotted both the difference

| u^{*} (t) - y (t) |

and the horizontal line

ξ (t) = ε M_{2 e}^{0.7, ζ}

, for

t \in [1, 2 e]

Figure 7. We plotted both the difference

| u^{*} (t) - y (t) |

and the horizontal line

ξ (t) = ε M_{2 e}^{0.7, ζ}

, for

t \in [1, 2 e]

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

© 2023 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

Ben Makhlouf, A.; Mchiri, L.; Rhaima, M. Some Stability Results and Existence of Solutions for a Backward Differential Equation with Time Advance via ζ—Caputo Fractional Derivative. Axioms 2023, 12, 581. https://doi.org/10.3390/axioms12060581

AMA Style

Ben Makhlouf A, Mchiri L, Rhaima M. Some Stability Results and Existence of Solutions for a Backward Differential Equation with Time Advance via ζ—Caputo Fractional Derivative. Axioms. 2023; 12(6):581. https://doi.org/10.3390/axioms12060581

Chicago/Turabian Style

Ben Makhlouf, Abdellatif, Lassaad Mchiri, and Mohamed Rhaima. 2023. "Some Stability Results and Existence of Solutions for a Backward Differential Equation with Time Advance via ζ—Caputo Fractional Derivative" Axioms 12, no. 6: 581. https://doi.org/10.3390/axioms12060581

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

Article Menu

Some Stability Results and Existence of Solutions for a Backward Differential Equation with Time Advance via ζ—Caputo Fractional Derivative

Abstract

1. Introduction

2. Preliminaries and Definitions

3. Main Results

4. Numerical Illustration

5. Conclusions

Author Contributions

Funding

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

Share and Cite

Article Metrics

Article Access Statistics

Further Information

Guidelines

MDPI Initiatives

Follow MDPI