Mittag-Leffler-Hyers-Ulam stability for a first- and second-order nonlinear differential equations using Fourier transform

Arunachalam Selvam; Sriramulu Sabarinathan; Arumugam Ponmana Selvan

doi:10.1515/dema-2024-0033

Article Open Access

Mittag-Leffler-Hyers-Ulam stability for a first- and second-order nonlinear differential equations using Fourier transform

Arunachalam Selvam , Sriramulu Sabarinathan and Arumugam Ponmana Selvan

Published/Copyright: August 13, 2024

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From the journal Demonstratio Mathematica Volume 57 Issue 1

Abstract

In this article, we apply the Fourier transform to prove the Hyers-Ulam and Hyers-Ulam-Rassias stability for the first- and second-order nonlinear differential equations with initial conditions. Additionally, we extend the results to investigate the Mittag-Leffler-Hyers-Ulam and Mittag-Leffler-Hyers-Ulam-Rassias stability of these differential equations using the proposed method.

Keywords: Fourier transform; Hyers-Ulam and Hyers-Ulam-Rassias stability; Mittag-Leffler-Hyers-Ulam and Mittag-Leffler-Hyers-Ulam-Rassias stability; nonlinear differential equations

MSC 2010: 34K20; 26D10; 39B82; 34A40; 39A30

1 Introduction

The study of stability for various functional equations was invented by a famous mathematician, Ulam [1], in 1940. He raised the question concerning the stability of functional equations: “Give conditions for a linear function near an approximately linear function to exist.” The first solution was brilliantly answered to the problem of Ulam for Cauchy additive functional equations based on Banach spaces by Hyers [2] in 1941. Rassias [3] later provided a generalized solution to Ulam’s question for approximately linear mappings in 1978. During these days, numerous mathematicians have contributed to the advancement of Ulam’s problem, addressing various functional equations in different spaces and directions [4–15].

Ulam’s problem has recently been extended by replacing functional equations with differential equations in the following form:

ζ ( g , φ , φ ′ , φ ″ , … , φ ( n ) ) = 0 ,

which is said to have Hyers-Ulam stability if for a given ε > 0 and a function φ such that

∣ ζ ( g , φ , φ ′ , φ ″ , … , φ ( n ) ) ∣ ≤ ε ,

there exists some φ a of the differential equation such that ∣ φ ( s ) − φ a ( s ) ∣ ≤ H ( ε ) and

lim ε → 0 H ( ε ) = 0 .

Oblaza seems to be the first author to investigate the Hyers-Ulam stability of linear differential equations [16,17]. Then, Alsina and Ger [18] published their work, which exhibits the Hyers-Ulam stability of the linear differential equation ψ ′ ( ℓ ) = λ ψ ( ℓ ) . Jung derived the general setting for Hyers-Ulam stability of the first-order linear differential equations in [19–22].

The generalized Hyers-Ulam stability of higher-order linear differential equations using the Laplace transform method was proved by Alqifiary and Jung [23] in 2014. In [24], the authors proved the Mittag-Leffler-Hyers-Ulam stability of the first- and second-order linear differential equations using the Fourier transform method [25,26]. The Hyers-Ulam stability of differential equations is being studied right now [27–36], and the research is still going on [37,38].

In [39], the authors investigated the existence, uniqueness, and Hyers-Ulam stability of the solution derived for the coupled system of fractional differential equations in Caputo sense with nonlinear p -Laplacian operator. In 2018, by nonlinear Leray-Schauder-type alternative and Banach’s fixed point theorem, they investigated the existence and uniqueness of solutions and also proved the Hyers-Ulam stability for the coupled system of fractional differential equations with the nonlinear p -Laplacian operator and Riemann-Liouville integral boundary conditions in [40]. In [41], the authors studied the general class of nonlinear singular fractional differential equations with p -Laplacian for the existence and uniqueness of a positive solution and the Hyers-Ulam stability [42–44].

Recently, the Ulam stability of the various types of differential equations of first-order, second-order, and higher-orders are investigated using Laplace transforms in [45,46], Aboodth transform in [47,48], and Mahgoub transform in [49–51].

Motivated and inspired by the existing literature, the main goal of this article is to implement the Fourier transform technique, we establish the Hyers-Ulam and Hyers-Ulam-Rassias stability, Mittag-Leffler-Hyers-Ulam, and Mittag-Leffler-Hyers-Ulam-Rassias stability of the following first- and second-order nonlinear differential equations:

(1) u φ ′ ( ℓ ) + v φ ( ℓ ) = n ( ℓ , φ ( ℓ ) ) ,

(2) φ ( 0 ) = α 0 ,

and

(3) u φ ″ ( ℓ ) + v φ ′ ( ℓ ) + w φ ( ℓ ) = n ( ℓ , φ ( ℓ ) , φ ′ ( ℓ ) ) ,

(4) φ ( 0 ) = α 0 , φ ′ ( 0 ) = α 1 ,

for any ℓ > 0 . Here, u , v , and w represent the given constants in F , φ ( ℓ ) is a continuously differentiable function, and n ( ℓ , φ ( ℓ ) ) and n ( ℓ , φ ( ℓ ) , φ ′ ( ℓ ) ) denote the given nonlinear functions.

2 Preliminaries

In this section, we refer to a few basic definitions, notations, and theorems that are useful throughout this article to prove the main results. Throughout this article, F denotes either the real field R or the complex field C .

Definition 2.1

For every function p : ( 0 , ∞ ) → F of exponential order, we define the Fourier transform of p by

ℱ { p ( ℓ ) } = P ( s ) = ∫ − ∞ ∞ p ( ℓ ) e − i ℓ s d ℓ .

Then, at the point of continuity of p , we have

p ( ℓ ) = 1 2 π ∫ − ∞ ∞ P ( s ) e − i ℓ s d s ,

which is said to be an inverse Fourier transform, and it is denoted by ℱ − 1 . We also introduce the convolution of two functions.

Definition 2.2

Let p and q be two functions both Lebesgue integrable on ( − ∞ , + ∞ ) . Let S denote the set of s to which the Lebesgue integral

r ( s ) = ∫ − ∞ ∞ p ( ℓ ) q ( s − ℓ ) d ℓ

exists. This integral defines a function r on S , which is said to be a convolution of p and q . We also denote r = p ⁎ q for convolution.

Definition 2.3

A Mittag-Leffler function of one parameter is defined as

E ν ( ℓ ) = ∑ k = 0 ∞ ℓ k Γ ( ν k + 1 ) ,

where ℓ , ν ∈ C and R e ( ν ) > 0 . If we put ν = 1 , then the aforementioned equation becomes

E 1 ( ℓ ) = ∑ k = 0 ∞ ℓ k Γ ( k + 1 ) = ∑ k = 0 ∞ ℓ k k = e ℓ ,

where E ν ( ℓ ) is the Mittag-Leffler function of one parameter.

Definition 2.4

Generalization by E ν ( ℓ ) is defined in the function

E ν , ϑ ( ℓ ) = ∑ k = 0 ∞ ℓ k Γ ( ν k + ϑ ) ,

where ℓ , ν , ϑ ∈ C , R e ( ν ) > 0 and R e ( ϑ ) > 0 .

Definition 2.5

Let φ : ( 0 , ∞ ) → F be a continuously differentiable function that satisfies

∣ u φ ′ ( ℓ ) + v φ ( ℓ ) − n ( ℓ , φ ( ℓ ) ) ∣ ≤ ε ,

for any ℓ > 0 . An initial value problem (IVP) (1) with initial condition (2) is Hyers-Ulam stable; if for all ε > 0 , then there exist H > 0 and ψ : ( 0 , ∞ ) → F , which is satisfies the nonlinear differential equation (1) such that ∣ φ ( ℓ ) − ψ ( ℓ ) ∣ ≤ H ε , for any ℓ > 0 , where H is called Hyers-Ulam stability constant of (1).

Definition 2.6

Let φ : ( 0 , ∞ ) → F be a continuously differentiable function; if for each ε > 0 , there exists ϕ : ( 0 , ∞ ) → ( 0 , ∞ ) , which satisfies the inequality

∣ u φ ′ ( ℓ ) + v φ ( ℓ ) − n ( ℓ , φ ( ℓ ) ) ∣ ≤ ϕ ( ℓ ) ε ,

for any ℓ > 0 . An IVP (1) with initial condition (2) is Hyers-Ulam-Rassias stable; then, there exists H > 0 and ψ : ( 0 , ∞ ) → F , which satisfies the nonlinear differential equation (1) such that ∣ φ ( ℓ ) − ψ ( ℓ ) ∣ ≤ H ϕ ( ℓ ) ε , for any ℓ > 0 , where H is called Hyers-Ulam-Rassias stability constant of (1).

Definition 2.7

Let φ : ( 0 , ∞ ) → F be a continuously differentiable function that satisfies

∣ u φ ′ ( ℓ ) + v φ ( ℓ ) − n ( ℓ , φ ( ℓ ) ) ∣ ≤ ε E ν ( ℓ ν ) ,

where E ν ( ℓ ν ) is a Mittag-Leffler function, for any ℓ > 0 . An IVP (1) is Mittag-Leffler-Hyers-Ulam stable; if for all ε > 0 , then there exist H > 0 and ψ : ( 0 , ∞ ) → F , which is satisfies the nonlinear differential equation (1) such that ∣ φ ( ℓ ) − ψ ( ℓ ) ∣ ≤ H ε E ν ( ℓ ν ) , for any ℓ > 0 , where H is called Mittag-Leffler-Hyers-Ulam stability constant of (1).

Definition 2.8

Let φ : ( 0 , ∞ ) → F be a continuously differentiable function, if there exist ϕ : ( 0 , ∞ ) → ( 0 , ∞ ) , which satisfies

∣ u φ ′ ( ℓ ) + v φ ( ℓ ) − n ( ℓ , φ ( ℓ ) ) ∣ ≤ ϕ ( ℓ ) ε E ν ( ℓ ν ) ,

where E ν ( ℓ ν ) is a Mittag-Leffler function, for any ℓ > 0 . An IVP (1) is Mittag-Leffler-Hyers-Ulam-Rassias stable; if for all ε > 0 , then there exist H > 0 and ψ : ( 0 , ∞ ) → F , which is satisfies the nonlinear differential equation (1) such that ∣ φ ( ℓ ) − ψ ( ℓ ) ∣ ≤ H ϕ ( ℓ ) ε E ν ( ℓ ν ) , for any ℓ > 0 , where H is called Mittag-Leffler-Hyers-Ulam-Rassias stability constant of (1).

Definition 2.9

Let φ : ( 0 , ∞ ) → F be a twice continuously differentiable function that satisfies ∣ u φ ″ ( ℓ ) + v φ ′ ( ℓ ) + w φ ( ℓ ) − n ( ℓ , φ ( ℓ ) , φ ′ ( ℓ ) ) ∣ ≤ ε , for any ℓ > 0 . A nonlinear differential equation (3) with initial condition (4) is Hyers-Ulam stable, then for every ε > 0 , there exist H > 0 and ψ : ( 0 , ∞ ) → F , which satisfies the nonlinear differential equation (3) such that ∣ φ ( ℓ ) − ψ ( ℓ ) ∣ ≤ H ε , for any ℓ > 0 , where H is called Hyers-Ulam stability constant of (3).

Definition 2.10

Let φ : ( 0 , ∞ ) → F be a twice continuously differentiable function, if there exist ϕ : ( 0 , ∞ ) → ( 0 , ∞ ) , which satisfies

∣ u φ ″ ( ℓ ) + v φ ′ ( ℓ ) + w φ ( ℓ ) − n ( ℓ , φ ( ℓ ) , φ ′ ( ℓ ) ) ∣ ≤ ϕ ( ℓ ) ε ,

for any ℓ > 0 . An IVP (3) with initial condition (4) is Hyers-Ulam-Rassias stable, and then for all ε > 0 , there exist H > 0 and ψ : ( 0 , ∞ ) → F , which satisfies the nonlinear differential equation (3) such that ∣ φ ( ℓ ) − ψ ( ℓ ) ∣ ≤ H ϕ ( ℓ ) ε , for any ℓ > 0 , where H is called Hyers-Ulam-Rassias stability constant of (3).

Definition 2.11

Let φ : ( 0 , ∞ ) → F be a twice continuously differentiable function that satisfies

∣ u φ ″ ( ℓ ) + v φ ′ ( ℓ ) + w φ ( ℓ ) − n ( ℓ , φ ( ℓ ) , φ ′ ( ℓ ) ) ∣ ≤ ε E ν ( ℓ ν ) ,

where E ν ( ℓ ν ) is a Mittag-Leffler function, for any ℓ > 0 . An IVP (3) with initial condition (4) is Mittag-Leffler-Hyers-Ulam stable; if for all ε > 0 , then there exist H > 0 and ψ : ( 0 , ∞ ) → F , which satisfies the nonlinear differential equation (3) such that ∣ φ ( ℓ ) − ψ ( ℓ ) ∣ ≤ H ε E ν ( ℓ ν ) , for any ℓ > 0 , where H is called Mittag-Leffler-Hyers-Ulam stability constant of (3).

Definition 2.12

Let φ : ( 0 , ∞ ) → F be a twice continuously differentiable function, if there exist ϕ : ( 0 , ∞ ) → ( 0 , ∞ ) , which satisfies

∣ u φ ″ ( ℓ ) + v φ ′ ( ℓ ) + w φ ( ℓ ) − n ( ℓ , φ ( ℓ ) , φ ′ ( ℓ ) ) ∣ ≤ ϕ ( ℓ ) ε E ν ( ℓ ν ) ,

where E ν ( ℓ ν ) is a Mittag-Leffler function, for any ℓ > 0 . An IVP (3) with initial condition (4) has Mittag-Leffler-Hyers-Ulam-Rassias stability; if for each ε > 0 , then there exist H > 0 and ψ : ( 0 , ∞ ) → F , which satisfies the nonlinear differential equation (3) such that ∣ φ ( ℓ ) − ψ ( ℓ ) ∣ ≤ H ϕ ( ℓ ) ε E ν ( ℓ ν ) , for any ℓ > 0 , where H is called Mittag-Leffler-Hyers-Ulam-Rassias stability constant of (3).

3 Ulam stability of first-order nonlinear IVP

In the following theorems, we will prove various types of Ulam stability of the first-order nonlinear equation (1) with initial conditions (2).

Theorem 3.1

Let u and v be constants in F . For every ε > 0 , there is a constant H > 0 such that φ : ( 0 , ∞ ) → F , which is a continuously differentiable function, and it satisfies the inequality

(5) ∣ u φ ′ ( ℓ ) + v φ ( ℓ ) − n ( ℓ , φ ( ℓ ) ) ∣ ≤ ε ,

for any ℓ > 0 , there exists ψ : ( 0 , ∞ ) → F satisfying (1) such that ∣ φ ( ℓ ) − ψ ( ℓ ) ∣ ≤ H ε , for any ℓ > 0 .

Proof

Assume that φ : ( 0 , ∞ ) → F is a continuously differentiable function that satisfies inequality (5). Let us choose a function p : ( 0 , ∞ ) → F as follows:

(6) p ( ℓ ) = u φ ′ ( ℓ ) + v φ ( ℓ ) − n ( ℓ , φ ( ℓ ) ) ,

for any ℓ > 0 . In view of (5), we have ∣ p ( ℓ ) ∣ ≤ ε . Applying Fourier transform to (6), we obtain that

ℱ { p ( ℓ ) } = ℱ { u φ ′ ( ℓ ) + v φ ( ℓ ) − n ( ℓ , φ ( ℓ ) ) } = u ℱ { φ ′ ( ℓ ) } + v ℱ { φ ( ℓ ) } − ℱ { n ( ℓ , φ ( ℓ ) ) } P ( ξ ) = ( v − i ξ u ) Φ ( ξ ) − ℱ { n ( ℓ , φ ( ℓ ) ) }

(7) Φ ( ξ ) = P ( ξ ) + ℱ { n ( ℓ , φ ( ℓ ) ) } v − i ξ u .

Thus,

(8) ℱ { φ ( ℓ ) } = Φ ( ξ ) = P ( ξ ) + ℱ { n ( ℓ , φ ( ℓ ) ) } v − i ξ u .

Now, choosing Q ( ξ ) = ℱ { q ( ℓ ) } = 1 v − i ξ u , then we have

ℱ { q ( ℓ ) } = 1 v − i ξ u ⇒ q ( ℓ ) = ℱ − 1 1 v − i ξ u .

We will set the solution

ψ ( ℓ ) = e − v u ℓ + { n ( ℓ , ψ ( ℓ ) ) } ⁎ q ( ℓ ) .

Then, using Fourier transform on ψ ( ℓ ) , we will have

ℱ { ψ ( ℓ ) } = Ψ ( ξ ) = ∫ − ∞ ∞ e − v u ℓ e i s ℓ d ℓ + ℱ { n ( ℓ , ψ ( ℓ ) ) } v − i ξ u ,

(9) Ψ ( ξ ) = ℱ { n ( ℓ , ψ ( ℓ ) ) } v − i ξ u ,

where ∫ − ∞ ∞ e − v u ℓ e i s ℓ d ℓ = 0 . Now,

ℱ { u ψ ′ ( ℓ ) + v ψ ( ℓ ) } = − i ξ u Ψ ( ξ ) + v Ψ ( ξ ) = ℱ { n ( ℓ , ψ ( ℓ ) ) } ,

we have ℱ { u ψ ′ ( ℓ ) + v ψ ( ℓ ) } = ℱ { n ( ℓ , ψ ( ℓ ) ) } . Since ℱ is linear and one-to-one operator, u ψ ′ ( ℓ ) + v ψ ( ℓ ) = n ( ℓ , ψ ( ℓ ) ) . Hence, ψ ( ℓ ) is a solution of the differential equation (1). Now, using (8) and (9),

ℱ { φ ( ℓ ) } − ℱ { ψ ( ℓ ) } = P ( ξ ) + ℱ { n ( ℓ , φ ( ℓ ) ) } v − i ξ u − ℱ { n ( ℓ , ψ ( ℓ ) ) } v − ( i ξ u ) = P ( ξ ) v − i ξ u = P ( ξ ) × Q ( ξ ) ⇒ ℱ { φ ( ℓ ) − ψ ( ℓ ) } = ℱ { p ( ℓ ) × q ( ℓ ) } .

Since the operator ℱ is one-to-one linear operator, φ ( ℓ ) − ψ ( ℓ ) = p ( ℓ ) × q ( ℓ ) . Taking modulus on both sides ∣ p ( ℓ ) ∣ ≤ ε , we have

∣ φ ( ℓ ) − ψ ( ℓ ) ∣ = ∣ p ( ℓ ) × q ( ℓ ) ∣ = ∫ 0 ℓ p ( s ) q ( ℓ − s ) d s , ≤ ∣ p ( ℓ ) ∣ ∫ 0 ℓ q ( ℓ − s ) d s ≤ ε ∫ 0 ℓ q ( ℓ − s ) d s ≤ H ε ,

for any ℓ > 0 , where H = ∫ 0 ℓ q ( ℓ − s ) d s and the integral exists for each value of ℓ . Therefore, the nonlinear differential equation (1) is Hyers-Ulam stable.□

Corollary 3.2

Let u and v be constants in F . For every ε > 0 , there exists a real constant H > 0 such that φ : ( 0 , ∞ ) → F being a continuously differentiable function with ϕ : ( 0 , ∞ ) → ( 0 , ∞ ) being an integrable function that satisfies the following inequality:

(10) ∣ u φ ′ ( ℓ ) + v φ ( ℓ ) − n ( ℓ , φ ( ℓ ) ) ∣ ≤ ϕ ( ℓ ) ε ,

for any ℓ > 0 , there exists some ψ : ( 0 , ∞ ) → F of (1) such that ∣ φ ( ℓ ) − ψ ( ℓ ) ∣ ≤ H ϕ ( ℓ ) ε , for any ℓ > 0 .

Proof

Choosing a function p : ( 0 , ∞ ) → F as follows:

(11) p ( ℓ ) = u φ ′ ( ℓ ) + v φ ( ℓ ) − n ( ℓ , φ ( ℓ ) ) ,

for any ℓ > 0 . Assume that φ : ( 0 , ∞ ) → F is a continuously differentiable function that satisfies inequality (10). In view of (10), we have ∣ p ( ℓ ) ∣ ≤ ϕ ( ℓ ) ε . Using Fourier transform to equation (11), we have

(12) Φ ( ξ ) = P ( ξ ) + ℱ { n ( ℓ , φ ( ℓ ) ) } v − i ξ u .

Using the same technique in Theorem 3.1, we can easily reach the rest of the proof. Also, one can easily reach at

∣ φ ( ℓ ) − ψ ( ℓ ) ∣ = ∣ p ( ℓ ) × q ( ℓ ) ∣ = ∫ 0 ℓ p ( s ) q ( ℓ − s ) d s , ≤ ∣ p ( ℓ ) ∣ ∫ 0 ℓ q ( ℓ − s ) d s ≤ ϕ ( ℓ ) ε ∫ 0 ℓ q ( ℓ − s ) d s ≤ H ϕ ( ℓ ) ε ,

for any ℓ > 0 , where H = ∫ 0 ℓ q ( ℓ − s ) d s and the integral exists for each value of ℓ . Therefore, the nonlinear differential equation (1) is Hyers-Ulam-Rassias stable.□

Now, we will establish the Mittag-Leffler-Hyers-Ulam stability of the differential equation (1).

Theorem 3.3

Let u and v be the constants in F . For all ε > 0 and a real constant H > 0 such that φ : ( 0 , ∞ ) → F is a continuously differentiable function that satisfies

(13) ∣ u φ ′ ( ℓ ) + v φ ( ℓ ) − n ( ℓ , φ ( ℓ ) ) ∣ ≤ ε E ν ( ℓ ν ) ,

where E ν ( ℓ ν ) is a Mittag-Leffler function; for any ℓ > 0 , there exists some ψ : ( 0 , ∞ ) → F , which satisfies the differential equation (1) such that ∣ φ ( ℓ ) − ψ ( ℓ ) ∣ ≤ H ε E ν ( ℓ ν ) , for any ℓ > 0 .

Proof

Assume that φ : ( 0 , ∞ ) → F is a continuously differentiable function that satisfies inequality (13). Let us define a function p : ( 0 , ∞ ) → F as follows:

(14) p ( ℓ ) = u φ ′ ( ℓ ) + v φ ( ℓ ) − n ( ℓ , φ ( ℓ ) ) ,

for any ℓ > 0 . In view of (13), we have ∣ p ( ℓ ) ∣ ≤ ε E ν ( ℓ ν ) . Taking Fourier transform to equation (14), we have

(15) ℱ { φ ( ℓ ) } = Φ ( ξ ) = P ( ξ ) + ℱ { n ( ℓ , φ ( ℓ ) ) } v − i ξ u .

Now, taking Q ( ξ ) = ℱ { q ( ℓ ) } = 1 ( v − i ξ u ) , then we have

ℱ { q ( ℓ ) } = 1 v − i ξ u ⇒ q ( ℓ ) = ℱ − 1 1 v − i ξ u .

We define a solution

ψ ( ℓ ) = e − v u ℓ + { n ( ℓ , ψ ( ℓ ) ) } ⁎ q ( ℓ ) .

Then, by applying Fourier transform on both sides, we obtain

ℱ { ψ ( ℓ ) } = Ψ ( ξ ) = ∫ − ∞ ∞ e − v u ℓ e i s ℓ d ℓ + ℱ { n ( ℓ , ψ ( ℓ ) ) } v − i ξ u ,

(16) Ψ ( ξ ) = ℱ { n ( ℓ , ψ ( ℓ ) ) } v − i ξ u ,

where ∫ − ∞ ∞ e − v u ℓ e i s ℓ d ℓ = 0 . Now,

ℱ { u ψ ′ ( ℓ ) + v ψ ( ℓ ) } = − i ξ u Ψ ( ξ ) + v Ψ ( ξ ) = ( v − i ξ u ) Ψ ( ξ ) = ℱ { n ( ℓ , ψ ( ℓ ) ) } .

We have ℱ { u ψ ′ ( ℓ ) + v ψ ( ℓ ) } = ℱ { n ( ℓ , ψ ( ℓ ) ) } . Since ℱ is linear and one-to-one operator,

u ψ ′ ( ℓ ) + v ψ ( ℓ ) = n ( ℓ , ψ ( ℓ ) ) .

Hence, ψ ( ℓ ) is a solution to the differential equation (1). Now, using (15) and (16),

ℱ { φ ( ℓ ) } − ℱ { ψ ( ℓ ) } = P ( ξ ) + ℱ { n ( ℓ , φ ( ℓ ) ) } v − i ξ u − ℱ { n ( ℓ , ψ ( ℓ ) ) } v − ( i ξ u ) = P ( ξ ) 1 v − i ξ u = P ( ξ ) × Q ( ξ ) ⇒ ℱ { φ ( ℓ ) − ψ ( ℓ ) } = ℱ { p ( ℓ ) × q ( ℓ ) } .

Since the operator ℱ is one-to-one linear operator, φ ( ℓ ) − ψ ( ℓ ) = p ( ℓ ) × q ( ℓ ) . Taking modulus on both sides and using the inequality ∣ p ( ℓ ) ∣ ≤ ε E ν ( ℓ ν ) , we have

∣ φ ( ℓ ) − ψ ( ℓ ) ∣ = ∣ p ( ℓ ) × q ( ℓ ) ∣ = ∫ 0 ℓ p ( s ) q ( ℓ − s ) d s , ≤ ∣ p ( ℓ ) ∣ ∫ 0 ℓ q ( ℓ − s ) d s ≤ E ν ( ℓ ν ) ε ∫ 0 ℓ q ( ℓ − s ) d s ≤ H ε E ν ( ℓ ν ) ,

for any ℓ > 0 , where H = ∫ 0 ℓ q ( ℓ − s ) d s and the integral exists for each value of ℓ . Therefore, the nonlinear differential equation (1) has the Mittag-Leffler-Hyers-Ulam stability.□

Corollary 3.4

Let u and v be the constants in F . For every ε > 0 and H > 0 such that let φ : ( 0 , ∞ ) → F be a continuously differentiable function ϕ : ( 0 , ∞ ) → ( 0 , ∞ ) , which satisfies

(17) ∣ u φ ′ ( ℓ ) + v φ ( ℓ ) − n ( ℓ , φ ( ℓ ) ) ∣ ≤ ϕ ( ℓ ) ε E ν ( ℓ ν ) ,

where E ν ( ℓ ν ) is a Mittag-Leffler function, for any ℓ > 0 , there exists some ψ : ( 0 , ∞ ) → F of (1) such that ∣ φ ( ℓ ) − ψ ( ℓ ) ∣ ≤ H ϕ ( ℓ ) ε E ν ( ℓ ν ) , for any ℓ > 0 .

Proof

We define a function p : ( 0 , ∞ ) → F as follows:

(18) p ( ℓ ) = u φ ′ ( ℓ ) + v φ ( ℓ ) − n ( ℓ , φ ( ℓ ) ) ,

for any ℓ > 0 . We have ∣ p ( ℓ ) ∣ ≤ ϕ ( ℓ ) ε E ν ( ℓ ν ) . Using the same technique in Theorem 3.3, we will have

(19) Φ ( ξ ) = P ( ξ ) + ℱ { n ( ℓ , φ ( ℓ ) ) } v − i ξ u ,

and using the inequality ∣ p ( ℓ ) ∣ ≤ ϕ ( ℓ ) ε E ν ( ℓ ν ) , we have

∣ φ ( ℓ ) − ψ ( ℓ ) ∣ = ∣ p ( ℓ ) × q ( ℓ ) ∣ ≤ H ϕ ( ℓ ) ε E ν ( ℓ ν ) ,

for any ℓ > 0 , where H = ∫ 0 ℓ q ( ℓ − s ) d s and the integral exists for each value of ℓ . Therefore, the nonlinear IVP (1) has Mittag-Leffler-Hyers-Ulam-Rassias stability.□

4 Ulam stability of second-order nonlinear IVP

In the following theorems, we will investigate various types of Ulam stability of the second-order nonlinear equation (3) with initial conditions (4).

Theorem 4.1

Let u, v, and w are constants in F . For every ε > 0 and a real constant H > 0 such that φ : ( 0 , ∞ ) → F is a twice continuously differentiable function that satisfies the inequality

(20) ∣ u φ ″ ( ℓ ) + v φ ′ ( ℓ ) + w φ ( ℓ ) − n ( ℓ , φ ( ℓ ) , φ ′ ( ℓ ) ) ∣ ≤ ε ,

for any ℓ > 0 , there exists some ψ : ( 0 , ∞ ) → F , which satisfies the IVP (3) such that ∣ φ ( ℓ ) − ψ ( ℓ ) ∣ ≤ H ε , for any ℓ > 0 .

Proof

Let φ : ( 0 , ∞ ) → F be a twice continuously differentiable function that satisfies inequality (20). We define a function p : ( 0 , ∞ ) → F as follows:

(21) p ( ℓ ) = u φ ″ ( ℓ ) + v φ ′ ( ℓ ) + w φ ( ℓ ) − n ( ℓ , φ ( ℓ ) , φ ′ ( ℓ ) ) ,

for any ℓ > 0 . Thus, ∣ p ( ℓ ) ∣ ≤ ε . Using Fourier transform to (21),

ℱ { p ( ℓ ) } = u ℱ { φ ″ ( ℓ ) } + v ℱ { φ ′ ( ℓ ) } + w ℱ { φ ( ℓ ) } − ℱ { n ( ℓ , φ ( ℓ ) , φ ′ ( ℓ ) ) } P ( ξ ) = ( − u ξ 2 − i ξ v + w ) Φ ( ξ ) − ℱ { n ( ℓ , φ ( ℓ ) , φ ′ ( ℓ ) ) }

(22) Φ ( ξ ) = P ( ξ ) + ℱ { n ( ℓ , φ ( ℓ ) , φ ′ ( ℓ ) ) } − u ξ 2 − i ξ v + w .

Since for every u , v and w are the constants in F , there exists a constant λ , μ ∈ F such that λ + μ = v u and λ μ = w u where λ ≠ μ ,

− u ξ 2 − i ξ v + w = u ( i ξ − λ ) ( i ξ − μ ) .

Thus,

(23) ℱ { φ ( ℓ ) } = Φ ( ξ ) = P ( ξ ) + ℱ { n ( ℓ , φ ( ℓ ) , φ ′ ( ℓ ) ) } u ( i ξ − λ ) ( i ξ − μ ) .

Now, taking Q ( ξ ) = ℱ { q ( ℓ ) } = 1 u ( i ξ − λ ) ( i ξ − μ ) , then we have

q ( ℓ ) = ℱ − 1 1 u ( i ξ − λ ) ( i ξ − μ ) .

We set

ψ ( ℓ ) = λ e − λ ℓ − μ e − μ ℓ λ − μ + { n ( ℓ , ψ ( ℓ ) , ψ ′ ( ℓ ) ) } ⁎ q ( ℓ ) .

Then, Fourier transform to ψ ( ℓ ) , we obtain

ℱ { ψ ( ℓ ) } = Ψ ( ξ ) = ∫ − ∞ ∞ λ e − λ ℓ − μ e − μ ℓ λ − μ e i s ℓ d ℓ + ℱ { n ( ℓ , ψ ( ℓ ) , ψ ′ ( ℓ ) ) } u ( i ξ − λ ) ( i ξ − μ ) ,

(24) Ψ ( ξ ) = ℱ { n ( ℓ , ψ ( ℓ ) , ψ ′ ( ℓ ) ) } u ( i ξ − λ ) ( i ξ − μ ) ,

where ∫ − ∞ ∞ λ e − λ ℓ − μ e − μ ℓ λ − μ e i s ℓ d ℓ = 0 . Now,

ℱ { u ψ ″ ( ℓ ) + v ψ ′ ( ℓ ) + w ψ ( ℓ ) } = ( − i ξ ) 2 u Ψ ( ξ ) + ( − i ξ ) v Ψ ( ξ ) + w Ψ ( ξ ) = ( − ξ 2 u − i ξ v + w ) Ψ ( ξ ) = ℱ { n ( ℓ , ψ ( ℓ ) , ψ ′ ( ℓ ) ) } .

We have ℱ { u ψ ″ ( ℓ ) + v ψ ′ ( ℓ ) + w ψ ( ℓ ) } = ℱ { n ( ℓ , ψ ( ℓ ) , ψ ′ ( ℓ ) ) } because ℱ is one-to-one and linear operator; thus, u ψ ″ ( ℓ ) + v ψ ′ ( ℓ ) + w ψ ( ℓ ) = n ( ℓ , ψ ( ℓ ) , ψ ′ ( ℓ ) ) . Hence, ψ ( ℓ ) a solution to the differential equation (3). Now, using (23) and (24),

ℱ { φ ( ℓ ) } − ℱ { ψ ( ℓ ) } = P ( ξ ) + ℱ { n ( ℓ , φ ( ℓ ) , φ ′ ( ℓ ) ) } u ( i ξ − λ ) ( i ξ − μ ) − ℱ { n ( ℓ , ψ ( ℓ ) , ψ ′ ( ℓ ) ) } u ( i ξ − λ ) ( i ξ − μ ) = P ( ξ ) u ( i ξ − λ ) ( i ξ − μ ) = P ( ξ ) × Q ( ξ ) ⇒ ℱ { φ ( ℓ ) − ψ ( ℓ ) } = ℱ { p ( ℓ ) × q ( ℓ ) } .

Since the operator ℱ is one-to-one linear operator, φ ( ℓ ) − ψ ( ℓ ) = p ( ℓ ) × q ( ℓ ) . Taking modulus on both sides and using the inequality ∣ p ( ℓ ) ∣ ≤ ε , we have

for any ℓ > 0 , where H = ∫ 0 ℓ q ( ℓ − s ) d s and the integral exists for each value of ℓ . Therefore, the nonlinear differential equation (3) has Hyers-Ulam stability.□

Corollary 4.2

Let u, v, and w be the constants in F . For every ε > 0 , and a real constant H > 0 such that let φ : ( 0 , ∞ ) → F be a twice continuously differentiable function ϕ : ( 0 , ∞ ) → ( 0 , ∞ ) , which satisfies the inequality

(25) ∣ u φ ″ ( ℓ ) + v φ ′ ( ℓ ) + w φ ( ℓ ) − n ( ℓ , φ ( ℓ ) , φ ′ ( ℓ ) ) ∣ ≤ ϕ ( ℓ ) ε ,

for any ℓ > 0 , there exists some ψ : ( 0 , ∞ ) → F of the differential equation (3) such that ∣ φ ( ℓ ) − ψ ( ℓ ) ∣ ≤ H ϕ ( ℓ ) ε , for any ℓ > 0 .

Proof

Let us choose a function p : ( 0 , ∞ ) → F as follows:

(26) p ( ℓ ) = u φ ″ ( ℓ ) + v φ ′ ( ℓ ) + w φ ( ℓ ) − n ( ℓ , φ ( ℓ ) , φ ′ ( ℓ ) ) ,

for any ℓ > 0 . Let φ : ( 0 , ∞ ) → F be a twice continuously differentiable function that satisfies inequality (30). So, ∣ p ( ℓ ) ∣ ≤ ϕ ( ℓ ) ε . Taking Fourier transform to (25), we have

(27) Φ ( ξ ) = P ( ξ ) + ℱ { n ( ℓ , φ ( ℓ ) , φ ′ ( ℓ ) ) } − u ξ 2 − i ξ v + w .

Using the same technique in Theorem 4.1, we will have

∣ φ ( ℓ ) − ψ ( ℓ ) ∣ = ∫ 0 ℓ p ( s ) q ( ℓ − s ) d s ≤ ∣ p ( ℓ ) ∣ ∫ 0 ℓ q ( ℓ − s ) d s ≤ ϕ ( ℓ ) ε ∫ 0 ℓ q ( ℓ − s ) d s ≤ H ϕ ( ℓ ) ε .

Using the inequality ∣ p ( ℓ ) ∣ ≤ ϕ ( ℓ ) ε , for any ℓ > 0 , where H = ∫ 0 ℓ q ( ℓ − s ) d s and the integral exists for each value of ℓ . Therefore, the nonlinear differential equation (3) has the Hyers-Ulam-Rassias stability.□

Theorem 4.3

Let u, v, and w be constants in F . For every ε > 0 , there is a real constant H > 0 such that φ : ( 0 , ∞ ) → F is a twice continuously differentiable function that satisfies

(28) ∣ u φ ″ ( ℓ ) + v φ ′ ( ℓ ) + w φ ( ℓ ) − n ( ℓ , φ ( ℓ ) , φ ′ ( ℓ ) ) ∣ ≤ ε E ν ( ℓ ν ) ,

where E ν ( ℓ ν ) is a Mittag-Leffler function for all ℓ > 0 , then there exists some ψ : ( 0 , ∞ ) → F satisfying (3) such that ∣ φ ( ℓ ) − ψ ( ℓ ) ∣ ≤ H ε E ν ( ℓ ν ) .

Proof

Let us define a function p : ( 0 , ∞ ) → F as follows:

(29) p ( ℓ ) = u φ ″ ( ℓ ) + v φ ′ ( ℓ ) + w φ ( ℓ ) − n ( ℓ , φ ( ℓ ) , φ ′ ( ℓ ) ) ,

for any ℓ > 0 . Then, in view of (28), we obtain ∣ p ( ℓ ) ∣ ≤ ε E ν ( ℓ ν ) . Applying Fourier transform to equation (29), we obtain that

(30) Φ ( ξ ) = P ( ξ ) + ℱ { n ( ℓ , φ ( ℓ ) , φ ′ ( ℓ ) ) } − u ξ 2 − i ξ v + w .

Since for every u , v and w are the constants in F , there exists a constant λ , μ ∈ F such that λ + μ = v u and λ μ = w u where λ ≠ μ , then,

− u ξ 2 − i ξ v + w = u ( i ξ − λ ) ( i ξ − μ ) .

Thus,

(31) ℱ { φ ( ℓ ) } = Φ ( ξ ) = P ( ξ ) + ℱ { n ( ℓ , φ ( ℓ ) , φ ′ ( ℓ ) ) } u ( i ξ − λ ) ( i ξ − μ ) .

Now, taking Q ( ξ ) = ℱ { q ( ℓ ) } = 1 u ( i ξ − λ ) ( i ξ − μ ) , then we have

q ( ℓ ) = ℱ − 1 1 u ( i ξ − λ ) ( i ξ − μ ) .

Setting

ψ ( ℓ ) = λ e − λ ℓ − μ e − μ ℓ λ − μ + { n ( ℓ , ψ ( ℓ ) , ψ ′ ( ℓ ) ) } ⁎ q ( ℓ ) ,

then Fourier transform on both sides, we obtain

ℱ { ψ ( ℓ ) } = Ψ ( ξ ) = ∫ − ∞ ∞ λ e − λ ℓ − μ e − μ ℓ λ − μ e i s ℓ d ℓ + ℱ { n ( ℓ , ψ ( ℓ ) , ψ ′ ( ℓ ) ) } u ( i ξ − λ ) ( i ξ − μ ) ,

(32) Ψ ( ξ ) = ℱ { n ( ℓ , ψ ( ℓ ) , ψ ′ ( ℓ ) ) } u ( i ξ − λ ) ( i ξ − μ ) ,

where ∫ − ∞ ∞ λ e − λ ℓ − μ e − μ ℓ λ − μ e i s ℓ d ℓ = 0 . Now, we can easily see that

ℱ { u ψ ″ ( ℓ ) + v ψ ′ ( ℓ ) + w ψ ( ℓ ) } = ℱ { n ( ℓ , ψ ( ℓ ) , ψ ′ ( ℓ ) ) } .

Since ℱ is linear and one-to-one operator, u ψ ″ ( ℓ ) + v ψ ′ ( ℓ ) + w ψ ( ℓ ) = n ( ℓ , ψ ( ℓ ) , ψ ′ ( ℓ ) ) . Hence, ψ ( ℓ ) a solution of the IVP (3). Now, using (31) and (32)

ℱ { φ ( ℓ ) − ψ ( ℓ ) } = ℱ { p ( ℓ ) × q ( ℓ ) } .

Since the operator ℱ is one-to-one linear operator, so φ ( ℓ ) − ψ ( ℓ ) = p ( ℓ ) × q ( ℓ ) . Using the inequality ∣ p ( ℓ ) ∣ ≤ ε E ν ( ℓ ν ) and simplifying, we obtain

∣ φ ( ℓ ) − ψ ( ℓ ) ∣ ≤ ε E ν ( ℓ ν ) ∫ 0 ℓ q ( ℓ − s ) d s ≤ H ε E ν ( ℓ ν ) ,

for any ℓ > 0 , where H = ∫ 0 ℓ q ( ℓ − s ) d s and the integral exists for each value of ℓ . Therefore, the second-order nonlinear IVP (3) has Mittag-Leffler-Hyers-Ulam stability.□

Corollary 4.4

Let u, v, and w be the constants in F . For every ε > 0 and H > 0 such that let φ : ( 0 , ∞ ) → F be a twice continuously differentiable function ϕ : ( 0 , ∞ ) → ( 0 , ∞ ) , which satisfies

(33) ∣ u φ ″ ( ℓ ) + v φ ′ ( ℓ ) + w φ ( ℓ ) − n ( ℓ , φ ( ℓ ) , φ ′ ( ℓ ) ) ∣ ≤ ϕ ( ℓ ) ε E ν ( ℓ ν ) ,

where E ν ( ℓ ν ) is a Mittag-Leffler function, for any ℓ > 0 , there exists some ψ : ( 0 , ∞ ) → F of (3) such that

∣ φ ( ℓ ) − ψ ( ℓ ) ∣ ≤ H ϕ ( t ) ε E ν ( ℓ ν ) ,

for any ℓ > 0 .

Proof

Choosing a function p : ( 0 , ∞ ) → F as follows:

(34) p ( ℓ ) = u φ ″ ( ℓ ) + v φ ′ ( ℓ ) + w φ ( ℓ ) − n ( ℓ , φ ( ℓ ) , φ ′ ( ℓ ) ) ,

for any ℓ > 0 . So, ∣ p ( ℓ ) ∣ ≤ ϕ ( ℓ ) ε E ν ( ℓ ν ) . Using the same technique in Theorem 4.3, we have

∣ φ ( ℓ ) − ψ ( ℓ ) ∣ ≤ H ϕ ( t ) ε E ν ( ℓ ν ) ,

for each ℓ > 0 . Therefore, the nonlinear differential equation (3) has Mittag-Leffler-Hyers-Ulam-Rassias stability.□

5 Conclusion

In this article, we have established the various types of Hyers-Ulam stabilities and Mittag-Leffler-Hyers-Ulam stability of first- and second-order nonlinear differential equations with initial conditions using the Fourier transform method.

Acknowledgements

Not applicable.

Funding information: Not applicable.
Author contributions: The authors equally conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.
Conflict of interest: The authors declare that they have no competing interests.
Data availability statement: Not applicable.

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Received: 2021-11-07

Revised: 2024-03-22

Accepted: 2024-05-24

Published Online: 2024-08-13

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/dema-2024-0033

Keywords for this article

Fourier transform; Hyers-Ulam and Hyers-Ulam-Rassias stability; Mittag-Leffler-Hyers-Ulam and Mittag-Leffler-Hyers-Ulam-Rassias stability; nonlinear differential equations

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