# Hyers-Ulam-Rassias-Wright Stability for Fractional Oscillation Equation

Authors: Zahra Eidinejad; Reza Saadati
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

In this paper, we consider the nonhomogeneous fractional delay oscillation equation with orderσ and introduce a class of control functions, i.e., Wright functions. Next, we apply the Cădariu-Radu method to prove the existence of a unique solution and Hyers-Ulam-Rassias-Wright stability of the fractional delay oscillation equation. At the end of the article, by an example, we show the application of the obtained results.


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## Body

## 1. Introduction

From the past until now, fractional calculus has gained considerable popularity and importance due to its many applications in various and wide scientific and engineering fields (see [1, 2]). We consider the following nonhomogeneous fractional delay oscillation equation:(1)D0σmϱ=−ϑmϱ−μ+gϱ+£−1msϱ,ϱ∈B≔0,B,(2)mϱ=ψϱ,m′ϱ=ψ′ϱ,−μ≤ϱ≤0,where(1)
g:B×ℝn⟶ℝn is an integrable function, mϱ∈ℝn,ϑ∈ℝn×n, and ψ∈C2−μ,0,ℝn denotes constant matrix; also, μ>0 is a fixed time and B=Nμ for a fixed N∈1,2,….(2)
D0σ is the Caputo fractional derivative with ƒ∈1,2 defined by(3)D0σmϱ=∫0ϱϱ−ℷ1−σΓ2−σm″ℷdℷ,if the integral exists.(3)
The classical Laplace transform is defined by the integral formula(4)£gs=∫0∞gϱe−sϱdϱ,where gϱ is absolutely integrable on 0,∞. Let J0σ denote the Riemann–Liouville fractional integral operator of order σ∈1,2. Assume the Laplace transforms of J0σgϱ,D0σgϱ exist for ϱ≤0 and the Laplace transform of gϱ−b exists for ϱ≤b. Then, we have(i)
–£J0σgϱs=s−σ£gϱs,(ii)
£g1ϱs=e−bs∫−b0e−ϱsϱdϱ+e−bs£gϱs,(iii)
–£g1ϱs=e−bs∫−b0e−ϱsgϱdϱ+e−bs£gϱs,where £−1 is the Laplace inverse transform.Exact solutions for the differential equation using the exponential function forσ=1 and the matrix sine and matrix cosine for σ=2 are obtained in [3, 4]. In 2018, Li and Wang obtained the exact solution of equation (1) for 0<σ<1 using the Mittag–Leffler type matrix function [5–7]. In the second section of the paper, we present the required definitions and concepts. In the third section, we use the alternative fixed point method and apply the Wright function as a control function to investigate the Hyers-Ulam-Rassias-Wright stability for equation (1). Finally, by an example, we show the application of the obtained results.


## 2. Preliminaries

In this section, we present several facts that we need to get the desired results. Since the control function used in this article is a Wright function, we start this section with a definition of the Wright function.Definition 1.
The Wright function is defined by the following series representation:(5)ZWσ,κq=∑ℓ=0∞qℓℓ!Γσℓ+κ,for σ>−1,κ>0,q∈ℝ. It is an entire function of order 1/1+σ, which has been known also as generalized Bessel (or Bessel Maitland) function [8, 9].Since we use two functions called the sine-type Mittag–Leffler and the cosine-type Mittag–Leffler and these two functions are derived from the extension of Mittag–Leffler function, we first provide a definition of the Mittag–Leffler function, then we state the main definitions.Definition 2.
The special function(6)Eσk=∑ℓ=0∞kℓΓσℓ+1,σ>0,k∈ℝn×n,and its general form(7)Eσ,κk=∑ℓ=0∞kℓΓσℓ+κ,σ,κ>0,k∈ℝn×n.Definition 3.
The cosine-type Mittag–Leffler function of a polynomial of degreepσ, Cσμϱ:ℝ⟶ℝn, for n∈ℕ and σ∈ℝ+ on the interval [(p− 1) μ,pμ) and for ϱ=pμ,p=0,1,2,…,N defined by(8)Cσμϱ≔Θ,−∞<ϱ<μ,I,−μ<ϱ<0,⋯I−ϑϱσΓσ+1+ϑ2ϱ−μ2σΓ2σ+1+⋯+−1pϑpϱ−p−1ϱpσΓpσ+1,p−1μ≤ϱ<kμ,.that I denotes a unit matrix and Θ denotes a zero matrix.Definition 4.
The sine-type Mittag–Leffler function of a polynomial of degreepσ+1, Sσμϱ:ℝ⟶ℝn, for n∈ℕ and σ∈ℝ+ on the interval [(p− 1) μ,pμ) and for ϱ=pμ,p=0,1,2,…,N defined by(9)Sσμϱ≔Θ,−∞<ϱ<μ,ϱ−μI,−μ<ϱ<0,⋯ϱ+μI−ϑϱσ+1Γσ+2+ϑ2ϱ−μ2σ+1Γ2σ+2+⋯+−1pϑpϱ−p−1μpσ+1Γpσ+2,p−1μ≤ϱ<pμ,that I denotes a unit matrix and Θ denotes a zero matrix.For the homogeneous fractional oscillation system,(10)D0σmϱ=−ϑmϱ−μ,ϱ∈B.We present the following Lemma from [10].Lemma 1.
ForCσμϱ and Sσμϱ, we find(i)
d/dϱSσμϱ=Cσμϱ for all ϱ∈ℝ.(ii)
Cσμϱ is a solution of (1), ϱ∈B.(iii)
Sσμϱ is a solution of (1), ϱ∈B.Theorem 5.
If the functionm is a solution of equation (1) with initial conditions (2), then the function m is defined as follows:(11)mϱ=sσI+ϑe−sμ£mϱs−sσ−1ψ0,(12)−sσ−2ψ′0+ϑe−sμ∫−μ0e−sμψϱdϱ−£gϱs.Proof.
Take the Laplace transform to both sides of (1) and according to Laplace’s definitions, we get(13)sσℒmϱs−sσ−1m0−sσ−2m′0=−ϑe−sμ∫−μ0e−szyzdz+ℒmϱs+ℒgϱs+mϱ.
We solve the above equation forℒmϱs; we get(14)sσI+ϑe−sμℒmϱs=sσ−1ψ0+sσ−2ψ′0−ϑe−sμ∫−μ0e−szψzdz+ℒgϱs+mϱ,then(15)mϱ=sσI+ϑe−sμℒmϱs−sσ−1ψ0−sσ−2ψ′0+ϑe−sμ∫−μ0e−szψzdz−ℒgϱs.The following alternative fixed point theorem was proved by Diaz-Margolis [11]; for more results on fixed point theory and application, we refer to [12–14].Theorem 6.
LetG be a set with a complete 0,∞ valued metric d. For m,m¯∈GZm,m¯∈G, consider the self-map ℧ on G such that the following relation holds:(16)d℧m,℧m¯≤γdm¯,m,where ϱ<1 is a Lipschitz constant. Let m∈G. Therefore, one of the following two conditions is established:(i)
d℧em,℧e+1m=∞,∀e∈ℕ,or(ii)
We can finde0∈ℕ such that d℧em,℧e+1m<∞,∀e≥e0,which implies that(1)
The fixed pointm¯∗ of ℧ is the convergence point of the sequence ℧em;(2)
In the set,ℭ∗=m¯∈G∣d℧e0m,m¯<∞,m¯∗ is the unique fixed point of ℧;(3)
1−γdm¯,m¯∗≤dm¯,℧m¯ for every m¯∈G.In this article, we use the norms=∑i=1ℓsi,s∈ℝn×1 and the matrix norm.In the sequel, we provide a definition of Hyers-Ulam-Rassais-Wright stability; for more details and result, we refer to [15–18].Definition 7.
We say that equation (1) is Hyers-Ulam-Rassais-Wright stable on B with respect to Wσ,κϱ, if there exists E>0 such that for each ℶ>0 and for each solution m¯ϱ satisfying inequality(17)D0σm¯ϱ+ϑm¯ϱ−μ−gϱ−£−1m¯sϱ≤ℶWσ,κϱ,ϱ∈B,there exists a solution mϱ of the problem (1) such that(18)m¯ϱ−mϱ≤GℶWσ,κϱ,for every ϱ∈B.Next, using the Diaz-Margolis theorem we prove the Hyers-Ulam-Rassais-Wright stability for equation (1).


## 3. Hyers-Ulam-Rassias-Wright Stability of the Fractional Oscillation Equation

In this section, we investigate the existence of a unique solution and Hyers-Ulam-Rassais-Wright stability for equation (1).Theorem 8.
Consider the positive constantϰ, such that 0<ϰ<1. Consider the control function Wσ,κ:B⟶ℝ, such that(19)£Wσ,κsϱ≤ϰWσ,κϱ,and(20)sσI+ϑe−sμ≤1.If for each ϱ∈B, m¯∈CB,ℝn is a differentiable function satisfying(21)D0σm¯ϱ+ϑm¯ϱ−μ−gϱ−£−1m¯sϱ≤ℶWσ,κϱ,ϱ∈B,then there is a unique solution m∈CB,ℝnZm∈CB,ℝn to equation (1), such that(22)m¯ϱ−mϱ≤ℸℶWσ,κϱ,where ℸ=ϰ/1−ϰ. Then, equation (1) has Hyers-Ulam-Rassais-Wright stability property.Proof.
Define the setB as follows:(23)ℭ=m:B⟶ℝn,mis continuous,and consider the mapping d:ℭ×ℭ⟶0,∞, which is defined as follows:(24)dm,m¯=inA∈0,∞:mϱ−m¯ϱ≤AWσ,κϱ,ϱ∈B.Step 1. We show thatℭ,d is 0,∞-valued complete metric space. If for functions m,m¯∈ℭ, we have dm,m¯>dm,v+dv,m¯, then there exists ϱ0∈B such that mϱ0−m¯ϱ0>dm,v+dv,m¯Wσ,κϱ0. Then, according to the definition d, we have(25)mϱ0−m¯ϱ0>mϱ0−vϱ0+vϱ0−m¯ϱ0,which is a contradiction.Now, we prove thatℭ,d is a 0,∞-valued complete. Let mn be a Cauchy sequence in ℭ,d. Using d, ∀ℶ>0, ∃Nℶ∈ℕ, ∀m,n≥Nε, and∀ϱ∈B, we have(26)mnϱ−mmϱ<ℶWσ,κϱ.Since the functionWσ,κ is continuous to the compact set B, due to the inequality equation (26), the sequence mn uniformly converges to a continuous function of ℭ such as m. Then, again, using the definition of d, the following relation is established for us, wherein for ℶ>0, ∃Nℶ∈ℕ, n≥Nℶ, ϱ∈ℭ,(27)mnϱ−mϱ<ℶWσ,κϱ.It should be noted that we have assumed here thatm tends to infinity. As a result, dmn,m≤ℶ, and that means ℭ,d is 0,∞-valued complete and the first step is over.Now, according to the main equation, we define the mapping℧:ℭ⟶ℭ, as follows:(28)℧mϱ=£D0σmϱ+ϑmϱ−μ−gϱs=sσI+ϑe−sμ£mϱs−sσ−1ψ0−sσ−2ψ′0+ϑe−sμ∫−μ0e−sμψϱdϱ−£gϱs.Step 2. We show that℧ is a contractional mapping. If for the functions m,m¯∈ℭ and the coefficient Amm¯∈0,∞, we assume that dm,m¯≤Amm¯, the following inequality for ϱ∈B is established for us:(29)mϱ−m¯ϱ<Amm¯Wσ,κϱ,and using (19) and (20), we have(30)℧mϱ−℧m¯ϱ=‖sσI+ϑe−sμ£mϱs−sσ−1ψ0−sσ−2ψ′0+ϑe−sμ∫−μ0e−sμψϱdϱ−£gϱs−sσI+ϑe−sμ£m¯ϱs+sσ−1ψ0+sσ−2ψ′0ϑe−sμ∫−μ0e−sμψϱdϱ+£gϱs=sσI+ϑe−sμ£mϱs−sσI+ϑe−sμ£m¯ϱs=sσI+ϑe−sμ£mϱ−m¯ϱs≤sσI+ϑe−sμ£mϱ−m¯ϱs≤sσI+ϑe−sμ£Amm¯Wσ,κsϱ≤Amm¯sσI+ϑe−sμ£Wσ,κsϱ≤Amm¯sσI+ϑe−sμϰWσ,κϱ=Amm¯ϰWσ,κϱ.Thus, we have(31)d℧m,℧m¯≤ϰdm,m¯.Since,0<ϰ<1, therefore ℧ is a contractional mapping.Step 3. We will show thatd℧m¯,m¯<∞. From inequality (21), we have(32)℧m¯ϱ−m¯ϱ=£D0σm¯ϱ+ϑm¯ϱ−μ−gϱ−£−1m¯sϱ=sσI+ϑe−sμ£m¯ϱs−sσ−1ψ0−sσ−2ψ′0+ϑe−sμ∫−μ0e−sμψϱdϱ−£gϱs−m¯ϱ≤ℶ£Wσ,κsϱ≤ℶϰWσ,κϱ.Consequently,(33)d℧m¯,m¯≤ℶϰ<∞,ϰ<1.Therefore, all the conditions of Theorem6 are true. Then, we have(1)
The sequence℧em converges to a fixed point such as m.(2)
The unique elementm is in the set ℭ∗=m¯∈ℭ:d℧m¯,m¯<∞ and is the unique fixed point; it means ℧m=m or equivalently(34)mϱ=sσI+ϑe−sμ£mϱs−sσ−1ψ0,(35)−sσ−2ψ′0+ϑe−sμ∫−μ0e−sμψϱdϱ−£gϱs.Sincem is a continuous function and according to Theorem 5, we have(36)D0σmϱ=−ϑmϱ−μ+gϱ+£−1mϱs,ϱ∈B≔0,B.(3)
According to the inequality (33), we get(37)dm¯,m≤11−ϰd℧m¯,m¯≤ℶϰ1−ϰWσ,κϱ.
Thus, equation (1) has Hyers-Ulam-Rassais-Wright stability property.
Now, we show the uniqueness of the obtained point. For convenience, we consider(38)d=ℶϰ1−ϰ,and let t be another differentiable function satisfying equation (26) and this means that the following equation holds:(39)D0σtϱ=−ϑtϱ−μ+gϱ+£−1tsϱ,ϱ∈B≔0,B.
Now, we show thatt is a fixed point of ℧ and t∈ℭ∗. Using equation (39), we have ℧t=t. In the sequel, we show that d℧m¯,t<∞. For m¯∈ℭ, dm¯,t<d. Using equation (39), we have(40)‖℧m¯ϱ−tϱ‖=£D0σm¯ϱ+ϑm¯ϱ−μ−gϱs−tϱ=‖sσI+ϑe−sμ£m¯ϱs−sσ−1ψ0−sσ−2ψ′0+ϑe−sμ∫−μ0e−sμψϱdϱ−£gϱs−sσI+ϑe−sμ£tϱs+sσ−1ψ0+sσ−2ψ′0ϑe−sμ∫−μ0e−sμψϱdϱ+£gϱs‖=sσI+ϑe−sμ£m¯ϱs−sσI+ϑe−sμ£tϱs=sσI+ϑe−sμ£m¯ϱ−tϱs≤sσI+ϑe−sμ£‖m¯ϱ−tϱ‖s≤‖sσI+ϑe−sμ£dWσ,κsϱ‖≤dsσI+ϑe−sμϰWσ,κϱ≤dϰWσ,κϱ,which means(41)d℧m¯,t≤ϰd<∞.Example 1.
Consider the following nonhomogeneous fractional delay oscillation equation:(42)D01/2mϱ=−ϑmϱ−17+ϱ−1−1/4sinϱ−1641+ϱ−11+ϱ+£−1msϱ,(43)mϱ=m1ϱ,m2ϱTψϱ=3ϱ,4ϱ2T,−μ≤ϱ≤0,where ϰ=1/4,σ=1/2,μ=1/7,B=3/5 and ϑ=3/5002/5. Assume that the following condition is true for the given continuous function:(44)£W1/2,κsϱ≤14W1/2,κϱ.
Forϱ∈B, let m¯∈CB,ℝn be a continuous function satisfying(45)D01/2m¯ϱ+ϑm¯ϱ−17−ϱ−1−1/4sinϱ−1641+ϱ−11+ϱ−£−1msϱ≤ℶW1/2,κϱ,ϱ∈B,therefore, according to the definitions given, m¯ holds in the following inequality:(46)‖m¯ϱ−sσI+ϑe−sμ£m¯ϱs−sσ−1ψ0−sσ−2ψ′0+ϑe−sμ∫−μ0e−sμψϱdϱ−£ϱ−1−1/4sinϱ−1641+ϱ−11+ϱs‖≤ℶ£W1/2,κsϱ≤14ℶW1/2,κϱ,where ψϱ=3ϱ,4ϱ2T. Then, there is a unique solution m∈CB,ℝn for equation (1), such that(1)
m is the fixed point of the operator ℧, which is defined as follows:(47)℧mϱ=sσI+ϑe−sμ£mϱs−sσ−1ψ0−sσ−2ψ′0+ϑe−sμ∫−μ0e−sμψϱdϱ−£ϱ−1−1/4sinϱ−1641+ϱ−11+ϱs.(2)
The fixed pointm is unique.(3)
m satisfies in equation (42).
Then,(48)dm,m¯≤ℸℶ,or(49)m¯ϱ−mϱ≤ℸℶW1/2,κϱ,where ℸ=1/3 [19–23].



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