# The Unique Periodic Solution of Abel’s Differential Equation

Authors: Ni Hua
Journal: Journal of Mathematics
 (2020)

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## Abstract

In this paper, the existence of a periodic solution for Abel’s differential equation is obtained first by using the fixed-point theorem. Then, by constructing the Lyapunov function, the uniqueness and stability of the periodic solution of the equation are obtained.


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## Body

## 1. Introduction

The nonlinear Abel-type first-order differential equation(1)dxdt=atx3+btx2+ctx+dt,plays an important role in many physical and technical applications [1, 2]. The mathematical properties of equation (1) have been intensively investigated in the mathematical and physical literature [3–15].Recently, Ni [16] supposed that γ=γt is a periodic particular solution of Abel’s differential equation under some conditions, then by means of the transformation method and the fixed-point theory, the author presented an alternative method of generating the other periodic solutions of Abel’s differential equation. Ni and Tian [17] got the sufficient conditions which guarantee the existence of three periodic solutions for Abel’s differential equation.All the time, the existence, uniqueness, and stability of the periodic solution of differential equations have always been an important research hotspot in the field of differential equations. So, in this paper, equation (1) is considered and a new criterion is given to judge the existence, uniqueness, and stability of the periodic solution of equation (1) by using the fixed-point theorem and constructing the Lyapunov function, and some new results are obtained.The rest of the paper is arranged as follows: in Section2, some lemmas and abbreviations are introduced to be used later; in Section 3, the existence and stability of the unique periodic solution of equation (1) are obtained; in Section 4, two examples are given to verify the main results of the paper; the paper ends with a short conclusion.


## 2. Preliminaries

In this section, some definitions, lemmas, and abbreviations are given, which will be used to prove the main results.Definition 1.
(see [18]). Suppose ft is an ω-periodic continuous function on R, then(2)af,λ=∫0ωfte−iλtdt,must exist, where af,λ is called the Fourier coefficient of ft and λ, such that af,λ≠0, is called the Fourier index of ft; there is a countable set Λf, when λ∈Λf and af,λ≠0, and as long as λ∉Λf, there must be af,λ=0, where Λf is called the exponential set of ft.Definition 2.
(see [18]). A set of real numbers composed of linear combinations of integer coefficients of elements in Λf is called a module or a frequency module of ft, which is denoted as modf, that is,(3)modf=μμ=∑j=1Nnjλj,nj,N∈Z+,N≥1,λj∈Λf.Lemma 1.
(see [19]). Consider the following equation:(4)dxdt=atx+bt,where atandbt are the ω-periodic continuous functions, and if ∫0ωatdt≠0, then equation (4) has a unique ω-periodic continuous solution ηt,modη⊆modat,bt and ηt can be written as follows:(5)ηt=∫−∞te∫statdτbsds,∫0ωatdt<0,−∫t+∞e∫statdτbsds,∫0ωatdt>0.Lemma 2.
(see [20]). Suppose that an ω-periodic function sequence fnt is convergent uniformly on any compact set of R, ft is an ω-periodic function, and modfn⊆modfn=1,2,⋯,n, then fnt is convergent uniformly on R.Lemma 3.
(see [21]). Suppose V is a metric space, C is a convex closed set of V, and its boundary is ∂C, if T:V⟶V is a continuous compact mapping, such that T∂C⊆C, then T has at least a fixed point on C.For the sake of convenience, suppose thatft is an ω-periodic continuous function on R, then it is denoted that(6)fM=supt∈0,ωft,fL=inft∈0,ωft.


## 3. Main Results

In this section, the existence, uniqueness, and stability of the periodic solution of equation (1) are studied.Theorem 1.
Consider equation (1), and at,bt,ct, and dt are all ω-periodic continuous functions on R, suppose that the following conditions hold:(7)H1at<0,H2b2t−3atct<0,then, equation (1) has a unique ω-periodic continuous solution which is uniformly asymptotic stable.Proof.
Let(8)ft,x=atx3+btx2+ctx+dt.
ByH1 and (8), it follows that(9)ft,x⟶+∞x⟶−∞,ft,x⟶−∞x⟶+∞.
By zero point theorem, there is at least a zero pointγt∈−∞,+∞ such that(10)ft,γt=atγ3t+btγ2t+ctγt+dt=0.
ByH1 and H2, it follows that(11)fx′t,x=3atx2+2btx+ct<0,thus, ft,x has a unique zero point γtγt∈−∞,+∞ such that (10) holds. Since at,bt,ct, and dt are all continuous functions on R, and from (10) and (11), it follows that γt is a continuous function on R.
By (10), it follows that(12)ft+ω,γt+ω=at+ωγ3t+ω+bt+ωγ2t+ω+ct+ωγt+ω+dt+ω=0.
ft,x is ω-periodic on t, that is,(13)ft,γt+ω=atγ3t+ω+btγ2t+ω+ctγt+ω+dt=0.
Ifγt is not an ω-periodic function, then(14)γt+ω≡γtt∈R,thus, ft,x has two zero points, γt and γt+ω, and it is a contradiction to the fact that ft,x has a unique zero point γt on −∞,+∞. Therefore,(15)γt+ω≡γtt∈R,that is to say, γt is an ω-periodic continuous function on R.
In addition,(16)ft,x=ft,x−ft,γt=fx′t,γt+θx−γtx−γt0<θ=θx<1=x−γt3atγt+θx−γt2+2btγt+θx−γt+ct.
Comparing the coefficients of the third power ofx of the left and right hands of (16), the following is obtained:(17)θ=33.
By (16), let x=0, and then γt satisfies(18)23−4atγ3t+233−1btγ2t−ctγt=dt,thus, equation (1) becomes(19)dxdt=x−γt3atγt+33x−γt2+2btγt+33x−γt+ct=x−γtatx2+23at1−33γt+23bt3x+3at1−332γ2t+2bt1−33γt+ct.
By simple computation andH2, it is obtained that the discriminant of the quadratic polynomial of one variable satisfies the following:(20)Δ=23at1−33γt+23bt32−4at3at1−332γ2t+2bt1−33γt+ct=4b2t−3atct3a2t<0.
Next, the existence of the unique periodic solutionϕt of equation (1) is proven by using the fixed-point theorem.
Suppose(21)S=φt∈CR,Rφt+ω=φt.
Given anyφt,ψt∈S, the distance is defined as follows:(22)ρφ,ψ=supt∈0,ωφt−ψt,thus, S,ρ is a complete metric space. A convex closed set B of S is taken as follows:(23)B=φt∈Sφt+ω=φt,γL≤φt≤γM,modφ⊆moda,b,c,d.
Given anyφt∈B, consider the following equation:(24)dxdt=x−γtatφ2t+23at1−33γt+23bt3φt+3at1−332γ2t+2bt1−33γt+ct=ftx−γt=ftx−ftγ,where(25)atφ2t+23at1−33γt+23bt3φt+3at1−332γ2t+2bt1−33γt+ct≜ft.
It is easy to see thatft is an ω-periodic continuous function, and by H1 and (20),(26)ft<0,and ft is bounded above and below by some negative constants −μand−λ for all t∈0,ω and any φt∈B, that is,(27)−μ≤ft≤−λ.
Thus,(28)∫0ωftdt<0.
By Lemma1, equation (24) has a unique ω-periodic continuous solution as follows:(29)ηt=−∫−∞te∫stfτdτfsγsds,modη⊆modft,ftγt.
By (18), it follows that(30)modγt⊆moda,b,c,d.
By (25), it follows that(31)modft⊆moda,b,c,γ,modftγt⊆moda,b,c,γ,thus, it follows that(32)modft⊆moda,b,c,d,modftγt⊆moda,b,c,d.
Hence,(33)modη⊆moda,b,c,d.
By (23), (28), and (29), the following is obtained:(34)ηt=−∫−∞te∫stfτdτfsγsds≥−γL∫−∞te∫stfτdτfsds=γL∫−∞te∫stfτdτd∫stfτdτ=γLe∫stfτdτ−∞t=γL1−e∫−∞tfτdτ=γL,ηt=−∫−∞te∫stfτdτfsγsds≤−γM∫−∞te∫stfτdτfsds=γM∫−∞te∫stfτdτd∫stfτdτ=γMe∫stfτdτ−∞t=γM1−e∫−∞tfτdτ=γM,thus ηt∈B.
Define a mapping as follows:(35)Tφt=ηt=−∫−∞te∫stfτdτfsγsds,thus, if given any φt∈B, then Tφt∈B; hence T:B⟶B.
Now, it is proven that the mappingT is a compact mapping.
Consider any sequenceφnt⊆Bn=1,2,⋯, then it follows that(36)γL≤φnt≤γM,modφn⊆moda,b,c,dn=1,2,⋯.
On the other hand,Tφnt=xφnt satisfies(37)dxφntdt=xφnt−γtatφn2t+23at1−33γt+23bt3φnt+3at1−332γ2t+2bt1−33γt+ct=fntxφnt−γt=fntxφnt−fntγt,where(38)atφn2t+23at1−33γt+23bt3φnt+3at1−332γ2t+2bt1−33γt+ct≜fnt.
It is easy to see thatfnt is an ω-periodic continuous function, and by (23), (25), (27), (36), and (38), the following is obtained:(39)−μ≤fnt≤−λ,thus,(40)dxφntdt≤2μγM,modxφnt⊆moda,b,c,d,hence, dxφnt/dt is uniformly bounded; therefore, xφnt is uniformly bounded and equicontinuous on R. By the theorem of Ascoli–Arzela, for any sequence xφnt⊆B, there exists a subsequence (also denoted by xφnt) such that xφnt is convergent uniformly on any compact set of R. By (40), combined with Lemma 2, xφnt is convergent uniformly on R, that is to say, T is relatively compact on B.
Next, it is proven thatT is a continuous mapping.
Supposeφnt⊆B,φt∈B and(41)φnt⟶φtn⟶∞.
By (35),(42)Tφnt−Tφt=∫−∞te∫stfnτdτfnsγsds−∫−∞te∫stfτdτfsγsds=∫−∞te∫stfnτdτfns−fsγsds+∫−∞te∫stfnτdτ−e∫stfτdτfsγsds=∫−∞te∫stfnτdτfns−fsγsds+∫−∞teξ∫stfnτ−fτdτfsγsds≤∫−∞te∫stfnτdτγsds+∫−∞teξ∫stdτfsγsdsρfn,f,where ξ is between ∫stfnτdτ and ∫stfτdτ, thus ξ≤−λt−s; hence,(43)Tφnt−Tφt≤∫−∞te−λt−sγsds+∫−∞te−λt−st−sfsγsdsρfn,f≤∫−∞te−λt−sγMds+∫−∞te−λt−st−sμγMdsρfn,f=γMλ+μγMλ2ρfn,f=λ+μγMλ2ρfn,f.
By (41) and the above inequality, it follows that(44)Tφnt⟶Tφtn⟶∞,therefore, T is continuous. By (35), T∂B⊆B. According to Lemma 3, T has at least a fixed point on B and the fixed point is the periodic continuous solution ϕt of equation (1), and(45)γL≤ϕt≤γM.
Define a Lyapunov function as follows:(46)Vt,xt=xt−ϕt2,where xt is the unique solution with an initial value xt0=x0 of equation (1), and differentiating both sides of (46) along the solution of equation (1), the following is obtained:(47)dVt,xtdt=2xt−ϕtdxtdt−dϕtdt=2xt−ϕtatx3t−ϕ3t+btx2t−ϕ2t+ctxt−ϕt=2xt−ϕt2atx2t+xtϕt+ϕ2t+btxt+ϕt+ct=2xt−ϕt2atx2t+atϕt+btxt+atϕ2t+btϕt+ct≤2xt−ϕt24atatϕ2t+btϕt+ct−atϕt+bt24at=2xt−ϕt23a2tϕ2t+2atbtϕt+4atct−b2t4at≤xt−ϕt22at12a2t4atct−b2t−4a2tb2t12a2t=23atct−b2t3atxt−ϕt2.
ByH2, there is a positive number δ such that 3atct−b2t≥δ>0; hence,(48)dVt,xtdt≤2δ3aLxt−ϕt2,therefore, the periodic solution ϕt of equation (1) is uniformly asymptotic stable; thus,(49)xt−ϕt⟶0t⟶+∞,so xt cannot be a periodic solution of equation (1), and equation (1) has a unique periodic solution ϕt which is uniformly asymptotic stable.
This is the end of the proof of Theorem1.
Similarly, the following theorem is obtained.Theorem 2.
Consider equation (1), and at,bt,ct, and dt are all ω-periodic continuous functions on R, suppose that the following conditions hold:(50)H1at>0,H2b2t−3atct<0,then, equation (1) has a unique ω-periodic continuous solution which is unstable.


## 4. Two Examples

In this section, two examples are given to verify the main results of the paper.Example 1.
Consider the following equation:(51)dxdt=−2+sintx3+1+costx2−8+costx+2+sint,here at=−2+sint,  bt=1+cost,  ct=−8+cost,  and dt=2+sint.
By simple computation, it is obtained that(52)at<0,b2t−3atct<0.
Clearly, equation (51) satisfies all the conditions of Theorem 1. It follows from Theorem 1 that equation (51) has a unique 2π-periodic solution ϕt, which is uniformly asymptotic stable.
Clearly, by the graph of the solution curve, given any initial valuex0=x0(e.g., x0=0), the solution curve of equation (51) tends to the curve of the periodic solution γt (see Figure 1).Figure 1
The solution curve of equation (51) with an initial value x0=0.Example 2.
Consider the following equation:(53)dxdt=2+sintx3+1+costx2+8+costx+2+sint,here at=2+sint,  bt=1+cost,  ct=8+cost,  and dt=2+sint.
By simple computation, it is obtained that(54)at>0,b2t−3atct<0.
Clearly, equation (53) satisfies all the conditions of Theorem 2. It follows from Theorem 2 that equation (53) has a unique 2π-periodic solution ϕt, which is unstable.
Clearly, by the graph of the solution curve, given any initial valuex0=x0(e.g., x0=1), the solution curve of equation (53) tends to ∞ at some time t (see Figure 2).Figure 2
The solution curve of equation (53) with an initial value x0=1.


## 5. Conclusion

In this paper, Abel’s differential equation (1) is considered and a simple criterion to judge the existence and stability of the unique periodic solution of the equation is obtained.In fact, conditionH2 of Theorems 1 or 2 implies that condition at≠0; thus, the following is obtained.Corollary 1.
Consider equation (1), and at,bt,ct, and dt are all ω-periodic continuous functions on R, suppose that the following condition holds:(55)H1b2t−3atct<0,then, equation (1) has a unique ω-periodic continuous solution.It is easy to see that whenbt≡0,atct>0, Theorems 1 and 2 are also true; thus, the following corollary is obtained.Corollary 2.
Consider the following Abel’s type equation:(56)dxdt=atx3+ctx+dt,where at,ct, and dt are all the ω-periodic continuous functions on R, suppose that the following condition holds:(57)H1atct>0,then, equation (56) has a unique ω-periodic continuous solution which is uniformly asymptotic stable if at<0 (unstable if at<0).Whendt≡0, the following is obtained.Corollary 3.
Consider the following Abel’s type equation:(58)dxdt=atx3+btx2+ctx,where at,bt, and ct are all the ω-periodic continuous functions on R, suppose that the following condition holds:(59)H1b2t−3atct<0,then, equation (58) has no nonzero periodic continuous solution, and the zero solution is uniformly asymptotic stable if at<0 (unstable if at>0).



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