# A Criterion for the Existence of the Unique Periodic Solution of One-Dimensional Periodic Differential Equation

Authors: Ni  Hua
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we discuss one-dimensional differential equation withω-period. By using the fixed point theory, the existence of a periodic solution is obtained; by using the second Lyapunov method, the uniqueness and stability of the periodic solution are obtained.


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## Body

## 1. Introduction

Consider the first-order nonautonomous differential equation:(1)dxdt=ft,x,where t,x∈ℝ and ft,x is continuous and is an ω-periodic function on t.As we all know, the first-order differential equation (1) is widely used to establish mathematical models in many fields, such as physics, biology, economy, and medicine. Because the nonautonomous differential equations have important practical applications, it has been a research subject of many scientists for a long time. The main problems studied are the existence, stability, and number of periodic solutions (see [1–8]).In [9], the authors gave several methods to study periodic solutions of one-dimensional ω-periodic differential equations, including Poincare’s map, the stability of zero solution and its multiplicity, normal form, and averaging method. They summarized and improved some existing results and obtained some new conclusions as well.As we all know, the existence, uniqueness, and stability of periodic solutions of differential equations have always been an important research hotspot in the field of differential equations (see [10–20]). However, the above literatures are basically about the study of periodic solutions of specific equations, rather than the study of general differential equations.In [21], the existence of a periodic solution for Abel’s differential equation is obtained first by using the fixed-point theorem. By constructing the Lyapunov function, the uniqueness, and stability of the periodic solution of the equation are obtained.Stimulated by the works of [21], in this paper, we consider more general differential equation (1). By using the fixed point theorem and constructing Lyapunov function, we give a new criterion for the existence, uniqueness, and stability of the periodic solutions of the equation (1). These results generalize some related results in some literatures.


## 2. Some Lemmas and Abbreviations

Lemma 1 (see [22]).
Consider the equation:(2)dxdt=atx+bt,where at and bt are ω-periodic continuous functions on ℝ. If ∫0ωatdt≠0, then (2) has a unique ω-periodic continuous solution ηt,modηt⊆modat,bt, and ηt can be written as(3)ηt=∫−∞te∫staμdμbsds,∫0ωatdt<0,−∫t+∞e∫staμdμbsds.∫0ωatdt>0.Lemma 2 (see [23]).
Assume that anω-periodic sequence fnt is convergent uniformly on any compact set of ℝ, ft is an ω-periodic function, and modfn⊆modfn=1,2,…; then, fnt is convergent uniformly on ℝ.Lemma 3 (see [24]).
AssumeV is a metric space and ℂ is a convex closed set of V; 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 convenience, assume thatft is an ω-periodic continuous function on ℝ, we denote(4)fM=supt∈0,ωft,fL=inft∈0,ωft.


## 3. Main Results

In this section, firstly, we use the mean value theorem to transform equation (1) into an equivalent equation. Then, we use the fixed point theorem to get the existence of periodic solution of equation (1). Finally, we use Lyapunov function method to obtain the uniqueness and stability of the periodic solution.Theorem 1.
Consider equation (1), f:ℝ×ℝ→ℛ is continuous, ft+ω,x=ft,x, and fx′ exists and is a continuous function, and rt is an ω-periodic continuous function on ℝ. Assume that the following conditions hold:(5)H1ft,rt=0,H2fx′t,x≤0≡0,then (1) equation (1) has a unique ω-periodic continuous solution Φt and (2) Φt is globally attractive.Proof.
(1)
ByH1 and differential mean value theorem, equation (1) is transformed into the equation:(6)dxdt=ft,x=ft,x−ft,rt=fx′t,rt+θ1xx−rtx−rt,0<θ1x<1.Noting thatf:ℝ×ℝ⟶ℝ is continuous, ft+ω,x=ft,x, fx′ exists and is a continuous function, and from (6), it follows that fx′t,rt+θ1xx−rt is continuous and(7)fx′t+ω,rt+ω+θ1xx−rt+ω=fx′t,rt+θ1xx−rt.Defining(8)S=ψt∈Cℝ,ℝ|ψt+ω=ψt,∀φt,ψt∈S, the distance is defined as follows:(9)ρφ,ψ=supt∈0,ωφt−ψt,and thus,S,ρ is a complete metric space.Take a convex closed setB of S as follows:(10)B=ψt∈S|rL≤ψt≤rM,modψ⊆modf,∀ψt∈B, and consider the equation:(11)dxdt=fx′t,rt+θ1ψtψt−rtx−rt=fx′t,rt+θ1ψtψt−rtx−fx′t,rt+θ1ψtψt−rtrt=fxψ′tx−fxψ′trt,where(12)fx′t,rt+θ1ψtψt−rt≜fxψ′t.It follows from (7), (10), and (12) that fxψ′t is an ω-periodic continuous function on t∈ℝ; thus, there is a positive number δ such that(13)fx′t,rt+θ1ψtψt−rt≤δ.It follows fromH2 that(14)∫0ωfxψ′tdt<0.Sincefxψ′t and rt are ω-periodic continuous functions, it follows(15)fxψ′trtis anω-periodic continuous function. It follows from (14) and Lemma 1 that equation (11) has a unique ω-periodic continuous solution as follows:(16)ηt=−∫−∞te∫stfxψ′μdμfxψ′srsds,(17)modη⊆modfxψ′t,fxψ′trt.It follows from (6), (10), and (11) that(18)modfxψ′t⊆modf,modfxψ′trt⊆modf,and hence, we have(19)modη⊆modf.It follows from (10), (14), and (16) that(20)ηt≥−rL∫−∞te∫stfxψ′μdμfxψ′sds=rL∫−∞te∫stfxψ′μdμd∫stfxψ′μdμ=rLe∫stfxψ′μdμ−∞t=rL1−e∫−∞tfxψ′μdμ−∞<t<+∞=rL,ηt≤−rM∫−∞te∫stfxψ′μdμfxψ′sds=rM∫−∞te∫stfxψ′μdμd∫stfxψ′sdμ=rMe∫stfxψ′μdμ−∞t=rM1−e∫−∞tfxψ′μdμ−∞<t<+∞=rM,and therefore,ηt∈B.Define a mapping as follows:(21)Tψt=−∫−∞te∫stfxψ′μdμfxψ′srsds,and thus, if given anyψt∈B, then Tψt∈B; hence, T:B⟶B.Now, we prove that the mappingT is a compact mapping.Consider any sequenceψnt⊆Bn=1,2,…; then, it follows(22)rL≤ψnt≤rM,modψn⊆modf,n=1,2,….On the other hand,Tψnt=xψnt satisfies(23)dxψntdt=fx′t,rt+θ2ψntψnt−rtxψnt−fx′t,rt+θ2ψntψnt−rtrt=fxψn′txψnt−fxψn′trt,where(24)fx′t,rt+θ2ψntψnt−rt≜fxψn′t.Sincef:ℝ×ℝ⟶ℝ is continuous, ft+ω,x=ft,x, fx′ exists and is a continuous function, and from (23) and ψnt⊆Bn=1,2,…, it follows that fxψn′t is an ω-periodic continuous function on t∈ℝ.It follows from (10), (13), (22), and (24) that(25)fx′t,rt+θ2ψntψnt−rt≤δ.Thus, we have(26)dxψntdt≤2fxψn′trM≤2δrM,(27)modxψnt⊆modf.Hence,dxψnt/dt is uniformly bounded; therefore, xψnt is uniformly bounded and equicontinuous on ℝ. By the theorem of Ascoli-Arzela, for any sequence xψnt⊆B, there exists a subsequence (also denoted by xψnt) such that xψnt is convergent uniformly on any compact set of ℝ. By (27), combined with Lemma 2, xψnt is convergent uniformly on ℝ, that is to say, T is relatively compact on B.Next, we prove thatT is a continuous mapping.Assumeψnt⊆B,ψt∈B, and(28)ψnt⟶ψt,n⟶∞.Sinceft,x has first-order continuous partial derivative on x, it follows from (28) that(29)fx′t,rt+θ2ψntψnt−rt⟶fx′t,rt+θ1ψtψt−rt,n⟶∞,that is,(30)fxψn′t⟶fxψ′t,n⟶∞.It follows from (21) that(31)Tψnt−Tψt=∫−∞te∫stfxψn′μdμfxψn′srsds−∫−∞te∫stfxψ′μdμfxψ′srsds=∫−∞te∫stfxψn′μdμfxψn′s−fxψ′srsds+∫−∞te∫stfxψn′μdμ−e∫stfxψ′dμfxψ′srsds=∫−∞0e∫s+ttfxψn′μdμfxψn′t+s−fxψ′t+srt+sds+∫−∞0e∫s+ttfxψn′μdμ−e∫s+ttfxψ′μdμfxψ′t+srt+sds=∫−∞0e∫s+ttfxψn′μdμfxψn′t+s−fxψ′t+srt+sds+∫−∞0eξ∫s+ttfxψn′μ−fxψ′μdμfxψ′t+srt+sds≤∫−∞0e∫s+ttfxψn′μdμrt+sds+∫−∞0eξ∫s+ttdμfxψ′s+trs+tdsρfxψn′,fxψ′=∫−∞−T0e∫t+stfxψn′μdμrt+sds+∫−T00e∫t+stfxψn′μdμrt+sds+∫−∞−T0eξ∫t+stdμfxψ′s+trs+tds+∫−T00eξ∫t+stdμfxψ′s+trs+tdsρfxψn′,fxψ′,whereξ is between ∫s+ttfxψn′μdμ and ∫s+ttfxψ′μdμ.Sincefxψn′tandfxψ′t are ω-periodic continuous functions on t∈ℝ, it follows(32)lims⟶−∞1−s∫s0fxψn′μdμ=1ω∫0ωfxψn′μdμ≜−λ1,lims⟶−∞1−s∫s0fxψ′μdμ=1ω∫0ωfxψ′μdμ≜−λ2.Thus, there is aT0>0(also, assume it is T0 in the above) when s≤−T0, such that(33)−1s∫t+stfxψn′μdμ+λ1<λ12,−1s∫t+stfxψ′μdμ+λ2<λ22,that is,(34)∫t+stfxψn′μdμ<λ1s2,∫t+stfxψ′μdμ<λ2s2.Set(35)λ0s2=maxλ1s2,λ2s2.Hence, we have(36)Tψnt−Tψt≤∫−∞−T0eλ0s/2rMds+T0eδT0rM−∫−∞−T0eλ0s/2sδrMds+T02eδT0rMρfxψn′,fxψ′=rM2λ0e−λ0T0/2+T0eδT0+4λ02e−λ0T0/2δ+T02eδT0δρfxψn′,fxψ′.It follows from (30) and above inequality that(37)Tψnt⟶Tψt,n⟶∞.Therefore,T is continuous. By (21), easy to see, T∂B⊆B. According to Lemma 3, T has at least a fixed point on B, the fixed point is the ω-periodic continuous solution Φt of equation (1), and(38)rL≤Φt≤rM.(2)
Construct a Lyapunov function as follows:(39)Vt,xt=xt−Φt2,where xt is the unique solution with initial value xt0=x0 of equation (1) and Φt is the periodic solution of equation (1).
Differentiating both sides of (39) along the solution of equation (1), we have(40)dVt,xtdt=2xt−Φtdxtdt−dΦtdt=2xt−Φtft,xt−ft,Φt=2xt−Φtfx′t,ζxt−Φt=2fx′t,ζxt−Φt2,where ζ is between xt and Φt.
ByH2, it follows(41)xt−Φt⟶0,t⟶+∞.
Thus,xt cannot be a periodic solution of equation (1), and it is proved that equation (1) has a unique ω-periodic continuous solution Φt which is globally attractive.
This is the end of the proof of Theorem1.Theorem 2.
Consider equation (1), f:ℝ×ℝ⟶ℝ is continuous and ft+ω,x=ft,x, and fx′ exists and is a continuous function; rt is an ω-periodic continuous function on ℝ. Assume that the following conditions hold:(42)H1ft,rt=0,H2fx′t,x≥0≡0.
Then, (1) equation (1) has a unique ω-periodic continuous solution Φt and (2) Φt is unstable.Proof.
(1)
ByH1 and differential mean value theorem, equation (1) is transformed into the equation(43)dxdt=ft,x=ft,x−ft,rt=fx′t,rt+θ1xx−rtx−rt,0<θ1x<1.Noting thatf:ℝ×ℝ⟶ℝ is continuous, ft+ω,x=ft,x, fx′ exists and is a continuous function, and from (43), it follows that fx′t,rt+θ1xx−rt is continuous and(44)fx′t+ω,rt+ω+θ1xx−rt+ω=fx′t,rt+θ1xx−rt.Defining(45)S=ψt∈Cℝ,ℝ|ψt+ω=ψt,∀φt,ψt∈S, the distance is defined as follows:(46)ρφ,ψ=supt∈0,ωφt−ψt.Thus,S,ρ is a complete metric space.Take a convex closed setB of S as follows:(47)B=ψt∈S|rL≤ψt≤rM,modψ⊆modf,∀ψt∈B, and consider the equation(48)dxdt=fx′t,rt+θ1ψtψt−rtx−rt=fx′t,rt+θ1ψtψ−rtx−fx′t,rt+θ1ψtψ−rtrt=fxψ′tx−fxψ′trt,where(49)fx′t,rt+θ1ψtψ−rt≜fxψ′t.It follows from (44), (47), and (49) that fxψ′t is an ω-periodic continuous function on t∈ℝ; thus, there is a positive number δ such that(50)fx′t,rt+θ1ψtψt−rt≤δ.It follows fromH2 that(51)∫0ωfxψ′tdt>0.Sincefxψ′t and rt are ω-periodic continuous functions, it follows(52)fxψ′trtis anω-periodic continuous function. It follows from (51) and Lemma 1 that equation (48) has a unique ω-periodic continuous solution as follows:(53)ηt=∫t+∞e∫stfxψ′μdμfxψ′srsds,(54)modη⊆modfxψ′t,fxψ′trt.It follows from (43), (47), and (48) that(55)modfxψ′t⊆modf,modfxψ′trt⊆modf.Hence, we have(56)modη⊆modf.It follows from (47), (51), and (53) that(57)ηt≥rL∫t+∞e∫stfxψ′μdμfxψ′sds=−rL∫t+∞e∫stfxψ′μdμd∫stfxψ′μdμ=−rLe∫stfxψ′μdμt+∞=−rLe∫+∞tfxψ′μdμ−1,−∞<t<+∞=rL,ηt≤rM∫t+∞e∫stfxψ′μdμfxψ′sds=−rM∫t+∞e∫stfxψ′μdμd∫stfxψ′μdμ=−rMe∫stfxψ′μdμt+∞=−rMe∫+∞tfx′μdμ−1,−∞<t<+∞=rM.Therefore,ηt∈B.Define a mapping as follows:(58)Tψt=∫t+∞e∫stfxψ′μdμfxψ′srsds.Thus, if given anyψt∈B, then Tψt∈B, and hence, T:B⟶B.Now, we prove that the mappingT is a compact mapping.Consider any sequenceψnt⊆Bn=1,2,…, and then, it follows(59)rL≤ψnt≤rM,modψn⊆modf,n=1,2,….On the other hand,Tψnt=xψnt satisfies(60)dxψntdt=fx′t,rt+θ2ψntψnt−rtxψnt−fx′t,rt+θ2ψntψnt−rtrt=fxψn′txψnt−fxψn′trt,where(61)fx′t,rt+θ2ψntψnt−rt≜fxψn′t.Sincef:ℝ×ℝ⟶ℝ is continuous, ft+ω,x=ft,x, fx′ exists and is a continuous function, and from (60) and ψnt⊆Bn=1,2,…, it follows that fxψn′t is an ω-periodic continuous function on t∈ℝ.It follows from (47), (50), (59), and (61) that(62)fx′t,rt+θ2ψntψnt−rt≤δ.Thus, we have(63)dxψntdt≤2fxψn′trt≤2δrM,(64)modxψnt⊆modf.Hence,dxψnt/dt is uniformly bounded, and therefore, xψnt is uniformly bounded and equicontinuous on ℝ. By the theorem of Ascoli-Arzela, for any sequence xψnt⊆B, there exists a subsequence (also denoted by xψnt) such that xψnt is convergent uniformly on any compact set of ℝ. By (64), combined with Lemma 2, xψnt is convergent uniformly on ℝ, that is to say, T is relatively compact on B.Next, we prove thatT is a continuous mapping.Assumeψnt⊆B,ψt∈B,(65)ψnt⟶ψt,n⟶∞.Sinceft,x has first-order continuous partial derivative on x, we have(66)fx′t,rt+θ2ψntψnt−rt⟶fx′t,rt+θ1ψtψt−rt,n⟶∞,that is,(67)fxψn′t⟶fxψ′t,n⟶∞.It follows from (58) that(68)Tψnt−Tψt=∫t+∞e∫stfxψn′μdμfxψn′srsds−∫t+∞e∫stfxψ′μdμfxψ′srsds=∫t+∞e∫stfxψn′μdμfxψn′s−fxψ′srsds+∫t+∞e∫stfxψn′μdμ−e∫stfxψ′μdμfxψ′srsds=∫0+∞e∫s+ttfxψn′μdμfxψn′t+s−fxψ′t+srt+sds+∫0+∞e∫s+ttfxψn′μdμ−e∫s+ttfxψ′μdμfxψ′t+srt+sds=∫0+∞e∫s+ttfxψn′μdμfxψn′t+s−fxψ′t+srt+sds+∫0+∞eξ∫s+ttfxψn′μ−fxψ′μdμfxψ′t+srt+sds≤∫0+∞e∫s+ttfxψn′μdμrt+sds+∫0+∞eξ∫s+ttdμfxψ′s+trs+tdsρfxψn′,fxψ′=∫0T0e∫t+stfxψn′μdμrt+sds+∫T0+∞e∫t+stfxψn′μdμrt+sds+∫0T0eξ∫t+stdμfxψ′s+trs+tds+∫T0+∞eξ∫t+stdμfxψ′s+trs+tdsρfxψn′,fxψ′,whereξ is between ∫s+ttfxψn′μdμ and ∫s+ttfxψ′μdμ.Sincefxψn′μandfxψ′μ are ω-periodic continuous functions on ℝ, it follows(69)lims⟶+∞1s∫0sfxψn′μdt=1ω∫0ωfxψn′μdt≜λ1,lims⟶+∞1s∫0sfxψ′μdt=1ω∫0ωfxψ′μdt≜λ2,and thus, there is aT0>0 (also, assume it is T0 in the above), when s≥T0, such that(70)−1s∫t+stfxψn′μdt−λ1<λ12,−1s∫t+stfxψ′μdt−λ2<λ22,that is,(71)∫t+stfxψn′μdt<−λ1s2,∫t+stfxψ′μdt<−λ2s2.Set(72)−λ0s2=max−λ1s2,−λ2s2,and hence, we have(73)Tψnt−Tψt≤T0eδT0rM+∫T0+∞e−λ0s/2rMds+T02eδT0δrM+∫T0+∞e−λ0s/2sδrMdsρfxψn′,fxψ′=rMT0eδT0+2λ0e−λ0T0/2+T02eδT0δ+4λ02e−λ0T0/2δρfxψn′,fxψ′.It follows from (67) and the above inequality that(74)Tψnt⟶Tψt,n⟶∞.Therefore,T is continuous. By (58), easy to see, T∂B⊆B. According to Lemma 3, T has at least a fixed point on B, the fixed point is the ω-periodic continuous solution Φt of equation (1), and(75)rL≤Φt≤rM.(2)
Construct a Lyapunov function as follows:(76)Vt,xt=xt−Φt2,where xt is the unique solution with initial value xt0=x0 of equation (1) and Φt is the periodic solution of equation (1).
Differentiating both sides of (76) along the solution of equation (1), we have(77)dVt,xtdt=2xt−Φtdxtdt−dΦtdt=2xt−Φtft,xt−ft,Φt=2xt−Φtfx′t,ζxt−Φt=2fx′t,ζxt−Φt2,where ζ is between xt and Φt.
ByH2, it follows(78)xt−Φt⟶+∞,t⟶+∞,and thus, xt cannot be a periodic solution of equation (1), and it is proved that equation (1) has a unique ω-periodic continuous solution Φt which is unstable.
This is the end of the proof of Theorem2.


## 4. Examples

The following examples show the feasibility of our main results.Example 1.
Consider the equation(79)dxdt=−ex+esint,where(80)ft,x=−ex+esint,ft,sint=0,fx′t,x=−ex≤0≡0,and hence, equation (79) satisfies all the conditions of Theorem 1. It follows from Theorem 1 that equation (79) has a unique 2π-periodic solution rt which is globally attractive.
Clearly, according to the solution curve of equation (79), if given any initial value x0=x0(e.g., x0=0.2), then the solution curve of equation (79) does tend to the curve of the periodic solution rt (see Figure 1).Figure 1
The curve of the solution of equation (79) with initial value x0=0.2.Example 2.
Consider the equation(81)dxdt=x5+x3−cos5t−cos3t,where(82)ft,x=x5+x3−cos5t−cos3t,ft,cost=0,fx′t,x=5x4+3x2≥0≡0.
Hence, equation (81) satisfies all the conditions of Theorem 2. It follows from Theorem 2 that equation (81) has a unique 2π-periodic solution rt which is unstable.
Clearly, according to the solution curve of equation (81), if given any initial value x0=x0(e.g., x0=0.6), then the solution curve of equation (81) arrives at enough large−∞ at some time t∗ (see Figure 2).Figure 2
The curve of the solution of equation (81) with initial value x0=0.6.


## 5. Conclusion

In this paper, a simple criterion for the existence, uniqueness, and stability of the periodic solution of one-dimensional periodic differential equations is given. This criterion can be applied to many one-dimensional periodic differential equations.



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