# The Existence ofn Periodic Solutions on One Element n-Degree Polynomial Differential Equation

Authors: Ni Hua; Tian Li-Xin
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

This paper deals with a class of one elementn-degree polynomial differential equations. By the fixed point theory, we obtain n periodic solutions of the equation. This paper generalizes some related conclusions of some papers.


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## Body

## 1. Introduction

Consider the following one elementn-degree polynomial differential equation:
(1)dxdt=∑i=0naitxi,n∈N+,where aiti=0,1,2,⋯,n is the ω-periodic continuous functions on R. When n=1, equation (1) is a linear differential equation. With regard to the periodic solution of the equation, we propose the following:Proposition 1 (see [1]).
Consider the following:(2)dxdt=a1tx+a0t,where a1t and a0t are ω-periodic continuous functions on R; if ∫0ωa1tdt≠0, then equation (2) has a unique ω-periodic continuous solution ηt, modη⊆moda1t,a0t, and ηt can be written as follows:
(3)ηt=∫−∞te∫sta1τdτa0sds,∫0ωa1tdt<0,−∫t+∞e∫sta1τdτa0sds,∫0ωa1tdt>0.Whenn=2, equation (1) is Riccati’s equation. Riccati’s equation plays an important role in fluid mechanics and in the theory of elastic vibration. There are many studies on this equation [2–9], and there is also a proposition about the periodic solutions of Riccati’s equation, as follows:Proposition 2 (see [2]).
Consider the following equation:(4)dxdt=a2tx2+a1tx+a0t,where a2t, a1t, and a0t are all ω-periodic continuous functions on R. Suppose that the following conditions hold:
(5)H1a2t≠0,H2a12t−4a2ta0t>0,H3supt∈0,ω−a1t2a2t−a1t2a2t2−a0ta2t<inft∈0,ω−a1t2a2t+a1t2a2t2−a0ta2t,then equation (4) has exactly two ω-periodic continuous solutions.Whena1t≡0, in [10], the author obtained the existence and more accurate range of two periodic solutions of equation (4) by means of the fixed point theorem.It is easy for us to guess under what conditions is equation (1) satisfied, and are there existing n periodic solutions of equation (1)?In this paper, we consider then-degree polynomial differential equation for the special case of equation (1) as follows:
(6)dxdt=atx−γ1t⋯x−γit⋯x−γnt,and we give a new criterion to judge the existence of n periodic solutions on equation (6); these conclusions generalize the relevant conclusions of References [1, 2, 10].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 of n periodic solutions on equation (6) is obtained. We end this paper with a short conclusion.


## 2. Some Lemmas and Abbreviations

Lemma 3 (see [11]).
Suppose 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 4 (see [12]).
SupposeV 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 a fixed point on C.For the sake of convenience, suppose thatft is an ω-periodic continuous function on R; we denote
(7)fM=supt∈0,ωft,fL=inft∈0,ωft.


## 3. Periodic Solutions of the Polynomial Differential Equation

In this section, we discuss the existence ofn periodic solutions of equation (6).Theorem 5.
Consider equation (6), at,γiti=1,2,⋯,n are all ω-periodic continuous functions on R; suppose that the following conditions hold:
(8)H1at≠0,∀t∈0,ω,H2γiM<γi+1L,i=1,2,⋯,n−1,then equation (6) has exactly n ω-periodic continuous solutions Φiti=1,2,⋯,n, and
(9)γiL≤Φit≤γiM,i=1,2,⋯,n.Proof.
ByH1, it follows at>0 or at<0. In order to avoid repetition, we only prove the case of at>0. As the proof of the existence of every periodic solution is the same, for the sake of simplicity, we only prove the existence of the n-th periodic solution Φnt of equation (6).Here, we will divide the proof into two steps.(1)
We prove the existence ofn periodic solutions of equation (6). Suppose
(10)S=φt∈CR,R∣φt+ω=φt,given any φt,ψt∈S, the distance is defined as follows:
(11)ρφ,ψ=supt∈0,ωφt−ψt.
Thus,S,ρ is a complete metric space. Take a convex closed set Bn of S as follows:
(12)Bn=φt∈SγnL≤φt≤γnM,modφ⊆moda,γ1,⋯,γn.
Given anyφt∈Bn, consider the following:
(13)dxdt=atφt−γ1t⋯φt−γit⋯x−γnt=ftx−γnt=ftx−ftγnt.
Here(14)ft=atφt−γ1t⋯φt−γit⋯φt−γn−1t.
ByH1, H2, and equation (12), we get that
(15)0<aLγnL−γ1M⋯γnL−γiM⋯γnL−γn−1M≤ft≤aMγnM−γ1L⋯γnM−γiL⋯γnM−γn−1L,hence, we have
(16)∫0ωftdt>0,and because at,γiti=1,⋯,n are ω-periodic continuous functions on R, it follows that ft,ftγnt are ω-periodic continuous functions on R, by equation (16). According to Proposition 1, equation (13) has a unique ω-periodic continuous solution as follows:
(17)ηt=∫t+∞e∫stfτdτfsγnsds,and
(18)modη⊆modft,ftγnt.
By equations (12) and (14), it follows that
(19)modft⊆moda,γ1,⋯,γn,modftγnt⊆moda,γ1,⋯,γn,hence we have
(20)modη⊆moda,γ1,⋯,γn.
By equations (12), (15), and (17), we get
(21)ηt≥γnL∫t+∞e∫stfτdτfsds=−γnL∫t+∞e∫stfτdτd∫stfτdτ=−γnLe∫stfτdτt+∞=−γnLe∫+∞tfτdτ−1=γnL,and
(22)ηt≤γnM∫t+∞e∫stfτdτfsds=−γnM∫t+∞e∫stfτdτd∫stfτdτ=−γnMe∫stfτdτt+∞=−γnMe∫+∞tfτdτ−1=γnM,hence, ηt∈Bn.
Define a mapping as follows:(23)Tφt=∫t+∞e∫stfτdτfsγnsds.
Thus, if given anyφt∈Bn, then Tφt∈Bn, hence T:Bn⟶Bn.
Now, we prove that the mappingT is a compact mapping.
Consider any sequenceφkt⊆Bnk=1,2,⋯, then it follows that
(24)γnL≤φkt≤γnM,modφk⊆moda,γ1,⋯,γn,k=1,2,⋯,on the other hand, Tφkt=xφkt satisfies
(25)dxφktdt=atφkt−γ1t⋯φkt−γit⋯xφkt−γnt.
Thus, we have(26)dxφktdt≤2aMγnM−γ1L⋯γnM−γiL⋯γnM−γn−1LγnM,modxφkt⊆moda,γ1,⋯,γn,hence dxφkt/dt is uniformly bounded; therefore, xφkt is uniformly bounded and equicontinuous on R, by the theorem of Ascoli-Arzela, for any sequence xφkt⊆Bn, there exists a subsequence (also denoted by xφkt) such that xφkt is convergent uniformly on any compact set of R, by equation (26), combined with Lemma 3, xφkt is convergent uniformly on R, that is to say, T is relatively compact on Bn.
Next, we prove thatT is a continuous mapping.
Supposeφkt⊆Bn,φt∈Bn, and
(27)φkt⟶φt·k⟶∞.
Denote(28)fkt=atφkt−γ1t⋯φkt−γit⋯φkt−γn−1t,then we have
(29)fkt⟶ft,k⟶∞,and
(30)0<aLγnL−γ1M⋯γnL−γiM⋯γnL−γn−1M≤fkt≤aMγnM−γ1L⋯γnM−γiL⋯γnM−γn−1L.
By equation (23), we have
(31)Tφkt−Tφt=∫t+∞e∫stfkτdτfksγnsds−∫t+∞e∫stfτdτfsγnsds=∫t+∞e∫stfkτdτfks−fsγnsds+∫t+∞e∫stfkτdτ−e∫stfτdτfsγnsds=∫t+∞e∫stfkτdτfks−fsγnsds+∫t+∞eξ∫stfkτ−fτdτfsγnsds≤∫t+∞e∫stfkτdτγnsds+∫t+∞eξ∫stdτfsγnsdsρfk,f,where ξ is between ∫stfkτdτ and∫stfτdτ; thus, ξ is between
(32)aLγnL−γ1M⋯γnL−γiM⋯γnL−γn−1Mt−s,and
(33)aMγnM−γ1L⋯γnM−γiL⋯γnM−γn−1Lt−s,hence we have
(34)Tφkt−Tφt≤∫t+∞eaLγnL−γ1M⋯γnL−γiM⋯γnL−γn−1Mt−sγnsds+∫t+∞eaLγnL−γ1M⋯γnL−γiM⋯γnL−γn−1Mt−ss−tfsγnsdsρfk,f≤γnMaLγnL−γ1M⋯γnL−γiM⋯γnL−γn−1M+γnMγnM−γ1L⋯γnM−γiL⋯γnM−γn−1LaLγnL−γ1M⋯γnL−γiM⋯γnL−γn−1M2ρfk,f.
By equation (29) and the above inequality, it follows that
(35)Tφkt⟶Tφtk⟶∞,hence, T is continuous; therefore, T:Bn⟶Bn is a continuous compact mapping, and by equation (23), it is easy to see, T∂Bn⊆Bn; according to Lemma 4, T has a fixed point on Bn, and the fixed point is the ω-periodic continuous solution Φnt of equation (6), and
(36)γnL≤Φnt≤γnM.
Similarly, we can prove the existence of the periodic solutionsΦiti=1,2,⋯,n−1 of equation (6), and we have
(37)γiL≤Φit≤γiMi=1,2,⋯⋯,n−1.
(2)
We prove that equation (6) has exactly n periodic solutions.
Let us discuss the possible range ofxt of equation (6); we divide the initial value xt0=x0 into the following parts:
(38)x0∈−∞,γ1L,γ1L,γ1M,γ1M,γ2L,γ2L,γ2M,γ2M,γ3L,⋯,γn−1M,γnL,γnL,γnM,γnM,+∞.
We will only prove the following cases. For the sake of convenience, supposen is an even number.Remark 1.
Ifn is an odd number, the proof is similar, and we omit it here.Let(39)ft,x=atx−γ1t⋯x−γit⋯x−γnt.
(i)
Ifx0∈−∞,γ1L
Consider equation (6). We have dx/dtt0,x0=ft0,x0>0. Thus, xt may stay at −∞,γ1L or enter into γ1L,γ1M at some time t. If xt stays at −∞,γ1L, then dx/dt=ft,x>0. Thus, xt cannot be a periodic solution of equation (6). If xtenters into γ1L,γ1M at some time t, then there is not a t1t1>t0 such that xt1=xt0=x0; thus xt can also not be a periodic solution of equation (6).
(ii)
Ifx0∈γ1L,γ1M, then equation (6) has an ω-periodic continuous solution xt=Φ1t with initial value xt0=Φ1t0
Asft,γ1=ft,γ2=0, by differential mean value theorem, it follows that
(40)fx′t,ξ1t=0,γ1t<ξ1t<γ2t.
By equation (39), we have
(41)fx′t,γ1t<0,(42)fx′t,γ2t>0.
Note that(43)γ1L≤Φ1t≤γ1M.
By equations (41) and (43), it follows that
(44)fx′t,Φ1t<0.
Now, suppose that there is anotherω-periodic continuous solution Ψ1t of equation (6) which satisfies
(45)γ1L≤Ψ1t≤γ1M.
Becauseft,x is a polynomial function with continuous partial derivatives to x, equation (6) satisfies the existence and uniqueness of solutions to initial value problems of differential equations, thus
(46)Φ1t−Ψ1t>0∀t∈R.
By equations (41) and (45), it follows that
(47)fx′t,Ψ1t<0.
Consider the following equation:(48)dΦ1t−Ψ1tdt=ft,Φ1t−ft,Ψ1t=fx′t,Ψ1t+θΦ1t−Ψ1tΦ1t−Ψ1t,0<θ<1.
Thus, we have(49)Φ1t−Ψ1t=Φ10−Ψ10e∫0tfx′s,Ψ1s+θΦ1s−Ψ1sds.
By equations (43) and (45), it follows that
(50)γ1L≤Ψ1t+θΦ1t−Ψ1t≤γ1M.
By equations (41) and (50), it follows that
(51)fx′t,Ψ1t+θΦ1t−Ψ1t<0.
By equations (49) and (51), it follows that
(52)Φ1t−Ψ1t⟶0t⟶+∞,
By equations (46) and (52), this is a contradiction, thus Ψ1t cannot be a periodic solution of equation (6), that is to say, equation (6) has exactly a unique ω-periodic continuous solution Φ1t which satisfies γ1L≤Φ1t≤γ1M.
(iii)
Ifx0∈γ1M,γ2L
Consider equation (6). We have dx/dtt0,x0=ft0,x0<0. Thus, xt may stay at γ1M,γ2L or enter into γ1L,γ1M at some time t. If xt stays at γ1M,γ2L, we have dx/dt=ft,x<0, then xt cannot be a periodic solution of equation (6). If xt enters into γ1L,γ1M at some time t, then there is not a t1t1>t0 such that xt1=xt0=x0; thus, xt can also not be a periodic solution of equation (6).
Similarly, ifx0∈γiL,γiM,i=2,3,⋯,n, we can prove that equation (6) has exactly a unique ω-periodic continuous solution Φit which satisfies γiL≤Φit≤γiMi=2,3,⋯,n.
Similarly, ifx0∈γiM,γi+1L,i=2,3,⋯,n, then the solution xt of equation (6) with an initial value xt0=x0 cannot be a periodic solution of equation (6).
(iv)
Ifx0∈γnM,+∞
Consider equation (6). We have dx/dtt0,x0=ft0,x0>0. Thus, xt may stay at ∈γn,+∞ or xt⟶+∞,t⟶+∞. If xt stays at γnM,+∞, we have dx/dt=ft,x>0, then xt cannot be a periodic solution of equation (6). If xt⟶+∞,t⟶+∞, then xt can also not be a periodic solution of equation (6).
To sum up, equation (6) has exactly n ω-periodic continuous solutions Φiti=1,2,⋯,n which satify
(53)γiL≤Φit≤γiMi=1,2,⋯,n.This is the end of the proof of Theorem5.Remark 2.
Ifn=1, Theorem 5 is exactly Proposition 1.Remark 3.
Ifn=2, Theorem 5 is exactly Proposition 2.


## 4. Conclusion

Consider the following Riccati’s differential equation:(54)dxdt=atx2+btx+ct,about the periodic solutions on equation (54), there is a conclusion as follows:Theorem 6 (see [13]).
Consider equation (54); at, bt, and ct are all ω-periodic continuous functions on R. Suppose that the following conditions hold:
(55)H1at>0,then equation (54) has at most two ω-periodic continuous solutions.What conditions are the coefficient functions of the equation satisfied? The equation has exactly two periodic solutions. Proposition2 gives the answer.Consider the following Abel’s differential equation:(56)dxdt=atx3+btx2+ctx+dt,about the periodic solutions on equation (56), we have the following result.Theorem 7 (see [13]).
Consider equation (56), at, bt, ct, and dt are all ω-periodic continuous functions on R. Suppose that the following conditions hold:
(57)H1at>0,then equation (56) has at most three ω-periodic continuous solutions.We cannot help but ask: What conditions are the coefficient functions of the equation satisfied? The equation has exactly three periodic solutions. We show the following answer.Corollary 8.
Consider the following Abel’s differential equation:(58)dxdt=atx−γ1tx−γ2tx−γ3t.Here, at,γiti=1,2,3 are all ω-periodic continuous functions on R. Suppose that the following conditions hold:
(59)H1at≠0,H2γ1M<γ2L≤γ2M<γ3L,then equation (58) has exactly three ω-periodic continuous solutions Φ1t, Φ2t, and Φ3t, and
(60)γ1L≤Φ1t≤γ1M,γ2L≤Φ2t≤γ2M,γ3L≤Φ3t≤γ3M.Forn≥3,n∈N+, equation (1) has not always the most n periodic solutions (see [13]). But in this paper, we obtain a new criterion for the existence of n periodic solutions of periodic equation (6); the size range of n periodic solutions is also given. It can be said that this paper is a generalization of the conclusions of the related articles on periodic solutions.



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