# Affine-Periodic Solutions for Dissipative Systems

Authors: Yu Zhang; Xue Yang; Yong Li
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

As generalizations of Yoshizawa’s theorem, it is proved that
a dissipative affine-periodic system admits affine-periodic solutions. This
result reveals some oscillation mechanism in nonlinear systems.


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## Body

## 1. Introduction

Consider the system(1)x′=f(t,x),′=ddt,
where f:ℝ1×ℝn→ℝn is continuous and ensures the uniqueness of solutions with respect to initial values. Fix T>0. The system (1) is said to be T-periodic if f(t+T,x)=f(t,x) for all (t,x)∈ℝ1×ℝn. For this T-periodic system, a major problem is to seek the existence of T-periodic solutions. Actually, some physical systems also admit the certain affine-periodic invariance. For example, let Q∈GL(n), and
(2)f(t+T,x)=Qf(t,Q-1x),∀(t,x).
This affine-periodic invariance exhibits two characters: periodicity in time and symmetry in space. Obviously, when Q=id, the invariance is just the usual periodicity; when Q=-id, the invariance implies the usual antisymmetry in space. When Q∈SO(n), the invariance shows the rotating symmetry in space. Hence, the invariance also reflects some properties of solutions in geometry. Now, (2) is said to possess the affine-periodic structure. For this affine-periodic system, we are concerned with the existence of affine-periodic solutions x(t) with
(3)x(t+T)=Qx(t),∀t.In the qualitative theory, it is a basic result that the dissipative periodic systems admit the existence of periodic solutions. The related topics had ever captured the main field in periodic solutions theory from the 1960s to the 1990s. For some litratures, see, for example, [1–12].In the present paper, we will see whether (1) admits affine-periodic solutions or not if (1) is affine-dissipative. Here, (1) is said to be affine-dissipative if Q-mx(t+mT) are ultimately bounded. Our main result is the following.Theorem 1.
LetQ∈GL(n). If the system (1) is Q-affine-periodic, that is,
(4)f(t+T,x)=Qf(t,Q-1x),
and affine-dissipative, then it admits a Q-affine-periodic solution x*(t); that is,
(5)x*(t+T)=Qx*(t),∀t.The paper is organized as follows. In Section2, we use the asymptotic fixed-point theorem, for example, Horn’s fixed-point theorem to prove Theorem 1. Section 3 deals with the case of functional differential equations, where an anagolous version is given and the proof is sketched. Finally, in Section 4, we illustrate some applications.


## 2.  Proof of Theorem1

In order to prove Theorem1, we first recall some preliminaries.Lemma 2 (Horn’s fixed-point theorem [13]).
LetX be a Banach space, and let S0⊂S1⊂S2⊂X be convex sets, where S0 is compact, S1 relatively open with respect to S2, and S2 closed. Assume that P:S0→X is continuous and satisfies
(6)Pj(S-1)⊂S2,j=0,1,…,N-1,Pj(S-1)⊂S0,j=N,…,2N-1.
Then, P has a fixed point in S0.The following is a usual definition.Definition 3.
The system (1) is said to be dissipative or ultimately bounded, if there is B0>0 and for any B>0, there are M=M(B)>0 and L=L(B)>0 such that for |x0|≤B,
(7)|x(t,x0)|≤M,∀t∈[0,L],|x(t,x0)|≤B0,∀t∈[L,∞),
where x(t,x0) denotes the solution of (1) with the initial value x(0)=x0.For the affine-periodic system (1), we have the following.Definition 4.
The system (1) is said to be Q-affine-dissipative, if there is B0>0 and for any B>0, there are M=M(B)>0 and L=L(B)>0 such that
(8)|x(t,x0)|≤M,∀t∈[0,L],|Q-mx(t+mT,x0)|≤B0,∀t∈[L,∞),m∈ℤ+1,
whenever |x0|≤B.Proof of Theorem1.
Define the mapP:ℝn→ℝn by
(9)P(x0)=Q-1x(T,x0),∀x0∈ℝn,
and set
(10)S0={y∈ℝn:|y|≤B0},S1={y∈ℝn:|y|<B1},S2={y∈ℝn:|y|≤B2},
where
(11)B1=B0+1,B2=sup{|Q-mx(mT,x0)|:m∈{0,…,N},|x0|≤B0+1}+B0+2,N=[L(B1)]+1.
By uniqueness and the affine periodicity of f(t,x), Q-mx(t+mT,x0) is still the solution of (1) for each m∈ℤ+1. Therefore,
(12)Pi(x0)=Q-ix(iT,x0),i=0,1,….
It follows from (8) that
(13)Pj(S-1)⊂S2,j=0,…,N-1,Pj(S-1)⊂S0,j=N,…,2N-1.
Thus, Horn’s fixed-point theorem implies that P has a fixed point x-0 in S0; that is, Q-1x(T,x-0)=x-0. Also, uniqueness yields
(14)Q-1x(t+T,x-0)=x(t,x-0),⟺x(t+T,x-0)=Qx(t,x-0),∀t.
This completes the proof of Theorem 1.


## 3. A Version to Functional Differential Equations

Consider the functional differential equation (FDE)(15)x′=F(t,xt),
where F:ℝ1×ℂ→ℝn is continuous, takes any bounded set in ℂ to a bounded set in ℝn, and ensures the uniqueness of solutions with respect to initial values, where ℂ=ℂ([-r,0],ℝn), xt(s)=x(t+s), and  s∈[-r,0]. Moreover, F is Q-affine-periodic; that is,
(16)F(t+T,φ)=QF(t,Q-1φ),∀(t,φ)∈ℝ1×ℂ.Definition 5.
The system (15) is said to be Q-affine-dissipative; if there is B0>0 and for any B>0, there are M=M(B)>0 and L=L(B)>0 such that
(17)|x(t,φ)|≤M,∀t∈[0,L],|Q-mx(t+mT,φ)|≤B0,∀t∈[L,∞),
whenever ∥φ∥=max[-r,0]|φ(s)|≤B; here, x(t,φ) denotes the solution of (15) at initial value x0=φ.We are in position to state another main result.Theorem 6.
If the system (15) is Q-affine-periodic-dissipative, then it admits a Q-affine-periodic solution x(t); that is,
(18)x(t+T)=Qx(t),∀t.Proof.
Define the mapP:ℂ→ℂ by
(19)P(φ)=Q-1xT(·,φ),∀φ∈ℂ,
and set
(20)S0={φ∈ℂ:∥φ∥≤B0,|φ(s1)-φ(s2)|≤h|s1-s2|,∀s1,s2∈[-r,0]},S1={φ∈ℂ:∥φ∥<B1,|φ(s1)-φ(s2)|<h1|s1-s2|,∀s1,s2∈[-r,0]},S2={φ∈ℂ:∥φ∥≤B2,|φ(s1)-φ(s2)|≤h2|s1-s2|,∀s1,s2∈[-r,0]},
where
(21)h=sup{|F(t,φ)|:t∈ℝ1,∥φ∥≤B0},h1=sup{|F(t,φ)|:t∈ℝ1,∥φ∥≤B0+1},B1=B0+1,(22)B2=sup{∥Q-mxmT(·,φ)∥:m∈{0,1,…,N},∥φ∥≤B0+1}+B0+2,h2=sup{|F(t,φ)|:t∈ℝ1,∥φ∥≤B2},
where N=[L(B1)+r]+2. Then, (17) and the constructions imply that
(23)Pj(S-1)⊂S2,j=0,…,N-1,Pj(S-1)⊂S0,j=N,…,2N-1.
Hence, P has a fixed point φ*∈S0 via Horn’s theorem. The uniqueness implies that x(t,φ*) is the desired affine-periodic solution of (15). The proof is complete.


## 4. Some Applications

First, we observe a simple example to show the meanings of affine-periodic solutions.Example 7.
Consider the equation(24)x′+2x=e-t.
Put f(t,x)=-2x+e-t. The general solution of (24) is
(25)x(t)=e-2tc+e-t(cisanyconstant).
Obviously, for given τ>0,
(26)f(t+τ,x)=e-τf(t,eτx),
and any solution x(t) satisfies
(27)|(e-τ)-mx(t+mτ)|=|emτe-2(t+mτ)c+emτe-(t+mτ)|≤e-(2t+mτ)|c|+1≤e-2t|c|+1,
which implies that (24) is e-τ-periodic-dissipative. By Theorem 1, (24) has an e-τ-affine-periodic solution. This solution is just x(t)=e-t and different from the usual periodic solutions!As usual, Lyapunov’s method is flexible in studying the existence of affine-periodic solutions. The following results illustrate applications in this aspect.Theorem 8.
Assume that there exists a Lyapunov’s functionV:ℝ+1×ℝn→ℝ+1 such that(i)
V(t,x) is of ℂ1;(ii)
V′(t,x)≤-W(t,x), |x|≥M>0, where W(t,x) is continuous in ℝ+1×{|x|≤M}, and W(t,x)≥α>0, |x|≥M;(iii)
Uniformly int,
(28)liminf|x|→∞V(t,x)>sup{V(t,x):t∈ℝ+1,|x|≤M}.Then, the system (1) has a Q-affine-periodic solution.Proof.
Letx(t,x0) denote the solution of (1) with the initial value x(0)=x0. Put
(29)K=sup{V(t,x):t∈ℝ+1,|x|≤M},G={x∈ℝn:V(t,x)≤K}.
By assumption (iii), G is bounded and closed. In the following, we will prove that for each B>0, there are M=M(B)>0 and N=N(B)>0 such that
(30)|x(t,x0)|≤M,∀t∈[0,N],x(t,x0)∈G,∀t≥N,
whenever |x0|≤B.
In fact, given thatx0∈ℝn, |x0|>M implies on the maximal interval [0,L) that |x(t,x0)|>M; we have
(31)0≤V(t,x(t,x0))≤V(0,x0)-∫0tW(s,x(s,x0))ds≤V(0,x0)-αt.
This shows that there is t1∈(0,∞) such that
(32)|x(t,x0)|>M,∀t∈[0,t1),|x(t1,x0)|=M.
Note that
(33)V(t,x(t,x0))≤V(t1,x(t1,x0)),if|x(t1,x0)|≥M,
which together with the construction of G yields
(34)x(t,x0)∈G,t∈[t1,∞).
If |x0|<M, and there is a t-∈(0,∞) such that
(35)|x(t,x0)|<M,t∈(0,t-),|x(t-,x0)|=M,
then we also have
(36)x(t,x0)∈G,∀t∈[t-,∞).
Of course, in case of |x0|=M, we have
(37)x(t,x0)∈G,∀t∈[0,∞).
Taking these cases into account, we choose
(38)N=t1.
Now, the existence of affine-periodic solutions is an immediate consequence. The proof is complete.Theorem 9.
Assume that(39)〈x,f(t,x)〉≤-a(t)|x|2,
where a∈Loc(ℝ+1) satisfies
(40)∫0∞a(s)ds=∞,∫0∞a-(s)ds<∞.
Then, (1) has an affine-periodic solution.Proof.
Let(41)V(t,x)=12|x|2.
Then,
(42)V′(t,x)=〈x,f(t,x)〉≤-2a(t)V(t,x)⟹V(t,x(t,x0))≤e-∫0t‍2a(s)dsV(0,x0),∀t≥0.
By assumption, ∫0∞‍a(s)ds=∞, there is t1∈(0,∞) such that
(43)e-∫0t1‍2a(s)ds12|x|2≤1.
Thus,
(44)V(t,x(t,x0))≤e-∫t1∞‍2a-(s)ds,∀t≥t1,⟹|x(t,x0)|≤2e-∫t1∞‍a-(s)ds,∀t≥t1.
By Theorem 1, (1) has an affine-periodic solution. This finishes the proof.Example 10.
Consider the system(*)±x′=±|x|2βx+(e-12πΘt)≡f(t,x),
where β≥0;x∈ℂn; Θ=(θ1,θ2,…,θn)T, θi>0, i=1,2,…,n. Let
(45)Q=e-12πΘT,T>0.
Then
(46)f(t+T,x)=Qf(t,Q-1x).
In the following, we only consider the case (*)-. Otherwise, set t→-t for (*)+. Take V(t,x)=(1/2)|x|2. Notice that for |x|≥2=M, α=2,
(47)V′(t,x)=〈x,f(t,x)〉=xTf-(t,x)=-|x|2β+2+xTe--12πΘt≤-|x|2β+2+|x|≤-|x|=-W(t,x)≤-α.
Hence, by Theorem 8, (*)- has a Q-affine T-periodic solution. Now, if leting p/q be a reduced fraction and θiT=p/q, i=1,2,…,n, then the Q-affine T-periodic solutions are just q-subharmonic ones; if ΘT∈ℚn (the set of rational vectors), then there is a K such that these affine T-periodic solutions are K-periodic ones; if ΘT∈ℝn∖ℚn, then these solutions are quasiperiodic ones with frequency ΘT.



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