# (Q, T)-affine-periodic solutions and Pseudo (Q, T)-affine-periodic solutions for Dynamic Equations on Time Scales

Authors: Ruichao Guo; Xiaomeng Jiang; Hongren Wang
Journal: Journal of Mathematics
 (2022)

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## Abstract

The aim of this paper is to study the existence ofQ,T-affine-periodic solutions for affine-periodic systems on time scales of the type xΔt=Atxt+ft and xΔt=Atxt+gt,xt,t∈T, assuming that corresponding homogeneous equation of this system admits exponential dichotomy. The result is also extended to the case of pseudo Q,T-affine-periodic solutions. The main approaches are based on the Banach contraction mapping principle, but certain technical aspects on time scales are more complicated.


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## Body

## 1. Introduction

In the mathematical theory of dynamical systems, exponential dichotomy plays a significant role in the analysis of nonautonomous dynamical systems. Exponential dichotomy was first studied by Lyapunov and Poincaré in the late nineteen century. Perron [1] developed the exponential dichotomy of linear differential equations, particularly, it has attracted wide attention in the field of stable and unstable invariant manifolds, and therefore has attracted much attention, see, for example, Coppel [2], Chow and Leiva [3], and Fink [4], and references therein. Li [5] studied analogous results for nonautonomous discrete time dynamical systems. Pözsche [6] introduced the notion of the exponential dichotomy in the calculus on measure chains. Chu et al. [7] introduced a definition for nonuniform dichotomy spectrum and proved the reducibility result.As we know, discrete time systems and continuous systems play important role in theory and application. The theory of time scales is a recent subject of research, which was introduced by Stefan Hilger in 1988 in order to unified continuous time and discrete time dynamic systems (see [8]). A time scale is a nonempty closed subset of real numbers, denoted by T. Since the theory of time scales can also describe continuous and discrete hybrid processes, it has an important role to model realistic problems, for instance, in the study of finance, economic, population models, neural networks, quantum physics, and technology. See [9–17] for more details.Since Kepler and Newton studied the motion of celestial bodies, the research of periodic solutions has a long history. As we know, the problem of existence and uniqueness of a periodic solution of differential equations has been the main research topic of dynamic systems. Recently, the concept of affine periodicity has been introduced by Li et al. [18–23], which are not only periodic in time, but also symmetric in space. This solution is a kind of periodic Q=Il, antiperiodic Q=Il, and rotation periodic (Q∈Ol), which are discussed in many papers like [24, 25]. For general affine matrix Q, Li et al. [19, 20, 26] obtained the existence of affine-periodic solutions for several kinds of affine-periodic systems.Recently, the qualitative properties of solutions of dynamic equations on time scales have been attracting of several researchers [27], specially concerning their periodicity [28–30]. On the contrary, Li and Wang in [31, 32] obtained almost periodicity on time scales. C. Lizama and L. G. Mesquita (see [33]) introduced the existence of almost automorphic on time scales. After that, Wang and Li proved the existence of (Q, T)-affine-periodic solutions on time scales via topological degree theory in [34].As we know, exponential dichotomy is one of the most important methods and tools in the study of periodic solutions of difference equations and differential equations. C. Cheng and Y. Li considered nonhomogeneous linear differential equations and semilinear differential equations and proved the existence of (Q, T)-affine-periodic solutions in [35]. Inspired by the above discussion, the main purpose of this paper is to investigate the existence of Q,T-affine-periodic solutions and pseudo Q,T-affine-periodic solutions when the corresponding homogeneous linear equations have exponential dichotomy on time scales.The rest of this paper is arranged as follows: in Section2, we give some preliminaries concerning the theory of time scales. We introduce the definitions of related concepts and some properties for dynamic equations on time scales. In Section 3, we first present some concepts for (Q, T)-affine-periodic solutions on time scales. We show the existence of (Q, T)-affine-periodic solutions of first-order linear dynamic equations on time scales and prove the existence and uniqueness of (Q, T)-affine-periodic solutions for semilinear dynamic equations on time scales. In Section 4, we obtain the pseudo Q,T-affine-periodic solutions for semilinear dynamic equations on time scales.


## 2. Preliminaries

In this section, we will introduce some basic notations, definitions, and results concerning the calculus on time scales which can be found in [36, 37].A time scaleT is a closed and nonempty subset of ℝ. It follows that forward jump operator σ:T⟶T, backward jump operator ρ:T⟶T, and the graininess function μ:T⟶[0,∞), respectively, as follows:(1)σt=infs∈T:s>t,ρt=sups∈T:s<t,μt=σt−t.In this definition, we supplementinf∅=supT and sup∅=infT. If σt>t, σt=t, ρt<t, ρt=t, we say that the point t∈T is right-scattered, right-dense, left-scattered, left-dense, respectively. If T has a left-scattered maximum M, Tκ=T\M. Otherwise, Tκ=T. Throughout this paper, we use a,bT to denote a closed interval restricted on T. That is, a,bT=t∈T;a⩽t⩽b.Definition 1.
Assumeu:T⟶ℝnandt∈Tκ. The delta derivative of the vectoruattisuΔtwith the property if for given anyε>0, there is a neighborhoodU⊂Toftsuch that(2)|uσt−us−uΔtσt−s|≤ε|σt−s|,for all s∈U.Throughout this paper, we useCrdT=Crdℝn=CrdT,ℝn to denote the set of rd-continuous functions. We use ℛ=ℛT=ℛT,ℝ to denote the set of all regressive and rd-continuous functions from T⟶ℝ.Next we will introduce the definition of periodic time scale.Definition 2 (see [38]).
We say that a time scaleTis called T-periodic if there existsT>0such thatt∈Timpliest±T∈T. ForT≠ℝ, the smallest positiveTis called the period of the time scale.In what follows, we present the definition about the generalized exponential functionept,s. For more details, see [36, 37].Definition 3.
Givenp∈ℛ, the exponential function is defined by(3)ept,s=exp∫st1μτlog(1+pτμτ)Δτ,μτ>0,exp∫stpτΔτ,μτ=0,where log is the principal logarithm function.Suppose thatp,q∈ℛ. Then p⊕qt and ⊖pt can be defined as follows:(4)p⊕qt=pt+qt+μtptqtfor allt∈Tk.(5)⊖pt=−pt1+μtptfor allt∈Tk.Definition 4 (see [33]).
AssumeAtisn×n rd-continuous matrix-valued function on T. We say that the linear dynamic system(6)xΔt=Atxt,admits an exponential dichotomy onTif there exist a projectionP on ℝn and positive constants K and α satisfying(7)|XtPX−1s|≤Ke⊖αt,s,s,t∈T,t≥s,|XtI−PX−1s|≤Ke⊖αs,t,s,t∈T,t≥s,whereXtis a fundamental solution matrix of (6) andI is the identity matrix.The following theorem will be essential to our main result, for more details see [[36], Theorem 2.39].Theorem 1.
Ifp∈ℛanda,b,c∈T, then(8)epc,·Δ=−pepc,·σ.(9)∫abptepc,σtΔt=epc,a−epc,b.The following result shows thate⊖pt,s is a bounded function (see [[33], Theorem 2.14]).Lemma 1.
Ifp>0,t>0, thene⊖pt,s≤1fort,s∈Tsuch thatt>s.


## 3. Exponential Dichotomy andQ,T-Affine-Periodic Solutions of Dynamic Equations on Time Scales

We start by introducing definitions ofQ,T-affine-periodic systems on time scales.Consider theQ,T-affine-periodic dynamic system on time scales(10)xΔ=ft,x,where f:T×ℝn⟶ℝn is rd-continuous, T is a T-periodic time scale and ensures the uniqueness of solutions with respect to initial value (for more details in [37]). We always assume Q∈GLℝn in this paper.Definition 5 (see [38]).
The system (10) is said to be aQ,Taffine-periodic system, if there existsQsuch that(11)ft+T,x=Qft,Q−1x.holds for all t,x∈T×ℝn.Definition 6 (see [38]).
A functionx:T⟶ℝnis called a solution of (10) ifx∈y:y∈CT,ℝn,yΔ∈CrdT,ℝnandxtsatisfies (10) for all t∈T.Definition 7 (see [38]).
A functionx:T⟶ℝnis said to be an (Q, T)-affine-periodic solution of (Q, T)-affine-periodic system (10) ifxσtis a solution of (10) and for any t∈T:(12)xσt+T=Qxσt.Remark 1.
Iftis right-dense, then functionx:T⟶ℝnis an (Q, T)-affine-periodic solution of (Q, T)-affine-periodic system (10) providedxtis a solution of (10) and for anyt∈T,(13)xt+T=Qxt.


### 3.1.Q,T-Affine-Periodic Solutions of First-Order Linear Dynamic Equations on Time Scales

Consider the linear nonhomogeneous dynamic equation on time scales(14)xΔt=Atxt+ft,where f:T⟶ℝn, A:T⟶ℝn×n, T is a T-periodic time scale, and its associated homogeneous equation is as follows:(15)xΔt=Atxt.Theorem 2.
LetAt,ftbeQ,Taffine-periodic onT, i.e.,At+T=QAtQ−1,ft+T=Qft. Also, suppose that linear dynamic equation (15) has an exponential dichotomy with projection P. Then nonhomogeneous linear differential equation (14) has a Q,T-affine-periodic solution.Proof.
By exponential dichotomy of (15), we have(16)xt=∫−∞tXtPX−1σsfsΔs−∫t+∞XtI−PX−1σsfsΔs.
We only need to prove thatx:T⟶ℝn is Q,T-affine-periodic, i.e., xσt+T=Qxσt. Let Ψt=Q−1Xt+TX−1TQ.
By theQ,T affine periodicity of At and Theorem 1.16 in [36], we have the following:(a)
Ift is a right-scattered point, then we have(17)ΨΔt=Ψσt−Ψtμt=Q−1Xσt+TX−1TQ−Q−1Xt+TX−1TQμt=Q−1Xσt+T−Xt+TμtX−1TQ=Q−1XΔt+TX−1TQ=AtΨt.(b)
Ift is a right-dense point, then we have(18)ΨΔt=limΔt⟶0Ψt+Δt−ΨtΔt=limΔt⟶0Q−1Xt+Δt+TX−1TQ−Q−1Xt+TX−1TQΔt=Q−1limΔt⟶0Xt+Δt+T−Xt+TΔtX−1TQ=Q−1XΔt+TX−1TQ=Q−1At+TXt+TX−1TQ=Q−1QAtQ−1Xt+TX−1TQ=AtΨt.
By (a) and (b), we see thatΨt is the solution of (15) with the initial value Ψ0=I. By the existence and uniqueness theorem on time scales (see [[36], Section 8.3]), we get Ψt=Xt. Thus(19)Xt+T=QXtQ−1XT.
By (19) and variable substitution, we get for all t∈T,(20)xσt+T=∫−∞σt+TXσt+TPX−1σsfsΔs−∫σt+T+∞Xσt+TI−PX−1σsfsΔs=∫−∞σtXσt+TPX−1σs+Tfs+TΔs−∫σt+∞Xσt+TI−PX−1σs+Tfs+TΔs=∫−∞σtQXσtQ−1XTP(QX(σt)Q−1XT)−1QfsΔs−∫σt+∞QXσtQ−1XTI−P(QX(σt)Q−1XT)−1QfsΔs=∫−∞σtQXσtQ−1XTPX−1TQX−1σtQ−1QfsΔs−∫σt+∞QXσtQ−1XTI−PX−1TQX−1σtQ−1QfsΔs.
Due to the exchangeability of matrix multiplication, i.e.,(21)Q−1XTP=PQ−1XT),(22)Q−1XTI−P=I−PQ−1XT,we have(23)xσt+T=Q∫−∞σtXσtPX−1σsfsΔs−∫σt+∞XσtI−PX−1σsfsΔs.
Thus, we confirm thatxσt+T=Qxσt and we get the desired result.


### 3.2.Q,T-Affine-Periodic Solutions for Semilinear Dynamic Equations on Time Scales

Consider the following semilinear dynamic equation:(24)xΔt=Atxt+ft,x,where A:T⟶ℝn×n and f:T⟶ℝn, T is a T-periodic time scale, At,ft,x are Q,T-affine-periodic, i.e.,(25)At+T=QAtQ−1,ft+T,x=Qft,Q−1x.Now, we introduce a definition about the solution of (24).Lemma 2.
If the linear system (15) admits exponential dichotomy, then the system (24) has a solutionx:T⟶ℝnas follows:(26)xt=∫−∞tXtPX−1σsfs,xsΔs−∫t+∞XtI−PX−1σsfs,xsΔs.We present an existence and uniqueness result ofQ,T-affine-periodic solution of (24).Theorem 3.
Consider the equation (24). LetAt,ft,xbeQ,Taffine-periodic,ft,xbe bounded. Assume that the following conditions hold:(i)
Equation (15) admits exponential dichotomy onT.(ii)
There exist positive constantsNandγsuch that0<L<γ/2N2+ν˜γ.(iii)
For everyx,y∈ℝnandt∈T,(27)|ft,x−ft,y|≤L|x−y|,whereν˜=supt∈T|νt|.Then, the system (24) has a unique solution which isQ,T-affine-periodic.Proof.
Firstly, we consider the nonhomogeneous linear equation:(28)xΔt=Atxt+ft,yt,where yt is an rd-continuous function.
By Lemma1, equation (28) has the following bounded solution:(29)xt=∫−∞tXtPX−1σsfs,ysΔs−∫t+∞XtI−PX−1σsfs,ysΔs.
We prove thatxt is Q,T-affine-periodic if yt is Q,T-affine-periodic:(30)xσt+T=∫−∞σt+TXσt+TPX−1σsfs,ysΔs−∫σt+T+∞Xσt+TI−PX−1σsfs,ysΔs.
Lets=r+T. By the Q,T-affine periodicity of ft,y, yt and (19), we obtain(31)xσt+T=∫−∞σtXσt+TPX−1σr+Tfr+T,yr+TΔs−∫σt+∞Xσt+TI−PX−1σr+Tfr+T,yr+TΔs=∫−∞σtQXσtQ−1XTP(QX(σr)Q−1XT)−1Qfr,yrΔs−∫σt+∞QXσtQ−1XTI−P(QX(σr)Q−1XT)−1Qfr,yrΔs=∫−∞σtQXσtPX−1σrfr,yrΔs−∫σt+∞QXσtI−PX−1σrfr,yrΔs.
Thus we getxσt+T=Qxσt which means that xt is Q,T-affine-periodic.
LetQ∈GLn and(32)CT=y:T⟶ℝn:yσt+T=Qyσt,for allt∈T,and define the norm as y=supt∈0,TT|yt|.
Define an operatorH:CT⟶CT as follows:(33)Hyt=∫−∞tXtPX−1σsfs,ysΔs−∫t+∞XtI−PX−1σsfs,ysΔs.
It is easy to see thatCT is a Banach space with norm ·. H is obviously well defined.
Now, let us prove thatH is a contraction. We claim that there exists a fixed point of H in CT. For any y,z∈CT and t∈T, by Theorem 1 and Lemma 1, we have(34)Hy·−Hz·=∫−∞tXtPX−1σsfs,y−fs,zΔs−∫t+∞XtI−PX−1σsfs,y−fs,zΔs≤supt∈0,TT∫−∞tNe⊖γt,σsL|ys−zs|Δs+∫t+∞Ne⊖γσs,tL|ys−zs|Δs≤1⊖γNe⊖γt,t−Ne⊖γt,+∞L‖y−z‖+∫t+∞Neγt,σsL‖y−z‖Δs≤1⊖γN−Ne⊖γt,+∞L‖y−z‖+1γN−Neγt,+∞L‖y−z‖≤1γ/1+ν˜γN−Ne⊖γt,−∞L‖y−z‖+1γN−Neγt,+∞L‖y−z‖≤1+ν˜γγN|+|Ne⊖γt,+∞L‖y−z‖+1γN|+|Ne⊖γt,+∞L‖y−z‖≤L‖y−z‖2N1+ν˜γγ+2Nγ=L2N2+ν˜γγ‖y−z‖≤‖y−z‖.
ThusHy· is a contraction mapping on CT. By the Banach fixed point theorem, H has a unique fixed point x⋆t∈CT, which is the unique Q,T-affine-periodic solution of (24).


## 3.1.Q,T-Affine-Periodic Solutions of First-Order Linear Dynamic Equations on Time Scales

Consider the linear nonhomogeneous dynamic equation on time scales(14)xΔt=Atxt+ft,where f:T⟶ℝn, A:T⟶ℝn×n, T is a T-periodic time scale, and its associated homogeneous equation is as follows:(15)xΔt=Atxt.Theorem 2.
LetAt,ftbeQ,Taffine-periodic onT, i.e.,At+T=QAtQ−1,ft+T=Qft. Also, suppose that linear dynamic equation (15) has an exponential dichotomy with projection P. Then nonhomogeneous linear differential equation (14) has a Q,T-affine-periodic solution.Proof.
By exponential dichotomy of (15), we have(16)xt=∫−∞tXtPX−1σsfsΔs−∫t+∞XtI−PX−1σsfsΔs.
We only need to prove thatx:T⟶ℝn is Q,T-affine-periodic, i.e., xσt+T=Qxσt. Let Ψt=Q−1Xt+TX−1TQ.
By theQ,T affine periodicity of At and Theorem 1.16 in [36], we have the following:(a)
Ift is a right-scattered point, then we have(17)ΨΔt=Ψσt−Ψtμt=Q−1Xσt+TX−1TQ−Q−1Xt+TX−1TQμt=Q−1Xσt+T−Xt+TμtX−1TQ=Q−1XΔt+TX−1TQ=AtΨt.(b)
Ift is a right-dense point, then we have(18)ΨΔt=limΔt⟶0Ψt+Δt−ΨtΔt=limΔt⟶0Q−1Xt+Δt+TX−1TQ−Q−1Xt+TX−1TQΔt=Q−1limΔt⟶0Xt+Δt+T−Xt+TΔtX−1TQ=Q−1XΔt+TX−1TQ=Q−1At+TXt+TX−1TQ=Q−1QAtQ−1Xt+TX−1TQ=AtΨt.
By (a) and (b), we see thatΨt is the solution of (15) with the initial value Ψ0=I. By the existence and uniqueness theorem on time scales (see [[36], Section 8.3]), we get Ψt=Xt. Thus(19)Xt+T=QXtQ−1XT.
By (19) and variable substitution, we get for all t∈T,(20)xσt+T=∫−∞σt+TXσt+TPX−1σsfsΔs−∫σt+T+∞Xσt+TI−PX−1σsfsΔs=∫−∞σtXσt+TPX−1σs+Tfs+TΔs−∫σt+∞Xσt+TI−PX−1σs+Tfs+TΔs=∫−∞σtQXσtQ−1XTP(QX(σt)Q−1XT)−1QfsΔs−∫σt+∞QXσtQ−1XTI−P(QX(σt)Q−1XT)−1QfsΔs=∫−∞σtQXσtQ−1XTPX−1TQX−1σtQ−1QfsΔs−∫σt+∞QXσtQ−1XTI−PX−1TQX−1σtQ−1QfsΔs.
Due to the exchangeability of matrix multiplication, i.e.,(21)Q−1XTP=PQ−1XT),(22)Q−1XTI−P=I−PQ−1XT,we have(23)xσt+T=Q∫−∞σtXσtPX−1σsfsΔs−∫σt+∞XσtI−PX−1σsfsΔs.
Thus, we confirm thatxσt+T=Qxσt and we get the desired result.


## 3.2.Q,T-Affine-Periodic Solutions for Semilinear Dynamic Equations on Time Scales

Consider the following semilinear dynamic equation:(24)xΔt=Atxt+ft,x,where A:T⟶ℝn×n and f:T⟶ℝn, T is a T-periodic time scale, At,ft,x are Q,T-affine-periodic, i.e.,(25)At+T=QAtQ−1,ft+T,x=Qft,Q−1x.Now, we introduce a definition about the solution of (24).Lemma 2.
If the linear system (15) admits exponential dichotomy, then the system (24) has a solutionx:T⟶ℝnas follows:(26)xt=∫−∞tXtPX−1σsfs,xsΔs−∫t+∞XtI−PX−1σsfs,xsΔs.We present an existence and uniqueness result ofQ,T-affine-periodic solution of (24).Theorem 3.
Consider the equation (24). LetAt,ft,xbeQ,Taffine-periodic,ft,xbe bounded. Assume that the following conditions hold:(i)
Equation (15) admits exponential dichotomy onT.(ii)
There exist positive constantsNandγsuch that0<L<γ/2N2+ν˜γ.(iii)
For everyx,y∈ℝnandt∈T,(27)|ft,x−ft,y|≤L|x−y|,whereν˜=supt∈T|νt|.Then, the system (24) has a unique solution which isQ,T-affine-periodic.Proof.
Firstly, we consider the nonhomogeneous linear equation:(28)xΔt=Atxt+ft,yt,where yt is an rd-continuous function.
By Lemma1, equation (28) has the following bounded solution:(29)xt=∫−∞tXtPX−1σsfs,ysΔs−∫t+∞XtI−PX−1σsfs,ysΔs.
We prove thatxt is Q,T-affine-periodic if yt is Q,T-affine-periodic:(30)xσt+T=∫−∞σt+TXσt+TPX−1σsfs,ysΔs−∫σt+T+∞Xσt+TI−PX−1σsfs,ysΔs.
Lets=r+T. By the Q,T-affine periodicity of ft,y, yt and (19), we obtain(31)xσt+T=∫−∞σtXσt+TPX−1σr+Tfr+T,yr+TΔs−∫σt+∞Xσt+TI−PX−1σr+Tfr+T,yr+TΔs=∫−∞σtQXσtQ−1XTP(QX(σr)Q−1XT)−1Qfr,yrΔs−∫σt+∞QXσtQ−1XTI−P(QX(σr)Q−1XT)−1Qfr,yrΔs=∫−∞σtQXσtPX−1σrfr,yrΔs−∫σt+∞QXσtI−PX−1σrfr,yrΔs.
Thus we getxσt+T=Qxσt which means that xt is Q,T-affine-periodic.
LetQ∈GLn and(32)CT=y:T⟶ℝn:yσt+T=Qyσt,for allt∈T,and define the norm as y=supt∈0,TT|yt|.
Define an operatorH:CT⟶CT as follows:(33)Hyt=∫−∞tXtPX−1σsfs,ysΔs−∫t+∞XtI−PX−1σsfs,ysΔs.
It is easy to see thatCT is a Banach space with norm ·. H is obviously well defined.
Now, let us prove thatH is a contraction. We claim that there exists a fixed point of H in CT. For any y,z∈CT and t∈T, by Theorem 1 and Lemma 1, we have(34)Hy·−Hz·=∫−∞tXtPX−1σsfs,y−fs,zΔs−∫t+∞XtI−PX−1σsfs,y−fs,zΔs≤supt∈0,TT∫−∞tNe⊖γt,σsL|ys−zs|Δs+∫t+∞Ne⊖γσs,tL|ys−zs|Δs≤1⊖γNe⊖γt,t−Ne⊖γt,+∞L‖y−z‖+∫t+∞Neγt,σsL‖y−z‖Δs≤1⊖γN−Ne⊖γt,+∞L‖y−z‖+1γN−Neγt,+∞L‖y−z‖≤1γ/1+ν˜γN−Ne⊖γt,−∞L‖y−z‖+1γN−Neγt,+∞L‖y−z‖≤1+ν˜γγN|+|Ne⊖γt,+∞L‖y−z‖+1γN|+|Ne⊖γt,+∞L‖y−z‖≤L‖y−z‖2N1+ν˜γγ+2Nγ=L2N2+ν˜γγ‖y−z‖≤‖y−z‖.
ThusHy· is a contraction mapping on CT. By the Banach fixed point theorem, H has a unique fixed point x⋆t∈CT, which is the unique Q,T-affine-periodic solution of (24).


## 4. PseudoQ,T-Affine-Periodic Solutions of Dynamic Equations on Time Scales

In this section, we consider dynamic equation on time scales given by(35)xΔt=Atxt+ft,x,where A:T⟶ℝn×n and f:T⟶ℝn, T is a T-periodic time scale, At,ft,x are Q,T-affine-periodic, i.e.,(36)At+T=QAtQ−1,ft+T,x=Qft,Q−1x,and its associated homogeneous equation is as follows:(37)xΔt=Atxt.ForT>0, define(38)CT(T,ℝn)=φ∈Crd(T,ℝn):φ(t+T)=Qφt) for allt∈T,C0T,ℝn=φ∈CrdT,ℝn:lim|t|⟶+∞|φt|=0,PAP0T,ℝn=φ∈CrdT,ℝn:limr⟶+∞12r∫−rrφtΔt=0.Definition 8.
A functionφ∈CrdT×D,ℝnis called asymptotically (Q, T)-affine-periodic solution if there existf∈CTT,ℝn,g∈C0T,ℝnsuch thatφ=f+g. Denote byAAPTT,ℝnthe set of asymptotically (Q, T)-affine-periodic solution.Definition 9.
A functionφ∈CrdT×D,ℝnis called pseudo (Q, T)-affine-periodic solution if there existf∈CTT,ℝn,g∈PAP0T,ℝnsuch thatφ=f+g. Denote byPAPTT,ℝnthe set of pseudo (Q, T)-affine-periodic solution.Remark 2.
Note thatfandgare uniquely determined.Remark 3.
It is obvious thatCTT,ℝn⊂AAPTT,ℝn⊂PAPTT,ℝn.Now, we present an existence and uniqueness result of pseudoQ,T-affine-periodic solution of (35).Theorem 4.
Consider equation (35). LetTbe aTperiodic time scale andf∈CrdT×ℝn,ℝnbe pseudoQ,T-affine-periodic on time scales. Suppose also the following conditions are fulfilled:(i)
A∈ℛT,ℝn×nis aQ,T-affine-periodic matrix function on time scales.(ii)
(37) has a exponential dichotomy onTwith projectionPand positive constantsMandγ.(ii)
There exists a constant0<Lf<γ/2M2+ν˜γsuch that(39)ft,x−ft,y≤Lfx−y,for every x,y∈ℝn and t∈T, where ν˜=supt∈T|νt|.Then, the system (35) has a unique pseudoQ,Taffine-periodic solution.Proof.
By Lemma2, equation (35) has the following bounded solution:(40)xt=∫−∞tXtPX−1σsfs,xsΔs−∫t+∞XtI−PX−1σsfs,xsΔs.
Define an operatorH:PAPTT,ℝn⟶PAPTT,ℝn as follows:(41)Hxt=∫−∞tXtPX−1σsfs,xsΔs−∫t+∞XtI−PX−1σsfs,xsΔs.
Let usprove that H is well defined. In fact, for x∈PAPTT,ℝn, we can rewrite xt≡x1t+x2t, where x1t∈CTT,ℝn and x2t∈PAP0.
Therefore, we get(42)Hxt=∫−∞tXtPX−1σsfs,xs−fs,x1sΔs+∫−∞tXtPX−1σsfs,x1sΔs−∫t+∞XtI−PX−1σsfs,x1sΔs−∫t+∞XtI−PX−1σsfs,xs−fs,x1sΔs.
Let(43)It=∫−∞tXtPX−1σsfs,x1sΔs−∫t+∞XtI−PX−1σsfs,x1sΔs,(44)Ut=∫−∞tXtPX−1σs[fs,xs−fs,x1sΔs−∫t+∞XtI−PX−1σsfs,xs−fs,x1sΔs.
Sincex1 is Q,T-affine-periodic, it is not difficult to show that It is Q,T-affine-periodic. We prove that It is Q,T-affine-periodic if x1t is Q,T-affine-periodic.(45)Iσt+T=∫−∞σt+TXσt+TPX−1σsfs,x1sΔs−∫σt+T+∞Xσt+TI−PX−1σsfs,x1sΔs.
Lets=r+T. By the Q,T-affine periodicity of ft,x, x1t and (19), we obtain(46)Iσt+T=∫−∞σt+TXσt+TPX−1σsfsΔs−∫σt+T+∞Xσt+TI−PX−1σsfsΔs=∫−∞σtXσt+TPX−1σs+Tfs+TΔs−∫σt+∞Xσt+TI−PX−1σs+Tfs+TΔs=∫−∞σtQXσtQ−1XTPQXσtQ−1XT−1QfsΔs−∫σt+∞QXσtQ−1XTI−PQXσtQ−1XTT−1QfsΔs=∫−∞σtQXσtQ−1XTPX−1TQX−1σtQ−1QfsΔs−∫σt+∞QXσtQ−1XTI−PX−1TQX−1σtQ−1QfsΔs.
Thus we getIσt+T=QIσt which means that It is Q,T-affine-periodic.
Then we need to prove that(47)limr⟶+∞12r∫−rrUtΔt=0.
By theexponential dichotomy and the Lipchitz condition, we have(48)12r∫−rrUtΔt≤12r∫−rr∫−∞tMe⊖rt,σsLf|x2s|ΔsΔt+12r∫−rr∫t+∞Me⊖rt,σsLf|x2s|ΔsΔt=MLf∫−∞te⊖rt,σs12r|x2sΔs|Δt+MLf∫t+∞e⊖rt,σs12r|x2s|ΔsΔt=MLf∫−∞te⊖rt,σsΦrsΔt+∫t+∞e⊖rt,σsΦrsΔt,where(49)Φrs=12r∫−rrx2sΔs.
Sincex2t∈PAP0, hence Φrs⟶0 as r⟶+∞. And e⊖rt,σs is integrable on −∞,t),[t,+∞, respectively. From Lebesque dominated convergence theorem, it follows that(50)limr⟶+∞MLf∫−∞te⊖rt,σsΦrsΔt+∫t+∞e⊖rt,σsΦrsΔt=0.
Then(51)limr⟶+∞12r∫−rrUtΔt=0.
HenceH is well defined. It remains to prove that H is a contraction. For any y,z∈PAPTT,ℝn and t∈T, by Theorem 1 and Lemma 1, we have(52)Hz·−Hy·=∫−∞tXtPX−1σsfs,z−fs,yΔs−∫t+∞XtI−PX−1σsfs,z−fs,yΔs≤supt∈0,TT∫−∞tMe⊖γt,σsLf|ys−zs|Δs+∫t+∞Me⊖γσs,tLf|ys−zs|Δs≤1⊖γMe⊖γt,−∞−Me⊖γt,tLf‖z−y‖+∫t+∞Meγt,σsLf‖z−y‖Δs≤1⊖γMe⊖γt,−∞−MLf‖z−y‖+1γM−Meγt,+∞Lf‖z−y‖≤1γ/1+ν˜γe⊖γt,−∞−MLf‖z−y‖+1γM−Meγt,+∞Lf‖z−y‖≤1+ν˜γγM|+|Me⊖γt,−∞Lf‖z−y‖+1γM|+|Me⊖γt,+∞L‖z−y‖≤Lf‖z−y‖2M1+ν˜γγ+2Mγ=Lf2M2+ν˜γγ‖z−y‖≤‖y−z‖.
ThusHy· is a contraction mapping on CT. By the Banach fixed point theorem, H has a unique fixed point x⋆t∈PAPT, which is the unique pseudo Q,T affine-periodic solution of (35).



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