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

The aim of this paper is to study the existence of ( Q,T ) -aﬃne-periodic solutions for aﬃne-periodic systems on time scales of the type x Δ ( t ) � A ( t ) x ( t ) + f ( t ) and x Δ ( t ) � A ( t ) x ( t ) + g ( t,x ( t )) , 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


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. e 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][10][11][12][13][14][15][16][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][19][20][21][22][23], which are not only periodic in time, but also symmetric in space.
is solution is a kind of periodic (Q � I l ), antiperiodic (Q � I l ), and rotation periodic (Q ∈ O(l)), which are discussed in many papers like [24,25]. For general affine matrix Q, Li et al. [19,20,26] obtained the existence of affineperiodic 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][29][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. e rest of this paper is arranged as follows: in Section 2, 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.

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 scale T 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: In this definition, we supplement inf∅ � supT and Definition 1. Assume u: T ⟶ R n and t ∈ T κ . e delta derivative of the vector u at t is u Δ (t) with the property if for given any ε > 0, there is a neighborhood U ⊂ T of t such that for all s ∈ U.
roughout this paper, we use C rd (T) � C rd (R n ) � C rd (T, R n ) to denote the set of rd-continuous functions. We use R � R(T) � R(T, 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 scale T is called In what follows, we present the definition about the generalized exponential function e p (t, s). For more details, see [36,37].
where log is the principal logarithm function.
Suppose that p, q ∈ R. en (p ⊕ q)(t) and (⊖p)(t) can be defined as follows: Definition 4 (see [33]). Assume A(t) is n × n rd-continuous matrix-valued function on T. We say that the linear dynamic system admits an exponential dichotomy on T if there exist a projection P on R n and positive constants K and α satisfying where X(t) is a fundamental solution matrix of (6) and I is the identity matrix.
e following theorem will be essential to our main result, for more details see [ [36], eorem 2.39].

Exponential Dichotomy and (Q, T)-Affine-Periodic Solutions of Dynamic Equations on Time Scales
We start by introducing definitions of (Q, T)-affine-periodic systems on time scales. Consider the (Q, T)-affine-periodic dynamic system on time scales where f: T × R n ⟶ R n 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 ∈ GL(R n ) in this paper.
Definition 5 (see [38]). e system (10) is said to be a (Q, T) affine-periodic system, if there exists Q such that holds for all (t, x) ∈ T × R n .
Definition 6 (see [38]). A function x: T ⟶ R n is called a solution of (10) if x ∈ y: y ∈ C(T, R n ), y Δ ∈ C r d (T, R n ) and x(t) satisfies (10) for all t ∈ T.
Definition 7 (see [38]). A function x: T ⟶ R n is said to be an (Q, T)-affine-periodic solution of (Q, T)-affine-periodic system (10) if x(σ(t)) is a solution of (10) and for any t ∈ T: is a solution of (10) and for any t ∈ T,

(Q, T)-Affine-Periodic Solutions of First-Order Linear
Dynamic Equations on Time Scales. Consider the linear nonhomogeneous dynamic equation on time scales where f: T ⟶ R n , A: T ⟶ R n×n , T is a T-periodic time scale, and its associated homogeneous equation is as follows: . Also, suppose that linear dynamic equation (15) has an exponential dichotomy with projection P. en nonhomogeneous linear differential equation (14) has a (Q, T)-affine-periodic solution.
Proof. By exponential dichotomy of (15), we have We only need to prove that x: By the (Q, T) affine periodicity of A(t) and eorem 1.16 in [36], we have the following: (a) If t is a right-scattered point, then we have (b) If t is a right-dense point, then we have By (a) and (b), we see that Ψ(t) is the solution of (15) with the initial value Ψ(0) � I. By the existence and uniqueness theorem on time scales (see [ [36], Section 8.3]), we get Ψ(t) � X(t). us By (19) and variable substitution, we get for all t ∈ T, Due to the exchangeability of matrix multiplication, i.e., we have us, we confirm that x(σ(t) + T) � Qx(σ(t)) and we get the desired result.

(Q, T)-Affine-Periodic Solutions for Semilinear Dynamic
Equations on Time Scales. Consider the following semilinear dynamic equation: where A: T ⟶ R n×n and f: T ⟶ R n , T is a T-periodic time scale, A(t), f(t, x) are (Q, T)-affine-periodic, i.e., Now, we introduce a definition about the solution of (24). (15) admits exponential dichotomy, then the system (24) has a solution x: T ⟶ R n as follows:

Journal of Mathematics
We present an existence and uniqueness result of (Q, T)-affine-periodic solution of (24).
Let Q ∈ GL(n) and and define the norm as ‖y‖ � sup t∈[0,T] T |y(t)|. Define an operator H: C T ⟶ C T as follows: It is easy to see that C T is a Banach space with norm ‖ · ‖. H is obviously well defined.

Journal of Mathematics
Now, let us prove that H is a contraction. We claim that there exists a fixed point of H in C T . For any y, z ∈ C T and t ∈ T, by eorem 1 and Lemma 1, we have us H(y)(·) is a contraction mapping on C T . By the Banach fixed point theorem, H has a unique fixed point x ⋆ (t) ∈ C T , which is the unique (Q, T)-affine-periodic solution of (24).

Pseudo (Q, T)-Affine-Periodic Solutions of Dynamic Equations on Time Scales
In this section, we consider dynamic equation on time scales given by where A: T ⟶ R n×n and f: and its associated homogeneous equation is as follows: For T > 0, define Denote by AAP T (T, R n ) the set of asymptotically (Q, T)-affine-periodic solution.
Remark 2. Note that f and g are uniquely determined.

Remark 3. It is obvious that
Now, we present an existence and uniqueness result of pseudo (Q, T)-affine-periodic solution of (35).

Theorem 4.
Consider equation (35). Let T be a T periodic time scale and f ∈ C r d (T × R n , R n ) be pseudo for every x, y ∈ R n and t ∈ T, where ] � sup t∈T |](t)|.
Then, the system (35) has a unique pseudo (Q, T) affineperiodic solution.
Proof. By Lemma 2, equation (35) has the following bounded solution:

(40)
Define an operator H: PAP T (T, R n ) ⟶ PAP T (T , R n ) as follows: Let us prove that H is well defined. In fact, for Let Let s � r + T. By the (Q, T)-affine periodicity of f(t, x), x 1 (t) and (19), we obtain us we get I(σ(t) + T) � QI(σ(t)) which means that en we need to prove that By the exponential dichotomy and the Lipchitz condition, we have Hence H is well defined. It remains to prove that H is a contraction. For any y, z ∈ PAP T (T, R n ) and t ∈ T, by eorem 1 and Lemma 1, we have (52) us H(y)(·) is a contraction mapping on C T . By the Banach fixed point theorem, H has a unique fixed point x ⋆ (t) ∈ PAP T , which is the unique pseudo (Q, T) affineperiodic solution of (35).

Data Availability
Supporting the results of the study can be found in the paper.

Conflicts of Interest
e authors declare that they have no conflicts of interest.