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

Guo, Ruichao; Jiang, Xiaomeng; Wang, Hongren

doi:https://doi.org/10.1155/2022/6874460

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Abstract Introduction Preliminaries Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 6874460 | https://doi.org/10.1155/2022/6874460

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

Ruichao Guo,¹Xiaomeng Jiang,²and Hongren Wang³

Academic Editor: Serkan Araci

Received27 Dec 2021

Revised09 Feb 2022

Accepted12 Feb 2022

Published31 Mar 2022

Abstract

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

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 . 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 , antiperiodic , and rotation periodic , which are discussed in many papers like [24, 25]. For general affine matrix , 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 -affine-periodic solutions and pseudo -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 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 -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 scale is a closed and nonempty subset of . It follows that forward jump operator , backward jump operator , and the graininess function , respectively, as follows:

In this definition, we supplement and . If , , , , we say that the point is right-scattered, right-dense, left-scattered, left-dense, respectively. If has a left-scattered maximum , . Otherwise, . Throughout this paper, we use to denote a closed interval restricted on . That is, .

Definition 1. Assumeand. The delta derivative of the vectoratiswith the property if for given any, there is a neighborhoodofsuch thatfor all .

Throughout this paper, we use to denote the set of rd-continuous functions. We use to denote the set of all regressive and rd-continuous functions from .

Next we will introduce the definition of periodic time scale.

Definition 2 (see [38]). We say that a time scaleis called T-periodic if there existssuch thatimplies. For, the smallest positiveis called the period of the time scale.

In what follows, we present the definition about the generalized exponential function . For more details, see [36, 37].

Definition 3. Given, the exponential function is defined bywhere is the principal logarithm function.

Suppose that . Then and can be defined as follows:

Definition 4 (see [33]). Assumeis rd-continuous matrix-valued function on . We say that the linear dynamic systemadmits an exponential dichotomy onif there exist a projection on and positive constants and satisfyingwhereis a fundamental solution matrix of (6) and is the identity matrix.

The following theorem will be essential to our main result, for more details see [[36], Theorem 2.39].

Theorem 1. Ifand, then

The following result shows that is a bounded function (see [[33], Theorem 2.14]).

Lemma 1. If, thenforsuch that.

3. Exponential Dichotomy and -Affine-Periodic Solutions of Dynamic Equations on Time Scales

We start by introducing definitions of -affine-periodic systems on time scales.

Consider the -affine-periodic dynamic system on time scaleswhere is rd-continuous, is a -periodic time scale and ensures the uniqueness of solutions with respect to initial value (for more details in [37]). We always assume in this paper.

Definition 5 (see [38]). The system (10) is said to be aaffine-periodic system, if there existssuch thatholds for all .

Definition 6 (see [38]). A functionis called a solution of (10) ifandsatisfies (10) for all .

Definition 7 (see [38]). A functionis said to be an (Q, T)-affine-periodic solution of (Q, T)-affine-periodic system (10) ifis a solution of (10) and for any :

Remark 1. Ifis right-dense, then functionis an (Q, T)-affine-periodic solution of (Q, T)-affine-periodic system (10) providedis a solution of (10) and for any,

3.1. -Affine-Periodic Solutions of First-Order Linear Dynamic Equations on Time Scales

Consider the linear nonhomogeneous dynamic equation on time scaleswhere , , is a -periodic time scale, and its associated homogeneous equation is as follows:

Theorem 2. Let,beaffine-periodic on, i.e.,. Also, suppose that linear dynamic equation (15) has an exponential dichotomy with projection P. Then nonhomogeneous linear differential equation (14) has a -affine-periodic solution.

Proof. By exponential dichotomy of (15), we haveWe only need to prove that is -affine-periodic, i.e., . Let .
By the affine periodicity of and Theorem 1.16 in [36], we have the following:(a)If is a right-scattered point, then we have(b)If is a right-dense point, then we haveBy (a) and (b), we see that is the solution of (15) with the initial value . By the existence and uniqueness theorem on time scales (see [[36], Section 8.3]), we get . ThusBy (19) and variable substitution, we get for all ,Due to the exchangeability of matrix multiplication, i.e.,we haveThus, we confirm that and we get the desired result.

3.2. -Affine-Periodic Solutions for Semilinear Dynamic Equations on Time Scales

Consider the following semilinear dynamic equation:where and , is a -periodic time scale, are -affine-periodic, i.e.,

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 solutionas follows:

We present an existence and uniqueness result of -affine-periodic solution of (24).

Theorem 3. Consider the equation (24). Letbeaffine-periodic,be bounded. Assume that the following conditions hold:(i)Equation (15) admits exponential dichotomy on.(ii)There exist positive constantsandsuch that.(iii)For everyand, where .

Then, the system (24) has a unique solution which is-affine-periodic.

Proof. Firstly, we consider the nonhomogeneous linear equation:where is an rd-continuous function.
By Lemma 1, equation (28) has the following bounded solution:We prove that is -affine-periodic if is -affine-periodic:Let . By the -affine periodicity of , and (19), we obtainThus we get which means that is -affine-periodic.
Let andand define the norm as .
Define an operator as follows:It is easy to see that is a Banach space with norm . is obviously well defined.
Now, let us prove that is a contraction. We claim that there exists a fixed point of in . For any and , by Theorem 1 and Lemma 1, we haveThus is a contraction mapping on . By the Banach fixed point theorem, has a unique fixed point , which is the unique -affine-periodic solution of (24).

4. Pseudo -Affine-Periodic Solutions of Dynamic Equations on Time Scales

In this section, we consider dynamic equation on time scales given bywhere and , is a -periodic time scale, are -affine-periodic, i.e.,and its associated homogeneous equation is as follows:

For , define

Definition 8. A functionis called asymptotically (Q, T)-affine-periodic solution if there exist,such that. Denote bythe set of asymptotically (Q, T)-affine-periodic solution.

Definition 9. A functionis called pseudo (Q, T)-affine-periodic solution if there exist,such that. Denote bythe set of pseudo (Q, T)-affine-periodic solution.

Remark 2. Note thatandare uniquely determined.

Remark 3. It is obvious that.

Now, we present an existence and uniqueness result of pseudo -affine-periodic solution of (35).

Theorem 4. Consider equation (35). Letbe aperiodic time scale andbe pseudo-affine-periodic on time scales. Suppose also the following conditions are fulfilled:(i)is a-affine-periodic matrix function on time scales.(ii)(37) has a exponential dichotomy onwith projectionand positive constantsand.(ii)There exists a constantsuch thatfor every and , where .

Then, the system (35) has a unique pseudoaffine-periodic solution.

Proof. By Lemma 2, equation (35) has the following bounded solution:Define an operator as follows:Let us prove that is well defined. In fact, for , we can rewrite , where and .
Therefore, we getLetSince is -affine-periodic, it is not difficult to show that is -affine-periodic. We prove that is -affine-periodic if is -affine-periodic.Let. By the -affine periodicity of , and (19), we obtainThus we get which means that is -affine-periodic.
Then we need to prove thatBy the exponential dichotomy and the Lipchitz condition, we havewhereSince, hence as . And is integrable on , respectively. From Lebesque dominated convergence theorem, it follows thatThenHence is well defined. It remains to prove that is a contraction. For any and , by Theorem 1 and Lemma 1, we haveThus is a contraction mapping on . By the Banach fixed point theorem, has a unique fixed point , which is the unique pseudo affine-periodic solution of (35).

Data Availability

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

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by NSFC Grant (No.12001224), Education Department of Jilin Province 13th Five-Year Plan to support scientific research projects (JJKH20200137KJ), and Science and Technology Department of Jilin Province funds (20210101153JC).

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Copyright © 2022 Ruichao Guo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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