Almost Periodic Solutions And Spectrum Analysis For Differential Equations With Piecewise Constant Argument

The theory of almost periodic functions was mainly created by the Danish mathe-matician H. Bohr during 1925-1926. Almost periodic functions, with a superior spatialstructure, are a generalization of periodic functions. Moreover, in real life, almost periodicphenomenon is more common than periodic phenomenon. From the development processof almost periodic functions, we know that the broader study of almost periodic functionsand the application of almost periodic type functions in equations, which mainly focus onthe discussions of the existence, uniqueness, stability of almost periodic type solutions ofequations, are the main directions.Differential equations with piecewise constant argument were proposed and studiedfirstly by K. Cooke, J. Wiener, S. Shah, et. al. Such equations combine the propertiesof differential equations and difference equations, and usually describe hybrid dynamicalsystems, have important applications in the control theory, biological model theory andhyperbolic dynamic systems. For these reasons, more and more people are concernedabout the existence and uniqueness of almost periodic type solutions of differential equa-tions with piecewise constant argument.In this paper, we study the existence, uniqueness and the spectrum property of almostperiodic solutions of two kind of differential equations with piecewise constant argument,i.e. differential equations with piecewise constant argument [·] and differential equationswith piecewise constant argument [·+2 1]. The main work is as follows:Firstly, the existence of almost periodic weak solutions and almost periodic solu-tions of differential equations with piecewise constant argument [·+2 1] are studied. Undera special condition, the existence and uniqueness of almost periodic type solutions forthese equations already have been studied in detail. If remove this special condition,there is no paper yet available to study the existence of almost periodic type solutions ofsuch equations. The first part of this paper is to study the existence of almost periodicweak solutions and solutions of such equations without the special condition. First ofall, we obtain difference equations from differential equations, using some property ofshift operators, the existence of almost periodic sequence solutions of difference equa-tions is obtained, then, we construct almost periodic weak solutions and solutions of this equations piecewise by almost periodic sequence solutions, therefore, the existence ofalmost periodic weak solutions, almost periodic solutions are proved. Meanwhile, someexamples are given to show that these solutions may not be unique, and also illustrate therelationship between weak solution and solution.Secondly, although, there are many conclusions about the existence and uniquenessof almost periodic type solutions for differential equations with piecewise constant argu-ment [·+2 1], there is no result about the spectrum property of almost periodic solutions ofsuch equations. The second part of this paper is to study the spectrum property of almostperiodic solutions for such equations. We firstly give the expression of almost periodicsequence solutions of difference equations corresponding to such equations, and the ex-pression of almost periodic solutions of such equations. By means of an approximationtheorem, the spectrum property of almost periodic sequence solutions is obtained, basedon this, the spectrum property of almost periodic solutions is obtained.Thirdly, the existence and uniqueness of almost periodic solutions for differentialequations with piecewise constant argument [·] are prove, furthermore, the expression ofalmost periodic solutions is obtained. Based on this, the spectrum property of the almostperiodic solutions is studied. As for the existence and uniqueness of almost periodic so-lutions of such equations, there have been literatures published on this, different methodsare used in this paper. As for the spectrum property of almost periodic solutions of suchequations, there already has a literature concerning this, however, some counterexamplesare constructed to show the results obtained in the literature are not correct. We correct it.