Partial differential equations generally have many different solutions a x u 2 2 2. Linear differential equations with almost periodic coefficients. This book is composed of 10 chapters that present important equations and the special functions they generate, ranging from mathieus equation to the intractable ellipsoidal wave equation. The equation is of first orderbecause it involves only the first derivative dy dx and not. Stepanov also introduced the notion of almost periodic function in view of the. Buy almost periodic solutions of impulsive differential equations lecture notes in mathematics.
Almost periodic functions in metric spaces harmonic analysis of almost periodic functions arithmetic properties of almost periods generalisation of the uniqueness theorem n almost periodic functions weakly almost periodic functions a theorem concerning the integral and certain questions of harmonic analysis stability in the sense. Over the last years, the question of studying almost periodic functions in robotics 1521, dynamic. Asymptotically almost periodic solutions of functionally di. Almost periodic solutions of impulsive differential equations. Instructors solutions manual partial differential equations. Stamov 2012, paperback at the best online prices at ebay. Semiperiodic solutions of difference and differential. Pdf almost periodic functions, bohr compactification, and.
On the bohr transform of almost periodic solutions for some differential equations in abstract spaces article pdf available in international journal of mathematics and mathematical sciences 279. The besicovitchdoss cnalmost periodic functions and solutions of abstract inhomogenous volterra integrodifferential equations belong. This book discusses almost periodic and almost automorphic solutions to abstract integro differential volterra equations that are degenerate in time, and in particular equations whose solutions are governed by degenerate solution operator families with removable singularities at. Almost periodic functions, bohr compactification, and. This idea will be integral to what well be doing in the remainder of this chapter and in the next chapter as we discuss one of the basic solution methods for partial differential equations. Differential equations with discontinuous delay 3 on cp, with a, b almost automorphic matrix valued functions and f a z almost auto morphic function. In section4, under matched spaces, through the basic properties of d almost periodic functions, some fundamental results of homogeneous and nonhomogeneous dynamic equations are established in which a suf. Mit opencourseware makes the materials used in the teaching of almost all of mits subjects available on the web, free of charge. Linear differential equations with almost periodic. Weakly asymptotically almost periodic functions 6 4.
The spaces of semi periodic sequences and functions are examined in the relationship to the closely related notions of almost periodicity, quasi. Laplaces equation recall the function we used in our reminder. The bohr compactification is shown to be the natural setting for studying almost periodic functions. For the cases of partial abstract functional differential equations and. Using ergodicity of functions, we prove the existence and uniqueness of. Asymptotically almost periodic solutions of operator equations.
In addition, sufficient conditions for their asymptotic stability are obtained by means of. In this paper we are concerned with the almost periodicity of solutions of the differential equation where is a linear, closed operator on a. Periodic, quasiperiodic, almost periodic, almost automorphic. By means of the contraction mapping principle, we establish some criteria for the existence and uniqueness of almost periodic solutions for this. Volume 259, issue 8, 15 october 2015, pages 42024228. Almost periodic and almost automorphic solutions to. In this paper we study the existence of almost periodic solutions of an autonomous neutral functional di. Almost periodic and pseudoalmost periodic solutions to. Quasi periodic functions possess the following properties. Evidently, the sum of these two is zero, and so the function ux,y is a solution of the partial differential equation. Differential and integral equations project euclid. Almost periodic differential equations springerlink. Some remarks on almost periodic functions in view of the lebesgue measure with applications to linear differential equations dariusz bugajewski and adam nawrocki adam mickiewicz university, faculty of mathematics and computer science ul. Pdf almost periodic functions, bohr compactification.
We consider a system of linear differential equations 1 x atx ddt where x is an n dimensional column vector and 40 is an nxn matrix whose elements are continuous periodic functions of a real variable. General firstorder differential equations and solutions a firstorder differential equation is an equation 1 in which. Composition of piecewise pseudo almost periodic functions. Periodic functions periodic functions are functions which repeat. Almost periodic functions and applications to dynamic. In the present book a systematic exposition of the results related to almost periodic solutions of impulsive differential equations is given and the potential for their application is illustrated. Almost periodic functions and differential equations book. Applying this method, we prove that in any neighbourhood of an almost periodic skewhermitian linear differential system there exists a system which does not possess a nontrivial almost periodic solution.
Almost periodic functions, bohr compactification, and differential equations springerlink. We analyze the existence of almost periodic respectively, almost automorphic, recurrent solutions of a linear nonhomogeneous differential or difference equation in a banach space, with almost periodic respectively, almost automorphic, recurrent coefficients. Favard separation method for almost periodic stochastic differential. We apply these composition theorems to investigate the existence of piecewise pseudo almost periodic mild solutions to abstract impulsive differential equations. The existence of almost periodic and asymptotically almost periodic solutions is one of the most attractive topics in the qualitative theory of differential equations due to their signi. The theory of almost periodic functions was founded by bohr in 19241926 8, 9, 10, and. We apply our results to transport models and to the wave equation. Asymptotically almost periodic solutions of differential equations. Pdf on the bohr transform of almostperiodic solutions. A consequence of the definition of the classes of almost periodic functions through the concept of closure is the approximation theorem. A scale of almost periodic functions spaces corduneanut, c. Discontinuous almost periodic type functions, almost. Under some conditions we prove that one of the following alternatives is fulfilled. The concept was first studied by harald bohr and later generalized by vyacheslav stepanov, hermann weyl and abram samoilovitch besicovitch, amongst others.
The need of considering z almost automorphic functions appears explic itly, even if f is almost automorphic, in the solution of system 1. Asymptotically almost periodic functions in the sobolev spaces 1 4. Coincidence degree methods in almost periodic differential equations qi, liangping and yuan, rong, topological methods in nonlinear analysis, 2017. Then, we consider a class of neutral functional dynamic equations with stepanov almost periodic terms on time scales in a banach space. This interesting monograph written by a known specialist in the field is addressed to a wide audience, not. Almost periodic sequence solutions of a discrete predatorprey system with beddingtondeangelis functional response du, zengji and li. In this article, by using schauders fixed point theorem, we study the existence of almost periodic solutions for abstract impulsive differential equations. Besicovitch almost periodic solutions for a class of. Almost periodic solutions of impulsive differential equations 2047 by gani t. In this paper we study the existence of almost periodic solutions for linear retarded functional differential equations with finite delay and values in a banach space. We relate the existence of almost periodic solutions with the stabilization of distributed control systems.
Applications to partial differential equations are also given. As an application, the existence and uniqueness results are formulated for semi periodic solutions of quasilinear difference and differential equations. Asymptotically almost periodic and almost periodic. Almost periodic and almost automorphic solutions of linear differential equations. Pdf almost periodic and almost automorphic solutions of. View the article pdf and any associated supplements and figures for a period of 48 hours. In this connection, interest has arisen in the study of almost periodic solutions of differential equations and differential equations with almost periodic coef. The theory of almost periodic functions has been developed in connection with prob lems of differential equations, stability theory, dynamical. The final topic that we need to discuss here is that of orthogonal functions.
On the almost periodic solutions of differential equations. Ergodicity and asymptotically almost periodic solutions of. We present one modifiable method for constructing almost periodic functions in. Existence and stability of almost periodic solutions for. Almost periodic solutions of impulsive differential. If you want to learn differential equations, have a look at differential equations for engineers if your interests are matrices and elementary linear algebra, try matrix algebra for engineers if you want to learn vector calculus also known as multivariable calculus, or calculus three, you can sign up for vector calculus for engineers. Quasiperiodic function encyclopedia of mathematics. Asymptotically almost periodic solutions to some classes of secondorder functional differential equations diagana, toka, henriquez, hernan, and hernandez, eduardo, differential and integral equations, 2008. Almost periodic solutions for first order differential equations. In this paper, we introduce the concept of piecewise pseudo almost periodic functions on a banach space and establish some composition theorems of piecewise pseudo almost periodic functions.
Once bohr established his fundamental theorem, he was able to show that any continuous almost periodic function is the limit of a uniformly convergent sequence of trigonometric polynomials. Almostperiodic function encyclopedia of mathematics. Asymptotic almost periodic functions and other weaker conditions. Differential equations department of mathematics, hong. For the differential equation, on a hilbert space, we find the necessary and sufficient conditions that the abovementioned equation has a unique almost periodic solution. With more than 2,400 courses available, ocw is delivering on. Almost periodic functions, bohr compactification, and differential equations article pdf available in milan journal of mathematics 661. In mathematics, an almost periodic function is, loosely speaking, a function of a real number that is periodic to within any desired level of accuracy, given suitably long, welldistributed almost periods. Over the last years, the question of studying almost periodic functions in. Almost periodic solution for neutral functional dynamic. We first propose a concept of almost periodic functions in the sense of stepanov on time scales. We denote by apr the set of all almost periodic functions in the sense of bohr on r. Almost periodic and almost automorphic solutions of linear.
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