To facilitate this we give a new composition theorem of weighted pseudoalmost periodic functions. Prouse, almost periodic functions and functional equations, van nostrandreinhold, new york, ny, usa, 1971. Almost periodic solution for a delayed lotkavolterra system. Ircub\sblcubarcub,xdollar if and only if it is the sum of an almost periodic function and an ergodic perturbation. There are almost periodic functions which are not mean periodic. Composition theorems of stepanov almost periodic functions. A childs garden of almost periodic functions caltech math. Almost periodic functions on time scales and applications.
Asymptotically almost periodic solution to a class of volterra difference equations. Almost periodic solutions for a weakly dissipated hybrid system. Every value of p 0 is a period and so it has no minimal period. Robust almost periodic dynamics for interval neural networks with mixed timevarying delays and discontinuous. In this work, we give some theorems on mild weighted pseudoalmost periodic solutions for some abstract semilinear differential equations with uniform continuity. Almostperiodic function on a group encyclopedia of mathematics. Almost periodic function a function whose value is approximately repeated when its argument is increased by properly selected constants the almost periods. Permanence and almost periodic solutions of a discrete ratiodependent leslie system with time delays and feedback controls yu, gang and lu, hongying, abstract and applied analysis, 2012.
We study meromorphic functions in a strip almost periodic with respect to the spherical metric. Bohr, who developed the notion of a uniformly almostperiodic function in his study of dirichlet series. As such, it has been a challenge for constructive mathematicians to find a natural treatment of them. Grande, hierarchy of almostperiodic function spaces, rendiconti di matematica, vol. Almost periodic functions on locally compact groups. The most important examples are the trigonometric functions, which repeat over intervals of 2. Elementary properties of almost periodic functions almost periodic functions are a generalization of the periodic functions which leave intact the property of completeness of fourier series. There are several inequivalent definitions of almost periodic functions. Eigenvalues, almost periodic functions, and the derivative of an integral.
Newest almostperiodicfunction questions mathoverflow. Introduction finding a natural constructive treatment of the theory of almost periodic functions has long been a challenge for constructive mathematics, see 2, 5 and 7. Uniformly almost periodic functions and almost periodic. Computing with almost periodic functions computing with almost periodic functions moody, r. Almost periodic solutions for some convolution equations.
Request pdf basic properties of almost periodic functions in section 2. Pdf almost periodic functions, bohr compactification, and. A first motivation for the study of almost periodic functions is the set of the various kinds of combining. Pdf almost periodic functions, bohr compactification. The theory of almost periodic functions was created and developed in its main features by bohr as a generalization of pure periodicity. A decomposition theorem is given showing that a function is in dollarlcub\cal paprcublcub\rm j\. Almost periodicity is a structural property of functions, which is invariant with respect to the operations of addition and multiplication, and also in some cases with respect to division, differentiation, integration, and other limiting processes. Almost periodic solutions for some convolution equations in. The concept was first studied by harald bohr and later generalized by vyacheslav stepanov, hermann weyl and abram samoilovitch besicovitch, amongst others. Most periodic functions have a minimal period, which is often called either the period or the base period. Gibson, onthe almost periodicity oftrigonometric polynomials inconstructive mathematics, nederl. Asymptotically almost periodic solution to a class of. Pdf a various types of almost periodic functions on banach.
Grande, hierarchy of almost periodic function spaces, rendiconti di matematica, vol. By using the theory of exponential dichotomy on time scales and fixed point theory based on monotone operator, some simple conditions are obtained for the existence and uniqueness of positive almost periodic solution of the system. In fact by truncating the series for the riemann zeta function. You have remained in right site to begin getting this info. A function, f, from z to c is called almost periodic in bochner sense if and only if f is bounded and fnn. The theory of almost periodic functions was initiated by h. The almost periodic functions form a natural example of a nonseparable normed space. Almost periodic function article about almost periodic. Almostperiodic functions and functional equations springerlink. An ap function on the line 00 functions in the sense defined here. Based on the theory of calculus on time scales, by using the properties of almost periodic functions and constructing a suitable lyapunov functional, sufficient conditions which guarantee the existence of a unique globally attractive positive almost periodic solution of the system are obtained. Get a printable copy pdf file of the complete article 360k, or click on a page image below to browse page by page. Here, too, are a number of monographs on the subject, most notably by l. 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.
Composition of pseudo almost periodic functions and cauchy problems with operator of non dense domain. Almostperiodic function article about almostperiodic. More precisely, a continuous function fx defined for all real values of 5 x is called almost periodic if for every. This paper is concerned with a singlespecies system with feedback control on time scales. The more vague and general notion of quasiperiodicity has even less to do with quasiperiodic functions in the mathematical sense. Almost periodic solution for a delayed lotkavolterra. Download file pdf almost periodic type functions and ergodicity recognizing the quirk ways to acquire this books almost periodic type functions and ergodicity is additionally useful. For example, sin t has minimal period is 2 it follows from this that the minimal period for sin2t is the only exception is the constant function. Pseudo almost periodic functions and their applications. Full text full text is available as a scanned copy of the original print version. If the inline pdf is not rendering correctly, you can download the pdf file here. In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. Stepanov almost periodic functions edit the space s p of stepanov almost periodic functions for p. Bohrs famous theory of almost periodic functions 4, if to arbitrary groups, and to show that it gives just the maximum range over which the fundamental results of fro.
Pdf this paper is a survey of the basic properties of various type of almost periodic functions. The almost periodic functions on the real line can be characterized as uniform limits of trigonometric functions. Prouse, almostperiodic functions and functional equations, van nostrandreinhold, new york, ny, usa, 1971. Periodic functions 2 now consider the functions y cos2t y sin2t the graphs of these functions are obtained from those of cos tand sin by compressing them along the xaxis by a factor of 2. I was wondering whether a similar definition exists on the complex plane a. Eigenvalues, almost periodic functions, and the derivative of.
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. There are meanperiodic functions which are not almost periodic. This paper is concerned with a delayed lotkavolterra system on time scales. Links to pubmed are also available for selected references. Existence of almost periodic solutions of discrete time equations. Brom, a constructive version of the fundamental theorem of the theory of almost periodic functions, thesis, university of california, san diego, 1974. Abstractin this paper, we first investigate some basic properties of remotely almost periodic functions and then show the existence and uniqueness of remotely almost periodic solutions to systems of differential equations with piecewise constant argument.
Heteroclinic motions joining almost periodic solutions for. When time scale or, our definition of the uniformly almost periodic functions is equivalent to the classical definitions of uniformly almost periodic functions and the uniformly almost periodic sequences, respectively. 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. Note that we obtain the most important case if x is a banach space, and that the theory of almost periodic functions of real variable with values in a banach space, given by bochner 3, is in its essential lines similar to the theory of classical almost periodic functions.
The relation between a pseudo almost periodic function and its almost periodic component is discussed. Almostperiodic function encyclopedia of mathematics. Almost periodic functions, constructive mathematics, topological groups. The first half of the book lays the groundwork, noting the basic properties of almost periodic functions, while the second half of this work addresses applications whose main emphasis is on the solvability of ordinary or partial differential equations in the class of almost periodic functions. The bohr almost periodic functions are essentially the same as continuous functions on the bohr compactification of the reals. Based on the theory of calculus on time scales, by using the properties of almost periodic functions and constructing a suitable lyapunov functional, sufficient conditions which guarantee the existence of a unique globally attractive positive almost periodic solution of the system are.
Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity. Almostperiodic function on a group encyclopedia of. Almost periodic solutions of a singlespecies system with. Serial femtosecond crystallography on in vivogrown crystals drives elucidation of mosquitocidal cyt1aa bioactivation cascade. Get a printable copy pdf file of the complete article 365k, or click on a page image below to browse page by page. Almost periodic functions, bohr compactification, and differential equations article pdf available in milan journal of mathematics 661. Then we get a complete description of zeros and poles for this. Get a printable copy pdf file of the complete article 284k, or click on a page image below to browse page by page. An ap function on the line 00 functions which are not mean periodic. Generalized asymptotically almost periodic and generalized asymptotically almost automorphic solutions of abstract multiterm fractional differential. Robust almost periodic dynamics for interval neural networks with mixed time. Hierarchy of almostperiodic function spaces dipartimento di. A lmost periodic functions stepanov, weyl and besicov. Eigenvalues, almost periodic functions, and the derivative.
His interest was initially in finite dirichlet series. We also use the result for sequences to construct an almost periodic function whose. An almostperiodic solution of hasegawawakatani equations with vanishing resistivity volume 146 issue 5 shintaro kondo skip to main content we use cookies to distinguish you from other users and to provide you with a better experience on our websites. Mean sensitive, mean equicontinuous and almost periodic functions for dynamical systems. A function fx in with complex values is called right almost periodic r.
Basic properties of almost periodic functions request pdf. Asymptotically almost periodic solution to a class of volterra difference. Then, based on these, we study the existence and uniqueness of almost periodic solutions and derive some fundamental. I was wondering whether a similar definition exists on the complex plane a locally. Full text is available as a scanned copy of the original print version. Some properties of weighted pseudo almost periodic functions zheng, zheming, ding, huisheng, and nguerekata, gaston m. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a pdf plugin installed and enabled in.
1449 499 1481 568 1581 760 1409 551 593 967 715 1476 900 1271 1125 1239 521 724 156 310 200 1240 1413 1563 717 761 1606 1596 1520 139 1469 1570 674 1571 652 1143 1180 1199 846 1237 1462 355 759 1138 858