Composition of piecewise pseudo almost periodic functions and. The bounded solution of equation 9 is given by pseudo almost periodic solutions in a banach space since for all t 2 0, then, x is defined and furthermore, we have llxll 5 llfll t mexp c t s ds, i 03 5 llfll iwndt lll 5 llfll r, 24 is a solution of equation 9. In 12, the authors proposed a concept of weighted pseudo almost automorphic functions wpaa and completeness and a composition theorem of the function space formed by wpaa were obtained this generalizes weighted pseudo almost periodic functions 16. The theory can be used, for example, to derive some of the recently obtained results in timefrequency anal. Then using the theories of resolvent operators and fixed point theorem, we investigate the existence and. Table of contents for pseudo almost periodic functions in banach spaces toka diagana. In this paper, we will present a solution to the fundamental problem. We need to introduce two new notions of pseudo almost periodicity that we will use in the sequel. We shall see that the space of the pseudo almost periodic measures is a locally convex space with respect to a certain topology. Let be a banach space and be the collection of bounded continuous functions from to with the norm. The relation between a pseudo almost periodic function and its almost periodic component is discussed. The space b p of besicovitch almost periodic functions for p. The class of such functions will be denoted by pap z,w.
Some extensions of bohrs concept have been introduced, most notably by a. Weighted pseudo almost periodic solutions for some neutral partial functional differential equations. A various types of almost periodic functions on banach. Almost periodic functions on a locally compact abelian group.
We use the contraction mapping principle to show the existence and the uniqueness of. Introduction let x,k k be a banach space and let a. We systematically explore the properties of these functions in banach space including composition theorems. Discontinuous generalized doublealmostperiodic functions. In this article, we establish some new composition theorems on measure pseudo almost automorphic functions via measure theory. Y are pseudo almost periodic functions under the assumption that f is lipschitzian with respect to the second variable. If one quotients out a subspace of null functions, it can be identified with the space of l p functions on the bohr compactification of the reals. The obtained compositions theorems generalize those established under the wellknown lipschitz conditions or the classical uniformly continuous conditions. He showed that these functions include certain earlier generalizations of the notion of almost periodic functions. Hence, the space of the weighted pseudo almostperiodic functions.
The working tools are based on the fixed point theorems, the fractional powers of operators and fractional calculus. We prove that the space of continuous periodic functions is a set of first category in the space of almost periodic functions, and we also show that the space of almost periodic functions is a set of first category in the space of almost automorphic functions. The class of weighted pseudoalmost periodic functions with values in banach spaces was introduced by t. Introduction the concept of pseudo almost periodicity is a natural genelization of almost periodicity. Actually, the completeness of the space of these functions is worthy to be studied deeply by new ideas. Pseudo almost periodic functions in banach spaces nova science publishers, inc. Maniar, composition of pseudoalmost periodic functions and cauchy problems with operator of nondense domain, ann. Pdf a various types of almost periodic functions on banach. Hierarchy of almostperiodic function spaces dipartimento di. Nguerekata, pseudo almost periodic mild solutions to hyperbolic evolution equations in abstract intermediate banach spaces. Almost automorphic type and almost periodic type functions. The argument is similar in spirit but more subtle than the one used to prove that p wi is a banach space. Ircub\sblcubarcub,xdollar is a banach space, and in the case of scalarvalued functions, is a dollarc\spdollaralgebra. Pseudo almost periodic solutions for some differential equations in a banach space.
We develop a theory of almost periodic elements in banach algebras and present an abstract version of a noncommutative wieners lemma. This book provides the reader with a comprehensive and accessible introduction to the concepts of almost periodicity and pseudo almost periodicity as well as their applications to differential equations, partial differential equations, integral equations, and partial neutral functional differential equations with delay on banach spaces. S 41, marrakesh 2 department of mathematics, faculty of sciences semlalia, b. Composition of pseudo almostperiodic functions and semilinear. Composition of piecewise pseudo almost periodic functions. As an application, we obtain conditions for the existence of weighted pseudo almost periodic solutions for a class of neutral functional di. Bibliographic record and links to related information available from the library of congress catalog. In section 2, we state some facts on pseudo almost periodic functions with values in a banach space.
In this paper we study a new class of functions, which we call. The existence of almost periodic, asymptotically almost periodic, almost automorphic, asymptotically almost automorphic, and pseudo almost periodic solutions is one of the most attracting topics in the qualitative theory of differential equations due to their significance and applications in physics, mechanics, mathematical biology, control theory, and others. Pseudoalmost periodic and pseudoalmost automorphic. The proof of this theorem is based on the banachs fixed point theorem. The main results are devoted to the study of various types of weighted pseudo almost periodic automorphic properties of convolution products. Firstly, we obtain an equivalent definition for weighted pseudo almost periodic functions, which shows a close relationship between asymptotically almost periodic functions and weighted pseudo almost periodic functions.
Existence and control of weighted pseudo almost periodic. In section 3, we study the existence and uniqueness of. In this paper, we analyze the existence and uniqueness of generalized weighted pseudo almost automorphic solutions of abstract volterra integrodifferential inclusions in banach spaces. Further, we prove that the function space wpdap 0 affiliated to wpdap is a translation almost closed. The range of a pseudo almost periodic function is bounded, but not relatively compact. Regular article composition of pseudo almostperiodic functions. In 103, zhang gave a composition theorem of pseudo almost periodic functions in.
Mar 17, 2015 in this work we prove some existence and uniqueness results for pseudo almost periodic and pseudo almost automorphic solutions to a class of semilinear differential equations in hilbert spaces using theoretical measure theory. A various types of almost periodic functions on banach spaces. We prove the uniqueness of decomposition of the weighted pseudo almost periodic functions with values in banach spaces and with weights in a new weight function set, and show that the new weight set is larger than all the existing weight function sets which ensures the uniqueness of such decomposition in literature. We denote by the set of all such functions it is wellknown that is a closed subspace of, and thus is a banach space under the supremum norm definition 1. This book presents a comprehensive introduction to the concepts of almost periodicity, asymptotic almost periodicity, almost automorphy, asymptotic almost automorphy, pseudoalmost periodicity, and pseudoalmost automorphy as well as their recent generalizations. Their combined citations are counted only for the first article. We prove that the set conformed by these functions is a banach space with a suitable norm. A function is called stepanovlike pseudo almost periodic if it can be decomposed as with and.
In this paper, for the impulsive fractional integrodifferential equations involving caputo fractional derivative in banach space, we investigate the existence and uniqueness of a pseudo almost periodic pcmild solution. Stepanovlike pseudo almost periodic functions on time scales. In this section, we recall some basis definitions and lemmas of bohr almost periodic functions and stepanovs almost periodic functions which are used throughout this paper. Almost automorphic functions and almost periodic functions in abstract spaces kluwer, new york. Stepanovlike asymptotical almost periodic functions and. Existence and uniqueness of pseudo almost automorphic. Some important remarks with concrete examples are also presented. As applications, we establish some sufficient criteria for. Weighted pseudo almost periodic functions and applications. In many vibration problems, it is very important to know precisely the bounds of the stabilityinstability frequencies and the associated amplitude ranges.
Weighted pseudo almost periodic functions and applications to semilinear evolution equations zhang, jun, xiao, tijun, and liang, jin, abstract and applied analysis, 2012 asymptotically almost automorphic solutions to nonautonomous semilinear evolution equations ding, h. Y,x denote the collection of all xvalued bounded continuous functions respectively, the class of jointly bounded. Table of contents for pseudo almost periodic functions in. Liu pointed that the properties of the almost periodic functions do not always hold in the set of pseudo almost periodic functions and given an example. We first give a solution to a key problem concerning the completeness of the space of weighted pseudo almost periodic functions and then establish a new composition theorem with respect to these functions. The main technique is based upon some appropriate composition theorems combined with the banach contraction mapping principle and the method of the invariant subspaces. Pdf weighted pseudo almostperiodic functions and applications.
The proof of this theorem is based on the banach s fixed point theorem. Existence and uniqueness of pseudo almost periodic solutions. Pseudo almost periodic solutions for some differential. Ams proceedings of the american mathematical society. Pseudo almost periodicity of fractional integrodifferential. In addition, we apply our composition theorem to study the existence and uniqueness of pseudo almost periodic solutions to a class of abstract semilinear evolution equation in a banach space. Furthermore, we show several properties of this class of functions as the. This book presents a comprehensive introduction to the concepts of almost periodicity, asymptotic almost periodicity, almost automorphy, asymptotic almost automorphy, pseudo almost periodicity, and pseudo almost automorphy as well as their recent generalizations. Several interesting and new properties of weighted pseudo almost periodic functions are established. The space s p of stepanov almost periodic functions for p. Banach spaces rather fragmented, maybe you could say it is underdeveloped, but one can argue that linear approximations are often used for considering nonlinear problems. A complete quasinormed vector space is called a quasi banach space.
The class of weighted pseudo almost periodic functions with values in banach spaces was introduced by t. Section 4 is dedicated to exhibiting some applications. Discontinuous generalized doublealmostperiodic functions on. This space is stable to translations and contains the space of the pseudo almost periodic functions which are uniformly continuous. First we present a property of the pseudo almost periodic functions. A vector space with an associated quasinorm is called a quasinormed vector space. Regarding the theory of operators in banach spaces it should be. It is the closure of the trigonometric polynomials under the norm. The problems involved in banach spaces are of different types. Exponential dichotomy and eberleinweak almost periodic. Based on this, the concepts of weighted pseudo double almost periodic functions wpdap in banach spaces and a translation almost closed set are introduced. Weighted pseudo almostperiodic functions and applications. We introduce almost periodic functions in pseudometric spaces using a.
Spaces of measurable functions lpspaces, orlicz spaces, kothe function spaces, lorentz spaces, rearrangement invariant spaces, ideal spaces, etc. An almost periodic noncommutative wieners lemma radu balan and ilya krishtal abstract. Here k denotes the field of real numbers or complex numbers and i is a closed and bounded interval a, b. Moreover, we prove an existence theorem for the weighted pseudo almost periodic mild solution to the semilinear evolution. This paper investigates the solvability and control of some weighted pseudo almost periodic solutions of abstract nonlinear vibration differential systems. Throughout this paper, we always assume that x, y are banach spaces. Under the assumption of uniformly continuity, we establish a composition theorem of pseudo almostperiodic functions in general banach spaces. Many further definitions of generalized spaces of almost periodic functions are present in. Diagana in 17 2006, while the class of weighted pseudo almost automorphic functions with values in banach spaces was introduced by j. The theory of almost periodic functions was introduced in the literature around 19241926 with the pioneering work of the danish mathematician bohr 25. Diagana in 17 2006, while the class of weighted pseudoalmost automorphic functions with values in banach spaces was introduced by j.
Almost periodic and almost automorphic problems of dynamic equations on time scales were. Some properties of weighted pseudo almost periodic functions. Then we use this result to answer a question about weakly almost periodic functions. Weighted pseudo almostperiodic functions and applications to. Pseudo almost periodic functions, almost periodic functions, exponentially stable evolution operator. Applications include the existence of pseudo almost periodic solutions to the transport and heat equations with delay. It contains the space of bohr almost periodic functions. Moreover, some remarks on weighted pseudo almostperiodic functions are given.
However, there are few papers concerned with pseudo almost periodic functions on impulsive systems. In particular, they proved in 3 that the space paa r,x. Contents data are machine generated based on prepublication provided by the publisher. So, one relies on the fact that the linear problems are relatively tractable, and on the theory we will consider.
Composition of pseudo almost periodic functions and cauchy problems with operator of non dense domain. Hence, the space of the weighted pseudo almostperiodic functions may not be a banach space under the usual supremum norm. Almost automorphic type and almost periodic type functions in. Motivated by the above, our main propose of this paper is to introduce the concept of piecewise pseudo almost periodic functions on a banach space and establish some composition theorems of piecewise pseudo almost periodic functions. Composition theorems of stepanov almost periodic functions. This collection includes pseudo periodic, pseudo anti periodic, pseudo bloch periodic, and unbounded functions.
Pdf a various types of almost periodic functions on. This means that weighted pseudo almost periodic functions are good generalizations of the zhangas pseudo almost periodic functions. Pseudo almost periodic functions and their applications. Abstract in this work, we give a generalization, to banach spaces. Then we state sufficient conditions for the closeness of the sum of two closed subspaces of the banach space of bounded functions and apply this result on various pseudo almost periodic spaces and pseudo almost automorphic spaces. The space of continuous periodic functions is a set of. Weighted pseudo almost periodic solutions to a class of. We establish a composition theorem of stepanov almost periodic functions, and, with its help, a composition theorem of stepanovlike pseudo almost periodic functions is obtained. Then we define piecewisecontinuous almost periodic functions with double periodicity. Pseudo almost periodic functions in banach spaces book. Throughout this paper, we let x, y be banach spaces and bc r, x the.
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