Semi-periodic solutions of difference and differential equations
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RESEARCH
Open Access
Semi-periodic solutions of difference and differential equations Jan Andres1* and Denis Pennequin2 *
Correspondence: [email protected] Department of Mathematical Analysis, Faculty of Science, Palacký University, 17. listopadu 12, Olomouc, 771 46, Czech Republic Full list of author information is available at the end of the article
Abstract
1
The spaces of semi-periodic sequences and functions are examined in the relationship to the closely related notions of almost-periodicity, quasi-periodicity and periodicity. Besides the main theorems, several illustrative examples of this type are supplied. As an application, the existence and uniqueness results are formulated for semi-periodic solutions of quasi-linear difference and differential equations. MSC: 34C15; 34C27; 34K14; 39A10; 42A16; 42A75 Keywords: semi-periodic sequences; semi-periodic functions; semi-periodic solutions; difference equations; differential equations
Introduction In [], it is observed that although the set of periodic sequences forms a linear space, its uniform closure is not the space of almost-periodic sequences but of semi-periodic sequences. In fact, the space of semi-periodic sequences was shown there to be Banach. The whole Sections I., I. in [] and Sections II., II. in [] are devoted to semi-periodic continuous functions, called there limit periodic functions (cf. also [, p.]). This class was shown there to be identical with the one of uniformly almost-periodic functions with one-term Q-base and, in case of integral one-term base, it reduces to the one of purely periodic functions. For some more references concerning limit periodic functions, see, e.g., [, ]. In fact, limit periodic functions were already considered by Bohr in , as pointed out in [, p.]. In the following section, we define analogously to [] the class of semi-periodic continuous functions (with values in a Banach space) and show that it is the same as the class of limit periodic functions considered in [, ] (see Theorem below). Let us note that many different notions with the same name (i.e., semi-periodic), like functions satisfying Floquet boundary conditions (see, e.g., [, ]) or those describing Bloch waves (see, e.g., [], and the references therein), exist in the literature (cf. also [, ]). Hence, after giving a definition of semi-periodic functions, which is analogous to [], we prove that the uniform closure of the set of periodic functions is again the one of semiperiodic functions. Unlike in the discrete case, the space of semi-periodic functions is, however, not linear and so not Banach. In order to clarify transparently the position of semi-periodic sequences and functions in the hierarchy of closely related spaces, we decided to illustrate it by means of Venn’s diagrams. Thus, the spaces of almost-periodic, semi-periodic, quasi-periodic and periodic functions and sequences and some of their © 2012 Andres and Pennequin; licensee Springer. This is an Open Access article distributed under the terms of the Creative
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