Open Access

Existence of Pseudo Almost Automorphic Solutions for the Heat Equation with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq1_HTML.gif -Pseudo Almost Automorphic Coefficients

Boundary Value Problems20092009:182527

DOI: 10.1155/2009/182527

Received: 12 March 2009

Accepted: 3 July 2009

Published: 17 August 2009

Abstract

We obtain the existence of pseudo almost automorphic solutions to the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq2_HTML.gif -dimensional heat equation with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq3_HTML.gif -pseudo almost automorphic coefficients.

1. Introduction

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq4_HTML.gif be an open bounded subset with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq5_HTML.gif boundary https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq6_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq7_HTML.gif be the space square integrable functions equipped with its natural https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq8_HTML.gif topology. Of concern is the study of pseudo almost automorphic solutions to the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq9_HTML.gif -dimensional heat equation with divergence terms
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ1_HTML.gif
(1.1)
where the symbols https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq10_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq11_HTML.gif stand, respectively, for the first- and second-order differential operators defined by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ2_HTML.gif
(1.2)

and the coefficients https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq12_HTML.gif are https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq13_HTML.gif -pseudo almost automorphic.

To analyze (1.1), our strategy will consist of studying the existence of pseudo almost automorphic solutions to the class of partial hyperbolic differential equations
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ3_HTML.gif
(1.3)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq14_HTML.gif is a sectorial linear operator on a Banach space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq15_HTML.gif whose corresponding analytic semigroup https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq16_HTML.gif is hyperbolic; that is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq17_HTML.gif the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq18_HTML.gif are arbitrary linear (possibly unbounded) operators on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq19_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq20_HTML.gif are https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq21_HTML.gif -pseudo almost automorphic for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq22_HTML.gif and jointly continuous functions.

Indeed, letting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq23_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq24_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq25_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq26_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq27_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq28_HTML.gif , one can readily see that (1.1) is a particular case of (1.3).

The concept of pseudo almost automorphy, which is the central tool here, was recently introduced in literature by Liang et al. [1, 2]. The pseudo almost automorphy is a generalization of both the classical almost automorphy due to Bochner [3] and that of pseudo almost periodicity due to Zhang [46]. It has recently generated several developments and extensions. For the most recent developments, we refer the reader to [1, 2, 79]. More recently, in Diagana [7], the concept of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq29_HTML.gif -pseudo almost automorphy (or Stepanov-like pseudo almost automorphy) was introduced. It should be mentioned that the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq30_HTML.gif -pseudo almost automorphy is a natural generalization of the notion of pseudo almost automorphy.

In this paper, we will make extensive use of the concept of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq31_HTML.gif -pseudo almost automorphy combined with the techniques of hyperbolic semigroups to study the existence of pseudo almost automorphic solutions to the class of partial hyperbolic differential equations appearing in (1.3) and then to the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq32_HTML.gif -dimensional heat equation (1.1).

In this paper, as in the recent papers [1012], we consider a general intermediate space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq33_HTML.gif between https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq34_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq35_HTML.gif . In contrast with the fractional power spaces considered in some recent papers by Diagana [13], the interpolation and Hölder spaces, for instance, depend only on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq36_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq37_HTML.gif and can be explicitly expressed in many concrete cases. Literature related to those intermediate spaces is very extensive; in particular, we refer the reader to the excellent book by Lunardi [14], which contains a comprehensive presentation on this topic and related issues.

Existence results related to pseudo almost periodic and almost automorphic solutions to the partial hyperbolic differential equations of the form (1.3) have recently been established in [12, 1518], respectively. Though to the best of our knowledge, the existence of pseudo almost automorphic solutions to the heat equation (1.1) in the case when the coefficients https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq38_HTML.gif are https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq39_HTML.gif -pseudo almost automorphic is an untreated original problem and constitutes the main motivation of the present paper.

2. Preliminaries

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq40_HTML.gif be two Banach spaces. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq41_HTML.gif (resp., https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq42_HTML.gif ) denote the collection of all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq43_HTML.gif -valued bounded continuous functions (resp., the class of jointly bounded continuous functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq44_HTML.gif ). The space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq45_HTML.gif equipped with the sup norm https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq46_HTML.gif is a Banach space. Furthermore, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq47_HTML.gif (resp., https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq48_HTML.gif ) denotes the class of continuous functions from https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq49_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq50_HTML.gif (resp., the class of jointly continuous functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq51_HTML.gif ).

The notation https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq52_HTML.gif stands for the Banach space of bounded linear operators from https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq53_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq54_HTML.gif equipped with its natural topology; in particular, this is simply denoted https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq55_HTML.gif whenever https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq56_HTML.gif .

Definition 2.1 (see [19]).

The Bochner transform https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq57_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq58_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq59_HTML.gif of a function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq60_HTML.gif is defined by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq61_HTML.gif

Remark 2.2.
  1. (i)

    A function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq62_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq63_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq64_HTML.gif , is the Bochner transform of a certain function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq65_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq66_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq67_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq68_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq69_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq70_HTML.gif .

     
  2. (ii)

    Note that if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq71_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq72_HTML.gif . Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq73_HTML.gif for each scalar https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq74_HTML.gif .

     

Definition 2.3.

The Bochner transform https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq75_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq76_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq77_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq78_HTML.gif of a function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq79_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq80_HTML.gif , with values in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq81_HTML.gif , is defined by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq82_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq83_HTML.gif .

Definition 2.4.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq84_HTML.gif . The space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq85_HTML.gif of all Stepanov bounded functions, with the exponent https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq86_HTML.gif , consists of all measurable functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq87_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq88_HTML.gif . This is a Banach space with the norm
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ4_HTML.gif
(2.1)

2.1. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq89_HTML.gif -Pseudo Almost Periodicity

Definition 2.5.

A function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq90_HTML.gif is called (Bohr) almost periodic if for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq91_HTML.gif there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq92_HTML.gif such that every interval of length https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq93_HTML.gif contains a number https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq94_HTML.gif with the property that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ5_HTML.gif
(2.2)

The number https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq95_HTML.gif above is called an https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq96_HTML.gif -translation number of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq97_HTML.gif , and the collection of all such functions will be denoted https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq98_HTML.gif .

Definition 2.6.

A function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq99_HTML.gif is called (Bohr) almost periodic in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq100_HTML.gif uniformly in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq101_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq102_HTML.gif is any compact subset https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq103_HTML.gif if for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq104_HTML.gif there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq105_HTML.gif such that every interval of length https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq106_HTML.gif contains a number https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq107_HTML.gif with the property that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ6_HTML.gif
(2.3)

The collection of those functions is denoted by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq108_HTML.gif .

Define the classes of functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq109_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq110_HTML.gif , respectively, as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ7_HTML.gif
(2.4)
and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq111_HTML.gif is the collection of all functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq112_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ8_HTML.gif
(2.5)

uniformly in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq113_HTML.gif .

Definition 2.7 (see [13]).

A function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq114_HTML.gif is called pseudo almost periodic if it can be expressed as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq115_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq116_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq117_HTML.gif . The collection of such functions will be denoted by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq118_HTML.gif .

Definition 2.8 (see [13]).

A function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq119_HTML.gif is said to be pseudo almost periodic if it can be expressed as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq120_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq121_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq122_HTML.gif . The collection of such functions will be denoted by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq123_HTML.gif .

Define https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq124_HTML.gif as the collection of all functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq125_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ9_HTML.gif
(2.6)

uniformly in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq126_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq127_HTML.gif is any bounded subset.

Obviously,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ10_HTML.gif
(2.7)

A weaker version of Definition 2.8 is the following.

Definition 2.9.

A function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq128_HTML.gif is said to be B-pseudo almost periodic if it can be expressed as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq129_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq130_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq131_HTML.gif . The collection of such functions will be denoted by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq132_HTML.gif .

Definition 2.10 (see [20, 21]).

A function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq133_HTML.gif is called https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq134_HTML.gif -pseudo almost periodic (or Stepanov-like pseudo almost periodic) if it can be expressed as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq135_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq136_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq137_HTML.gif . The collection of such functions will be denoted by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq138_HTML.gif .

In other words, a function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq139_HTML.gif is said to be https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq140_HTML.gif -pseudo almost periodic if its Bochner transform https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq141_HTML.gif is pseudo almost periodic in the sense that there exist two functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq142_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq143_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq144_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq145_HTML.gif .

To define the notion of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq146_HTML.gif -pseudo almost automorphy for functions of the form https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq147_HTML.gif , we need to define the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq148_HTML.gif -pseudo almost periodicity for these functions as follows.

Definition 2.11.

A function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq149_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq150_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq151_HTML.gif , is said to be https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq152_HTML.gif -pseudo almost periodic if there exist two functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq153_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq154_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq155_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq156_HTML.gif .

The collection of those https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq157_HTML.gif -pseudo almost periodic functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq158_HTML.gif will be denoted https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq159_HTML.gif .

2.2. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq160_HTML.gif -Almost Automorphy

The notion of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq161_HTML.gif -almost automorphy is a new notion due to N'Guérékata and Pankov [22].

Definition 2.12 (Bochner).

A function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq162_HTML.gif is said to be almost automorphic if for every sequence of real numbers https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq163_HTML.gif there exists a subsequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq164_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ11_HTML.gif
(2.8)
is well defined for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq165_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ12_HTML.gif
(2.9)

for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq166_HTML.gif .

Remark 2.13.

The function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq167_HTML.gif in Definition 2.12 is measurable but not necessarily continuous. Moreover, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq168_HTML.gif is continuous, then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq169_HTML.gif is uniformly continuous. If the convergence above is uniform in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq170_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq171_HTML.gif is almost periodic. Denote by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq172_HTML.gif the collection of all almost automorphic functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq173_HTML.gif . Note that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq174_HTML.gif equipped with the sup norm, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq175_HTML.gif , turns out to be a Banach space.

We will denote by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq176_HTML.gif the closed subspace of all functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq177_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq178_HTML.gif . Equivalently, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq179_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq180_HTML.gif is almost automorphic, and the convergences in Definition 2.12 are uniform on compact intervals, that is, in the Fréchet space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq181_HTML.gif . Indeed, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq182_HTML.gif is almost automorphic, then its range is relatively compact. Obviously, the following inclusions hold:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ13_HTML.gif
(2.10)

Definition 2.14 (see [22]).

The space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq183_HTML.gif of Stepanov-like almost automorphic functions (or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq184_HTML.gif -almost automorphic) consists of all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq185_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq186_HTML.gif . That is, a function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq187_HTML.gif is said to be https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq188_HTML.gif -almost automorphic if its Bochner transform https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq189_HTML.gif is almost automorphic in the sense that for every sequence of real numbers https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq190_HTML.gif there exists a subsequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq191_HTML.gif and a function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq192_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ14_HTML.gif
(2.11)

as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq193_HTML.gif pointwise on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq194_HTML.gif .

Remark 2.15.

It is clear that if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq195_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq196_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq197_HTML.gif -almost automorphic, then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq198_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq199_HTML.gif -almost automorphic. Also if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq200_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq201_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq202_HTML.gif -almost automorphic for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq203_HTML.gif . Moreover, it is clear that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq204_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq205_HTML.gif . Thus, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq206_HTML.gif can be considered as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq207_HTML.gif .

Definition 2.16.

A function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq208_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq209_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq210_HTML.gif , is said to be https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq211_HTML.gif -almost automorphic in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq212_HTML.gif uniformly in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq213_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq214_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq215_HTML.gif -almost automorphic for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq216_HTML.gif ; that is, for every sequence of real numbers https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq217_HTML.gif , there exists a subsequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq218_HTML.gif and a function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq219_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ15_HTML.gif
(2.12)

as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq220_HTML.gif pointwise on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq221_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq222_HTML.gif .

The collection of those https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq223_HTML.gif -almost automorphic functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq224_HTML.gif will be denoted by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq225_HTML.gif .

2.3. Pseudo Almost Automorphy

The notion of pseudo almost automorphy is a new notion due to Liang et al. [2, 9].

Definition 2.17.

A function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq226_HTML.gif is called pseudo almost automorphic if it can be expressed as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq227_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq228_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq229_HTML.gif . The collection of such functions will be denoted by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq230_HTML.gif .

Obviously, the following inclusions hold:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ16_HTML.gif
(2.13)

Definition 2.18.

A function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq231_HTML.gif is said to be pseudo almost automorphic if it can be expressed as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq232_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq233_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq234_HTML.gif . The collection of such functions will be denoted by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq235_HTML.gif .

A substantial result is the next theorem, which is due to Liang et al. [2].

Theorem 2.19 (see [2]).

The space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq236_HTML.gif equipped with the sup norm https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq237_HTML.gif is a Banach space.

We also have the following composition result.

Theorem 2.20 (see [2]).

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq238_HTML.gif belongs to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq239_HTML.gif and if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq240_HTML.gif is uniformly continuous on any bounded subset https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq241_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq242_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq243_HTML.gif , then the function defined by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq244_HTML.gif belongs to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq245_HTML.gif provided https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq246_HTML.gif .

3. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq247_HTML.gif -Pseudo Almost Automorphy

This section is devoted to the notion of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq248_HTML.gif -pseudo almost automorphy. Such a concept is completely new and is due to Diagana [7].

Definition 3.1 (see [7]).

A function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq249_HTML.gif is called https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq250_HTML.gif -pseudo almost automorphic (or Stepanov-like pseudo almost automorphic) if it can be expressed as
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ17_HTML.gif
(3.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq251_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq252_HTML.gif . The collection of such functions will be denoted by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq253_HTML.gif .

Clearly, a function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq254_HTML.gif is said to be https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq255_HTML.gif -pseudo almost automorphic if its Bochner transform https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq256_HTML.gif is pseudo almost automorphic in the sense that there exist two functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq257_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq258_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq259_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq260_HTML.gif

Theorem 3.2 (see [7]).

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq261_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq262_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq263_HTML.gif . In other words, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq264_HTML.gif .

Obviously, the following inclusions hold:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ18_HTML.gif
(3.2)

Theorem 3.3 (see [7]).

The space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq265_HTML.gif equipped with the norm https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq266_HTML.gif is a Banach space.

Definition 3.4.

A function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq267_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq268_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq269_HTML.gif , is said to be https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq270_HTML.gif -pseudo almost automorphic if there exists two functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq271_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ19_HTML.gif
(3.3)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq272_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq273_HTML.gif . The collection of those https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq274_HTML.gif -pseudo almost automorphic functions will be denoted by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq275_HTML.gif .

We have the following composition theorems.

Theorem 3.5.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq276_HTML.gif be a https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq277_HTML.gif -pseudo almost automorphic function. Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq278_HTML.gif is Lipschitzian in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq279_HTML.gif uniformly in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq280_HTML.gif ; that is there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq281_HTML.gif such
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ20_HTML.gif
(3.4)

for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq282_HTML.gif .

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq283_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq284_HTML.gif defined by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq285_HTML.gif belongs to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq286_HTML.gif .

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq287_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq288_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq289_HTML.gif . Similarly, let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq290_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq291_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq292_HTML.gif , that is,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ21_HTML.gif
(3.5)

for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq293_HTML.gif .

It is obvious to see that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq294_HTML.gif . Now decompose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq295_HTML.gif as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ22_HTML.gif
(3.6)
Using the theorem of composition of almost automorphic functions, it is easy to see that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq296_HTML.gif . Now, set
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ23_HTML.gif
(3.7)
Clearly, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq297_HTML.gif . Indeed, we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ24_HTML.gif
(3.8)
and hence for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq298_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ25_HTML.gif
(3.9)
Now using (3.5), it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ26_HTML.gif
(3.10)

Using the theorem of composition of functions of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq299_HTML.gif (see [13]) it is easy to see that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq300_HTML.gif .

Theorem 3.6.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq301_HTML.gif be an https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq302_HTML.gif -pseudo almost automorphic function, where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq303_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq304_HTML.gif . Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq305_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq306_HTML.gif are uniformly continuous in every bounded subset https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq307_HTML.gif uniformly for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq308_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq309_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq310_HTML.gif defined by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq311_HTML.gif belongs to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq312_HTML.gif .

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq313_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq314_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq315_HTML.gif . Similarly, let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq316_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq317_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq318_HTML.gif .

It is obvious to see that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq319_HTML.gif . Now decompose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq320_HTML.gif as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ27_HTML.gif
(3.11)
Using the theorem of composition of almost automorphic functions, it is easy to see that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq321_HTML.gif . Now, set
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ28_HTML.gif
(3.12)
We claim that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq322_HTML.gif . First of all, note that the uniformly continuity of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq323_HTML.gif on bounded subsets https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq324_HTML.gif yields the uniform continuity of its Bohr transform https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq325_HTML.gif on bounded subsets of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq326_HTML.gif . Since both https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq327_HTML.gif are bounded functions, it follows that there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq328_HTML.gif a bounded subset such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq329_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq330_HTML.gif . Now from the uniform continuity of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq331_HTML.gif on bounded subsets of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq332_HTML.gif , it obviously follows that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq333_HTML.gif is uniformly continuous on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq334_HTML.gif uniformly for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq335_HTML.gif . Therefore for every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq336_HTML.gif there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq337_HTML.gif such that for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq338_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq339_HTML.gif yield
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ29_HTML.gif
(3.13)
Using the proof of the composition theorem [2, Theorem 2.4], (applied to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq340_HTML.gif ) it follows
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ30_HTML.gif
(3.14)

Using the theorem of composition [2, Theorem 2.4] for functions of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq341_HTML.gif it is easy to see that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq342_HTML.gif .

4. Sectorial Linear Operators

Definition 4.1.

A linear operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq343_HTML.gif (not necessarily densely defined) is said to be sectorial if the following holds: there exist constants https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq344_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq345_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq346_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq347_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ31_HTML.gif
(4.1)

The class of sectorial operators is very rich and contains most of classical operators encountered in literature.

Example 4.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq348_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq349_HTML.gif be open bounded subset with regular boundary https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq350_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq351_HTML.gif be the Lebesgue space.

Define the linear operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq352_HTML.gif as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ32_HTML.gif
(4.2)

It can be checked that the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq353_HTML.gif is sectorial on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq354_HTML.gif .

It is wellknown that [14] if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq355_HTML.gif is sectorial, then it generates an analytic semigroup https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq356_HTML.gif , which maps https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq357_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq358_HTML.gif and such that there exist https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq359_HTML.gif with
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ33_HTML.gif
(4.3)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ34_HTML.gif
(4.4)
Throughout the rest of the paper, we suppose that the semigroup https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq360_HTML.gif is hyperbolic; that is, there exist a projection https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq361_HTML.gif and constants https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq362_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq363_HTML.gif commutes with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq364_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq365_HTML.gif is invariant with respect to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq366_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq367_HTML.gif is invertible, and the following hold:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ35_HTML.gif
(4.5)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ36_HTML.gif
(4.6)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq368_HTML.gif and, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq369_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq370_HTML.gif .

Recall that the analytic semigroup https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq371_HTML.gif associated with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq372_HTML.gif is hyperbolic if and only if
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ37_HTML.gif
(4.7)

see details in [23, Proposition 1.15, page 305]

Definition 4.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq373_HTML.gif . A Banach space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq374_HTML.gif is said to be an intermediate space between https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq375_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq376_HTML.gif , or a space of class https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq377_HTML.gif , if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq378_HTML.gif , and there is a constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq379_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ38_HTML.gif
(4.8)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq380_HTML.gif is the graph norm of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq381_HTML.gif .

Concrete examples of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq382_HTML.gif include https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq383_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq384_HTML.gif , the domains of the fractional powers of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq385_HTML.gif , the real interpolation spaces https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq386_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq387_HTML.gif , defined as the space of all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq388_HTML.gif such
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ39_HTML.gif
(4.9)
with the norm
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ40_HTML.gif
(4.10)

the abstract Hölder spaces https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq389_HTML.gif as well as the complex interpolation spaces https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq390_HTML.gif ; see Lunardi [14] for details.

For a hyperbolic analytic semigroup https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq391_HTML.gif , one can easily check that similar estimations as both (4.5) and (4.6) still hold with the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq392_HTML.gif -norms https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq393_HTML.gif . In fact, as the part of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq394_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq395_HTML.gif is bounded, it follows from (4.6) that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ41_HTML.gif
(4.11)
Hence, from (4.8) there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq396_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ42_HTML.gif
(4.12)
In addition to the above, the following holds:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ43_HTML.gif
(4.13)
and hence from (4.5), one obtains
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ44_HTML.gif
(4.14)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq397_HTML.gif depends on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq398_HTML.gif . For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq399_HTML.gif , by (4.4) and (4.8),
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ45_HTML.gif
(4.15)
Hence, there exist constants https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq400_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq401_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ46_HTML.gif
(4.16)

5. Existence of Pseudo Almost Automorphic Solutions

This section is devoted to the search of an almost automorphic solution to the partial hyperbolic differential equation (1.3).

Definition 5.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq402_HTML.gif . A bounded continuous function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq403_HTML.gif is said to be a mild solution to (1.3) provided that the function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq404_HTML.gif is integrable on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq405_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq406_HTML.gif is integrable on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq407_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq408_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ47_HTML.gif
(5.1)

for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq409_HTML.gif .

Throughout the rest of the paper we denote by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq410_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq411_HTML.gif the nonlinear integral operators defined by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ48_HTML.gif
(5.2)

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq412_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq413_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq414_HTML.gif Throughout the rest of the paper, we suppose that the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq415_HTML.gif is sectorial and generates a hyperbolic (analytic) semigroup https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq416_HTML.gif and requires the following assumptions.

(H.1)Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq417_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq418_HTML.gif , or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq419_HTML.gif , or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq420_HTML.gif , or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq421_HTML.gif . Moreover, we assume that the linear operators https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq422_HTML.gif are bounded.

(H.2)Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq423_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq424_HTML.gif be an S p -pseudo almost automorphic function in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq425_HTML.gif uniformly in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq426_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq427_HTML.gif be S p -pseudo almost automorphic in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq428_HTML.gif uniformly in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq429_HTML.gif . Moreover, the functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq430_HTML.gif are uniformly Lipschitz with respect to the second argument in the following sense: there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq431_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ49_HTML.gif
(5.3)

for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq432_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq433_HTML.gif .

In order to show that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq434_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq435_HTML.gif are well defined, we need the next lemma whose proof can be found in Diagana [12].

Lemma 5.2 (see [12]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq436_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ50_HTML.gif
(5.4)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ51_HTML.gif
(5.5)

The proof for the pseudo almost automorphy of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq437_HTML.gif is similar to that of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq438_HTML.gif and hence will be omitted.

Lemma 5.3.

Under assumptions (H.1)-(H.2), consider the function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq439_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq440_HTML.gif , defined by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ52_HTML.gif
(5.6)
for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq441_HTML.gif . If
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ53_HTML.gif
(5.7)

then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq442_HTML.gif .

Remark 5.4.

Note that the assumption https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq443_HTML.gif holds in several case. This is in particular the case when https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq444_HTML.gif .

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq445_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq446_HTML.gif , it follows that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq447_HTML.gif . Setting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq448_HTML.gif and using Theorem 3.5 it follows that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq449_HTML.gif . Moreover, using (5.5) it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ54_HTML.gif
(5.8)

and hence the function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq450_HTML.gif is integrable over https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq451_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq452_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq453_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq454_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq455_HTML.gif . Define, for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq456_HTML.gif the sequence of integral operators
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ55_HTML.gif
(5.9)

for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq457_HTML.gif .

Now letting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq458_HTML.gif , it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ56_HTML.gif
(5.10)
Using Hölder's inequality and the estimate (5.8), it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ57_HTML.gif
(5.11)
Using the assumption https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq459_HTML.gif , we then deduce from the well-known Weirstrass theorem that the series https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq460_HTML.gif is uniformly convergent on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq461_HTML.gif . Furthermore,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ58_HTML.gif
(5.12)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq462_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ59_HTML.gif
(5.13)

for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq463_HTML.gif

We claim that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq464_HTML.gif . Indeed, let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq465_HTML.gif be a sequence of real numbers. Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq466_HTML.gif , there exists a subsequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq467_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq468_HTML.gif and a function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq469_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ60_HTML.gif
(5.14)
Define
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ61_HTML.gif
(5.15)
Set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq470_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq471_HTML.gif . Then using both Hölder's inequality and (5.5), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ62_HTML.gif
(5.16)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq472_HTML.gif , as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq473_HTML.gif .

Obviously,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ63_HTML.gif
(5.17)
Similarly, we can prove that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ64_HTML.gif
(5.18)

Therefore the sequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq474_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq475_HTML.gif , and hence its uniform limit https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq476_HTML.gif .

Let us show that each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq477_HTML.gif . Indeed,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ65_HTML.gif
(5.19)
and hence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq478_HTML.gif , as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq479_HTML.gif . Furthermore, using the assumption https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq480_HTML.gif , we then deduce from the well-known Weirstrass theorem that the series
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ66_HTML.gif
(5.20)
is uniformly convergent on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq481_HTML.gif . Moreover,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ67_HTML.gif
(5.21)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq482_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ68_HTML.gif
(5.22)

for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq483_HTML.gif

Consequently the uniform limit https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq484_HTML.gif ; see [21, Lemma 2.5] . Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq485_HTML.gif is pseudo almost automorphic.

The proof for the almost automorphy of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq486_HTML.gif is similar to that of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq487_HTML.gif and hence will be omitted.

Lemma 5.5.

Under assumptions (H.1)-(H.2), consider the function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq488_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq489_HTML.gif , defined by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ69_HTML.gif
(5.23)

for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq490_HTML.gif .

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq491_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq492_HTML.gif .

Proof.

The proof is similar to that of Lemma 5.3 and hence omitted, though here we make use of the approximation (5.4) rather than (5.5).

Throughout the rest of the paper, the constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq493_HTML.gif denotes the bound of the embedding https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq494_HTML.gif , that is,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ70_HTML.gif
(5.24)

Theorem 5.6.

Under the previous assumptions and if assumptions (H.1)-(H.2) hold, then the evolution equation (1.3) has a unique pseudo almost automorphic solution whenever https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq495_HTML.gif is small enough, that is,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ71_HTML.gif
(5.25)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq496_HTML.gif .

Proof.

In https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq497_HTML.gif , define the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq498_HTML.gif by setting
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ72_HTML.gif
(5.26)

for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq499_HTML.gif .

As we have previously seen, for every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq500_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq501_HTML.gif . From previous assumptions one can easily see that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq502_HTML.gif is well defined and continuous. Moreover, from Theorem 3.5, Lemma 5.3, and Lemma 5.5 we infer that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq503_HTML.gif maps https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq504_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq505_HTML.gif . In particular, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq506_HTML.gif maps https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq507_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq508_HTML.gif . To complete the proof one has to show that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq509_HTML.gif has a unique fixedpoint. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq510_HTML.gif . It is routine to see that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ73_HTML.gif
(5.27)

Therefore, by the Banach fixed-point principle, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq511_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq512_HTML.gif has a unique fixed-point, which obviously is the only pseudo almost automorphic solution to (1.3).

6. Example

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq513_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq514_HTML.gif be an open bounded subset with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq515_HTML.gif boundary https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq516_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq517_HTML.gif equipped with its natural topology https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq518_HTML.gif .

Define the linear operator appearing in (1.3) as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ74_HTML.gif
(6.1)

The operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq519_HTML.gif defined above is sectorial and hence is the infinitesimal generator of an analytic semigroup https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq520_HTML.gif . Moreover, the semigroup https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq521_HTML.gif is hyperbolic as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq522_HTML.gif .

Throughout the rest of the paper, for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq523_HTML.gif , we take https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq524_HTML.gif equipped with its https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq525_HTML.gif -norm https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq526_HTML.gif . Moreover, we let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq527_HTML.gif and suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq528_HTML.gif . Letting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq529_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq530_HTML.gif , one easily sees that both operators are bounded from https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq531_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq532_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq533_HTML.gif .

We require the following assumption.

(H.3)Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq534_HTML.gif , let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq535_HTML.gif be an https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq536_HTML.gif -pseudo almost automorphic function in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq537_HTML.gif uniformly in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq538_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq539_HTML.gif be https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq540_HTML.gif -pseudo almost automorphic in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq541_HTML.gif uniformly in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq542_HTML.gif . Moreover, the functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq543_HTML.gif are uniformly Lipschitz with respect to the second argument in the following sense: there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq544_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ75_HTML.gif
(6.2)

for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq545_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq546_HTML.gif .

We have the following.

Theorem 6.1.

Under the previous assumptions including (H.3), then the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq547_HTML.gif -dimensional heat equation (1.1) has a unique pseudo almost automorphic solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq548_HTML.gif whenever https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq549_HTML.gif is small enough.

Classical examples of the above-mentioned functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq550_HTML.gif are given as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ76_HTML.gif
(6.3)

where the functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq551_HTML.gif are https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq552_HTML.gif -pseudo almost automorphic.

In this particular case, the corresponding heat equation, that is,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_Equ77_HTML.gif
(6.4)

has a unique pseudo almost automorphic solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq553_HTML.gif whenever https://static-content.springer.com/image/art%3A10.1155%2F2009%2F182527/MediaObjects/13661_2009_Article_831_IEq554_HTML.gif is small enough.

Authors’ Affiliations

(1)
Department of Mathematics, Howard University
(2)
Department of Mathematical Sciences, Florida Institute of Technology

References

  1. Liang J, Zhang J, Xiao T-J: Composition of pseudo almost automorphic and asymptotically almost automorphic functions. Journal of Mathematical Analysis and Applications 2008, 340: 1493–1499. 10.1016/j.jmaa.2007.09.065MATHMathSciNetView ArticleGoogle Scholar
  2. Xiao T-J, Liang J, Zhang J: Pseudo almost automorphic solutions to semilinear differential equations in banach spaces. Semigroup Forum 2008, 76(3):518–524. 10.1007/s00233-007-9011-yMATHMathSciNetView ArticleGoogle Scholar
  3. Bochner S: Continuous mappings of almost automorphic and almost periodic functions. Proceedings of the National Academy of Sciences of the United States of America 1964, 52: 907–910. 10.1073/pnas.52.4.907MATHMathSciNetView ArticleGoogle Scholar
  4. Zhang CY: Pseudo-almost-periodic solutions of some differential equations. Journal of Mathematical Analysis and Applications 1994, 151: 62–76.View ArticleGoogle Scholar
  5. Zhang CY: Pseudo almost periodic solutions of some differential equations. II. Journal of Mathematical Analysis and Applications 1995, 192: 543–561. 10.1006/jmaa.1995.1189MATHMathSciNetView ArticleGoogle Scholar
  6. Zhang CY: Integration of vector-valued pseudo-almost periodic functions. Proceedings of the American Mathematical Society 1994, 121: 167–174. 10.1090/S0002-9939-1994-1186140-8MATHMathSciNetView ArticleGoogle Scholar
  7. Diagana T: Existence of pseudo almost automorphic solutions to some abstract differential equations with -pseudo almost automorphic coefficients. Nonlinear Analysis: Theory, Methods & Applications 2009, 70(11):3781–3790. 10.1016/j.na.2008.07.034MATHMathSciNetView ArticleGoogle Scholar
  8. Ezzinbi K, Fatajou S, N'Guérékata GM: Pseudo almost automorphic solutions to some neutral partial functional differential equations in banach space. Nonlinear Analysis: Theory, Methods & Applications 2009, 71(1–2):674–684. 10.1016/j.na.2008.10.100View ArticleGoogle Scholar
  9. Liang J, N'Guérékata GM, Xiao T-J, Zhang J: Some properties of pseudo-almost automorphic functions and applications to abstract differential equations. Nonlinear Analysis: Theory, Methods & Applications 2009, 70(7):2731–2735. 10.1016/j.na.2008.03.061MATHMathSciNetView ArticleGoogle Scholar
  10. Boulite S, Maniar L, N'Guérékata GM: Almost automorphic solutions for hyperbolic semilinear evolution equations. Semigroup Forum 2005, 71: 231–240. 10.1007/s00233-005-0524-yMATHMathSciNetView ArticleGoogle Scholar
  11. Diagana T, N'Guérékata GM: Pseudo almost periodic mild solutions to hyperbolic evolution equations in intermediate banach spaces. Applicable Analysis 2006, 85(6–7):769–780. 10.1080/00036810600708499MATHMathSciNetView ArticleGoogle Scholar
  12. Diagana T: Existence of pseudo almost periodic solutions to some classes of partial hyperbolic evolution equations. Electronic Journal of Qualitative Theory of Differential Equations 2007, 2007(3):1–12.View ArticleGoogle Scholar
  13. Diagana T: Pseudo Almost Periodic Functions in Banach Spaces. Nova Science, New York, NY, USA; 2007.MATHGoogle Scholar
  14. Lunardi A: Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and Their Applications. Volume 16. Birkhäuser, Basel, Switzerland; 1995.Google Scholar
  15. Diagana T, Henríquez N, Hernàndez E: Almost automorphic mild solutions to some partial neutral functional-differential equations and applications. Nonlinear Analysis: Theory, Methods & Applications 2008, 69(5):1485–1493. 10.1016/j.na.2007.06.048MATHMathSciNetView ArticleGoogle Scholar
  16. Diagana T, N'Guérékata GM: Almost automorphic solutions to some classes of partial evolution equations. Applied Mathematics Letters 2007, 20(4):462–466. 10.1016/j.aml.2006.05.015MATHMathSciNetView ArticleGoogle Scholar
  17. Diagana T: Existence and uniqueness of pseudo-almost periodic solutions to some classes of partial evolution equations. Nonlinear Analysis: Theory, Methods & Applications 2007, 66(2):384–395. 10.1016/j.na.2005.11.031MATHMathSciNetView ArticleGoogle Scholar
  18. Diagana T: Existence of almost automorphic solutions to some neutral functional differential equations with infinite delay. Electronic Journal of Differential Equations 2008, 2008(129):1–14.MathSciNetGoogle Scholar
  19. Pankov A: Bounded and Almost Periodic Solutions of Nonlinear Operator Differential Equations, Mathematics and Its Applications (Soviet Series). Volume 55. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1990.Google Scholar
  20. Diagana T: Stepanov-like pseudo almost periodic functions and their applications to differential equations. Communications in Mathematical Analysis 2007, 3(1):9–18.MATHMathSciNetGoogle Scholar
  21. Diagana T: Stepanov-like pseudo-almost periodicity and its applications to some nonautonomous differential equations. Nonlinear Analysis: Theory, Methods & Applications 2009, 69(12):4277–4285.MathSciNetView ArticleGoogle Scholar
  22. N'Guérékata GM, Pankov A: Stepanov-like almost automorphic functions and monotone evolution equations. Nonlinear Analysis: Theory, Methods & Applications 2008, 68(9):2658–2667. 10.1016/j.na.2007.02.012MATHMathSciNetView ArticleGoogle Scholar
  23. Engel KJ, Nagel R: One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics. Volume 194. Springer, New York, NY, USA; 2000.Google Scholar

Copyright

© T. Diagana and R. P. Agarwal. 2009

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.