Pseudo Almost Periodic Functions And Their Applications

Abstract

The space of pseudo almost periodic functions on {dollar}{lcub}\\rm J\\!I{rcub}\\sb{lcub}a{rcub}{dollar} (denoted by {dollar}{lcub}\\cal PAP{rcub}({lcub}\\rm J\\!I{rcub}\\sb{lcub}a{rcub},X){dollar} when they are vector-valued and {dollar}{lcub}\\cal PAP{rcub}({lcub}\\rm J\\!I{rcub}\\sb{lcub}a{rcub}){dollar} when they are scalar-valued, {dollar}{lcub}\\rm J\\!I{rcub}\\sb{lcub}a{rcub}=\\lbrack a,\\infty){dollar} for {dollar}a\\in\\IR{dollar} and {dollar}{lcub}\\rm J\\!I{rcub}\\sb{lcub}a{rcub}=\\IR{dollar} for {dollar}a={lcub}-\\infty{rcub}{dollar}, X is a Banach space) is defined and its properties are studied. A decomposition theorem is given showing that a function is in {dollar}{lcub}\\cal PAP{rcub}({lcub}\\rm J\\!I{rcub}\\sb{lcub}a{rcub},X){dollar} if and only if it is the sum of an almost periodic function and an ergodic perturbation. The relation between a pseudo almost periodic function and its almost periodic component is discussed. It is shown that {dollar}{lcub}\\cal PAP{rcub}({lcub}\\rm J\\!I{rcub}\\sb{lcub}a{rcub},X){dollar} is a Banach space, and in the case of scalar-valued functions, is a {dollar}C\\sp*{dollar}-algebra. Fourier analysis is carried out on {dollar}{lcub}\\cal PAP{rcub}({lcub}\\rm J\\!I{rcub}\\sb{lcub}a{rcub},X){dollar}.;As applications of the general theory of {dollar}{lcub}\\cal PAP{rcub}({lcub}\\rm J\\!I{rcub}\\sb{lcub}a{rcub}){dollar}, the solutions for three types of differential equations--ordinary, nonlinear parabolic and boundary value problems for harmonic functions--are investigated.;A necessary and sufficient condition is given for the pseudo almost periodicity of the primitive of a function in {dollar}{lcub}\\cal PAP{rcub}({lcub}\\rm J\\!I{rcub}\\sb{lcub}a{rcub},X){dollar}. As a corollary of this, an open question is answered: a necessary and sufficient condition is given for the weak almost periodicity of the primitive of a weakly almost periodic function.;The theory of vector-valued means is developed. A formula is set up between a vector-valued mean and a scalar-valued mean. As an application of the theory of vector-valued means, a theorem is given to show that the space of vector-valued weakly almost periodic functions is admissible.