Asymptotic and pseudo almost periodicity of the convolution operator and applications to differential and integral equations
For \(f,g: \mathbb{R}\to\mathbb{R}\) which are Lebesgue measurable the convolution \(f*g\) is defined by \[ (f* g)(x)= \int_{\mathbb{R}} f(x- y)g(y)\,dy, \] whenever the integral exists. A function \(f\) in the Banach space \(BC(\mathbb{R})\) of all continuous bounded real functions from \(\mathbb{R}\) with sup-norm \((\|\;\|)\) is said to be almost periodic if for each \(\varepsilon> 0\) there exists \(l_\varepsilon> 0\) such that every interval of length \(l_\varepsilon\) contains a number \(\tau\) with \[ \| f_{-\tau}- f\|_\infty< \varepsilon, \] where \(f_{-\tau}(t)= f(t+\tau)\) for all \(t\in\mathbb{R}\). The family of such functions will be denoted by \(AP(\mathbb{R})\), \[ AP_0(\mathbb{R})= \Biggl\{f\in BC(\mathbb{R}): \lim_{T\to\infty}\, {1\over 2\tau} \int^T_{-\tau} |f(s)|\,ds= 0\Biggr\}. \] \(PAP(\mathbb{R})\) denotes the class of all functions \(f\in BC(\mathbb{R})\) such that \(f= h+ \phi\), where \(h\in AP(\mathbb{R})\) and \(\phi\in AP_0(\mathbb{R})\) and its members are called pseudo almost periodic functions. In this paper, the authors show that if \(f\in PAP(\mathbb{R})\) and \(g\in L^1(\mathbb{R})\) then \(f* g\in PAP(\mathbb{R})\). They also discuss first-order linear/semilinear or Volterra type differential equations admitting solutions in \(PAP(\mathbb{R})\).
scientific article; zbMATH DE number 5073447