Weighted pseudo almost periodic functions, convolutions and abstract integral equations

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• 作者：Aní ; bal Coronel^a ; ^{acoronel@ubiobio.cl" class="auth_mail" title="E-mail the corresponding author} ; Manuel Pinto^b ; ^{pintoj.uchile@gmail.cl" class="auth_mail" title="E-mail the corresponding author} ; Daniel Sep&uacute ; lveda^c ; ^{daniel.sep.oe@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Weighted pseudo almost periodic functions ; Pseudo almost periodic functions ; Integral equations ; Partial functional-differential equations
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：15 March 2016
• 年：2016
• 卷：435
• 期：2
• 页码：1382-1399
• 全文大小：456 K

文摘

This paper deals with a systematic study of the convolution operator Kf=f⁎k$\mathcal{K}f=f⁎k$ defined on weighted pseudo almost periodic functions space PAP(X,ρ)$\mathit{PAP}(\mathbb{X},\rho )$ and with k∈L¹(R)$k\in L^1(\mathbb{R})$. Upon making several different assumptions on k,f$k,f$ and ρ, we get five main results. The first two main results establish sufficient conditions on k and ρ   such that the weighted ergodic space PAP₀(X,ρ)${\mathit{PAP}}_0(\mathbb{X},\rho )$ is invariant under the operator K$\mathcal{K}$. The third result specifies a sufficient condition on all functions (k,f$k,f$ and ρ  ) such that the Kf∈PAP₀(X,ρ)$\mathcal{K}f\in {\mathit{PAP}}_0(\mathbb{X},\rho )$. The fourth result is a sufficient condition on the weight function ρ   such that PAP₀(X,ρ)${\mathit{PAP}}_0(\mathbb{X},\rho )$ is invariant under K$\mathcal{K}$. The hypothesis of the convolution invariance results allows to establish a fifth result related to the translation invariance of PAP₀(X,ρ)${\mathit{PAP}}_0(\mathbb{X},\rho )$. As a consequence of the fifth result, we obtain a new sufficient condition such that the unique decomposition of a weighted pseudo almost periodic function on its periodic and ergodic components is valid and also for the completeness of PAP(X,ρ)$\mathit{PAP}(\mathbb{X},\rho )$ with the supremum norm. In addition, the results on convolution are applied to general abstract integral and differential equations.