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Existence of $AP_r$-Almost Periodic Soutlions For Some Classes of Functional Differential Equations

Volume 15, Number 2 (2013), 47 - 55

Existence of $AP_r$-Almost Periodic Soutlions For Some Classes of Functional Differential Equations

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Abstract

This paper presents a couple of existence results, related to the classes of functional equations of the form $x+k\ast x=f$, or $\frac{d}{dt}[\dot{x}+k\ast x]=f$, with $f, x\in AP_r(R, \,{\cal{C}})=$ the space of almost periodic functions defined by
\[
AP_r(R,\, {\cal{C}})=\left\{f : f\simeq \sum_{j=1}^{\infty} f_j\,e^{i\lambda_j\,t},\,\,\,f_j\in {\cal{C}},\,\,\lambda_j\in R,\,\,\sum_{j=1}^{\infty}|f_j|^r < \infty\right\},
\]
the norm being given by $|f|_r= \left(\sum_{j=1}^{\infty}|f_j|^r\right)^{\frac{1}{r}}$, for each $r\in [1, 2]$. The convolution product $k\ast x$, $k\in L^1(R,\, {\cal{C}})$, $x\in AP_r(R,\, {\cal{C}})$ is defined by
\[
(k\ast x)(t)= \sum_{j=1}^{\infty} x_j\left( \int_R k(s)\,e^{-\lambda_j\,s}\,ds\right)\,e^{i\lambda_j\,t},
\]
where $x(t)\simeq \sum_{j=1}^{\infty} x_j\,e^{i\lambda_j\,t}$.