Existence of $AP_{r}$-Almost Periodic Solutions For Some Classes of Functional Differential Equations

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 $f, x\in AP_r(R, \,{\mathcal{C}})=$ the space of almost periodic functions defined by \[ AP_r(R,\, {\mathcal{C}})=\left\{f : f\simeq \sum_{j=1}^{\infty} f_j\,e^{i\lambda_j t},\,f_j\in {\mathcal{C}},\lambda_j\in R,\sum_{j=1}^{\infty}|f_j|^r \lt \infty\right\}, \] 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 $k\ast x$, $k\in L^1(R,\, {\mathcal{C}})$, $x\in AP_r(R,\, {\mathcal{C}})$ 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 $x(t)\simeq \sum_{j=1}^{\infty} x_j\,e^{i\lambda_j\,t}$.