0:|u(t)-qu(t-r)|>(1+q)\rho\} . \] By continuity, one can see that \[ |u(t_{0})-qu(t_{0}-r)|=(1+q)\rho, \] and there exists a positive constant $\varepsilon>0$ such that \[ |u(t)-qu(t-r)|>(1+q)\rho\quad\text{for }t\in(t_{0},t_{0}+\varepsilon). \] Using the variation-of-constants formula (\ref{var const}), we get that \[ |u(t_{0})-qu(t_{0}-r)|\leq e^{-t_{0}}( 1+q)\rho+ \int_{0}^{t_{0}}e^{-(t_{0}-s)} [|a_1 |_{\infty}|u(s+\theta)|d\theta+|f|_{\infty}] ds. \] Since $|u(t)-qu(t-r)|\leq(1+q)\rho$ for $t\leq t_{0}$, then \[ |u(t)|\leq(1+q)\rho+q| u(t-r)|\quad\text{for }t\in[-r,t_{0}] . \] $|\varphi|<\rho$, then we can see that \[ |u(t)|\leq\frac{1+q}{1-q}\rho\quad\text{for }t\in[-r,t_{0}], \] and \[ |u(t_{0})-qu(t_{0}-r)|\leq e^{-t_{0}}(1+q)\rho+(1-e^{-t_{0}})[ \frac{1+q}{1-q}|a_1 |_{\infty}\rho+|f|_{\infty}] . \] Using hypotheses (H8), we obtain \[ |u(t_{0})-qu(t_{0}-r)|\leq e^{-t_{0}}( 1+q)\rho+(1-e^{-t_{0}})(\beta(1+q) \rho+|f|_{\infty}), \] consequently, \begin{gather*} |u(t_{0})-qu(t_{0}-r)|\leq(1+q) \rho-(1-e^{-t_{0}})((1-\beta)(1+q)\rho-|f|_{\infty}), \\ |u(t_{0})-qu(t_{0}-r)|\leq( 1+q)\rho-(1-e^{-t_{0}})(1-\beta)<(1+q)\rho. \end{gather*} By continuity, there exists a positive $\varepsilon_{0}$ such that \[ |u(t)-qu(t-r)|<(1+q)\rho\quad\text{for }t\in(t_{0},t_{0}+\varepsilon_{0}), \] which gives a contradiction and we deduce that \[ |u(t)-qu(t-r)|\leq(1+q)\rho\quad\text{for }t\geq 0. \] Let $t\in[0,r]$. Then \[ |u(t)|\leq(1+q)\rho+q\rho \leq(1+q)(1+q)\rho, \] and for $t\in[ r,2r]$, \[ |u(t)|\leq(1+q)( 1+q+q^{2})\rho. \] We proceed by steps, then for $t\in[(n-1)r,nr]$, we have \[ |u(t)|\leq(1+q)( 1+q+q^{2}+\dots+q^{n})\rho. \] Consequently, \[ |u(t)|\leq(1+q)\rho \underset{n\geq 0}{\sum}q^{n}=\frac{1+q}{1-q}\rho\quad\text{for all }t\geq 0. \] Then, Equation (\ref{linear}) has a bounded solution $u$ on $\mathbb{R}^{+}$. By Theorem \ref{colat}, we deduce that Equation (\ref{linear}) has an $\omega$-periodic solution. \end{proof} \subsection*{Nonlinear case} We consider the nonlinear equation \begin{equation} \begin{gathered} \begin{aligned} \frac{\partial}{\partial t}[u(t,x)-qu(t-r,x)] &=\frac {\partial^{2}}{\partial x^{2}}[u(t,x)-qu(t-r,x)] +a_2 (t)g_1 (u(t-r,x)) \\ &\quad +h_2 (t,x) \quad\text{for }t\geq 0,\; x\in[0,\pi], \end{aligned}\\ [u(t,x)-qu(t-r,x)]_{x=0,\pi}=0\quad\text{for }t\geq 0, \\ u(\theta,x)=\varphi_{0}(\theta,x)\quad\text{for }\theta\in[-r,0],\; x\in[0,\pi], \end{gathered} \label{exanon} \end{equation} where $g_1 :\mathbb{R}\to\mathbb{R}$ is a Lipschitz continuous function and $a_2 $, $\varphi_{0}:[-r,0]\times[0,\pi]\to \mathbb{R}$ are continuous functions and $0r$ that \[ \label{enq2}\underset{s\in[ t-r,t]}{\sup}|h(s)|\leq\widetilde{a} e^{r-t}|\varphi|+\widetilde{b}\quad\text{for }\varphi\in C.\newline \] Using the estimate \eqref{est wu expo}, we obtain \[ |u_{t}(.,\varphi)|\leq ae^{-bt}|\varphi|+c\quad\text{for }t>r\,\; \varphi\in C, \] for some positive constants $a$, $b$ and $c$. Consequently, there exists a positive constant $\widetilde{K}$ such that \[ \limsup_{t\to+\infty} |u(t,\varphi)|<\widetilde {K}\quad\text{for }\varphi\in C, \] and we deduce that the solutions of \eqref{e0} are ultimately bounded. \end{proof} Consequently by Theorem \ref{theo de perio nonline}, we obtain the following result. \begin{proposition} Assume that (H9) and (H10) hold. Then \eqref{e0} has an $\omega$-periodic solution. \end{proposition} \subsection*{Acknowledgments} The authors would like to thank the anonymous referee for his/her careful reading of the original version. \begin{thebibliography}{00} \bibitem {adimezzin} M. Adimy and K. Ezzinbi, Local existence and linearized stability for partial functional differential equations, Dynamic Systems and Applications, Vol. 7, no. 3, 389-404, (1998). \bibitem {AdiEzz1} M. Adimy and K. 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