On Bounded Weak and Strong Solutions of Non Linear Differential Equations with and without Delay in Banach Spaces

Assume that $E$ is a Banach space, $B_{r}=\{x\in E:\Vert x\Vert \le r\}$ and $C([-d,0],B_{r})$ is the Banach space of continuous functions from $[-d,0]$ into $B_{r}$. Consider $f:\mathbf{R}^+\times E\to E$; $f^{d}:[0,T]\times C([-d,0],B_{r})\to E$; for each $t\in [0,T]$ the mapping $\theta_{t}\in C([-d,0],B_{r})$ is defined by $\theta_{t}x(s)= x(t+s)$, $s\in [-d,0]$ and let $A(t)$ be a linear operator from $E$ into itself. In this paper we give existence theorems for bounded weak and strong solutions of the nonlinear differential equation 26767 \dot{x}(t)=A(t)x+f(t,x),\qquad t\in \mathbf{R}^+, 26767 and we prove that, with certain conditions, the differential equation with delay 26767 \dot{x}(t)=L(t)x(t)+f^{d}(t,\theta_{t}x),\qquad \text{if}\quad t\in [0,T] \qquad\qquad(\mathrm{Q}) 26767 has at least one weak solution where $L(t)$ is a linear operator from $E$ into $E$. Moreover, under suitable assumptions, the problem $(\mathrm{Q})$ has a solution. Furthermore under a generalization of the compactness assumptions, we show that $(\mathrm{Q})$ has a solution too.