On theorems for weak solutions of nonlinear differential equations with and without delay in Banach spaces

In the present work we give an existence theorem for bounded weak solution of the differential equation $$\dot{x}(t)=A(t)x(t)+f(t,x(t)),t\ge 0$$ where $\{A(t):t\in I{\mathbb{R}}^{+}\}$ is a family of linear operators from a Banach space $E$ into itself, $B_r=\{x\in E:\Vert x\Vert \le r\}$ and $f:{\mathbb{R}}^{+}\times B_r\to E$ is weakly-weakly continuous. Furthermore, we give existence theorem for the differential equation with delay $$\dot{x}(t)=\hat{A}(t)x(t)+f^d(t,{\theta }_{t}x)\text{if}t\in [0,T],$$ where $T,d>0$, $C_{B_r}([-d,0])$ is the Banach space of continuous functions from $[-d,0]$ into $B_r$, $f_d:[0,T]\times C_{B_r}([-d,0])\to E$ weakly-weakly continuous function, $\hat{A}(t):[0,T]\to L(E)$ is strongly measurable and Bochner integrable operator on $[0,T]$ and ${\theta }_{t}x(s)=x(t+s)$ for all $s\in [-d,0]$.