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

• Volume: 47, Issue: 2
• ISSN: 2080-1211

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## Abstract

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In the present work we give an existence theorem for bounded weak solution of the differential equation $\stackrel{˙}{x}\left(t\right)=A\left(t\right)x\left(t\right)+f\left(t,x\left(t\right)\right),\phantom{\rule{1.0em}{0ex}}t\ge 0$ where $\left\{A\left(t\right):t\in I{ℝ}^{+}\right\}$ is a family of linear operators from a Banach space $E$ into itself, ${B}_{r}=\left\{x\in E:\parallel x\parallel \le r\right\}$ and $f:{ℝ}^{+}×{B}_{r}\to E$ is weakly-weakly continuous. Furthermore, we give existence theorem for the differential equation with delay $\stackrel{˙}{x}\left(t\right)=\stackrel{^}{A}\left(t\right)x\left(t\right)+{f}^{d}\left(t,{\theta }_{t}x\right)\phantom{\rule{1.0em}{0ex}}\text{if}\phantom{\rule{4pt}{0ex}}t\in \left[0,T\right],$ where $T,d0$, ${C}_{{B}_{r}}\left(\left[-d,0\right]\right)$ is the Banach space of continuous functions from $\left[-d,0\right]$ into ${B}_{r}$, ${f}_{d}:\left[0,T\right]×{C}_{{B}_{r}}\left(\left[-d,0\right]\right)\to E$ weakly-weakly continuous function, $\stackrel{^}{A}\left(t\right):\left[0,T\right]\to L\left(E\right)$ is strongly measurable and Bochner integrable operator on $\left[0,T\right]$ and ${\theta }_{t}x\left(s\right)=x\left(t+s\right)$ for all $s\in \left[-d,0\right]$.

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