if any one can help me how to prove this theorem.
Prove that for the IVP $$ \left\{ \begin{aligned} x′(t)&=cx(t)[1−x(t−r)]\\ x(\mu)&=\phi(\mu),\quad \mu \in [−r,0] \end{aligned} \right\| $$ has for all $\phi \in C([−r,0],\mathbb{R})$ with $\phi(0)>0$ a unique solution which exists and remains positive for all $t>0$.
You can start treating this as a problem $$ x'(t)=x(t)g(t),~~ x(0)=\phi(0), $$ where you know the solution formula by separation, $$ x(t)=\phi(0)\exp\left(\int_0^t g(s)\,ds\right). $$ Now insert $g(s)=c[1-x(t-r)]$ to get $$ x(t)=\phi(0)\exp\left(\int_{-r}^{t-r}c[1-x(s)] \,ds\right). $$ This should tell you anything you need.