Assume $g:[0,\infty) \to \mathbb R$ to be continuous and $$\int_{0}^{\infty} s|g(s)| \,\mathbb ds< \infty .$$ I want to find $\alpha>0$ such that the system of integral equations $$y_1(t)=1+\int_{t}^{\infty} (t-s)g(s)y_1(s) \,\mathbb ds$$ $$y_2(t)=t+\int_{\alpha} ^ {t} sg(s)y_2(s) \,\mathbb ds + \int_{t}^{\infty} tg(s)y_2(s) \,\mathbb ds $$ has a solution with these properties $$\lim_{t \to \infty } y_1(t) =1 $$ $$\lim_{t \to \infty} y_1'(t)=0$$ $$\lim_{t \to \infty } \frac{y_2(t)}{t} =1 $$ $$ \lim_{t \to \infty} y_2'(t)=1.$$
I could show these properties, but I do not understand why $\alpha$ is important and where it is used.