I have 2 types of integral equation:
Type 1: $Z(x) = f(x) + \alpha\int_{a}^{b}Z(x-\xi)d\xi$, where $a$ and $b$ are real numbers.
Type 2: $Z(x) = f(x) + \alpha\int_{a}^{x}Z(x-\xi)d\xi$
I would like to calculate $Z(\infty)$ in each case. In the first case it is trivial, since $$Z(\infty)=f(\infty) + \alpha\int_{a}^{b}Z(\infty)d\xi = f(\infty) + \alpha Z(\infty)(b-a)$$ and hence $$Z(\infty) = \dfrac{f(\infty)}{1-\alpha(b-a)}.$$
How to to calculate $Z(\infty)$ in the second case (Type 2)? I don't know how to justify $Z(\infty)$ in the integral term of Type 2 ?
Thanks