Answer Exercise 7.4.2 for the Volterra integral equation

where a ‰¤ t ‰¤ b.
Exercise 7.4.2
The Fredholm Alternative of Theorem 5.55 first appeared in the study of what are now known as Fredholm integral equations:

in which K(x, y) and f(x) are prescribed continuous functions. Explain how the integral equation is a linear system; i.e., describe the linear map L, its domain and target spaces, and prove linearity.

u(t)+ 1 K(ts)a(s) ds = f(r). n(x) +1 K(x, y) u(v) dv=f(x).