Some complementary bivariational principles for linear integral equations of Fredholm type

A simple method is described which can be used to generate complemen­tary bivariational principles yielding upper and lower bounds to the quantity Q ═ ∫sp(s) n0(s) ds, where p(s) is the (vector) solution of a linear integral equation of Fredholm type p0(s) = p(s) — λ ∫sK(s, s') p(s') ds' and n0(s) and p0(s) are given functions. The method involves a generaliza­tion, requiring two approximating functions, of results obtained from a study of the particular case n0(s) = p0(s), a classical variational problem occurring in transport theory and other fields of applied mathematics. The bounds are compared with those of other authors and some further generalizations are indicated.