Some complementary bivariational principles for linear integral equations of Fredholm type

Abstract

A simple method is described which can be used to generate complemen­tary bivariational principles yielding upper and lower bounds to the quantity Q ═ ∫ s p ( s ) n 0 ( s ) d s , where p ( s ) is the (vector) solution of a linear integral equation of Fredholm type p 0 ( s ) = p ( s ) — λ ∫ s K ( s, s' ) p ( s' ) d s' and n 0 ( s ) and p 0 ( s ) are given functions. The method involves a generaliza­tion, requiring two approximating functions, of results obtained from a study of the particular case n 0 ( s ) = p 0 ( 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.