Some Variational Principles for Integral Equations

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Abstract
Complementary variational principles are developed for the solution of Fredholm integral equations with symmetric positive-definite kernels. In particular, the theory is applied to linear equations of the type φ(r)=f(r)+λ∫K(r,s)φ(s)ds,and bounds are obtained for ∫fφdr. When λ is negative, the bounds are complementary upper and lower ones. When λ is positive, the bounds are one-sided, but an improvement is made on a result of Strieder and Prager [J. Math. Phys. 8, 514 (1967)]. A condition given by these authors for the existence of bounds does not seem to be strictly necessary, and alternative conditions are derived. Systematic improvement of bounds by iterative and scaling procedures is discussed.

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