Dual bivariational principles for linear problems

This paper presents dual bivariational principles which yield upper and lower bounds for 〈g, φ〉, where g is an arbitrary function and φ is the solution of the linear equation Aφ = f with general mixed boundary conditions. Variational principles associated with 〈f, φ〉 are taken as the starting-point, and the results generalize those of recent authors for linear integral equations.