ABSTRACT TREATMENT OF SOME FORWARD-BACKWARD PROBLEMS OF TRANSPORT AND SCATTERING

Two abstract methods are described for problem such as x( ∂u ∂t)(x, t) + Au(x, t) = f(x, t), u(x, 0) = g+(x), x > 0; u(x, t1) = -g-(x), x < 0, which occur, among other places, in transport and scattering theory. Here A is a positive linear operator which may be integral, differential, or integrodifferential. The problem is given an abstract formulation as an operator differential equation. Existence and uniqueness of solutions is proved by a variational method. A more constructive operator-theoretic approach is also given. Various applications are described. © 1979.