1. We describe, at first in a very formaI manner, our essential aim. n Let m be an op en subset of R, with boundary am. In m and on am we introduce, respectively, linear differential operators P and Qj' 0 i 'V. By "non-homogeneous boundary value problem" we mean a problem of the following type: let f and gj' 0 i 'v, be given in function space s F and G, F being a space" on m" and the G/ s spaces" on am"; j we seek u in a function space u/t "on m"...