On the completeness of root subspaces of boundary value problems for first order systems of ordinary differential equations

Abstract

The paper is concerned with the completeness problem of root functions of general boundary value problems for first order systems of ordinary differential equations. We introduce and investigate the class of weakly regular boundary conditions. We show that this class is much broader than the class of regular boundary conditions introduced by G.D. Birkhoff and R.E. Langer. Our main result states that the system of root functions of a boundary value problem is complete and minimal provided that the boundary conditions are weakly regular. Moreover, we show that in some cases the weak regularity of boundary conditions is also necessary for the completeness. Also we investigate the completeness for 2×2 Dirac type equations subject to irregular boundary conditions. Emphasize that our results are the first results on the completeness for general first order systems even in the case of regular boundary conditions.