On the Limit of Regular Dissipative and Self-Adjoint Boundary Value Problems with Nonseparated Boundary Conditions when an Interval Stretches to the Semiaxis

Abstract

For the symmetric differential system of the first order that contains a spectral parameter in Nevanlinna's manner the limit of regular boundary value problems with dissipative or accumulative nonseparated boundary conditions is studied when the interval stretches to the semiaxis. When for the considered system the case of the limit point takes place in one of the complex half-planes, we obtain the condition which guarantees the non-self-adjointness of the boundary condition at zero that corresponds to the limit boundary problem. This result is illustrated on the perturbed almost periodic systems. When the boundary condition in the prelimit regular problems is periodic, we show that the limit characteristic matrix is also the characteristic matrix on the whole axis if the coefficients of the system are extended in a certain way on the negative semiaxis. In the general case we find the condition when the convergence of characteristic matrixes implies the convergence of resolvents.