A Note on Bitsadze-Samarskii Type Nonlocal Boundary Value Problems: Well-Posedness

The Bitsadze-Samarskii type nonlocal boundary value problem {(d)2u(t)/dt(2) + Au(t) = f(t), 0 < t < 1, u(t)(0) = phi, u(t)(1) = beta u(t)(lambda) + psi, 0 <= lambda <= 1, vertical bar beta vertical bar <= 1 for the differential equation in a Hilbert space H with the self-adjoint positive definite operator A is considered. The well posedness of this problem in Holder spaces without a weight is established. The coercivity inequalities for the solution of the Neumann-Bitsadze-Samarskii type nonlocal boundary value problem for elliptic equations are obtained. The first order of accuracy difference scheme for the approximate solution of this problem is presented. The well posedness of this difference scheme in difference analogue of Holder spaces is established. In applications, the stability, the almost coercivity and the coercivity estimates for solutions of difference schemes for elliptic equations are obtained.