Differential Equations, Vol. 41, No. 7, 2005, pp. 1010–1018. Translated from Differentsial’nye Uravneniya, Vol. 41, No. 7, 2005, pp. 961–969. Original Russian Text Copyright c  2005 by Sapagovas, ˇ Stikonas. NUMERICAL METHODS On the Structure of the Spectrum of a Differential Operator with a Nonlocal Condition M. P. Sapagovas and A. D. ˇ Stikonas Institute of Mathematics and Computer Science, Vilnius, Lithuania Received March 4, 2005 1. INTRODUCTION The papers [1, 2], where problems for a parabolic equation with an integral condition instead of a usual boundary condition were considered, and the paper [3], where a nonclassical boundary value problem for an elliptic equation with a nonlocal condition of another type was stated and in- vestigated, were among the first papers that laid a foundation for the study of differential equations with a nonlocal condition. Later, there were numerous papers dealing with differential equations of various types with nonlocal conditions as well as their finite-difference counterparts. Eigenvalue problems for differential operators with a nonlocal condition are a relatively new trend in this field. The papers [4–9] study the eigenvalue problem for a one- or two-dimensional differential operator with special nonlocal conditions that involve only boundary points. The eigenvalue problem with a nonlocal Bitsadze–Samarskii condition was considered in [10, 11], and a problem with an integral condition was studied in [12–15]. In the present paper, we consider the spectral problem for a one- or two-dimensional differential operator with a nonlocal condition containing boundary as well as interior points. The presence of interior points in the nonlocal condition makes for a more diverse picture of the eigenvalue distribution for a differential operator. The main attention is paid to the eigenvalue structure depending on the parameter occurring in the nonlocal condition. The analysis of eigenvalues of a difference operator with a nonlocal condition permits one to analyze the stability of difference schemes [6–8] and justify the convergence of iterative methods for finite-difference equations [10] and is also of interest in itself. 2. STATEMENT OF THE PROBLEM Consider the two-dimensional spectral problem ∂ 2 u ∂x 2 + ∂ 2 u ∂y 2 + λu =0 (1) in the domain 0 <x< 1, 0 <y< 1 with the boundary conditions u(0,y)= u(x, 0) = u(x, 1) = 0 (2) and with the Bitsadze–Samarskii nonlocal condition u(1,y)= γu(ξ,y), (3) where γ and ξ are given real numbers and 0 <ξ< 1. We also consider the similar one-dimensional problem d 2 u dx 2 + λu =0, 0 <x< 1, (4) u(0) = 0, (5) u(1) = γu(ξ ). (6) 0012-2661/05/4107-1010 c  2005 Pleiades Publishing, Inc.