Nonlinear Analysis, Theory,Methods&Applications, Vol. 24, No. 2, pp. 251-256, 1995 Copyright © 1995ElsevierScience Ltd Pergamon Printed in Great Britain. All rights reserved 0362-546X/95 $9.50 + .00 0362-546X(94)E0003-Y WELL-POSED SOLVABILITY OF THE BOUNDARY VALUE PROBLEM FOR DIFFERENCE EQUATIONS OF ELLIPTIC TYPE ALLABEREN ASHYRALYEV Department of Mathematics, The Turkmen State University, Saparmyrat Turkmenbashy Shayoly 31, 744000 Ashgabat, Turkmenistan (Received 1 September 1991; received for publication 27 January 1994) Key words and phrases: Difference schemes, elliptic equations, stability estimates, coercive inequality. SECTION 1 A well-known widely used method for approximate solutions of various problems of mathematical physics is the method of difference schemes. The main characteristics of differences schemes are their accuracy and stability. Modern computers give the possibility of realizing highly accurate difference schemes. Therefore, construction and investigation of difference schemes of high accuracy for various types of boundary value problems of mathematical physics are very important problems. The present paper is devoted to the construction and investigation of difference schemes of high order accuracy for approximately solving the boundary value problem -v"(t) + Av(t) = f(t) (0 < t <_ 1), v(O) = Vo, v(1) = v~. (1) For differential equations in an arbitrary Banach space E with an unbounded positive operator A. It is known (see [1, 2]) that various boundary value problems for elliptical equations can be reduced to the boundary value problem (1). Investigation of two step difference scheme of high order accuracy for approximately solving the boundary value problem (1) has been studied in [3]. This difference scheme is constructed using the exact difference scheme for the solution of the boundary value problem (1) but with an operator A 1/2, instead of A. Therefore, the problem of substantiation of difference schemes is easier. We have established the stability and coercive stability of the difference scheme in the difference analogues of many Banach spaces of smooth functions. However, for practical realization of this difference scheme it is first necessary to construct an operator A1/2. This action is very difficult for a computer. Therefore, in spite of theoretical results the role of their application of a numerical solution for a boundary value problem is decreased. In the present paper we investigate solvability two steps of difference schemes for approximately solving the abstract boundary value problem (1) reproduced by the Taylor's decomposition in three points. It is based upon stability and coercive stability of this difference scheme. This difference scheme is constructed with operator A and, therefore, it is free from the defects of difference schemes which were investigated in [3]. However, for the substantiation of this difference scheme it is necessary to construct an operator B = B(r, A) (a considerably difficult formula) and to give certain estimates for the order of the resolvent of operator B. Therefore, we cannot investigate this scheme in the general case of the positive operator A. We establish the well-posed two step difference scheme of high accuracy for 251