Projection-iterative methods for a class of difference equations (Q2380361)

The paper deals with projection-iterative methods to solve operator equations of the form \(Au+ Bu= f\), where \(A\) is a Toeplitz operator in a Banach space \(l_p(N)\), \(B\) is considered as a perturbation of \(A\) and \(f\) is a known element. Solving such an equation by a projection method means to give two sequences \((P_n)\), \((Q_n)\) of projections on \(l_p(N)\) converging strongly to the identity operator, and pass to the approximate equation \[ Q_n(A+ B)P_n u_n= Q_nf,\qquad u_n\in P_n(l_p(N)). \] The projection method is convergent, if \(\lim u_n= u\) in \(l_p(N)\). The basic idea of the present paper is to combine projection and iterative procedures in order to obtain approximate methods, which are able to apply to nonelliptic cases. Convergence analysis is given for the proposed methods. The methods are illustrated by considering perturbed second-order difference equations. Equations corresponding to Jacobi and discrete Schrödinger operators are examined, too.