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The aim of this paper is to continue and extend the discussion in [9] concerning the 'numerically irrelevant' Solutions NIS' of finite difference approximations to certain nonlinear boundary value problems in the context of continuation methods or more generally and more appropriately, in the context of global topological perturbations of nonlinear eigen value problems. This latter technique äs a numerical device for the global numerical study of nonlinear eigenvalue and bifurcation problems has been set up, discussed and applied to various nonlinear problems in [9] and [11]. Our discussion here will make an essential use of these techniques and ideas and moreover, provide a mathematical foundation for the numerical procedures suggested in [9] which were designed to avoid NIS. Our present attempt and approach has been motivated by a recent numerical study of Bohl [4] and a recent paper of Ambrosetti and Hess [3]. The former deals with the existence and characterization of NIS of finite difference approximations to a boundary value problem of the type

Nonlinear elliptic boundary value problems versus their finite difference approximations: numerically irrelevant solutions.