International Journal of Mathematics and Mathematical Sciences
Volume 3 (1980), Issue 1, Pages 69-77
doi:10.1155/S0161171280000051
Existence theorem for the difference equation Yn+1−2Yn+Yn−1=h2f(yn)
Department of Physics-Mathematics, Université de Moncton, Moncton, N. B., Canada
Received 27 November 1978
Copyright © 1980 F. Weil. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
For the difference equation (Yn+1−2Yn+Yn−1)h2=f(Yn) sufficient conditions are shown such that for a given Y0 there is either a unique value of Y1 for which the sequence Yn strictly monotonically approaches a constant as n approaches infinity or a continuum interval of such values. It has been shown previously that the first alternative is related to the existence of a Peierls barrier energy in the dislocation model of Frenkel and Kontorova.