International Journal of Mathematics and Mathematical Sciences
Volume 3 (1980), Issue 1, Pages 69-77

Existence theorem for the difference equation Yn+12Yn+Yn1=h2f(yn)

F. Weil

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.


For the difference equation (Yn+12Yn+Yn1)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.