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)

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.