Asymptotic formulae for nonlinear functional difference equations

In this paper, we show asymptotic formulae for solutions of the nonlinear functional difference equation y(n + 1) = L(yn) + V(n,yn) + f(n,yn), where yn(θ) = (n + θ) for θ ε {−r, −r + 1, … , 0}, L, V (n, •) are linear applications, and f(n, •) is not necessarily linear defined from {−r, −r + 1, … ,0} to CN. We ask for a trichotomic spectral condition on L, ¦V(n, •)¦ → 0 as n → +∞, ¦V(n + 1, •) − V(n, •)¦ ε ℓ1, f(n, 0) = 0, and there is γ ε ℓ1 such that ¦f(n,x) − f(n,y)¦ ≤ γ(n)¦x − y¦.