Asymptotic properties of solutions to fourth order difference equations

Abstract

We consider equations of the form Δ 2 ( r n Δ 2 x n ) = a n f ( x σ ( n ) ) + b n , where a , b are sequences of real numbers, r is a sequence of positive real numbers, σ a sequence of integers. Let Y denote the space of all solutions of the equation Δ 2 ( r n Δ 2 y n ) = 0 , and let s be a fixed nonpositive real number. We present sufficient conditions under which for a given sequence y ∈ Y there exists a solution x with the asymptotic behavior x n = y n + o ( n s ) . Moreover, we establish conditions under which for a given solution x there exists a sequence y ∈ Y such that x has the above asymptotic behavior. The obtained results are applied to the study of solutions of discrete Euler–Bernoulli beam equation.