The nonlinear difference equation \[ xₙ₊₁-x_n=a_nφ _n(xσ (n))+b_n, {(E)}\] where (aₙ), (bₙ) are real sequences, φ ₙ\: R R, (σ (n)) is a sequence of integers and lim n ∞σ (n)=∞, is investigated. Sufficient conditions for the existence of solutions of this equation asymptotically equivalent to the solutions of the equation yₙ₊₁-yₙ=bₙ are given. Sufficient conditions under which for every real constant there exists a solution of equation () convergent to this constant are also obtained.
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Janusz Migda (2004) studied this question.