new results on the asymptotic behavior of solutions to a class of second order nonhomogeneous difference equations

we investigate the asymptotic behavior of solutions to the following system of second

order nonhomogeneous difference equation:

{u_n+1 − (1 + θ_n)u_n + θ_nu_n−1 ∈ c_nau_n + f_n n ≥ 1

u₀ = x, sup _n≥0 |u_n| < +∞

where a is a maximal monotone operator in a real hilbert space h, {c_n} and {θ_n} are positive real sequences and {f_n} is a sequence in h. we show the weak and strong convergence of solutions and their weighted averages to an element of a−1(0), which is the asymptotic center of the sequence {u_n}, under appropriate assumptions on the sequences {c_n}, {θ_n} and {f_n}. our results continue our previous work in djafari rouhani and khatibzadeh (2008, 2010) [30,31], by presenting some new results on the asymptotic behavior of solutions, including in particular a completely new strong convergence result, and extend some previous results by apreutesei (2003), morosanu (1979) and mitidieri and morosanu (1985–86)  to the nonhomogeneous case and without assuming a to have a nonempty zero set.