As a continuation of our previous work in Djafari Rouhani and Katibzadeh (2008) [1], we investigate the asymptotic behavior of solutions to the following system of second order nonhomogeneous difference equations
y =  ⎧ ⎪ ⎨
⎪ ⎩

u_{n+1}−(1+θ_{n})u_{n} +θ_{n}u_{n−1} ∈ c_{n}Au_{n} +f_{n} 
 
u_{0} = a ∈ H, 
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. With suitable conditions on A and the sequences {c_{n}},{θ_{n}} and {f_{n}}, we show the weak or strong convergence of {u_{n}} or its weighted average to an element of A^{−1}(0), which is also the asymptotic center of the sequence {u_{n}}, implying therefore in particular that the existence of a solution {u_{n}} implies that A^{−}1(0) ≠ \varnothing . Our result extend some previous results by Apreutesei (2007, 2003, 2003) [13,23,24], Morosanu (1988, 1979) [4,20], and Mitidieri and Morosanu (1985/86) [31], whose proofs use the assumption A^{−1}(0) ≠ \varnothing , as well as the authors Djafari Rouhani and Khatibzadeh (2008) [1](as mentioned there in the section on future directions), to the nonhomogeneous case with {θ_{n}} ≠ 1. We also present some applications of our results to optimization.
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