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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 \\
\begin{displaymath}
y = \left\{\begin{array}{ll}
u_{n+1}-(1+\theta_n)u_n +\theta_n{u_{n-1}}\in c_nAu_n +f_n & n\geq1 \\
u_0 = a\in H, \hspace*{0.5 cm} \sup\limits_{n\geq0} |u_n|<+\infty \\
\end{array} \right.
\end{displaymath}
where A is a maximal monotone operator in a real Hilbert space H, $\{c_n\}$ and $\{\theta_n\}$ are positive real sequences and $\{f_n\}$ is a sequence in H. With suitable conditions on A and the sequences $\{c_n\}$,$\{\theta_n\}$ 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)\neq\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)\neq\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 $\{\theta_n\}\neq1$. We also present some applications of our results to optimization.
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