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
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y = | ⎧
⎪
⎨
⎪
⎩
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un+1−(1+θn)un +θnun−1 ∈ cnAun +fn |
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u0 = a ∈ H, |
sup
n ≥ 0
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|un| < +∞ |
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where A is a maximal monotone operator in a real Hilbert space H, {cn} and {θn} are positive real sequences and {fn} is a sequence in H. With suitable conditions on A and the sequences {cn},{θn} and {fn}, we show the weak or strong convergence of {un} or its weighted average to an element of A−1(0), which is also the asymptotic center of the sequence {un}, implying therefore in particular that the existence of a solution {un} 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 {θn} ≠ 1. We also present some applications of our results to optimization.
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