Asymptotic behavior of bounded solutions to a class of second order nonhomogeneous difference equations of monotone type

As a continuation of our previous work in Djafari Rouhani and Khatibzadeh (2008) [1], we investigate the asymptotic behavior of solutions to the following system of second order nonhomogeneous difference equations {un+1−(1+θn)un+θnun−1∈cnAun+fnn≥1u0=a∈H,supn≥0∣un∣<+∞ where AA is a maximal monotone operator in a real Hilbert space HH, {cn}{cn} and {θn}{θn} are positive real sequences and {fn}{fn} is a sequence in HH. With suitable conditions on AA and the sequences {cn}{cn}, {θn}{θn} and {fn}{fn}, we show the weak or strong convergence of {un}{un} or its weighted average to an element of A−1(0)A−1(0), which is also the asymptotic center of the sequence {un}{un}, implying therefore in particular that the existence of a solution {un}{un} implies that A−1(0)≠0̸A−1(0)≠0̸. Our results extend some previous results by Apreutesei (2007, 2003, 2003) [13], [23] and [24], Morosanu (1988, 1979) [4] and [20], and Mitidieri and Morosanu (1985/86) [31], whose proofs use the assumption A−1(0)≠0̸A−1(0)≠0̸, 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{θn}≠1. We also present some applications of our results to optimization.