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: {un+1−(1+θn)un+θnun−1∈cnAun+fnn≥1u0=x,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. We show the weak and strong convergence of solutions and their weighted averages to an element of A−1(0)A−1(0), which is the asymptotic center of the sequence {un}{un}, under appropriate assumptions on the sequences {cn}{cn}, {θn}{θn} and {fn}{fn}. Our results continue our previous work in Djafari Rouhani and Khatibzadeh (2008, 2010) [30] and [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) [27] and [28], Morosanu (1979) [21] and Mitidieri and Morosanu (1985–86) [22] to the nonhomogeneous case and without assuming AA to have a nonempty zero set.