Convergence of solutions to a second order difference inclusion

Abstract

We investigate the asymptotic behavior of solutions to the following second order difference inclusion {ui+1−(1+θi)ui+θiui−1∈ciAui,i≥1u0=x,supi≥0|ui|<+∞, where A is a maximal monotone operator in a real Hilbert space H and {ci} and {θi} are positive real sequences. We show weak and strong convergence of solutions to an element of A−1(0), for general maximal monotone operator and when A=∂φ, where φ is a convex, proper and lower semicontinuous function. Our results extend and improve previous results by Morosanu (1979) [18], Mitidieri–Morosanu (1985–86) [23] (see also Morosanu (1988), [1, pp. 156–168]) and Aperutesei (2003) [21,22], as well as some recent works of Djafari Rouhani–Khatibzadeh (2010, 2011) [25,26] and Khatibzadeh (2011) [27] with more general assumptions on parameters {ci} and {θi} in homogeneous case.