Nonhomogeneous nonlinear oscillator with damping: asymptotic analysis in continuous and discrete time

We consider the following second order evolution equation modelling a nonlinear oscillator with damping$$\ddot{u} (t) + \gamma \dot u(t) + Au\left( t \right) = f\left( t \right),\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {{\rm{SEE}}} \right)$$where A is a maximal monotone and α-inverse strongly monotone operator in a real Hilbert space H. With suitable assumptions on γ and f(t) we show that A−1(0) ≠ ∅, if and only if (SEE) has a bounded solution and in this case we provide approximation results for elements of A−1(0) by proving weak and strong convergence theorems for solutions to (SEE) showing that the limit belongs to A−1(0). As a discrete version of (SEE), we consider the following second order difference equation$${u_{n + 1}} - {u_n} - {\alpha _n}\left( {{u_n} - {u_{n - 1}}} \right) + {\lambda _n}A{u_{n + 1}\ni} f\left( t \right),$$where A is assumed to be only maximal monotone (possibly multivalued). By using the results in [Djafari Rouhani B., Khatibzadeh H., On the proximal point algorithm, J. Optim. Theory Appl., 2008, 137, 411–417], we prove ergodic, weak and strong convergence theorems for the sequence un, and show that the limit is the asymptotic center of un and belongs to A−1(0). This again shows that A−1(0) ≠ ∅ if and only if un is bounded. Also these results solve an open problem raised in [Alvarez F., Attouch H., An inertial proximal method for maximal monotone operators via dicretization of a nonlinear oscillator with damping, Set Valued Anal., 2001, 9, 3–11], namely the study of the convergence results for the inexact inertial proximal algorithm. Our paper is motivated by the previous results in [Djafari Rouhani B., Asymptotic behaviour of quasi-autonomous dissipative systems in Hilbert spaces, J. Math. Anal. Appl., 1990, 147, 465–476; Djafari Rouhani B., Asymptotic behaviour of almost nonexpansive sequences in a Hilbert space, J. Math. Anal. Appl., 1990, 151, 226–235; Djafari Rouhani B., Khatibzadeh H., Asymptotic behavior of bounded solutions to some second order evolution systems, Rocky Mountain J. Math., 2010, 40, 1289–1311; Djafari Rouhani B., Khatibzadeh H., A strong convergence theorem for solutions to a nonhomogeneous second order evolution equation, J. Math. Anal. Appl., 2010, 363, 648–654; Djafari Rouhani B., Khatibzadeh H., Asymptotic behavior of bounded solutions to a class of second order nonhomogeneous evolution equations, Nonlinear Anal., 2009, 70, 4369–4376; Djafari Rouhani B., Khatibzadeh H., On the proximal point algorithm, J. Optim. Theory Appl., 2008, 137, 411–417] and significantly improves upon the results of [Attouch H., Maingé P. E., Asymptotic behavior of second-order dissipative evolution equations combining potential with non-potential effects, ESAIM Control Optim. Calc. Var., 2011, 17(3), 836–857], and [Alvarez F., Attouch H., An inertial proximal method for maximal monotone operators via dicretization of a nonlinear oscillator with damping, Set Valued Anal., 2001, 9, 3–11].