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

Behzad Djafari Rouhani; Mohsen Rahimi Piranfar

摘要

We consider the following second order evolution equation modelling a nonlinear oscillator with damping ü(t)+γu˙(t)+Au(t)=f(t), (SEE)$$ddot{u} (t) + gamma dot u(t) + Auleft( t right) = fleft( t right),,,,,,,,,,,,,,left( {{rm{SEE}}} right)$$ 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 un+1-un-αn(un-un-1)+λnAun+1∋f(t),$${u_{n + 1}} - {u_n} - {alpha _n}left( {{u_n} - {u_{n - 1}}} right) + {lambda _n}A{u_{n + 1}ni} fleft( t right),$$ 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 u n , and show that the limit is the asymptotic center of u n and belongs to A −1 (0). This again shows that A −1 (0) ≠ ∅ if and only if u n 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.

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

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