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

• Autorzy: Rouhani Behzad Djafari , Piranfar Mohsen Rahimi

Identyfikatory

DOI

10.1515/dema-2019-0028

• Języki publikacji: EN

Abstrakty

EN

We consider the following second order evolution equation modelling a nonlinear oscillator with damping ü(t) +𝛾 ů(t) + Au(t) = f(t), where A is a maximal monotone andα-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), 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].

Słowa kluczowe

EN

nonlinear oscillator with damping; inexact inertial proximal method; asymptotic behavior; maximal monotone operator; asymptotic center

PL

oscylator nieliniowy; zachowanie asymptotyczne; operator monotoniczny

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2019
• Tom: Vol. 52, nr 1
• Strony: 274--282
• Opis fizyczny: Bibliogr. 12 poz.

Twórcy

autor

Rouhani Behzad Djafari

• Department of Mathematical Sciences, University of Texas at El Paso, 500W. University Ave., El Paso, TX 79968, USA

autor

Piranfar Mohsen Rahimi

• Department of Mathematical Sciences, University of Texas at El Paso, 500 W. University Ave., El Paso, TX 79968, USA
• Department of Mathematics, Institute for Advanced Studies in Basic Sciences, P.O. Box 45195-1159, Zanjan, Iran

Bibliografia

• [1] Djafari Rouhani B., Asymptotic behaviour of quasi-autonomous dissipative systems in Hilbert spaces, J. Math. Anal. Appl., 1990, 147, 465-476
• [2] Djafari Rouhani B., Asymptotic behaviour of almost nonexpansive sequences in a Hilbert space, J. Math. Anal. Appl., 1990, 151, 226-235
• [3] Djafari Rouhani B., Khatibzadeh H., Asymptotic behavior of bounded solutions to some second order evolution systems, Rocky Mountain J. Math., 2010, 40, 1289-1311
• [4] 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
• [5] 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
• [6] Alvarez F., On the minimizing property of a second order dissipative system in Hilbert spaces, SIAM J. Control Optim., 2000, 38, 1102-1119
• [7] Attouch H., Goudou X., Redont P., The heavy ball with friction method, I. The continuous dynamical system: global exploration of the local minima of real-valued function by asymptotic analysis of a dissipative dynamical system, Commun. Contemp. Math., 2000, 2, 1-34
• [8] 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, 836-857
• [9] 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
• [10] Edelstein M., The construction of an asymptotic center with a fixed-point property, Bull. Amer. Math. Soc., 1972, 78, 206-208
• [11] Djafari Rouhani B., Khatibzadeh H., On the proximal point algorithm, J. Optim. Theory Appl., 2008, 137, 411-417
• [12] Brézis H., Lions P. L., Produits infinis de résolvantes, Israel J. Math., 1978, 29, 329-345

Uwagi

PL

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).