Rocky Mountain Journal of Mathematics

Asymptotic Behavior of Bounded Solutions to Some Second Order Evolution Systems

Behzad Djafari Rouhani and Hadi Khatibzadeh

Full-text: Open access

Article information

Source
Rocky Mountain J. Math., Volume 40, Number 4 (2010), 1289-1311.

Dates
First available in Project Euclid: 30 August 2010

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1283175799

Digital Object Identifier
doi:10.1216/RMJ-2010-40-4-1289

Mathematical Reviews number (MathSciNet)
MR2718815

Zentralblatt MATH identifier
1216.47108

Subjects
Primary: 47J35: Nonlinear evolution equations [See also 34G20, 35K90, 35L90, 35Qxx, 35R20, 37Kxx, 37Lxx, 47H20, 58D25] 34G25: Evolution inclusions
Secondary: 47H05: Monotone operators and generalizations 47H04: Set-valued operators [See also 28B20, 54C60, 58C06]

Keywords
Second order evolution equation monotone operator asymptotic behavior nonexpansive curve ergodic theorem

Citation

Rouhani, Behzad Djafari; Khatibzadeh, Hadi. Asymptotic Behavior of Bounded Solutions to Some Second Order Evolution Systems. Rocky Mountain J. Math. 40 (2010), no. 4, 1289--1311. doi:10.1216/RMJ-2010-40-4-1289. https://projecteuclid.org/euclid.rmjm/1283175799


Export citation

References

  • J.B. Baillon, Un théorème de type ergodique pour les contractions non linéaires dans un espace de Hilbert, C.R. Acad. Sci. Paris 280 (1975), A1511-A1514.
  • J.B. Baillon and H. Brézis, Une remarque sur 1e comportement asymptotique des semi-groupes non linéaires, Houston J. Math. 2 (1976), 5-7.
  • V. Barbu, Nonlinear semigroups and differential equations in Banach spaces, Noordhoff International Publishing, Leiden, 1976.
  • H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Math. Stud. 5, North-Holland Publishing Co., Amsterdam, 1973.
  • H. Brézis and F.E. Browder, Remarks on nonlinear ergodic theory, Adv. Math. 25 (1977), 165-177.
  • R.E. Bruck, On the strong convergence of an averaging iteration for the solution of operator equations involving monotone operators in Hilbert space, J. Math. Anal. Appl. 64 (1978), 319-327.
  • --------, Asymptotic convergence of nonlinear contraction semigroups in Hilbert spaces, J. Funct. Anal. 18 (1975), 15-26.
  • --------, Periodic forcing of solutions of a boundary value problem for a second order differential equation in Hilbert space, J. Math. Anal. Appl., 76 (1980), 159-173.
  • M. Edelstein, The construction of an asymptotic center with a fixed-point property, Bull. Amer. Math. Soc. 78 (1972), 206-208.
  • J. Garcia Falset, W. Kaczor, T. Kuczumow and S. Reich, Weak convergence theorems for asymptotically nonexpansive mappings and semigroups, Nonlinear Anal. 43 (2001), 377-401.
  • K. Goebel and W.A. Kirk, Topics in metric fixed point theory, Cambridge Stud. Adv. Math. 28, Cambridge University Press, Cambridge, 1990.
  • K. Goebel and S. Reich, Uniform convexity, hyperbolic geometry, and nonexpansive mappings, Monographs Textbooks Pure Appl. Math. 83, Marcel Dekker, Inc., New York, 1984.
  • W. Kaczor, T. Kuczumow and S. Reich, A mean ergodic theorem for nonlinear semigroups which are asymptotically nonexpansive in the intermediate sense, J. Math. Anal. Appl. 246 (2000), 1-27.
  • E. Mitidieri, Asymptotic behaviour of some second order evolution equations, Nonlinear Anal. 6 (1982), 1245-1252.
  • --------, Some remarks on the asymptotic behaviour of the solutions of second order evolution equations, J. Math. Anal. Appl. 107 (1985), 211-221.
  • G. Morosanu, Nonlinear evolution equations and applications, Editura Academiei Romane (and D. Reidel Publishing Company), Bucharest, 1988.
  • --------, Asymptotic behaviour of solutions of differential equations associated to monotone operators, Nonlinear Anal. 3 (1979), 873-883.
  • H. Okochi, A note on asymptotic strong convergence of nonlinear contraction semigroups, Proc. Japan Acad. Math. Sci. 56 (1980), 83-84.
  • I.E. Poffald and S. Reich, An incomplete Cauchy problem, J. Math. Anal. Appl. 113 (1986), 514-543.
  • B. Djafari Rouhani, Ergodic theorems for nonexpansive sequences in Hilbert spaces and related problems, Ph.D. thesis, Yale University, 1981%, part I, pp. 1-76.
  • --------, Asymptotic behaviour of quasi-autonomous dissipative systems in Hilbert spaces, J. Math. Anal. Appl. 147 (1990), 465-476.
  • --------, Asymptotic behaviour of almost nonexpansive sequences in a Hilbert space, J. Math. Anal. Appl. 151 (1990), 226-235.
  • --------, An ergodic theorem for sequences in a Hilbert space, Nonlinear Anal. Forum 4 (1999), 33-48.
  • L. Véron, Un exemple concernant le comportement asymptotique de la solution bornée de l'équation $d^2u/dt^2 \in \pa \vp (u)$, Monatsh. Math. 89 (1980), 57-67.
  • --------, Problèmes d'évolution du second ordre associés à des opérateurs monotones, C.R. Acad. Sci. Paris 278 (1974), 1099-1101.
  • --------, Equations d'évolution du second ordre associées à des opérateurs maximaux monotones, Proc. Roy. Soc. Edinburgh 75 (1975/76), 131-147.