Multiple solutions for a class of perturbed damped vibration problems

Volume 16, Issue 3, pp 351-363 http://dx.doi.org/10.22436/jmcs.016.03.05

Publication Date: September 15, 2016 Submission Date: February 25, 2016

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 2643 Downloads
• 4256 Views

Authors

Mohamad Reza Heidari Tavani - Department of Mathematics, Science and Research branch, Islamic Azad University, Tehran, Iran. Ghasem A. Afrouzi - Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran. Shapour Heidarkhani - Department of Mathematics, Faculty of Sciences, Razi University, 67149 Kermanshah, Iran.

Abstract

The existence of three distinct weak solutions for a class of perturbed damped vibration problems with nonlinear terms depending on two real parameters is investigated. Our approach is based on variational methods.

Share and Cite

Keywords

• Multiple solutions
• perturbed damped vibration problem
• critical point theory
• variational methods.

MSC

•  34C25
•  58E30
•  47H04

References

• [1] G. A. Afrouzi, S. Heidarkhani, S. Moradi , Perturbed elastic beam problems with nonlinear boundary conditions , Annal. Al. I. Cuza Univ. Math., (to appear),


• [2] F. Antonacci, P. Magrone, Second order nonautonomous systems with symmetric potential changing sign, Rend. Mat. Appl., 18 (1998), 367-379.
  • MathSciNet
  • Google Scholar
  • MATH


• [3] G. Bonanno, P. Candito , Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities, J. Differential Equations, 244 (2008), 3031-3059.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [4] G. Bonanno, G. DAgui, Multiplicity results for a perturbed elliptic Neumann problem , Abstr. Appl. Anal., 2010 (2010 ), 10 pages.
  • View Article
  • MathSciNet


• [5] G. Bonanno, R. Livrea, Periodic solutions for a class of second-order Hamiltonian systems, Electron. J. Differential Equations, 2005 (2005), 13 pages.
  • Google Scholar


• [6] G. Bonanno, R. Livrea, Multiple periodic solutions for Hamiltonian systems with not coercive potential , J. Math. Anal. Appl., 363 (2010), 627-638.
  • View Article
  • Google Scholar
  • MATH


• [7] G. Bonanno, R. Livrea , Existence and multiplicity of periodic solutions for second order Hamiltonian systems depending on a parameter , J. Convex Anal., 20 (2013), 1075-1094.
  • MathSciNet
  • Google Scholar
  • MATH


• [8] G. Bonanno, S. A. Marano , On the structure of the critical set of non-differentiable functions with a weak compactness condition , Appl. Anal., 89 (2010), 1-10.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [9] G. Chen, Nonperiodic damped vibration systems with asymptotically quadratic terms at infinity: infinitely many homoclinic orbits , Abstr. Appl. Anal., 2013 (2013 ), 7 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [10] G. Chen, Non-periodic damped vibration systems with sublinear terms at infinity: infinitely many homoclinic orbits , Nonlinear Anal., 92 (2013), 168-176.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [11] H. Chen, Z. He , New results for perturbed Hamiltonian systems with impulses , Appl. Math. Comput., 218 (2012), 9489-9497.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [12] G.-W. Chen, J. Wang, Ground state homoclinic orbits of damped vibration problems, Bound. Value Probl., 2014 (2014 ), 15 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [13] G. Cordaro , Three periodic solutions to an eigenvalue problem for a class of second order Hamiltonian systems , Abstr. Appl. Anal., 18 (2003), 1037-1045.
  • View Article
  • MathSciNet
  • Google Scholar


• [14] G. Cordaro, G. Rao, Three periodic solutions for perturbed second order Hamiltonian systems , J. Math. Anal. Appl., 359 (2009), 780-785.
  • View Article
  • MathSciNet
  • Google Scholar


• [15] G. DAgui, S. Heidarkhani, G. Molica Bisci, Multiple solutions for a perturbed mixed boundary value problem involving the one-dimensional p-Laplacian , Electron. J. Qual. Theory Diff. Eqns., 2013 (2013 ), 14 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [16] F. Faraci, Multiple periodic solutions for second order systems with changing sign potential, J. Math. Anal. Appl., 319 (2006), 567-578.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [17] F. Faraci, R. Livrea, Infinitely many periodic solutions for a second-order nonautonomous system , Nonlinear Anal., 54 (2003), 417-429.
  • View Article
  • MathSciNet
  • Google Scholar


• [18] J. R. Graef, S. Heidarkhani, L. Kong, Infinitely many solutions for a class of perturbed second-order impulsive Hamiltonian systems, Acta Appl. Math., 139 (2015), 81-94.


• [19] J. R. Graef, S. Heidarkhani, L. Kong , Nontrivial periodic solutions to second-order impulsive Hamiltonian systems, Electron. J. Differential Equations, 2015 (2015), 17 pages.
  • MathSciNet
  • Google Scholar
  • MATH


• [20] S. Heidarkhani, G. A. Afrouzi, M. Ferrara, G. Caristi, S. Moradi, Existence results for impulsive damped vibration systems , Bull. Malays. Math. Sci. Soc., 2016 (2016 ), 20 pages.
  • View Article
  • Google Scholar


• [21] J. Mawhin, M. Willem, Critical Point Theory and Hamiltonian Systems, Springer-Verlag, New York (1989)
  • Google Scholar


• [22] P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems , Proc. Roy. Soc. Edinburgh Sect. A, 114 (1990), 33-38.
  • View Article
  • Google Scholar


• [23] P. H. Rabinowitz, Variational methods for Hamiltonian systems, Handbook of Dynamical Systems vol. 1, Part A, 2002 (2002), 1091-1127.
  • View Article
  • Google Scholar


• [24] J. Sun, H. Chen, J. J. Nieto, M. Otero-Novoa, The multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects , Nonlinear Anal., 72 (2010), 4575-4586.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [25] C.-L. Tang, Periodic solutions of non-autonomous second order systems with \(\gamma\)-quasisubadditive potential, J. Math. Anal. Appl., 189 (1995), 671-675.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [26] C.-L. Tang, Periodic solutions for nonautonomous second order systems with sublinear nonlinearity , Proc. Amer. Math. Soc., 126 (1998), 3263-3270.
  • View Article
  • MathSciNet


• [27] C.-L. Tang, X.-P. Wu, Periodic solutions for a class of nonautonomous subquadratic second order Hamiltonian systems, J. Math. Anal. Appl., 275 (2002), 870-882.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [28] X. Wu, J. Chen, Existence theorems of periodic solutions for a class of damped vibration problems, Appl. Math. Comput., 207 (2009), 230-235.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [29] X. Wu, S. Chen, K. Teng, On variational methods for a class of damped vibration problems , Nonlinear Anal., 68 (2008), 1432-1441.
  • View Article
  • MathSciNet
  • Google Scholar


• [30] X. Wu, W. Zhang, Existence and multiplicity of homoclinic solutions for a class of damped vibration problems, Nonlinear Anal., 74 (2011), 4392-4398.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [31] J. Xiao, J. J. Nieto, Variational approach to some damped Dirichlet nonlinear impulsive differential equations, J. Franklin Inst., 348 (2011), 369-377.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [32] E. Zeidler, Nonlinear functional analysis and its applications, Vol. II: Linear monotone operators, Springer-Verlag, New York (1985)
  • View Article
  • MathSciNet


• [33] W. Zou , Variant fountain theorems and their applications, Manuscripta Math., 104 (2001), 343-358.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH