American Institute of Mathematical Sciences

Advanced
October  2014, 19(8): 2569-2580. doi: 10.3934/dcdsb.2014.19.2569

The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion

Piotr Kowalski 1,

1. 

Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warszawa, Poland

Received  December 2013 Revised  March 2014 Published  August 2014

We consider a differential inclusion which is somehow analogous to the classical Duffing's equation with Dirichlet boundary condition. We prove the existence of a solution using two-steps approach. Firstly we consider an auxiliary problem where we substitute $h := \frac{dx} {dt} \in L^2(0,1)$. Next, using the method of pseudomonotone and coercive operators, we prove the existence of a solution to the auxiliary problem. Finally we prove that under suitable assumptions, an iterative scheme converges to the solution of our inclusion, which appears to be unique.
Keywords: Dirichlet BVP, hemivariational inequalities, Duffing equation, differential inclusions., pseudomonotone operators.
Mathematics Subject Classification: Primary: 49K15; Secondary: 34B1.
Citation: Piotr Kowalski. The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2569-2580. doi: 10.3934/dcdsb.2014.19.2569
References:
[1]

P. Amster, Nonlinearities in a second order ODE, Electron. J. Differ. Equ., 6 (2001), 13-21.  Google Scholar

[2]

P. Amster and M. C. Mariani, A second order ODE with a nonlinear final condition, Electron. J. Differ. Equ., 75 (2001), 1-9.  Google Scholar

[3]

J. Aubin and A. Cellina, Differential Inclusions: Set-Valued Maps and Viability Theory, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4.  Google Scholar

[4]

M. Barboteu, K. Bartosz, P. Kalita and A. Ramadan, Analysis of a contact problem with normal compliance, Finite penetration and nonmonotone slip dependent friction, Communications in Contemporary Mathematics, 16 (2014), 1350016, 29 pp. doi: 10.1142/S0219199713500168.  Google Scholar

[5]

M. Galewski, On the Dirichlet problem for a Duffing type equation, E. J. Qualitative Theory of Diff. Equ., 15 (2011), 1-12.  Google Scholar

[6]

P. Holmes, A nonlinear oscillator with a strange attractor, Philosophical Transactions of the Royal Society A, 292 (1979), 419-448. doi: 10.1098/rsta.1979.0068.  Google Scholar

[7]

P. Holmes and D. Rand, Phase portraits and bifurcations of the non-linear oscillator $x'' +(\alpha+\gamma x^2 )x'+\beta x+\delta x^3 =0$, International Journal of Non-linear Mechanics, 15 (1980), 449-458.  Google Scholar

[8]

P. Holmes and D. Whitley, On the attracting set for Duffing's equation, II. A geometrical model for moderate force and damping, Physica D, 7 (1983), 111-123. doi: 10.1016/0167-2789(83)90121-5.  Google Scholar

[9]

P. J. Holmes and D. A. Rand, The bifurcations of Duffing's equation: An application of catastrophe theory, Journal of Sound and Vibration, 44 (1976), 237-253. doi: 10.1016/0022-460X(76)90771-9.  Google Scholar

[10]

W. Huang and Z. Shen, On a two-point boundary value problem of Duffing type equation with Dirichlet conditions. Appl. Math., 14 (1999), 131-136. doi: 10.1007/s11766-999-0018-x.  Google Scholar

[11]

J. Mawhin, The forced pendulum: A paradigm for nonlinear analysis and dynamical systems, Exposition. Math., 6 (1988), 271-287.  Google Scholar

[12]

S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities, Models and Analysis of Contact Problem, Springer, 2013. doi: 10.1007/978-1-4614-4232-5.  Google Scholar

[13]

F. C. Moon and P. J. Holmes, A magnetoelastic strange attractor, Journal of Sound and Vibration, 65 (1979), 275-296. doi: 10.1016/0022-460X(79)90520-0.  Google Scholar

[14]

F. C. Moon and P. J. Holmes, Addendum: A magnetoelastic strange attractor, Journal of Sound and Vibration, 69 (1980), 339 pp. Google Scholar

[15]

P. Tomiczek, Remark on Duffing equation with Dirichlet boundary condition, Electron. J. Differ. Equ., 81 (2007), 1-3.  Google Scholar

show all references

References:
[1]

P. Amster, Nonlinearities in a second order ODE, Electron. J. Differ. Equ., 6 (2001), 13-21.  Google Scholar

[2]

P. Amster and M. C. Mariani, A second order ODE with a nonlinear final condition, Electron. J. Differ. Equ., 75 (2001), 1-9.  Google Scholar

[3]

J. Aubin and A. Cellina, Differential Inclusions: Set-Valued Maps and Viability Theory, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4.  Google Scholar

[4]

M. Barboteu, K. Bartosz, P. Kalita and A. Ramadan, Analysis of a contact problem with normal compliance, Finite penetration and nonmonotone slip dependent friction, Communications in Contemporary Mathematics, 16 (2014), 1350016, 29 pp. doi: 10.1142/S0219199713500168.  Google Scholar

[5]

M. Galewski, On the Dirichlet problem for a Duffing type equation, E. J. Qualitative Theory of Diff. Equ., 15 (2011), 1-12.  Google Scholar

[6]

P. Holmes, A nonlinear oscillator with a strange attractor, Philosophical Transactions of the Royal Society A, 292 (1979), 419-448. doi: 10.1098/rsta.1979.0068.  Google Scholar

[7]

P. Holmes and D. Rand, Phase portraits and bifurcations of the non-linear oscillator $x'' +(\alpha+\gamma x^2 )x'+\beta x+\delta x^3 =0$, International Journal of Non-linear Mechanics, 15 (1980), 449-458.  Google Scholar

[8]

P. Holmes and D. Whitley, On the attracting set for Duffing's equation, II. A geometrical model for moderate force and damping, Physica D, 7 (1983), 111-123. doi: 10.1016/0167-2789(83)90121-5.  Google Scholar

[9]

P. J. Holmes and D. A. Rand, The bifurcations of Duffing's equation: An application of catastrophe theory, Journal of Sound and Vibration, 44 (1976), 237-253. doi: 10.1016/0022-460X(76)90771-9.  Google Scholar

[10]

W. Huang and Z. Shen, On a two-point boundary value problem of Duffing type equation with Dirichlet conditions. Appl. Math., 14 (1999), 131-136. doi: 10.1007/s11766-999-0018-x.  Google Scholar

[11]

J. Mawhin, The forced pendulum: A paradigm for nonlinear analysis and dynamical systems, Exposition. Math., 6 (1988), 271-287.  Google Scholar

[12]

S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities, Models and Analysis of Contact Problem, Springer, 2013. doi: 10.1007/978-1-4614-4232-5.  Google Scholar

[13]

F. C. Moon and P. J. Holmes, A magnetoelastic strange attractor, Journal of Sound and Vibration, 65 (1979), 275-296. doi: 10.1016/0022-460X(79)90520-0.  Google Scholar

[14]

F. C. Moon and P. J. Holmes, Addendum: A magnetoelastic strange attractor, Journal of Sound and Vibration, 69 (1980), 339 pp. Google Scholar

[15]

P. Tomiczek, Remark on Duffing equation with Dirichlet boundary condition, Electron. J. Differ. Equ., 81 (2007), 1-3.  Google Scholar

[1]

Yirong Jiang, Nanjing Huang, Zhouchao Wei. Existence of a global attractor for fractional differential hemivariational inequalities. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1193-1212. doi: 10.3934/dcdsb.2019216
[2]

Elimhan N. Mahmudov. Optimal control of evolution differential inclusions with polynomial linear differential operators. Evolution Equations & Control Theory, 2019, 8 (3) : 603-619. doi: 10.3934/eect.2019028
[3]

Zhenhai Liu, Van Thien Nguyen, Jen-Chih Yao, Shengda Zeng. History-dependent differential variational-hemivariational inequalities with applications to contact mechanics. Evolution Equations & Control Theory, 2020, 9 (4) : 1073-1087. doi: 10.3934/eect.2020044
[4]

Savin Treanţă. On a class of differential quasi-variational-hemivariational inequalities in infinite-dimensional Banach spaces. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021027
[5]

Gabriele Bonanno, Beatrice Di Bella. Fourth-order hemivariational inequalities. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 729-739. doi: 10.3934/dcdss.2012.5.729
[6]

Leszek Gasiński, Nikolaos S. Papageorgiou. Nonlinear hemivariational inequalities with eigenvalues near zero. Conference Publications, 2005, 2005 (Special) : 317-326. doi: 10.3934/proc.2005.2005.317
[7]

Leszek Gasiński. Existence results for quasilinear hemivariational inequalities at resonance. Conference Publications, 2007, 2007 (Special) : 409-418. doi: 10.3934/proc.2007.2007.409
[8]

Siegfried Carl. Comparison results for a class of quasilinear evolutionary hemivariational inequalities. Conference Publications, 2007, 2007 (Special) : 221-229. doi: 10.3934/proc.2007.2007.221
[9]

Mariusz Michta. On solutions to stochastic differential inclusions. Conference Publications, 2003, 2003 (Special) : 618-622. doi: 10.3934/proc.2003.2003.618
[10]

Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277
[11]

Thomas Lorenz. Mutational inclusions: Differential inclusions in metric spaces. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 629-654. doi: 10.3934/dcdsb.2010.14.629
[12]

Stanisław Migórski, Yi-bin Xiao, Jing Zhao. Fully history-dependent evolution hemivariational inequalities with constraints. Evolution Equations & Control Theory, 2020, 9 (4) : 1089-1114. doi: 10.3934/eect.2020047
[13]

Stanislaw Migórski, Anna Ochal. Navier-Stokes problems modeled by evolution hemivariational inequalities. Conference Publications, 2007, 2007 (Special) : 731-740. doi: 10.3934/proc.2007.2007.731
[14]

Yongjian Liu, Zhenhai Liu, Ching-Feng Wen. Existence of solutions for space-fractional parabolic hemivariational inequalities. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1297-1307. doi: 10.3934/dcdsb.2019017
[15]

O. Chadli, Z. Chbani, H. Riahi. Recession methods for equilibrium problems and applications to variational and hemivariational inequalities. Discrete & Continuous Dynamical Systems, 1999, 5 (1) : 185-196. doi: 10.3934/dcds.1999.5.185
[16]

Stanisław Migórski, Biao Zeng. Convergence of solutions to inverse problems for a class of variational-hemivariational inequalities. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4477-4498. doi: 10.3934/dcdsb.2018172
[17]

Stanislaw Migórski. A class of hemivariational inequalities for electroelastic contact problems with slip dependent friction. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 117-126. doi: 10.3934/dcdss.2008.1.117
[18]

Ludovic Dan Lemle. $L^1(R^d,dx)$-uniqueness of weak solutions for the Fokker-Planck equation associated with a class of Dirichlet operators. Electronic Research Announcements, 2008, 15: 65-70. doi: 10.3934/era.2008.15.65
[19]

Zhaosheng Feng, Goong Chen, Sze-Bi Hsu. A qualitative study of the damped duffing equation and applications. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1097-1112. doi: 10.3934/dcdsb.2006.6.1097
[20]

Zhibo Cheng, Jingli Ren. Periodic and subharmonic solutions for duffing equation with a singularity. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1557-1574. doi: 10.3934/dcds.2012.32.1557

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (40)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences