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

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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

Abstract
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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 and 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.
[2]	P. Amster and M. C. Mariani, A second order ODE with a nonlinear final condition, Electron. J. Differ. Equ., 75 (2001), 1-9.
[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.
[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.
[5]	M. Galewski, On the Dirichlet problem for a Duffing type equation, E. J. Qualitative Theory of Diff. Equ., 15 (2011), 1-12.
[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.
[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.
[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.
[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.
[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.
[11]	J. Mawhin, The forced pendulum: A paradigm for nonlinear analysis and dynamical systems, Exposition. Math., 6 (1988), 271-287.
[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.
[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.
[14]	F. C. Moon and P. J. Holmes, Addendum: A magnetoelastic strange attractor, Journal of Sound and Vibration, 69 (1980), 339 pp.
[15]	P. Tomiczek, Remark on Duffing equation with Dirichlet boundary condition, Electron. J. Differ. Equ., 81 (2007), 1-3.

show all references

References:

[1]	P. Amster, Nonlinearities in a second order ODE, Electron. J. Differ. Equ., 6 (2001), 13-21.
[2]	P. Amster and M. C. Mariani, A second order ODE with a nonlinear final condition, Electron. J. Differ. Equ., 75 (2001), 1-9.
[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.
[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.
[5]	M. Galewski, On the Dirichlet problem for a Duffing type equation, E. J. Qualitative Theory of Diff. Equ., 15 (2011), 1-12.
[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.
[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.
[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.
[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.
[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.
[11]	J. Mawhin, The forced pendulum: A paradigm for nonlinear analysis and dynamical systems, Exposition. Math., 6 (1988), 271-287.
[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.
[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.
[14]	F. C. Moon and P. J. Holmes, Addendum: A magnetoelastic strange attractor, Journal of Sound and Vibration, 69 (1980), 339 pp.
[15]	P. Tomiczek, Remark on Duffing equation with Dirichlet boundary condition, Electron. J. Differ. Equ., 81 (2007), 1-3.

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