# American Institute of Mathematical Sciences

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

 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.
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.

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##### 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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