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On the continuous dependence of solutions to a fractional Dirichlet problem. The case of saddle points
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 |
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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