Document Type : Research Paper
Author
Department of Mathematics, Faculty of Sciences, University of Monastir, P.O.Box 5019, Monastir, Tunisia.
Abstract
This article concerns the existence of fast homoclinic solutions for the following damped vibration system
\begin{equation*}
\frac{d}{dt}(P(t)\dot{u}(t))+q(t)P(t)\dot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0,
\end{equation*}
where $P,L\in C\left(\mathbb{R},\mathbb{R}^{N^{2}}\right)$ are symmetric and positive definite matrices, $q\in C\left(\mathbb{R},\mathbb{R}\right)$ and $W\in C^{1}\left(\mathbb{R}\times\mathbb{R}^{N},\mathbb{R}\right)$. Applying the Fountain Theorem and Dual Fountain Theorem, we prove the above system possesses two different sequences of fast homoclinic solutions when $L$ satisfies a new coercive condition and the potential $W(t,x)$ is combined nonlinearity.
Keywords
- Damped vibration systems
- Fast homoclinic solutions
- Variational methods
- Fountain Theorem
- Dual Fountain Theorem
Main Subjects
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[5] P. Chen and X.H. Tang, Fast homoclinic solutions for a class of damped vibration problems with sub-quadratic potentials, Math. Nachr., 286 (1) (2013), pp. 4-16.
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[7] L. Chu and Q. Zhang, Homoclinic solutions for a class of second order Hamiltonian systems with locally defined potentials, Nonlinear Anal., 75 (2012), pp. 3188-3197.
[8] C. Deng and D.L. Wu, Multiple homoclinic solutions for a class of nonhomogeneous Hamiltonian systems, Bound. Value Probl., 2018:56 (2018), pp. 1-10.
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[11] Z.Q. Han and M.H. Yang, Infinitely many solutions for second order Hamiltonian systems with odd nonlinearities, Nonlinear Anal. (TMA), 74 (7) (2011), pp. 2633-2646.
[12] M.R. HeidariTavani, Existence of periodic solutions for a class of Hamiltonian systems, Eurasian Math. J., 9 (2) (2018), pp. 22-33.
[13] M.R. Heidari Tavani, Existence results for a class of p-Hamiltonian systems, J. Math. Extension, 15 (2) (2021), pp. 1-15.
[14] M. Izydorek and J. Janczewska, Homoclinic solutions for a class of the second order Hamiltonian sysstems, J. Diff. Syst., 219 (2) (2005), pp. 375-389.
[15] J. Jiang, S. Lu and X. Lv, Homoclinic solutions for a class of second order Hamiltonian systems, Nonlinear Anal. (RWA), 13 (2012), pp. 176-185.
[16] W. Jiang and Q. Zhang, Multiple homoclinic solutions for superquadratic Hamiltonian systems, Electron. J. Diff. Equ., 66 (2016), pp. 1-12.
[17] F. Khelifi and M. Timoumi, Even homoclinic orbits for a class of damped vibration systems, Indigationes Math., 28 (2017), pp. 1111-1125.
[18] Y. Li and Q. Zhang, Existence and multiplicity of fast homoclinic solutions for a class of nonlinear second order nonautonomous systems in a weighted Sobolev space, J. Funct. Spaces, 2015, Article ID 495040 (2015), pp. 1-12.
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[20] X. Lin and X.H. Tang, Infinitely many homoclinic orbits for Hamiltonian systems with indefinite sign subquadratic potentials, Nonlinear Anal., 74 (2011), pp. 6314-6325.
[21] X. Lin and X.H. Tang, New conditions on homoclinic solutions for a subquadratic second order Hamiltonian system, Bound. Value Probl., 2015:111 (2015).
[22] C. Liu and Q. Zhang, Infinitely many homoclinic solutions for second order Hamiltonian systems, Nonlinear Anal., 72 (2010), pp. 894-903.
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[26] J. Sun and T-f. Wu, Homoclinic solutions for a second order Hamiltonian system with a positive semi-definite matrix, Chaos, Solit. Fract., 76 (2015), pp. 24-31.
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[28] X.H. Tang, Infinitely many homoclinic solutions for a second-order Hamiltonian system, Math. Nachr., 289 (1) (2016), pp. 116-127.
[29] C.L. Tang and Li-Li Wan, Existence of homoclinic orbits for second order Hamiltonian systems without (AR) condition, Nonlinear Anal., 74 (2011), pp. 5303-5313.
[30] X.H. Tang and L. Xiao, Homoclinic solutions for a class of second order Hamiltonian systems, Nonlinear Anal., 71 (2009), pp. 1140-1152.
[31] M. Timoumi, Existence and multiplicity of fast homoclinic solutions for a class of damped vibration problems, J. Nonlinear Funct. Anal., 2016, Article ID 9 (2016), pp. 1-22
[32] M. Timoumi, Ground state homoclinic orbits of a class of superquadratic damped vibration problems, Comm. Optim. Theory, 2017, Article ID 29 (2017), pp. 1-25.
[33] M. Timoumi, Infinitely many fast homoclinic solutions for a class of superquadratic damped vibration systems, J. Ellip. Parab. Systems, 6 (9) (2020), pp. 1-21.
[34] M. Timoumi, Infinitely many fast homoclinic solutions for a class of superquadratic damped vibration systems, J. Nonlinear Funct. Anal., 2020, Article ID 8 (2020), pp. 1-16.
[35] M. Timoumi, On ground state homoclinic orbits of a class of superquadratic damped vibration systems, Mediterr. J. Math., 15 (53) (2018), pp. 1-20.
[36] M. Timoumi and A. Chouikha, Infinitely many fast homoclinic solutions for damped vibration systems with locally defined potentials, Nonlinear Studies, 27 (1) (2020), pp. 1-19.
[37] J. Wei and J. Wang, Infinitely many homoclinic orbits for the second order Hamiltonian systems with general potentials, J. Math. Anal. Appl.,366 (2) (2010), pp. 694-699.
[38] X. Wu and W. Zhang, Existence and multiplicity of homoclinic solutions for a class of damped vibration problems, Nonlinear Anal., 74 (2011), pp. 4392-4398.
[39] R. Yuan and Z. Zhang, Fast homoclinic solutions for some second order non-autonomous systems, J. Math. Anal. Appl., 376 (2011), pp. 51-63.
[40] R. Yuan and Z. Zhang, Homoclinic solutions for some second order nonautonomous hamiltonian systems without the globally superquadratic condition, Nonlinear Anal., 72 (2010), pp. 1809-1819.
[41] Z. Zhang, Existence of homoclinic solutions for second order Hamiltonian systems with general potentials, J. Appl. Math. Comput., 44 (2014), pp. 263-272.
[42] W. Zou, Variant Fountain theorems and their applications, Manuscripta Math., 1004 (2001), pp. 343-358.
[2] G. Chen, Homoclinic orbits for second order Hamiltonian systems with asymptotically linear terms at infinity, Adv. Diff. syst., 2014:114 (2014), pp. 1-9.
[3] G.W. Chen, superquadratic or asymptotically quadratic Hamiltonian systems: ground state homoclinic orbits, Ann. Mat. Pura Appl., 194(3) (2013), pp. 903-918.
[4] G. Chen and Z. He, Infinitely many homoclinic solutions for a class of second order Hamiltonian systems, Adv. Diff. syst., 161 (2014), pp. 1-15.
[5] P. Chen and X.H. Tang, Fast homoclinic solutions for a class of damped vibration problems with sub-quadratic potentials, Math. Nachr., 286 (1) (2013), pp. 4-16.
[6] G. Chen and L. Zhang, Infinitely many homoclinic solutions for perturbed second-order Hamiltonian systems with subquadratic potentials, Electr. J. Qual. Theo. Diff. Eq., 9 (2020), pp. 1-23.
[7] L. Chu and Q. Zhang, Homoclinic solutions for a class of second order Hamiltonian systems with locally defined potentials, Nonlinear Anal., 75 (2012), pp. 3188-3197.
[8] C. Deng and D.L. Wu, Multiple homoclinic solutions for a class of nonhomogeneous Hamiltonian systems, Bound. Value Probl., 2018:56 (2018), pp. 1-10.
[9] Y. Ding, Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems, Nonlinear Anal., 25 (11) (1995), pp. 1095 -1113.
[10] S. Guo, Z. Liu and Z. Zhang, Homoclinic orbits for the second order Hamiltonian systems, Nonlinear Anal. (RWA), 36 (2017), pp. 116-138.
[11] Z.Q. Han and M.H. Yang, Infinitely many solutions for second order Hamiltonian systems with odd nonlinearities, Nonlinear Anal. (TMA), 74 (7) (2011), pp. 2633-2646.
[12] M.R. HeidariTavani, Existence of periodic solutions for a class of Hamiltonian systems, Eurasian Math. J., 9 (2) (2018), pp. 22-33.
[13] M.R. Heidari Tavani, Existence results for a class of p-Hamiltonian systems, J. Math. Extension, 15 (2) (2021), pp. 1-15.
[14] M. Izydorek and J. Janczewska, Homoclinic solutions for a class of the second order Hamiltonian sysstems, J. Diff. Syst., 219 (2) (2005), pp. 375-389.
[15] J. Jiang, S. Lu and X. Lv, Homoclinic solutions for a class of second order Hamiltonian systems, Nonlinear Anal. (RWA), 13 (2012), pp. 176-185.
[16] W. Jiang and Q. Zhang, Multiple homoclinic solutions for superquadratic Hamiltonian systems, Electron. J. Diff. Equ., 66 (2016), pp. 1-12.
[17] F. Khelifi and M. Timoumi, Even homoclinic orbits for a class of damped vibration systems, Indigationes Math., 28 (2017), pp. 1111-1125.
[18] Y. Li and Q. Zhang, Existence and multiplicity of fast homoclinic solutions for a class of nonlinear second order nonautonomous systems in a weighted Sobolev space, J. Funct. Spaces, 2015, Article ID 495040 (2015), pp. 1-12.
[19] X. Lin and X.H. Tang, Homoclinic solutions for a class of second order Hamiltonian systems, J. Math. Anal. Appl., 354 (2009), pp. 539-549.
[20] X. Lin and X.H. Tang, Infinitely many homoclinic orbits for Hamiltonian systems with indefinite sign subquadratic potentials, Nonlinear Anal., 74 (2011), pp. 6314-6325.
[21] X. Lin and X.H. Tang, New conditions on homoclinic solutions for a subquadratic second order Hamiltonian system, Bound. Value Probl., 2015:111 (2015).
[22] C. Liu and Q. Zhang, Infinitely many homoclinic solutions for second order Hamiltonian systems, Nonlinear Anal., 72 (2010), pp. 894-903.
[23] S. Lu, X. Lv and P. Yan, Existence of homoclinics for a class of Hamiltonian systems, Nonlinear Anal., 72 (2010), pp. 390-398.
[24] P.H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems, Proc. Roy. Soc. Edinburgh, 114 (1990), pp. 33-38.
[25] P.H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems, Math. Z., 206 (3) (1991), pp. 473-499.
[26] J. Sun and T-f. Wu, Homoclinic solutions for a second order Hamiltonian system with a positive semi-definite matrix, Chaos, Solit. Fract., 76 (2015), pp. 24-31.
[27] J. Sun and T-f. Wu, Multiplicity and concentration of homoclinic solutions for some second order Hamiltonian systems, Nonlinear Anal., 114 (2015), pp. 105-115.
[28] X.H. Tang, Infinitely many homoclinic solutions for a second-order Hamiltonian system, Math. Nachr., 289 (1) (2016), pp. 116-127.
[29] C.L. Tang and Li-Li Wan, Existence of homoclinic orbits for second order Hamiltonian systems without (AR) condition, Nonlinear Anal., 74 (2011), pp. 5303-5313.
[30] X.H. Tang and L. Xiao, Homoclinic solutions for a class of second order Hamiltonian systems, Nonlinear Anal., 71 (2009), pp. 1140-1152.
[31] M. Timoumi, Existence and multiplicity of fast homoclinic solutions for a class of damped vibration problems, J. Nonlinear Funct. Anal., 2016, Article ID 9 (2016), pp. 1-22
[32] M. Timoumi, Ground state homoclinic orbits of a class of superquadratic damped vibration problems, Comm. Optim. Theory, 2017, Article ID 29 (2017), pp. 1-25.
[33] M. Timoumi, Infinitely many fast homoclinic solutions for a class of superquadratic damped vibration systems, J. Ellip. Parab. Systems, 6 (9) (2020), pp. 1-21.
[34] M. Timoumi, Infinitely many fast homoclinic solutions for a class of superquadratic damped vibration systems, J. Nonlinear Funct. Anal., 2020, Article ID 8 (2020), pp. 1-16.
[35] M. Timoumi, On ground state homoclinic orbits of a class of superquadratic damped vibration systems, Mediterr. J. Math., 15 (53) (2018), pp. 1-20.
[36] M. Timoumi and A. Chouikha, Infinitely many fast homoclinic solutions for damped vibration systems with locally defined potentials, Nonlinear Studies, 27 (1) (2020), pp. 1-19.
[37] J. Wei and J. Wang, Infinitely many homoclinic orbits for the second order Hamiltonian systems with general potentials, J. Math. Anal. Appl.,366 (2) (2010), pp. 694-699.
[38] X. Wu and W. Zhang, Existence and multiplicity of homoclinic solutions for a class of damped vibration problems, Nonlinear Anal., 74 (2011), pp. 4392-4398.
[39] R. Yuan and Z. Zhang, Fast homoclinic solutions for some second order non-autonomous systems, J. Math. Anal. Appl., 376 (2011), pp. 51-63.
[40] R. Yuan and Z. Zhang, Homoclinic solutions for some second order nonautonomous hamiltonian systems without the globally superquadratic condition, Nonlinear Anal., 72 (2010), pp. 1809-1819.
[41] Z. Zhang, Existence of homoclinic solutions for second order Hamiltonian systems with general potentials, J. Appl. Math. Comput., 44 (2014), pp. 263-272.
[42] W. Zou, Variant Fountain theorems and their applications, Manuscripta Math., 1004 (2001), pp. 343-358.
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