Top

Journal of Applied Mathematics and Computing

Published in:

01-02-2014 | Original Research

Existence of homoclinic solutions for second order Hamiltonian systems with general potentials

Author: Ziheng Zhang

Published in: Journal of Applied Mathematics and Computing | Issue 1-2/2014

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

In this paper we are concerned with the existence of homoclinic solutions for the following second order non-autonomous Hamiltonian systems

$$ \ddot{q}-L(t)q+W_{q}(t,q)=0, $$

(HS)

where $W\in C^{1}(\mathbb{R}\times\mathbb{R}^{n},\mathbb{R})$ and $L\in C(\mathbb{R},\mathbb{R}^{n^{2}})$ is a symmetric and positive definite matrix for all $t\in\mathbb{R}$. Assuming that the potential W satisfies some weaken global Ambrosetti-Rabinowitz conditions and L meets the coercive condition, we show that (HS) has at least one nontrivial homoclinic solution via using the Mountain Pass Theorem. Some recent results in the literature are generalized and significantly improved.

previous article Analysis of a delayed predator-prey model with ratio-dependent functional response and quadratic harvesting

next article Global analysis of multi-strains SIS, SIR and MSIR epidemic models

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Alves, C.O., Carrião, P.C., Miyagaki, O.H.: Existence of homoclinic orbits for asymptotically periodic systems involving Duffing-like equation. Appl. Math. Lett. 16(5), 639–642 (2003) CrossRefMATHMathSciNet

Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14(4), 349–381 (1973) CrossRefMATHMathSciNet

Caldiroli, P., Montecchiari, P.: Homoclinic orbits for second order Hamiltonian systems with potential changing sign. Commun. Appl. Nonlinear Anal. 1(2), 97–129 (1994) MATHMathSciNet

Coti Zelati, V., Rabinowitz, P.H.: Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials. J. Am. Math. Soc. 4(4), 693–727 (1991) CrossRefMATHMathSciNet

Ding, Y.: Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems. Nonlinear Anal. 25(11), 1095–1113 (1995) CrossRefMATHMathSciNet

Ding, Y., Girardi, M.: Periodic and homoclinic solutions to a class of Hamiltonian systems with the potentials changing sign. Dyn. Syst. Appl. 2(1), 131–145 (1993) MATHMathSciNet

Ding, Y., Lee, C.: Homoclinics for asymptotically quadratic and superquadratic Hamiltonian systems. Nonlinear Anal. 71, 1395–1413 (2009) CrossRefMATHMathSciNet

Korman, P., Lazer, A.C.: Homoclinic orbits for a class of symmetric Hamiltonian systems. Electron. J. Differ. Equ. 01, 1–10 (1994) MathSciNet

Liu, X., Zhang, Y., Shi, H.: Homoclinic orbits and subharmonics for second order p-Laplacian difference equations. J. Appl. Math. Comput. (2013). doi:10.1007/s12190-013-0673-1

10.

Lv, Y., Tang, C.: Existence of even homoclinic orbits for a class of Hamiltonian systems. Nonlinear Anal. 67(7), 2189–2198 (2007) CrossRefMATHMathSciNet

11.

Lv, X., Jiang, J.: Existence of homoclinic solutions for a class of second-order Hamiltonian systems with general potentials. Nonlinear Anal., Real World Appl. 13(3), 1152–1158 (2012) CrossRefMATHMathSciNet

12.

Lv, X., Lu, S.: Homoclinic orbits for a class of second-order Hamiltonian systems without a coercive potential. J. Appl. Math. Comput. 39, 121–130 (2012) CrossRefMathSciNet

13.

Omana, W., Willem, M.: Homoclinic orbits for a class of Hamiltonian systems. Differ. Integral Equ. 5(5), 1115–1120 (1992) MATHMathSciNet

14.

Ou, Z., Tang, C.: Existence of homoclinic solution for the second order Hamiltonian systems. J. Math. Anal. Appl. 291, 203–213 (2004) CrossRefMATHMathSciNet

15.

Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Reg. Conf. Ser. in. Math., vol. 65. American Mathematical Society, Providence (1986)

16.

Rabinowitz, P.H., Tanaka, K.: Some results on connecting orbits for a class of Hamiltonian systems. Math. Z. 206(3), 473–499 (1991) CrossRefMATHMathSciNet

17.

Sun, J., Chen, H., Nieto, J.J.: Homoclinic solutions for a class of subquadratic second-order Hamiltonian systems. J. Math. Anal. Appl. 373(1), 20–29 (2011) CrossRefMATHMathSciNet

18.

Sun, J., Chen, H., Nieto, J.J.: Homoclinic orbits for a class of first-order nonperiodic asymptotically quadratic Hamiltonian systems with spectrum point zero. J. Math. Anal. Appl. 378(1), 117–127 (2011) CrossRefMATHMathSciNet

19.

Sun, J., Chen, H., Nieto, J.J.: On ground state solutions for some non-autonomous Schrödinger-Poisson systems. J. Differ. Equ. 252(5), 3365–3380 (2012) CrossRefMATHMathSciNet

20.

Sun, J., Chu, J., Feng, Z.: Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero. Discrete Contin. Dyn. Syst., Ser. A 33(8), 3807–3824 (2013) CrossRefMATHMathSciNet

21.

Wan, L., Tang, C.: Existence and multiplicity of homoclinic orbits for second order Hamiltonian systems without (AR) condition. Discrete Contin. Dyn. Syst., Ser. B 15, 255–271 (2011) MATHMathSciNet

22.

Wang, J., Xu, J., Zhang, F.: Homoclinic orbits for a class of Hamiltonian systems with superquadratic or asymptotically quadratic potentials. Commun. Pure Appl. Anal. 10, 269–286 (2011) CrossRefMATHMathSciNet

23.

Wang, J., Zhang, F., Xu, J.: Existence and multiplicity of homoclinic orbits for the second order Hamiltonian systems. J. Math. Anal. Appl. 366, 569–581 (2010) CrossRefMATHMathSciNet

24.

Wei, J., Wang, J.: Infinitely many homoclinic orbits for the second order Hamiltonian systems with general potentials. J. Math. Anal. Appl. 366, 694–699 (2010) CrossRefMATHMathSciNet

25.

Yang, J., Zhang, F.: Infinitely many homoclinic orbits for the second-order Hamiltonian systems with super-quadratic potentials. Nonlinear Anal., Real World Appl. 10, 1417–1423 (2009) CrossRefMATHMathSciNet

26.

Yuan, R., Zhang, Z.: Homoclinic solutions for a class of second order Hamiltonian systems. Results Math. 61, 195–208 (2012) CrossRefMATHMathSciNet

27.

Zhang, J., Lv, D., Tang, Y.: Homoclinic orbits for an infinite dimensional Hamiltonian system with periodic potential. J. Appl. Math. Comput. (2013). doi:10.1007/s12190-013-0685-x

28.

Zhang, Q., Liu, C.: Infinitely many homoclinic solutions for second order Hamiltonian systems. Nonlinear Anal. 72, 894–903 (2010) CrossRefMATHMathSciNet

29.

Zhang, Z., Yuan, R.: Homoclinic solutions for a class of non-autonomous subquadratic second order Hamiltonian systems. Nonlinear Anal. 71, 4125–4130 (2009) CrossRefMATHMathSciNet

30.

Zhang, Z., Yuan, R.: Homoclinic solutions for a class of asymptotically quadratic Hamiltonian systems. Nonlinear Anal., Real World Appl. 11, 4185–4193 (2010) CrossRefMATHMathSciNet

31.

Zou, W., Li, S.: Infinitely many homoclinic orbits for the second-order Hamiltonian systems. Appl. Math. Lett. 16, 1283–1287 (2003) CrossRefMATHMathSciNet

Title: Existence of homoclinic solutions for second order Hamiltonian systems with general potentials
Author: Ziheng Zhang
Publication date: 01-02-2014
Publisher: Springer Berlin Heidelberg
Published in: Journal of Applied Mathematics and Computing / Issue 1-2/2014
Print ISSN: 1598-5865
Electronic ISSN: 1865-2085
DOI: https://doi.org/10.1007/s12190-013-0692-y

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Other articles of this Issue 1-2/2014

Existence of solutions for nonlinear high-order fractional boundary value problem with integral boundary condition

Existence of positive solutions for the singular fractional differential equations

Oscillation of even order advanced type dynamic equations with mixed nonlinearities on time scales

A new composite scheme for two-layer shallow water flows with shocks

Iterative algorithm for solving a class of general Sylvester-conjugate matrix equation

Existence of positive solution for a sixth–order differential system with variable parameters

Premium Partner