Existence of Homoclinic Solutions for a Class of Nonlinear Second-order Problems

• Wei Yang ^[1] ; Ruyun Ma ^[1]
  1. [1] Northwest Normal University

     Northwest Normal University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº Extra 1, 2024
• Idioma: inglés
• DOI: 10.1007/s12346-024-01114-9
• Enlaces
  • Texto completo
• Resumen

  • We are concerned with the existence of homoclinic solutions for the nonlinear problems ⎧ ⎨ ⎩ u + ωu − ku = f (t, u, u ), t ∈ R, lim |t|→+∞ u(t) = 0, (P) where ω ∈ R, k > 0 are real constants, and f : R3 → R is an L1−Carathéodory function. Under some suitable conditions, the existence of homoclinic solutions for problem (P) and the corresponding coupled systems are provided. The proofs of the main results are based on the method of upper and lower solutions.

• Referencias bibliográficas
  • 1. Kivshar, Y.S.: Bright and dark spatial solitons in non-Kerr media. Opt. Quant. Electron. 30, 571–614 (1998)
  • 2. Bonheure, D., Torres, J.P.: Bounded and homoclinic-like solutions of a second-order singular differential equation. Bull. Lond. Math. Soc....
  • 3. Rabinowitz, P.H., Tanaka, K.: Some results on connecting orbits for a class of Hamiltonian systems. Math. Z. 206, 473–499 (1991)
  • 4. Izydorek, M., Janczewska, J.: Homoclinic solutions for nonautonomous second-order Hamiltonian systems with a coercive potential. J. Math....
  • 5. Tang, X.H., Lin, X.Y.: Homoclinic solutions for a class of second-order Hamiltonian systems. J. Math. Anal. Appl. 354, 539–549 (2009)
  • 6. Zhang, Z.H., Yuan, R.: Homoclinic solutions for a class of non-autonomous subquadratic second-order Hamiltonian systems. Nonlinear Anal....
  • 7. Djeball, S., Moussoui, T.: A class of second order BVPs on infinite intervals. Electron. J. Qual. Theory Differ. Equ. 2006, 1–19 (2006)
  • 8. Minhós, F., Carrasco, H.: Existence of homoclinic solutions for nonlinear second-order problems. Mediterr. J. Math. 13, 3849–3861 (2016)
  • 9. Goodarzi, Z., Mokhtarzadeh, M.R., Pournaki, M.R., Razani, A.: Note on periodic solutions of matrix Riccati differential equations. Appl....
  • 10. Mokhtarzadeh, M.R., Pournaki, M.R., Razani, A.: An existence-uniqueness theorem for a class of boundary value problems. Fixed Point Theory...
  • 11. Mokhtarzadeh, M.R., Pournaki, M.R., Razani, A.: A note on periodic solutions of Riccati equations. Nonlinear Dyn. 62, 119–125 (2010)
  • 12. Pournaki, M.R., Razani, A.: On the existence of periodic solutions for a class of generalized forced Liénard equations. Appl. Math. Lett....
  • 13. Goodarzi, Z., Razani, A., Mokhtarzadeh, M.R.: A periodic solution of the coupled matrix Riccati differential equations. Miskolc Math....
  • 14. Razani, A.: An existence theorem for ordinary differential equation in Menger probabilistic metric space. Miskolc Math. Notes 15, 711–716...
  • 15. Razani, A.: Non-existence of solution of Haraux–Weissler equation on a strictly starshped domain. Miskolc Math. Notes 24, 395–402 (2023)
  • 16. Grossinho, M.R., Minhós, F., Santos, A.I.: A note on a class of problems for a higher order fully nonlinear equation under one sided Nagumo...
  • 17. Minhós, F., de Sousa, R.: Existence of homoclinic solutions for nonlinear second-order coupled systems. J. Differ. Edu. 266, 1414–1428...
  • 18. Zhang, L., Ge, W.: Solvability of a second order boundary value problem on an unbounded domain. Appl. Math. E-Notes 10, 40–46 (2010)