Global Bifurcation of Nodal Solutions for Nonlinear n-th Order Boundary Value Problems

• Ruyun Ma ^[1] ; Xiaolei Tong ^[1]
  1. [1] Xidian University

     Xidian University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 25, Nº 3, 2026
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • We are concerned with the existence of nodal solutions for the n-th order boundary value problem ⎧⎪⎨ ⎪⎩ u(n)(t) + a(t) f (u(t)) = 0, t ∈ (0, π), u(i )(0) = 0, i ∈ {i1, ..., in−m}, u( j )(π) = 0, j ∈ {j1, ..., jm}, where n ≥ 2, 1 ≤ m ≤ n − 1, {i1, ..., in−m}, { j1, ..., jm} are subsets of the set of integers in {0, ..., n − 1}; a : [0, π] → (−1)mR − is continuous and does not vanish identically on any subinterval of [0, π]; f : R → R is a continuous function. The proof of our main result is based upon bifurcation techniques.

• Referencias bibliográficas
  • 1. Krajcinovic, D.: Sandwich beam analysis. J. Appl. Mech. 39, 773–778 (1972)
  • 2. Gupta, C.: Existence and uniqueness theorem for the bending of an elastic beam equation. Appl. Anal. 26, 289–304 (1988)
  • 3. Lazer, A., Mckenna, P.: Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis. SIAM...
  • 4. Baldwin, P.:Asymptotic estimates of the eigen values of a sixth-order boundary-value problem obtained by using global phase-integral methods....
  • 5. Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. Dover, New York (1981)
  • 6. Meyer, G.H.: Initial Value Methods for Boundary Value Problems. Academic Press, New York (1973)
  • 7. Naito,Y., Tanaka, S.: On the existence of multiple solutions of the boundary value problem for nonlinear second-order differential equations....
  • 8. Ma, R., Thompson, B.: Nodal solutions for nonlinear eigenvalue problems. Nonlinear Anal. 59(5), 707–718 (2004)
  • 9. Rynne, B.P.: Global bifurcation for 2mth-order boundary value problems and infinity many solutions of superlinear problems. J. Differential...
  • 10. Eloe, P.W., Henderson, J.: Singular nonlinear (k, n − k) conjugate boundary value problems. J. Differential Equations 133(1), 136–151...
  • 11. Zhang, X., Shao, Z., Zhong, Q.: Positive solutions for semipositone (k, n − k) conjugate boundary value problems with singularities on...
  • 12. Su, H.,Wei, Z.: Positive solutions to semipositone (k, n−k) conjugate eigenvalue problems. Nonlinear Anal. 69, 3190–3201 (2008)
  • 13. Webb, J.R.L.: Nonlocal conjugate type boundary value problems of higher order. Nonlinear Anal. 71, 1933–1940 (2009)
  • 14. Jiang, W., Zhang, J.: Positive solutions for (k, n − k) conjugate boundary value problems in Banach spaces. Nonlinear Anal. 71, 723–729...
  • 15. Yang, B.: Upper estimate for positive solutions of the (p, n − p) conjugate boundary value problem. J. Math. Anal. Appl. 390, 535–548...
  • 16. Lin, X., Jiang, D., Li, X.: Existence and uniqueness of solutions for singular (k, n − k) conjugate boundary value problems. Comput. Math....
  • 17. Dalbono, F.: Multiplicity of solutions for asymptotically linear n-th order boundary value problems. Topol. Methods Nonlinear Anal. 30(1),...
  • 18. Elias, U.: Eigenvalue problems for the equation Ly + p(x)y = 0. J. Differential Equations 29, 28–57 (1978)
  • 19. Ambrosetti, A., Hess, P.: Positive solutions of asymptotically linear elliptic eigenvalue problems. J. Math. Anal. Appl. 73(2), 411–422...
  • 20. Cabada, A., Fernández-Gómez, C.: Constant sign solutions of two-point fourth order problems. Appl. Math. Comput. 263, 122–133 (2015)
  • 21. A. Cabada, J. Cid, L. Sanchez, A.: Positivity and lower and upper solutions for fourth order boundary value problems. Nonlinear Anal....
  • 22. Li, Y.: Positive solutions of fourth-order periodic boundary value problems. Nonlinear Anal. 54, 1069– 1078 (2003)
  • 23. Ma, R., Wang, H., Elsanosi, M.: Spectrum of a linear fourth-order differential operator and its applications. Math. Nachr. 286, 1805–1819...
  • 24. Lu, Y., Ma, R.: Disconjugacy conditions and spectrum structure of clamped beam equations with two parameters. Commun. Pure Appl. Anal....
  • 25. Bi, Z., Liu, S.: Positive solutions for third-order boundary value problems with indefinite weight. Mediterr. J. Math. 20, 18 (2023)
  • 26. Rabinowitz, P.H.: Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 7, 487–513 (1971)