Opuscula Mathematica
Opuscula Math. 38, no. 4 (2018), 557-571

Linear Sturm-Liouville problems with Riemann-Stieltjes integral boundary conditions

Qingkai Kong
Thomas E. St. George
Abstract. We study second-order linear Sturm-Liouville problems involving general homogeneous linear Riemann-Stieltjes integral boundary conditions. Conditions are obtained for the existence of a sequence of positive eigenvalues with consecutive zero counts of the eigenfunctions. Additionally, we find interlacing relationships between the eigenvalues of such Sturm-Liouville problems and those of Sturm-Liouville problems with certain two-point separated boundary conditions.
Keywords: nodal solutions, integral boundary value problems, Sturm-Liouville problems, eigenvalues, matching method.
Mathematics Subject Classification: 34B10, 34B15.
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References ▼
  1. [1] Y. An, Global structure of nodal solutions for second-order \(m\)-point boundary value problems with superlinear nonlinearities, Bound. Value Probl. 2011, Art. ID 715836.
  2. [2] J. Chamberlain, L. Kong, Q. Kong, Nodal solutions of nonlocal boundary value problems, Mathematical Modelling and Analysis 14 (2009), 435-450.
  3. [3] J. Chamberlain, L. Kong, Q. Kong, Nodal solutions of boundary value problems with boundary conditions involving Riemann-Stieltjes integrals, Nonlinear Anal. 74 (2011), 2380-2387.
  4. [4] L.H. Erbe, Eigenvalue criteria for existence of positive solutions to nonlinear boundary value problems, Math. Comput. Modelling 32 (2000), 529-539.
  5. [5] F. Genoud, B.P. Rynne, Second order, multi-point problems with variable coefficients, Nonlinear Anal. 74 (2011), 7269-7284.
  6. [6] L. Kong, Q. Kong, Nodal solutions of second order nonlinear boundary value problems, Math. Proc. Camb. Phil. Soc. 146 (2009), 747-763.
  7. [7] L. Kong, Q. Kong, J.S.W. Wong, Nodal solutions of multi-point boundary value problems, Nonlinear Anal. 72 (2010), 382-389.
  8. [8] L. Kong, Q. Kong, M.K. Kwong, J.S.W. Wong, Linear Sturm-Liouville problems with multi-point boundary conditions, Math. Nachr. 286 (2013), 1167-1179.
  9. [9] Q. Kong, Existence and nonexistence of solutions of second-order nonlinear boundary value problems, Nonlinear Anal. 66 (2007), 2635-2651.
  10. [10] Q. Kong, T.E. St. George, Existence of Nodal solutions of boundary value problems with two multi-point boundary conditions, Dynamical Systems Appl. 23 (2014).
  11. [11] Q. Kong, T.E. St. George, Linear Sturm-Liouville problems with general homogeneous linear multi-point boundary conditions, Math. Nach. 2016, 1-12.
  12. [12] Q. Kong, A. Zettl, Eigenvalues of regular Sturm-Liouville problems, J. Differential Equations 131 (1996), 1-19.
  13. [13] Q. Kong, H. Wu, A. Zettl, Dependence of the nth Sturm-Liouville eigenvalue on the problem, J. Differential Equations 156 (1999), 328-354.
  14. [14] Q. Kong, H. Wu, A. Zettl, Limits of Sturm-Liouville eigenvalues when the interval shrinks to an end point, Proc. Roy. Soc. Edinburgh 138 A (2008), 323-338.
  15. [15] R. Ma, Nodal solutions for a second-order \(m\)-point boundary value problem, Czech. Math. J. 56(131) (2006), 1243-1263.
  16. [16] R. Ma, D. O'Regan, Nodal solutions for second-order \(m\)-point boundary value problems with nonlinearities across several eigenvalues, Nonlinear Anal. 64 (2006), 1562-1577.
  17. [17] Y. Naito, S. Tanaka, On the existence of multiple solutions of the boundary value problem for nonlinear second-order differential equations, Nonlinear Anal. 56 (2004), 919-935.
  18. [18] B.P. Rynne, Spectral properties and nodal solutions for second-order, \(m\)-point, boundary value problems, Nonlinear Anal. 67 (2007), 3318-3327.
  19. [19] B.P. Rynne, Spectral properties of second-order, multi-point, p-Laplacian boundary value problems, Nonlinear Anal. 72 (2010), 4244-4253.
  20. [20] B.P. Rynne, Linear, second-order problems with Sturm-Liouville-type multi-point boundary conditions, Electron. J. Differential Equns. 2012 (2012) 146, 1-21.
  21. [21] B.P. Rynne, Linear and nonlinear, second-order problems with Sturm-Liouville-type, multi-point boundary conditions, Nonlinear Anal. 136 (2016), 195-214.
  22. [22] X. Xu, Multiple sign-changing solutions for some \(m\)-point boundary-value problems, Electronic J. Diff. Eqns. 2004 (2004) 89, 1-14.
  23. [23] X. Xu, J. Sun, D. O'Regan, Nodal solutions for \(m\)-point boundary value problems using bifurcation methods, Nonlinear Anal. 68 (2008), 3034-3046.
  24. [24] A. Zettl, Sturm-Liouville theory, [in:] Mathematical Surveys and Monographs, vol. 121, American Mathematical Society, 2005.
About the authors ▼
Qingkai Kong
Northern Illinois University, Department of Mathematics, DeKalb, IL 60115, USA
Thomas E. St. George
Carroll University, Department of Mathematics, Waukesha, WI 53186, USA
About the article ▼
Communicated by P.A. Cojuhari.
Received: 2017-07-20.
Accepted: 2017-08-08.
Published online: 2018-04-11.
Cite this article as ▼
Qingkai Kong, Thomas E. St. George, Linear Sturm-Liouville problems with Riemann-Stieltjes integral boundary conditions, Opuscula Math. 38, no. 4 (2018), 557-571, https://doi.org/10.7494/OpMath.2018.38.4.557.
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