Opuscula Math. 38, no. 4 (2018), 557-571
https://doi.org/10.7494/OpMath.2018.38.4.557

Opuscula Mathematica

# 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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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. J. Chamberlain, L. Kong, Q. Kong, Nodal solutions of nonlocal boundary value problems, Mathematical Modelling and Analysis 14 (2009), 435-450.
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. L.H. Erbe, Eigenvalue criteria for existence of positive solutions to nonlinear boundary value problems, Math. Comput. Modelling 32 (2000), 529-539.
5. F. Genoud, B.P. Rynne, Second order, multi-point problems with variable coefficients, Nonlinear Anal. 74 (2011), 7269-7284.
6. L. Kong, Q. Kong, Nodal solutions of second order nonlinear boundary value problems, Math. Proc. Camb. Phil. Soc. 146 (2009), 747-763.
7. L. Kong, Q. Kong, J.S.W. Wong, Nodal solutions of multi-point boundary value problems, Nonlinear Anal. 72 (2010), 382-389.
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. Q. Kong, Existence and nonexistence of solutions of second-order nonlinear boundary value problems, Nonlinear Anal. 66 (2007), 2635-2651.
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. Q. Kong, T.E. St. George, Linear Sturm-Liouville problems with general homogeneous linear multi-point boundary conditions, Math. Nach. 2016, 1-12.
12. Q. Kong, A. Zettl, Eigenvalues of regular Sturm-Liouville problems, J. Differential Equations 131 (1996), 1-19.
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. 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. R. Ma, Nodal solutions for a second-order $$m$$-point boundary value problem, Czech. Math. J. 56(131) (2006), 1243-1263.
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. 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. B.P. Rynne, Spectral properties and nodal solutions for second-order, $$m$$-point, boundary value problems, Nonlinear Anal. 67 (2007), 3318-3327.
19. B.P. Rynne, Spectral properties of second-order, multi-point, p-Laplacian boundary value problems, Nonlinear Anal. 72 (2010), 4244-4253.
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. B.P. Rynne, Linear and nonlinear, second-order problems with Sturm-Liouville-type, multi-point boundary conditions, Nonlinear Anal. 136 (2016), 195-214.
22. X. Xu, Multiple sign-changing solutions for some $$m$$-point boundary-value problems, Electronic J. Diff. Eqns. 2004 (2004) 89, 1-14.
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. A. Zettl, Sturm-Liouville theory, [in:] Mathematical Surveys and Monographs, vol. 121, American Mathematical Society, 2005.
• 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
• Communicated by P.A. Cojuhari.
• Accepted: 2017-08-08.
• Published online: 2018-04-11.