×

Nodal solutions of multi-point boundary value problems. (English) Zbl 1195.34028

The authors study a nonlinear multipoint boundary value problem of the form \[ y''+ w(t) f(y)= 0,\qquad t\in (a,b), \]
\[ \cos\alpha\;y(\alpha)- \sin\alpha\;y')a_= 0, \]
\[ y(b)- \sum^m_{i=1} k_i y(\eta_i)= 0. \] Here \(zf(z)> 0\) for all \(z\neq 0\). The coefficient \(w(t)\) is positive. Sufficient conditions are obtained for the existence and nonexistence of nodal solutions, which have specific properties of zeroes of the solution and its first derivative. These sufficient conditions are formulated with the help of eigenvalues of an auxiliary Sturm-Liouville problem.

MSC:

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Agarwal, R.P.; Kiguradze, I., On multi-point boundary value problems for linear ordinary differential equations with singularities, J. math. anal. appl., 297, 131-151, (2004) · Zbl 1058.34012
[2] Clark, S.; Henderson, J., Uniqueness implies existence and uniqueness criterion for nonlocal boundary value problems for third order differential equations, Proc. amer. math. soc., 134, 3363-3372, (2006) · Zbl 1120.34010
[3] Graef, J.R.; Yang, B., Positive solutions to a multi-point higher order boundary value problem, J. math. anal. appl., 316, 409-421, (2006) · Zbl 1101.34004
[4] Henderson, J., Solutions of multipoint boundary value problems for second order equations, Dynam. systems appl., 15, 111-117, (2006) · Zbl 1104.34310
[5] Henderson, J.; Karna, B.; Tisdell, C.C., Existence of solutions for three-point boundary value problems for second order equations, Proc. amer. math. soc., 133, 1365-1369, (2005) · Zbl 1061.34009
[6] Kwong, M.K., The shooting method and multiple solutions of two/multi-point BVPs of second-order ODE, Electron. J. qual. theory differ. equ., 6, pp. 14, (2006) · Zbl 1117.34011
[7] M.K. Kwong, J.S.W. Wong, The Shooting Method and Non-homogeneous Multi-point BVPs of Second Order ODE, Boundary Value Problems 2007, pp. 16, Article ID 64012
[8] Webb, J.R.L.; Lan, K.Q., Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type, Topol. methods nonlinear anal., 27, 91-115, (2006) · Zbl 1146.34020
[9] Kong, Q., Existence and nonexistence of solutions of second-order nonlinear boundary value problems, Nonlinear anal., 66, 2635-2651, (2007) · Zbl 1119.34024
[10] Kong, L.; Kong, Q., Nodal solutions of second order nonlinear boundary value problems, Math. proc. Cambridge philos. soc., 146, 747-763, (2009) · Zbl 1189.34043
[11] Ma, R., Nodal solutions of second-order boundary value problems with superlinear or sublinear nonlinearities, Nonlinear anal., 66, 950-961, (2007) · Zbl 1113.34011
[12] Ma, R.; Thompson, B., Multiplicity results for second-order two-point boundary value problems with superlinear or sublinear nonlinearities, J. math. anal. appl., 303, 726-735, (2005) · Zbl 1075.34017
[13] Ma, R.; Thompson, B., Nodal solutions for nonlinear eigenvalue problems, Nonlinear anal., 59, 707-718, (2004) · Zbl 1059.34013
[14] Naito, Y.; Tanaka, S., On the existence of multiple solutions of the boundary value problem for nonlinear second-order differential equations, Nonlinear anal., 56, 919-935, (2004) · Zbl 1046.34038
[15] Ma, R., Nodal solutions for a second-order \(m\)-point boundary value problem, Czech. math. J., 56, 131, 1243-1263, (2006) · Zbl 1164.34329
[16] Ma, R.; O’Regan, D., Nodal solutions for second-order \(m\)-point boundary value problems with nonlinearities across several eigenvalues, Nonlinear anal., 64, 1562-1577, (2006) · Zbl 1101.34006
[17] Rynne, B.P., Spectral properties and nodal solutions for second-order, \(m\)-point, boundary value problems, Nonlinear anal., 67, 3318-3327, (2007) · Zbl 1142.34010
[18] Xu, X., Multiple sign-changing solutions for some \(m\)-point boundary-value problems, Electron. J. differential equations, 2004, 89, 1-14, (2004)
[19] Xu, X.; Sun, J.; O’Regan, D., Nodal solutions for \(m\)-point boundary value problems using bifurcation methods, Nonlinear anal., 68, 3034-3046, (2008) · Zbl 1141.34009
[20] P. Bailey, N. Everitt, A. Zettl, SLEIGN2 package for Sturm-Liouville problems. See web site www.math.niu.edu/SL2/
[21] Zettl, A., Sturm – liouville theory, () · Zbl 1074.34030
[22] Kong, Q.; Zettl, A., Eigenvalue of regular sturm – liouville problems, J. differential equations, 131, 1-19, (1996) · Zbl 0862.34020
[23] Kong, Q.; Wu, H.; Zettl, A., Limits of sturm – liouville eigenvalues when the interval shrinks to an end point, Proc. roy. soc. Edinburgh, 138 A, 323-338, (2008) · Zbl 1160.34022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.