## 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
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### References:

  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  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  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  Henderson, J., Solutions of multipoint boundary value problems for second order equations, Dynam. systems appl., 15, 111-117, (2006) · Zbl 1104.34310  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  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  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  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  Kong, Q., Existence and nonexistence of solutions of second-order nonlinear boundary value problems, Nonlinear anal., 66, 2635-2651, (2007) · Zbl 1119.34024  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  Ma, R., Nodal solutions of second-order boundary value problems with superlinear or sublinear nonlinearities, Nonlinear anal., 66, 950-961, (2007) · Zbl 1113.34011  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  Ma, R.; Thompson, B., Nodal solutions for nonlinear eigenvalue problems, Nonlinear anal., 59, 707-718, (2004) · Zbl 1059.34013  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  Ma, R., Nodal solutions for a second-order $$m$$-point boundary value problem, Czech. math. J., 56, 131, 1243-1263, (2006) · Zbl 1164.34329  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  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  Xu, X., Multiple sign-changing solutions for some $$m$$-point boundary-value problems, Electron. J. differential equations, 2004, 89, 1-14, (2004)  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  P. Bailey, N. Everitt, A. Zettl, SLEIGN2 package for Sturm-Liouville problems. See web site www.math.niu.edu/SL2/  Zettl, A., Sturm – liouville theory, () · Zbl 1074.34030  Kong, Q.; Zettl, A., Eigenvalue of regular sturm – liouville problems, J. differential equations, 131, 1-19, (1996) · Zbl 0862.34020  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
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