## On sign-changing solution for some three-point boundary value problems.(English)Zbl 1069.34019

The authors consider the second-order autonomous differential equation $y''(t)+f(y)=0 \eqno (1)$ with three-point boundary conditions $y(0)=0,\quad \alpha y(\eta)=y(1), \eqno (2)$ where $$\alpha, \eta \in (0,1)$$, $$f\in C(\mathbb R)$$ is strictly increasing and $$f(0)=0$$.
They suppose that $$\lambda_1<\lambda_2<\dots <\lambda_n <\dots$$ is the sequence of positive solutions of the equation $\sin \sqrt x = \alpha \sin \eta \sqrt x.$ Then, under the assumption that there exists $$n_0\in \mathbb N$$ such that $\lambda_{2n_0} < \lim_{x\to 0} {f(x)\over x} < \lambda _{2n_0+1},\quad \lim_{| x| \to \infty} {f(x)\over x} < 2(1-\alpha \eta),$ the authors prove the existence of a sign-changing solution of problem (1), (2). Moreover, they get at least one positive and at least one negative solution of (1), (2). Their proofs are based on the fixed-point index method.

### MSC:

 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations
Full Text:

### References:

 [1] Amann, H., Fixed point equation and nonlinear eigenvalue problems in ordered Banach spaces, SIAM rev., 18, 620-709, (1976) · Zbl 0345.47044 [2] Bortsch, T.; Wang, Z.-Q., On the existence of sign-changing solutions for semilinear Dirichlet problems, Topol. methods nonlinear anal., 7, 115-131, (1996) [3] Castra, A.; Cossion, J.; Neuberger, J.M., A sign-changing solution for a superlinear Dirichlet problem, Rocky mount. J. math., 27, 1041-1053, (1997) · Zbl 0907.35050 [4] Dajun, G.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic press Inc New York · Zbl 0661.47045 [5] Dancer, E.N.; Du, Y., Existence of sign-changing solutions for some semilinear problems with jumping nonlinearities at zero, Proc. roy. soc. Edinburgh, 124A, 1165-1176, (1994) · Zbl 0819.35054 [6] Feng, W.; Webb, J.R.L., Solvability of a m-point nonlinear boundary value problem with nonlinear growth, J. math. anal. appl., 212, 467-480, (1997) · Zbl 0883.34020 [7] Gupta, C.P., A sharper condition for the solvability of a m-point nonlinear boundary value problem, J. math. anal. appl., 205, 579-586, (1997) [8] Gupta, C.P.; Trofimchuk, Sergej I., Existence of a solution of a three-point boundary value problem and spectral radius of a related linear operator, Nonlinear anal., 34, 489-507, (1998) · Zbl 0944.34009 [9] Π’in, V.A.; Moiseev, E.I., Nonlocal boundary value problem of the second kind for a sturm – liouville operator, Differential equations, 23, 8, 979-987, (1987) · Zbl 0668.34024 [10] Krasnosel’skiľ, M.A.; Zabreľko, P.P., Geometrical methods of nonlinear analysis, (1984), Springer Berlin Heidelberg, New York, Tokyo [11] Liu, X., Nontrivial solutions of singular nonlinear m-point boundary value problems, J. math. anal. appl., 284, 576-590, (2003) · Zbl 1042.34031 [12] Ma, R., Existence of solutions of nonlinear m-point boundary-value problems, J. math. anal. appl., 256, 556-567, (2001) · Zbl 0988.34009 [13] Moshinsky, M., Sobre los problemas de condiciones a la frontiera en una dimension de caracteristicas discontinuas, Bol. soc. mat. mexicana, 7, 1C25, (1950) [14] Timoshenko, S., Theory of elastic stability, (1961), McGraw-Hill New York [15] Webb, J.R.L., Positive solutions of some three point boundary value problems via fixed point index, Nonlinear anal., 47, 4319-4332, (2001) · Zbl 1042.34527 [16] Xu, X., Positive solutions for singular m-point boundary value problems with positive parameter, J. math. anal. appl., 291, 352-367, (2004) · Zbl 1047.34016 [17] Xu, X., Multiplicity results for positive solutions of some semi-positone three-point boundary value problems, J. math. anal. appl., 291, 673-689, (2004) · Zbl 1056.34035
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.