## Sign-changing solutions to second-order integral boundary value problems.(English)Zbl 1151.34018

Summary: By using the fixed point index theory and Leray-Schauder degree theory, we consider the existence and multiplicity of sign-changing solutions to nonlinear second-order integral boundary value problem
$-u^{\prime\prime}(t)=f(u(t))\text{ for all }t\in [0,1]$ subject to
$u(0)=0\text{ and }u(1)=g\biggl(\int _0^1 u(s)\,ds\biggr),$
where $$f,g \in \mathcal C(\mathbb R,\mathbb R)$$. We obtain some new existence results concerning sign-changing solutions by computing eigenvalues and the algebraic multiplicities of the associated linear problem. If $$f$$ and $$g$$ satisfy certain conditions, then this problem has at least six different nontrivial solutions: two positive solutions, two negative solutions and two sign-changing solutions. Moreover, if $$f$$ and $$g$$ are also odd, then the problem has at least eight different nontrivial solutions, where are two positive, two negative and four sign-changing solutions.

### MSC:

 34B15 Nonlinear boundary value problems for ordinary differential equations 47N20 Applications of operator theory to differential and integral equations
Full Text:

### References:

 [1] Guo, D., Nonlinear functional analysis, (2001), Shandong Sci. & Tec. Press [2] Guo, D.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic Press New York · Zbl 0661.47045 [3] Krasnosel’skil˘, M.A.; Zabrel˘ko, P.P., Geometrical methods of nonlinear analysis, (1984), Springer Verlag Berlin [4] Li, F.; Liang, Z.; Zhang, Q.; Li, Y., On sign-changing solutions for nonlinear operator equations, J. math. anal. appl., 327, 1010-1028, (2007) · Zbl 1114.47052 [5] Pang, C.; Dong, W.; Wei, Z., Multiple solutions for fourth-order boundary value problems, J. math. anal. appl., 314, 464-476, (2006) · Zbl 1094.34012 [6] Xu, X., Multiple sign-changing solutions for some $$m$$-point boundary value problems, Electronic J. differential equations, 89, 1-14, (2004) [7] Xu, X.; Sun, J., On sign-changing solution for some three-point boundary value problems, Nonlinear anal., 59, 491-505, (2004) · Zbl 1069.34019 [8] Yang, Z., Positive solutions to a system of second-order nonlocal boundary value problems, Nonlinear anal., 62, 1251-1265, (2005) · Zbl 1089.34022 [9] Z. Yang, Existence of nontrivial solutions for a nonlinear Sturm-Liouville problem with integral boundary conditions, Nonlinear Anal. (2006), doi:10.1016/j.na.2006.10.044 [10] Yang, Z., Existence and nonexistence results for positive solutions of an integral boundary value problem, Nonlinear anal., 65, 1489-1511, (2006) · Zbl 1104.34017 [11] Yang, Z., Positive solutions of a second-order integral boundary value problem, J. math. anal. appl., 321, 751-765, (2006) · Zbl 1106.34014
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.