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
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References:

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