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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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