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Existence and multiplicity of solutions for fourth-order boundary value problems with parameters. (English) Zbl 1109.34015
Summary: We study the existence and multiplicity of the solutions for the fourth-order boundary value problem $u^{(4)}(t)+\eta u''(t)-\zeta u(t) =\lambda f\bigl(t,u(t) \bigr),\;0<t<1,\;u(0)=u(1)=u''(0)=u''(1)=0,\tag{BVP}$ where $$f:[0,1]\times\mathbb{R} \times\mathbb{R}$$ is continuous, $$\zeta,\eta \in\mathbb{R}$$ and $$\lambda\in\mathbb{R}^+$$ are parameters. By means of the idea of the decomposition of operators, and the critical point theory, we obtain that if the pair $$(\eta,\zeta)$$ is on the curve $$\zeta=-\eta^2/4$$ satisfying $$\eta<2\pi^2$$, then the above BVP has at least one, two, three, and infinitely many solutions for $$\lambda$$ being in different interval, respectively.

##### MSC:
 34B15 Nonlinear boundary value problems for ordinary differential equations
##### Keywords:
existence; multiple solutions; fourth-order BVP
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##### References:
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