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Zbl 0836.34023
Hernandez, Gaston E.; Manasevich, Raul
Existence and multiplicity of solutions of a fourth order equation. (English)
[J] Appl. Anal. 54, No.3-4, 237-250 (1994). ISSN 0003-6811; ISSN 1563-504X/e

We study the existence and multiplicity of solutions of the fourth order nonlinear boundary value problem $y^{(iv)} = \lambda f(x,y)$, $y(0) = y(1) = y''(0) = y''(1) = 0$, $f(x,y)$ is assumed to be positive, continuous and increasing in $y$.\par We prove that there exists a $\lambda^* > 0$ such that there is always a solution for $0 \le \lambda < \lambda^* $ $(\lambda^*$ can be $\infty)$. From our results it follows that if ${z \over f(x,z)}$ is bounded (for $x$ in some compact interval around ${1 \over 2})$ then there exists $\lambda_*$ such that there is no solution for $\lambda > \lambda_*$.\par We provide conditions on $f$ such that there are at least two solutions for $0 < \lambda < \lambda_*$. Degree theory is used to prove this result.
[G.E.Hernandez (Storrs)]

MSC 2000:: *34B15 Nonlinear boundary value problems of ODE

Keywords: existence; multiplicity; fourth order nonlinear boundary value problem

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Zentralblatt MATH Berlin [Germany]