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Zbl 1098.34012
Ma, Ruyun
Nodal solutions of boundary value problems of fourth-order ordinary differential equations.
(English)
[J] J. Math. Anal. Appl. 319, No. 2, 424-434 (2006). ISSN 0022-247X

Summary: We study the existence of nodal solutions of the fourth-order two-point boundary value problem $$y''''+\beta(t)y''=a(t)f(y),\ 0<t<1,\qquad y(0)=y(1)=y''(1)=0,$$ where $\beta\in C[0,1]$ with $\beta(t)<\pi^2$ on $[0,1]$, $a\in C[0,1]$ with $a\ge 0$ on $[0,1]$ and $a(t)\equiv 0$ on any subinterval of $[0,1]$, and $f\in C(\bbfR)$ satisfies $f(u)u>0$ for all $u\ne 0$. We give conditions on the ratio $f(s)/s$ at infinity and zero that guarantee the existence of nodal solutions. The proof of our main results is based upon bifurcation techniques.
MSC 2000:
*34B15 Nonlinear boundary value problems of ODE

Keywords: multiplicity results; eigenvalues; disconjugate; bifurcation methods; nodal solutions

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