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Multiple solutions of some fourth-order boundary value problems. (English) Zbl 1126.34013
The authors study the fourth order boundary value problem
$x^{(4)}=f(t,x),\quad t\in[0,1],\quad x(0)=x(1)=x''(0)=x''(1)=0, \tag{1}$
where $$f:[0,1]\times{\mathbb R}\to {\mathbb R}$$ is a continuous function. They assume $$f$$ to be Lipschitzian and strictly increasing in $$x$$, superlinear at infinity, and moreover that (1) has strict lower and upper solutions. Then they obtain the existence of four solutions; if $$f(t,0)\equiv0$$ they find a positive solution, a negative solution and a sign-changing solution. The setting is variational, using a version of a four-critical point theorem due to Z. Liu and J. Sun, see [J. Differ. Equations 172, No. 2, 257–299 (2001; Zbl 0995.58006)]. There are some innacuracies in the paper; namely the authors use a cone with empty interior, and they point out a generalization where $$x^{(4)}$$ is replaced with $$x^{(4)}+mx$$, ($$m>0$$) but this is possible only for a restricted range of values of $$m$$.

##### MSC:
 34B15 Nonlinear boundary value problems for ordinary differential equations 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces 47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
Zbl 0995.58006
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