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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
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