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Existence of $n$ solutions and/or positive solutions to a semipositone elastic beam equation. (English) Zbl 1113.34013
This paper considers nonlinear fourth order boundary value problems of the form $$u^{(4)}(t)=f(t,u(t),u''(t)),$$ $$u(0)=u(1)=u''(0)=u''(1)=0.$$ Under certain conditions on the nonlinearity $f$, it is proven that the problem has at least $n$ solutions, where $n$ is any positive integer, under additional conditions these solutions are positive. The proofs of these results are based on the Krasnosel’skii fixed point theorem for compact mappings in subsets of cones.

MSC:
34B15Nonlinear boundary value problems for ODE
34B18Positive solutions of nonlinear boundary value problems for ODE
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References:
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