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Existence and multiplicity of solutions of a fourth order equation. (English) Zbl 0836.34023

We study the existence and multiplicity of solutions of the fourth order nonlinear boundary value problem y (iv) =λf(x,y), y(0)=y(1)=y '' (0)=y '' (1)=0, f(x,y) is assumed to be positive, continuous and increasing in y.

We prove that there exists a λ * >0 such that there is always a solution for 0λ<λ * (λ * can be ). From our results it follows that if z f(x,z) is bounded (for x in some compact interval around 1 2) then there exists λ * such that there is no solution for λ>λ * .

We provide conditions on f such that there are at least two solutions for 0<λ<λ * . Degree theory is used to prove this result.

34B15Nonlinear boundary value problems for ODE