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Three symmetric positive solutions for Lidstone problems by a generalization of the Leggett-Williams theorem. (English) Zbl 0958.34020
The authors deal with the existence of three positive solutions to the so-called Lidstone BVP $$y^{(4)}=f(y,-y''), 0\le t\le 1, y(0)=y''(0)=y(1)=y''(1)=0.$$ These solutions are required to be symmetric in the sense that they satisfy $y(t)=y(1-t),$ for all $t\in[0,1].$ The paper is organized as follows: In section 2 the authors provide some facts from the theory of cones in Banach spaces and they state the so-called five functional fixed-point theorem (FFFPT) (a generalization of the Legget-Williams fixed-point theorem). Section 3 contains some basic properties which the corresponding Green function of the problem has. In the same section, by imposing growth conditions on the function $f$ and by using the FFFPT the authors state and prove the main existence results. Furthermore they give growth information for the second derivatives of the solutions. Finally, in section 4, the authors investigate the same question as above for the discrete fourth-order Lidstone BVP of the form $$\Delta^{4}y(t-2)=f(y(t),\ -\Delta(t-1)), a+2\le {t} \le {b+2},$$ under the conditions $y(a)=\Delta^{2}y(a)=0=\Delta^{2}y(b+2)=y(b+4).$

34B18Positive solutions of nonlinear boundary value problems for ODE
34B27Green functions
39A12Discrete version of topics in analysis
Full Text: EMIS EuDML