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Multiple positive solutions to a third-order discrete focal boundary value problem. (English) Zbl 1001.39022
The authors are concerned with the existence of three positive solutions (p.s.) to the third-order boundary \[ -\Delta^3x(t-k)+f(x(t))=0,\quad t\in[a+k,b+k] \] with boundary conditions \(x(a)=\Delta x(t_2)=\Delta^2 x(b+1)=0\) where \(f:\mathbb{R}\to\mathbb{R}\) is continuous, \(f\) is nonnegative for \(x\geq 0\) and \(k\in\{1,2\}\).
In a previous paper the authors together with A. C. Peterson [J. Comput. Appl. Math. 88, No. 1, 103-118 (1998; reviewed above)] imposed conditions on \(f\) to yield at least three p.s. to the enounced b.v.p. applying the fixed-point theorem (f.p.t.) of R. W. Leggett and L. R. Williams [Indiana Univ. Math. J. 28, 673-688 (1979; Zbl 0421.47033)]. R. I. Avery [Math. Sci. Res. Hot-Line 3, No. 7, 9-14 (1999; Zbl 0965.47038)] has given a generalization of this f.p.t. which is used in the present paper to prove the existence of three p.s.

MSC:
39A11 Stability of difference equations (MSC2000)
47H10 Fixed-point theorems
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