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Existence of multiple solutions for second-order discrete boundary value problems. (English) Zbl 1005.39014

The authors provide sufficient conditions for the existence of (at least) three positive solutions of the discrete two-point boundary value problem

Δ 2 y k-1 +f(k,y k ,Δy k-1 )=0,k=1,2,,n,
y 0 =0=y n ,

where Δy k :=y k+1 -y k is the usual forward difference operator (note a slightly modified notation used by the authors) and where f:{1,2,,n}× 2 is continuous. The proof is based on the existence of pairs of discrete lower solutions α 1 , α 2 and discrete upper solutions β 1 , β 2 that satisfy the inequalities α 1 α 2 and β 1 β 2 . This method is a discrete analog of the authors’ continuous-time results [J. Differ Equations 166, No. 2, 443-454 (2000; Zbl 1013.34017)]. However, in this paper the assumptions do not require α 2 β 1 and allow f being dependent on Δy k-1 . If f is a function of its second variable y k only, then the results of this paper are sharp and generalize those of R. J. Avery and A. C. Peterson [Panam. Math. J. 8, No. 3, 1-12 (1998; Zbl 0959.39006)]. This paper will be useful for researchers interested in two-point boundary value problems and/or their positive solutions.

39A11Stability of difference equations (MSC2000)
39A12Discrete version of topics in analysis
34B15Nonlinear boundary value problems for ODE