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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 $$ \Delta^2 y_{k-1} + f(k,y_k,\Delta y_{k-1})=0, \quad k=1,2,\dots,n, $$ $$ y_0=0=y_n, $$ where $\Delta y_k:=y_{k+1}-y_k$ is the usual forward difference operator (note a slightly modified notation used by the authors) and where $f:\{1,2,\dots,n\}\times{\Bbb R}^2\to{\Bbb R}$ is continuous. The proof is based on the existence of pairs of discrete lower solutions $\alpha_1$, $\alpha_2$ and discrete upper solutions $\beta_1$, $\beta_2$ that satisfy the inequalities $\alpha_1\leq\alpha_2$ and $\beta_1\leq\beta_2$. 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 $\alpha_2\leq\beta_1$ and allow $f$ being dependent on $\Delta y_{k-1}$. If $f$ is a function of its second variable $y_k$ only, then the results of this paper are sharp and generalize those of {\it R. J. Avery} and {\it 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.

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