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Double positive solutions of \((n,p)\) boundary value problems for higher order difference equations. (English) Zbl 0873.39008

The authors provide existence criteria for double positive solutions of the \((n,p)\) boundary value problem \(\Delta^n y+ F(k,y,\Delta y,\ldots,\Delta^{n-2}y)=G(k,y,\Delta y,\ldots,\Delta^{n-1} y)\) for \(n-1\leq k\leq N\), \(\Delta^i y(0)=0\) for \(0\leq i\leq n-2\), and \(\Delta^p y(N+n-p)=0\), where \(n\geq2\) and \(0\leq p\leq n-1\) is fixed. Upper and lower bounds for the two positive solutions are also established for a particular boundary value problem when \(n=2\). The importance of the obtained results is illustrated by several examples.
The technique used here depends on the fixed point theory of operators on a cone as developed by M. A. Krasnosel’skij [Positive solution of operator equations 1964; Zbl 0121.10604)] and D. Guo and V. Lakshmikantham [Nonlinear problems in abstract cones (1988; Zbl 0661.47045)].

MSC:

39A12 Discrete version of topics in analysis
39A10 Additive difference equations
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