Agarwal, R. P.; O’Regan, D. Multiple solutions for higher-order difference equations. (English) Zbl 0941.39003 Comput. Math. Appl. 37, No. 9, 39-48 (1999). Let \(T\in\{1,2, \dots\}\), \(I_0=\{0,1, \dots,T\}\) and \(y:I_n= \{0,1, \dots, T+n\}\to\mathbb{R}\). The authors discuss the \(n\)th \((n\geq 2)\) order discrete conjugate problem \[ (-1)^{n-p} \Delta^n y(k)=f \bigl(k,y(k)\bigr),\;k\in I_0, \]\[ \Delta^iy(0)=0,\;0\leq i\leq p-1\quad \text{(here }1\leq p\leq n-1), \] and the \(n\)th \((n\geq 2)\) order discrete \((n,p)\) problem \[ \begin{aligned} & \Delta^n y(k)+f\bigl(k,y(k)\bigr) =0,\;k\in i_0,\\ & \Delta^iy(0)=0,\;0\leq i\leq n-2,\\ & \Delta^p y(T+n-p)=0,\;0\leq p\leq n-1\;\text{ is fixed}.\end{aligned} \] The technique employed is based on existence principles, lower type inequalities and Krasnoselskij’s fixed point theorem in a cone. Reviewer: B.G.Pachpatte (Aurangabad) Cited in 1 ReviewCited in 12 Documents MSC: 39A10 Additive difference equations Keywords:multiple solutions; higher-order difference equations; discrete conjugate problem; existence; inequalities PDFBibTeX XMLCite \textit{R. P. Agarwal} and \textit{D. O'Regan}, Comput. Math. Appl. 37, No. 9, 39--48 (1999; Zbl 0941.39003) Full Text: DOI References: [1] Agarwal, R. P., Difference Equations and Inequalities (1992), Marcel Dekker: Marcel Dekker New York · Zbl 0784.33008 [2] Agarwal, R. P.; Henderson, J.; Wong, P. J.Y., On superlinear and sublinear \((n, p)\) boundary value problems for higher order difference equations, Nonlinear World, 4, 101-115 (1997) · Zbl 0902.39004 [3] R.P. Agarwal, D. O’Regan and P.J.Y. Wong, Positive solutions of differential, Difference and Integral Equations; R.P. Agarwal, D. O’Regan and P.J.Y. Wong, Positive solutions of differential, Difference and Integral Equations [4] Lasota, A., A discrete boundary value problem, Ann. Plon. Math., 20, 183-190 (1968) · Zbl 0157.15602 [5] Wong, P. J.Y.; Agarwal, R. P., Double positive solutions of \((n, p)\) boundary value problems for higher order difference euations, Computers Math. Applic., 32, 8, 1-21 (1996) · Zbl 0873.39008 [6] R.P. Agarwal and D. O’Regan, Singular discrete \((np\); R.P. Agarwal and D. O’Regan, Singular discrete \((np\) [7] Eloe, P. W., A generalization of concavity for finite differences, Computers Math. Applic., 36, 10-12, 109-113 (1998) · Zbl 0933.39038 [8] P.J.Y. Wong and R.P. Agarwal, Extensions of continuous and discrete inequalities due to Eloe and Henderson, Nonlinear Anal; P.J.Y. Wong and R.P. Agarwal, Extensions of continuous and discrete inequalities due to Eloe and Henderson, Nonlinear Anal · Zbl 0933.34008 [9] R.P. Agarwal and D. O’Regan, Discrete conjugate boundary value problems (to appear).; R.P. Agarwal and D. O’Regan, Discrete conjugate boundary value problems (to appear). [10] Erbe, L. H.; Hu, S.; Wang, H., Multiple positive solutions of some boundary value problems, Jour. Math. Anal. Appl., 184, 640-648 (1994) · Zbl 0805.34021 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.