Agarwal, R. P.; O’Regan, D. Boundary value problems for discrete equations. (English) Zbl 0890.39001 Appl. Math. Lett. 10, No. 4, 83-89 (1997). The paper presents some general properties of solutions of a nonlinear discrete boundary value problem \[ \Delta^2 y(i-1)+\mu f(i,y(i))=0,\quad i\in N,\quad y(0)=0,\quad y(T+1)=0,\tag{1} \] where \(\mu\geq 0\), \(T\in\{1,2,\ldots,T\}\), \(N^+=\{0,1,\ldots,T+1\}\), \(y:N^+\to\mathbb{R}\), and \(f:N\times\mathbb{R}\to\mathbb{R}\) is continuous. The main result is a discrete version of the generalized Gelfand problem. The authors’ aim here is to establish the existence of nonnegative solutions. The notion of upper and lower solutions of the boundary problem (1) is introduced. An easy argument (based on the Schauder fixed point theorem) guarantees that the problem has a solution which lies between the lower and the upper solution. The obtained results are the consequences of a new existence principle established in the first part of the paper. Reviewer: A.D.Mednykh (Novosibirsk) Cited in 44 Documents MSC: 39A10 Additive difference equations Keywords:nonlinear difference equations; boundary value problems; upper and lower solutions; nonnegative solutions PDFBibTeX XMLCite \textit{R. P. Agarwal} and \textit{D. O'Regan}, Appl. Math. Lett. 10, No. 4, 83--89 (1997; Zbl 0890.39001) Full Text: DOI References: [1] Agarwa, R. P., Difference Equations and Inequalities (1992), Marcel Dekker: Marcel Dekker New York [2] Agarwal, R. P., On boundary value problems for second order discrete systems, Appl. Anal., 20, 1-17 (1985) · Zbl 0596.39003 [3] Dugundji, J.; Granas, A., Fixed Point Theory, (Monografie Mat. (1982), PWN: PWN Warsaw) · Zbl 1025.47002 [4] R.P. Agarwal and D. O’Regan, A fixed point approach for nonlinear discrete boundary value problems. Computers Math. Applic.; R.P. Agarwal and D. O’Regan, A fixed point approach for nonlinear discrete boundary value problems. Computers Math. Applic. [5] Munkres, J. R., Topology (1975), Prentice Hall: Prentice Hall Englewood Cliffs, NJ · Zbl 0107.17201 [6] Henderson, J., Singular boundary value problems for difference equations, Dynamic Systems and Appl., 1, 271-282 (1992) · Zbl 0761.39002 [7] Lakshmikantham, V.; Trigiante, D., Theory of Difference Equations (1988), Academic Press: Academic Press San Diego · Zbl 0683.39001 [8] Lasota, A., A discrete boundary value problem, Ann. Polon. Math., 20, 183-190 (1968) · Zbl 0157.15602 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.