Ma, Ruyun; Ma, Huili Global structure of positive solutions for superlinear discrete boundary value problems. (English) Zbl 1233.39006 J. Difference Equ. Appl. 17, No. 9, 1219-1228 (2011). The article deals with the following Dirichlet problem \[ \Delta^2u(t - 1) + \lambda a(t)f(u(t)) = 0, \qquad u(0) = u(T + 1) = 0,\eqno(1) \] where \(a:\;\{0,1,\ldots,T + 1\} \to {\mathbb R}^+\), \(f \in C([0,\infty),[0,\infty))\), \(f(s) > 0\) for \(s > 0\). Let \(f_0 = \lim_{s \to 0} \, f(s)/s = \infty\) and \(f_\infty = \lim_{s \to \infty} \, f(s)/s \in [0,\infty]\), \(\Sigma\) be the closure of set of positive solutions of Problem (1). The main results are the following statements: If \(f_\infty = 0\) then there exists a component \(\zeta \subset \Sigma\) with \((0,0) \in \zeta\) and \(\text{Proj}|_{\mathbb R} \, \zeta = (0,\infty)\); moreover, Problem (1) has at least one positive solution for \(\lambda \in (0,\infty)\); If \(f_\infty \in (0,\infty)\) then there exists a component \(\zeta \subset \Sigma\) with \((0,0) \in \zeta\) and \(\text{Proj}|_{\mathbb R} \, \zeta \subset \bigg(0,{\frac{\lambda_{1}}{f_{\infty}}}\bigg)\) moreover, Problem (1) has at least one positive solution for \(\lambda \in \bigg(0, {\frac{\lambda_{1}}{f_{\infty}}}\bigg)\) If \(f_\infty = \infty\) then there exists a component \(\zeta \subset \Sigma\) with \((0,0) \in \zeta\) for which \(\text{Proj}|_{\mathbb R} \, \zeta\) is a bounded closed interval and \(\zeta\)approaches \((0,\infty)\) as \(\|u\| \to \infty\) moreover, there exist \(0 < \lambda_*, \lambda^* < \infty\) such that Problem (1) has at least two positive solution for \(\lambda \in (0,\lambda_*\) and no positive solutions for \(\lambda > \lambda^*\). The number \(\lambda_1\) is the first (smallest) eigenvalue for the linear problem \[ \Delta^2u(t - 1) + \lambda a(t)u(t) = 0, \qquad u(0) = u(T + 1) = 0. \] Reviewer: Peter Zabreiko (Minsk) Cited in 2 Documents MSC: 39A12 Discrete version of topics in analysis 39A22 Growth, boundedness, comparison of solutions to difference equations Keywords:difference equations; positive solutions; bifurcation of solutions; superlinear discrete boundary value problems PDFBibTeX XMLCite \textit{R. Ma} and \textit{H. Ma}, J. Difference Equ. Appl. 17, No. 9, 1219--1228 (2011; Zbl 1233.39006) Full Text: DOI References: [1] DOI: 10.1016/S0898-1221(98)80035-7 · Zbl 0933.39003 · doi:10.1016/S0898-1221(98)80035-7 [2] DOI: 10.1016/S0893-9659(97)00064-5 · Zbl 0890.39001 · doi:10.1016/S0893-9659(97)00064-5 [3] DOI: 10.1016/S0895-7177(97)00186-6 · Zbl 0889.39011 · doi:10.1016/S0895-7177(97)00186-6 [4] DOI: 10.1016/j.na.2006.07.010 · Zbl 1130.39017 · doi:10.1016/j.na.2006.07.010 [5] DOI: 10.1016/j.jmaa.2005.01.032 · Zbl 1076.39016 · doi:10.1016/j.jmaa.2005.01.032 [6] DOI: 10.1016/S0024-3795(00)00133-6 · Zbl 0956.39012 · doi:10.1016/S0024-3795(00)00133-6 [7] DOI: 10.1016/j.laa.2006.10.026 · Zbl 1121.39013 · doi:10.1016/j.laa.2006.10.026 [8] Ma R., Appl. Math. Comput. (2008) [9] DOI: 10.1016/j.na.2006.09.058 · Zbl 1129.39006 · doi:10.1016/j.na.2006.09.058 [10] DOI: 10.1016/0022-1236(71)90030-9 · Zbl 0212.16504 · doi:10.1016/0022-1236(71)90030-9 [11] Jirari A., Mem. Am. Math. Soc. pp 542– (1995) [12] Whyburn G.T., Topological Analysis (1958) · Zbl 0080.15903 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.