On eigenvalue intervals for discrete second order boundary value problems. (English) Zbl 1087.39009
Using cone theoretic techniques and sign properties of an associated Green’s function, the authors find eigenvalue intervals such that there exists at least one positive solution for the three-point discrete boundary value problem
\[
\Delta^2y(k-1) + \lambda h(k) f(y(k)) = 0, k \in \{1, 2, \dots, T\}
\]
\[ y(0) = 0, y(T+1) = a y(n), \] where \(T \geq 3\) is fixed, \(n \in \{2, \dots, T-1\}\), and \(a > 0\) is such that \(an < T + 1\).
\[ y(0) = 0, y(T+1) = a y(n), \] where \(T \geq 3\) is fixed, \(n \in \{2, \dots, T-1\}\), and \(a > 0\) is such that \(an < T + 1\).
Reviewer: Eric R. Kaufmann (Little Rock)
MSC:
| 39A11 | Stability of difference equations (MSC2000) |
| 39A12 | Discrete version of topics in analysis |
| 34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
| 34B15 | Nonlinear boundary value problems for ordinary differential equations |
| 34L05 | General spectral theory of ordinary differential operators |
Keywords:
eigenvalue intervals; positive solution; cones; Green’s function; three-point discrete boundary value problemReferences:
| [1] | Agarwal, R.P., O’Regan, D., Wong, P.J.Y. Positive solutions of differential, difference and integral equations. Kluwer Academic Publishers, Boston, 1999 |
| [2] | Elaydi, S.N. An Introduction to difference equations, second edition. Undergraduate Texts in Mathematics. Springer-Verlag, New York, 1999 |
| [3] | Eloe, P.W., Raffoul, Y., Reid, D.T. et al. Positive solutions of nonlinear functional difference equations. Comput. Math. Appl., 42:639–646 (2001) · Zbl 0998.39003 · doi:10.1016/S0898-1221(01)00183-3 |
| [4] | Erbe, L.H., Hu, S.C., Wang H.Y. Multiple positive solutions of some boundary value problems. J. Math. Anal. Appl., 184:640–648 (1994) · Zbl 0805.34021 · doi:10.1006/jmaa.1994.1227 |
| [5] | Guo, D.J., Lakshmikantham ,V. Nonlinear problems in abstract cones. Academic Press, San Diego, 1988 · Zbl 0661.47045 |
| [6] | Guo, Y.P., Ge W.G. Positive solutions for three-point boundary value problems with dependence on the first order derivative. J. Math. Anal. Appl., 290:291–301 (2004) · Zbl 1054.34025 · doi:10.1016/j.jmaa.2003.09.061 |
| [7] | Jiang, D.Q., Lin, H., Existence of positive solutions to (k, n k) conjugate boundary value problems. Kyushu J. Math., 53:115–125 (1998) · Zbl 0928.34020 · doi:10.2206/kyushujm.53.115 |
| [8] | Kong, L.J., Kong Q.K., Zhang B.G. Positive solutions of third-order functional difference equations. Comput. Math. Appl., 44:481–489 (2002) · Zbl 1057.39014 · doi:10.1016/S0898-1221(02)00170-0 |
| [9] | Ma, R.Y. Positive solutions of a nonlinear three-point boundary value problems. Electron J. Differential Equations., 34:1–8 (1999) · Zbl 0926.34009 |
| [10] | Ma, R.Y. Positive solutions for second order three-point boundary value problems. Appl. Math. Lett., 14:1–5 (2001) · Zbl 0989.34009 · doi:10.1016/S0893-9659(00)00102-6 |
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.
