zbMATH — the first resource for mathematics

Three-point boundary value problems for difference equations. (English) Zbl 1075.39015
The authors consider the discrete nonlinear difference equation \(\Delta^2x_{k-1} +f(x_k)=0\) together with a three point boundary condition \(x_0=0\), \(x_{n+1} = ax_\ell+b\). By means of Krasnoselskii’s fixed point theorem they prove results on (non-)existence and uniqueness of positive solutions. Finally they point out an application to a discrete model of heat conduction.

39A12 Discrete version of topics in analysis
34B15 Nonlinear boundary value problems for ordinary differential equations
39A11 Stability of difference equations (MSC2000)
Full Text: DOI
[1] Il’in, V.A.; Moiseev, E.I., Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator, Differential equations, 23, 8, 979-987, (1987) · Zbl 0668.34024
[2] Feng, W.; Webb, J.R.L., Solvability of an m-point boundary value problems with nonlinear growth, J. math. anal. appl., 212, 467-480, (1997) · Zbl 0883.34020
[3] Feng, W., On an m-point nonlinear boundary value problem, Nonlinear analysis TMA, 30, 6, 5369-5374, (1997) · Zbl 0895.34014
[4] Gupta, C.P., Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation, J. math. anal. appl., 168, 540-551, (1992) · Zbl 0763.34009
[5] Gupta, C.P., A sharper condition for the solvability of a three-point second order boundary value problem, J. math. anal. appl., 205, 579-586, (1997) · Zbl 0874.34014
[6] Gupta, C.P., A generalized multi-point boundary value problem for second order ordinary differential equations, Appl. math. computer, 89, 133-146, (1998) · Zbl 0910.34032
[7] Ma, R., Existence theorems for a second order m-point boundary value problem, J. math. anal. appl., 211, 545-555, (1997) · Zbl 0884.34024
[8] Ma, R., Positive solutions for second order three-point boundary value problems, Appl. math. lett., 14, 1, 1-5, (2001) · Zbl 0989.34009
[9] Ma, R., Positive solutions of a nonlinear three-point boundary value problem, Electronic journal of differential equations, 34, 1-8, (1999)
[10] Krasnoselskii, M.A., Positive solutions of operator equation, (1964), P. Noordhoff
[11] Henderson, J., Positive solutions for nonlinear difference equations, Nonlinear studies, 4, 29-36, (1997) · Zbl 0883.39002
[12] Agarwal, R.P.; Wong, P.J., Advanced topics in difference equations, (1997), Kluwer Academic Publishers Groningen, The Netherlands · Zbl 0878.39001
[13] Merdivenci, F., Two positive solutions of a boundary value problem for difference equations, J. difference equations appl., 1, 263-270, (1995) · Zbl 0854.39001
[14] Merdivenci, F., Green’s matrices and positive solutions of a discrete boundary value problem, Pan amer. math. J., 5, 25-42, (1995) · Zbl 0839.39002
[15] Anderson, D.; Avery, R.; Peterson, A., Three positive solutions to a discrete focal boundary value problem, J. computational and applied math., 88, 103-118, (1998) · Zbl 1001.39021
[16] Wong, P.J., Positive solutions of discrete (n,p) boundary value problems, Nonlinear analysis, TM&A, 30, 1, 377-388, (1997) · Zbl 0893.39001
[17] Agarwal, R.P.; Usmani, R.A., The formulation of invariant imbedding method to solve multipoint discrete boundary value problems, Appl. math. lett., 4, 4, 17-22, (1991) · Zbl 0724.65068
[18] Sheng, Q.; Agarwal, R.P., Existence and uniqueness of the solutions of nonlinear n-point boundary value problems, Nonlinear world, 2, 69-86, (1995) · Zbl 0810.34014
[19] Atici, F.; Peterson, A., Bounds for positive solutions for a focal boundary value problem, Advances in difference equations II, 36, 10/12, 99-107, (1998), Special Issue of Computers Math. Applic. · Zbl 0933.39040
[20] Guo, D.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic Press · Zbl 0661.47045
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.