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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.

MSC:

39A12 Discrete version of topics in analysis
34B15 Nonlinear boundary value problems for ordinary differential equations
39A11 Stability of difference equations (MSC2000)
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[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: 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
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