×

zbMATH — the first resource for mathematics

Existence and uniqueness results for discrete second-order periodic boundary value problems. (English) Zbl 1057.39008
The paper contains results on existence and uniqueness of a second order nonlinear difference equation \[ - \Delta^2 y(n-1) + q(n) y(n) = f(n,y(n)), \] \(\Delta\) the forward difference operator, with periodic boundary conditions. The methods used in the proofs are based on the notions of lower and upper solutions, Green’s functions and Brouwer’s fixed point theorem.

MSC:
39A12 Discrete version of topics in analysis
34B15 Nonlinear boundary value problems for ordinary differential equations
39A11 Stability of difference equations (MSC2000)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Agarwal, R.P.; O’Regan, D., Boundary value problems for discrete equations, Appl. math. lett., 10, 4, 83-89, (1997) · Zbl 0890.39001
[2] Agarwal, R.P.; Wong, P.J.Y., Advanced topics in difference equations, (1997), Kluwer Academic · Zbl 0914.39005
[3] Atici, F.M.; Guseinov, G.Sh., Positive periodic solutions for nonlinear difference equations with periodic coefficients, J. math. anal. appl., 232, 166-182, (1999) · Zbl 0923.39010
[4] Cabada, A., Extremal solutions for the difference φ-Laplacian problem with nonlinear functional boundary conditions, (), 593-601, (3-5) · Zbl 1001.39006
[5] Cabada, A., The method of lower and upper solutions for second, third, fourth, and higher order boundary value problems, J. math. anal. app., 185, 302-320, (1994) · Zbl 0807.34023
[6] Cabada, A.; Nieto, J.J., A generalization of the monotone iterative technique for nonlinear second order periodic boundary value problems, J. math. anal. app., 151, 181-189, (1990) · Zbl 0719.34039
[7] Cabada, A.; Otero-Espinar, V., Optimal existence results for nth order periodic boundary value difference problems, J. math. anal. app., 247, 67-86, (2000) · Zbl 0962.39006
[8] Zhuang, W.; Chen, Y.; Cheng, S.S., Monotone methods for a discrete boundary problem, Computers math. applic., 32, 12, 41-49, (1996) · Zbl 0872.39005
[9] Hristova, S.G.; Roberts, L.F., Monotone-iterative method of V. lakshmikantham for a periodic boundary value problem for a class of differential equations with “supremum”, Nonlinear anal., 44, 601-612, (2001) · Zbl 1003.34023
[10] Heikkilä, S.; Lakshmikantham, V., Monotone iterative techniques for discontinuous nonlinear differential equations, (1994), Marcel Dekker New York · Zbl 0804.34001
[11] Kannan, R.; Lakshmikantham, V., Existence of periodic solutions of nonlinear boundary value problems and the method of upper and lower solutions, App. anal., 17, 103-113, (1984) · Zbl 0532.34030
[12] Lakshmikantham, V.; Leela, S., Remarks on first and second order periodic boundary value problems, Nonlinear anal., 8, 281-287, (1984) · Zbl 0532.34029
[13] Leela, S., Monotone method for second order periodic boundary value problems, Nonlinear anal., 7, 349-355, (1983) · Zbl 0524.34023
[14] Nieto, J.J., Nonlinear second-order periodic boundary value problems, J. math. anal. appl., 130, 22-29, (1988) · Zbl 0678.34022
[15] Sedziwy, S., Nonlinear periodic boundary value problem for a second order ordinary differential equations, Nonlinear anal., 32, 7, 881-890, (1998) · Zbl 0934.34009
[16] Ma, R., Multiplicity of positive solutions for second-order three-point boundary value problems, Computers math. applic., 40, 2/3, 193-204, (2000) · Zbl 0958.34019
[17] Nieto, J.J., An abstract monotone iterative technique, Nonlinear anal., 28, 12, 1923-1933, (1997) · Zbl 0883.47058
[18] Pao, C.V., Monotone iterative methods for finite difference system of reaction-diffusion equations, Numer. math., 46, 571-586, (1985) · Zbl 0589.65072
[19] Wang, Y.M., Monotone methods for a boundary value problem of second-order discrete equation, Computers math. applic., 36, 6, 77-92, (1998) · Zbl 0932.65130
[20] Agarwal, R.P., Boundary value problems for higher order differential equations, (1986), World Scientific · Zbl 0598.65062
[21] Kelley, W.G.; Peterson, A.C., Difference equations: an introduction with applications, (1991), Academic Press New York · Zbl 0733.39001
[22] Dunford, N.; Schwartz, J., Linear operators, (part I), (1957), Interscience New York
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.