Optimal existence results for \(n\)th order periodic boundary value difference equations. (English) Zbl 0962.39006

The authors study the \(n\)-th order nonlinear difference equation \[ u(k+n)= f\bigl(k,u(k), \dots,u(k+n) \bigr),\;k\in\{0, \dots, N-1\}, \] with periodic boundary conditions. Supposing that there exist ordered lower and upper solutions, they obtain a solution for this problem.
The existence results are equivalent to finding the values of \(K_i\), \(i=1, \dots,n\), for which the linear operator \[ u(k+n)+ \sum^n_{i=0} K_iu(k+i) \] in the space of periodic functions is an inverse positive operator.
Moreover an expression of the Green function is obtained. Applications to the first and second order equations are included.


39A11 Stability of difference equations (MSC2000)
39A70 Difference operators
Full Text: DOI


[1] Agarwal, R. P., Focal Boundary Value Problems for Differential and Difference Equations. Focal Boundary Value Problems for Differential and Difference Equations, Mathematics and Its Applications, 436 (1998), Kluwer Academic: Kluwer Academic Dordrecht · Zbl 0914.34001
[2] Agarwal, R. P.; O’Regan, D., Multiple solutions for higher-order difference equations, Comput. Math. Appl., 37, 39-48 (1999) · Zbl 0941.39003
[3] Agarwal, R. P.; O’Regan, D.; Wong, P., Positive Solutions of Differential, Difference and Integral Equations (1999), Kluwer Academic: Kluwer Academic Dordrecht · Zbl 1157.34301
[4] Agarwal, R. P.; Wong, F., Upper and lower solutions method for higher-order discrete boundary value problems, Math. Inequal. Appl., 1, 551-557 (1998) · Zbl 0917.39002
[5] Cabada, A., The method of lower and upper solutions for second, third, fourth and higher order boundary value problems, J. Math. Anal. Appl., 185, 302-320 (1994) · Zbl 0807.34023
[6] Cabada, A.; Otero-Espinar, V.; Pouso, R. L., Existence and approximation of solutions for discontinuous first order difference problems with non-linear functional boundary conditions in the presence of lower and upper solutions, Comput. Math. Appl., 39, 21-33 (2000) · Zbl 0972.39002
[7] Elaydi, S. N., An Introduction to Difference Equations. An Introduction to Difference Equations, Undergraduate Texts in Mathematics (1996), Springer-Verlag: Springer-Verlag New York · Zbl 0840.39002
[8] Elaydi, S. N.; Harris, W. A., On the computation of \(A^N\), SIAM Rev., 40, 965-971 (1998) · Zbl 0913.65037
[9] Goldberg, S., Introduction to Difference Equations (1960), Wiley: Wiley New York
[10] Heikkilä, S.; Lakshmikantham, V., Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations (1994), Dekker: Dekker New York · Zbl 0804.34001
[11] Lakshmikantham, V.; Trigiante, D., Theory of Difference Equations: Numerical Methods and Applications. Theory of Difference Equations: Numerical Methods and Applications, Mathematics in Science and Engineering, 181 (1988), Academic Press: Academic Press Boston · Zbl 0683.39001
[12] Atici, F. Mendivenci; Guseinov, G. Sh., Positive periodic solutions for nonlinear difference equations with periodic coefficients, J. Math. Anal. Appl., 232, 166-182 (1999) · Zbl 0923.39010
[13] Nkashama, M. N., A generalized upper and lower solutions method and multiplicity results for nonlinear first-order ordinary differential equations, J. Math. Anal. Appl., 140, 381-395 (1989) · Zbl 0674.34009
[14] Zhuang, W.; Chen, Y.; Cheng, S. S., Monotone methods for a discrete boundary problem, Comput. Math. Appl., 32, 41-49 (1996) · Zbl 0872.39005
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