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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.

MSC:

39A11 Stability of difference equations (MSC2000)
39A70 Difference operators
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[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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