November  2017, 16(6): 1977-1988. doi: 10.3934/cpaa.2017097

Almost reducibility of linear difference systems from a spectral point of view

Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile

* Corresponding author

Received  September 2016 Revised  May 2017 Published  July 2017

Fund Project: This research has been partially supported by MATHAMSUD program (16-MATH-04 STADE) and FONDECYT Regular 1170968.

We prove that, under some conditions, a linear nonautonomous difference system is Bylov's almost reducible to a diagonal one whose terms are contained in the Sacker and Sell spectrum of the original system.

In the above context, we provide an example of the concept of diagonally significant system, recently introduced by Pötzsche. This example plays an essential role in the demonstration of our results.

Citation: Álvaro Castañeda, Gonzalo Robledo. Almost reducibility of linear difference systems from a spectral point of view. Communications on Pure and Applied Analysis, 2017, 16 (6) : 1977-1988. doi: 10.3934/cpaa.2017097
References:
[1]

B. AulbachN. Van Minh and P.P. Zabreiko, The concept of spectral dichotomy for linear difference equations, J. Math. Anal. Appl., 185 (1994), 275-287.  doi: 10.1006/jmaa.1994.1248.

[2]

B. Aulbach and S. Siegmund, A spectral theory for nonautonomous difference equations, in New Trends in Difference Equations, Temuco, Chile, 2000 (eds. J. López{Fenner and M. Pinto), Taylor and Francis, (2002), 45-55.

[3]

B. Aulbach and S. Siegmund, The dichotomy spectrum for noninvertible systems of linear difference equations, J. Difference Equ. Appl., 7 (2001), 895-913.  doi: 10.1080/10236190108808310.

[4]

F. Battelli and K. J. Palmer, Criteria for exponential dichotomy for triangular systems, J. Math. Anal. Appl., 428 (2015), 525-543.  doi: 10.1016/j.jmaa.2015.03.029.

[5]

B. F. Bylov, Almost reducible system of differential equations, Sibirsk. Mat. Zh., 3 (1963), 333-359 (Russian). 

[6]

S. Elaydi, An Introduction to Difference Equations 3rd edition, Springer-Verlag, New York, 2005.

[7]

I. GohbergM. A. Kaashoek and J. Kos, Classification of linear--time varying difference equations under kinematic similarity, Integral Equations Operator Theory, 25 (1996), 445-480.  doi: 10.1007/BF01203027.

[8]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, American Mathematical Society, Providence RI, 2011. doi: 10.1090/surv/176.

[9]

F. Lin, Spectrum set and contractible sets of linear differential equations, Chinese Ann. Math. Ser. A, 11 (1990), 111-120 (Chinese). 

[10]

F. Lin, Hartman's linearization on nonautonomous unbounded system, Nonlinear Anal., 66 (2007), 38-50.  doi: 10.1016/j.na.2005.11.007.

[11]

K. J. Palmer, Exponential dichotomies, the shadowing lemma and transversal homoclinic points, in Dynamics Reported (eds. U. Kirchgraber and H. -O. Walther), John Wiley & Sons, Ltd. , Chichester; B. G. Teubner, Stuttgart, (1988), 265-306.

[12]

G. Papaschinopoulos and J. Schinas, Criteria for an exponential dichotomy of difference equations, Czechoslovak Math. J., 35 (1985), 295-299.  doi: 10.1016/0022-247X(86)90216-7.

[13]

G. Papaschinopoulos, Dichotomies in terms of Lyapunov functions for linear difference equations, J. Math. Anal. Appl., 152 (1990), 524-535.  doi: 10.1016/0022-247X(90)90082-Q.

[14]

M. Pinto, Discrete Dichotomies, Comput. Math. Appl., 28 (1994), 259-270.  doi: 10.1016/0898-1221(94)00114-6.

[15]

C. Pötzsche, Dichotomy spectra of triangular equations, Discrete Contin. Dyn. Syst., 36 (2016), 423-450.  doi: 10.3934/dcds.2016.36.423.

[16]

R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems, J. Differential Equations, 27 (1978), 320-358.  doi: 10.1016/0022-0396(78)90057-8.

[17]

S. Siegmund, Block diagonalization of linear difference equations, J. Difference Equ. Appl., 8 (2002), 177-189.  doi: 10.1080/10236190211950.

show all references

References:
[1]

B. AulbachN. Van Minh and P.P. Zabreiko, The concept of spectral dichotomy for linear difference equations, J. Math. Anal. Appl., 185 (1994), 275-287.  doi: 10.1006/jmaa.1994.1248.

[2]

B. Aulbach and S. Siegmund, A spectral theory for nonautonomous difference equations, in New Trends in Difference Equations, Temuco, Chile, 2000 (eds. J. López{Fenner and M. Pinto), Taylor and Francis, (2002), 45-55.

[3]

B. Aulbach and S. Siegmund, The dichotomy spectrum for noninvertible systems of linear difference equations, J. Difference Equ. Appl., 7 (2001), 895-913.  doi: 10.1080/10236190108808310.

[4]

F. Battelli and K. J. Palmer, Criteria for exponential dichotomy for triangular systems, J. Math. Anal. Appl., 428 (2015), 525-543.  doi: 10.1016/j.jmaa.2015.03.029.

[5]

B. F. Bylov, Almost reducible system of differential equations, Sibirsk. Mat. Zh., 3 (1963), 333-359 (Russian). 

[6]

S. Elaydi, An Introduction to Difference Equations 3rd edition, Springer-Verlag, New York, 2005.

[7]

I. GohbergM. A. Kaashoek and J. Kos, Classification of linear--time varying difference equations under kinematic similarity, Integral Equations Operator Theory, 25 (1996), 445-480.  doi: 10.1007/BF01203027.

[8]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, American Mathematical Society, Providence RI, 2011. doi: 10.1090/surv/176.

[9]

F. Lin, Spectrum set and contractible sets of linear differential equations, Chinese Ann. Math. Ser. A, 11 (1990), 111-120 (Chinese). 

[10]

F. Lin, Hartman's linearization on nonautonomous unbounded system, Nonlinear Anal., 66 (2007), 38-50.  doi: 10.1016/j.na.2005.11.007.

[11]

K. J. Palmer, Exponential dichotomies, the shadowing lemma and transversal homoclinic points, in Dynamics Reported (eds. U. Kirchgraber and H. -O. Walther), John Wiley & Sons, Ltd. , Chichester; B. G. Teubner, Stuttgart, (1988), 265-306.

[12]

G. Papaschinopoulos and J. Schinas, Criteria for an exponential dichotomy of difference equations, Czechoslovak Math. J., 35 (1985), 295-299.  doi: 10.1016/0022-247X(86)90216-7.

[13]

G. Papaschinopoulos, Dichotomies in terms of Lyapunov functions for linear difference equations, J. Math. Anal. Appl., 152 (1990), 524-535.  doi: 10.1016/0022-247X(90)90082-Q.

[14]

M. Pinto, Discrete Dichotomies, Comput. Math. Appl., 28 (1994), 259-270.  doi: 10.1016/0898-1221(94)00114-6.

[15]

C. Pötzsche, Dichotomy spectra of triangular equations, Discrete Contin. Dyn. Syst., 36 (2016), 423-450.  doi: 10.3934/dcds.2016.36.423.

[16]

R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems, J. Differential Equations, 27 (1978), 320-358.  doi: 10.1016/0022-0396(78)90057-8.

[17]

S. Siegmund, Block diagonalization of linear difference equations, J. Difference Equ. Appl., 8 (2002), 177-189.  doi: 10.1080/10236190211950.

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