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August  2016, 9(4): 959-977. doi: 10.3934/dcdss.2016036

Recurrent equations with sign and Fredholm alternative

Juan Campos 1, , Rafael Obaya 2, and  Massimo Tarallo 3,

1. 

Departamento de Matemática Aplicada, Universidad de Granada, 18071 Granada
2. 

Departamento de Matemática Aplicada, Universidad de Valladolid, Paseo del Cauce 59, 47011 Valladolid
3. 

Università di Milano, via Saldini 50, 20133 Milano, Italy

Received  July 2015 Revised  December 2015 Published  August 2016

We prove that a Fredholm--type Alternative holds for recurrent equations with sign, extending a previous result by Cieutat and Haraux in [3]. Moreover, we show that this can be seen a particular case of [1] and we provide a solution to an interesting question raised by Hale in [6]. Finally we characterize the existence of exponential dichotomies also in the nonrecurrent case.
Keywords: spectral theory, bounded and recurrent solutions, Fredholm alternative., Favard theory, Recurrent and almost periodic linear equations.
Mathematics Subject Classification: Primary: 34C27; Secondary: 34B05, 34A3.
Citation: Juan Campos, Rafael Obaya, Massimo Tarallo. Recurrent equations with sign and Fredholm alternative. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 959-977. doi: 10.3934/dcdss.2016036
References:
[1]

J. Campos, R. Obaya and M. Tarallo, Favard theory for the adjoint equation and Fredholm Alternative,, preprint., ().   Google Scholar

[2]

J. Campos and M. Tarallo, Almost automorphic linear dynamics by Favard theory, J. Differential Equations, 256 (2014), 1350-1367. doi: 10.1016/j.jde.2013.10.018.  Google Scholar

[3]

P. Cieutat and A. Haraux, Exponential decay and existence of almost periodic solutions for some linear forced differential equations, Port. Math., 59 (2002), 141-159.  Google Scholar

[4]

H. Dym, Linear Algebra in Action, Graduate Studies in Mathematics, Vol. 78, AMS, Prividence, 2007. Google Scholar

[5]

J. Favard, Sur les equations différentielles linéairesà coefficients presque-périodiques, Acta Math. , 51 (1928), 31-81. doi: 10.1007/BF02545660.  Google Scholar

[6]

J. K. Hale, Ordinary Differential Equations, Pure and Applied Mathematics, Vol. XXI, Wiley-Interscience, New York, 1969.  Google Scholar

[7]

J. C. Lillo, Approximate similarity and almost periodic matrices, Proc. Amer. Math. Soc., 12 (1961), 400-407.  Google Scholar

[8]

R. Ortega and M. Tarallo, Almost periodic equations and conditions of Ambrosetti-Prodi type, Math. Proc. Camb. Phil. Soc., 135 (2003), 239-254. doi: 10.1017/S0305004103006662.  Google Scholar

[9]

K. J. Palmer, On bounded solutions of almost periodic linear differential systems, J. Math. Anal. Appl., 103 (1984), 16-25. doi: 10.1016/0022-247X(84)90152-5.  Google Scholar

[10]

R. J. Sacker and G. R. Sell, Existence of dichotomies and invariant splittings for linear differential systems. I, J. Differential Equations, 15 (1974), 429-458. doi: 10.1016/0022-0396(74)90067-9.  Google Scholar

[11]

R. J. Sacker and G. R. Sell, Existence of dichotomies and invariant splittings for linear differential systems. II, J. Differential Equations, 22 (1976), 478-496. doi: 10.1016/0022-0396(76)90042-5.  Google Scholar

[12]

R. J. Sacker and G. R. Sell, Existence of dichotomies and invariant splittings for linear differential systems. III, J. Differential Equations, 22 (1976), 497-522. doi: 10.1016/0022-0396(76)90043-7.  Google Scholar

[13]

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.  Google Scholar

[14]

M. Tarallo, Fredholm's alternative for a class of almost periodic linear systems, Discrete Contin. Dyn. Syst., 32 (2012), 2301-2313. doi: 10.3934/dcds.2012.32.2301.  Google Scholar

[15]

M. Tarallo, The Favard separation condition as a purely dimensional fact, J. Dyn. Diff. Equations, 25 (2013), 291-304. doi: 10.1007/s10884-013-9309-2.  Google Scholar

show all references

References:
[1]

J. Campos, R. Obaya and M. Tarallo, Favard theory for the adjoint equation and Fredholm Alternative,, preprint., ().   Google Scholar

[2]

J. Campos and M. Tarallo, Almost automorphic linear dynamics by Favard theory, J. Differential Equations, 256 (2014), 1350-1367. doi: 10.1016/j.jde.2013.10.018.  Google Scholar

[3]

P. Cieutat and A. Haraux, Exponential decay and existence of almost periodic solutions for some linear forced differential equations, Port. Math., 59 (2002), 141-159.  Google Scholar

[4]

H. Dym, Linear Algebra in Action, Graduate Studies in Mathematics, Vol. 78, AMS, Prividence, 2007. Google Scholar

[5]

J. Favard, Sur les equations différentielles linéairesà coefficients presque-périodiques, Acta Math. , 51 (1928), 31-81. doi: 10.1007/BF02545660.  Google Scholar

[6]

J. K. Hale, Ordinary Differential Equations, Pure and Applied Mathematics, Vol. XXI, Wiley-Interscience, New York, 1969.  Google Scholar

[7]

J. C. Lillo, Approximate similarity and almost periodic matrices, Proc. Amer. Math. Soc., 12 (1961), 400-407.  Google Scholar

[8]

R. Ortega and M. Tarallo, Almost periodic equations and conditions of Ambrosetti-Prodi type, Math. Proc. Camb. Phil. Soc., 135 (2003), 239-254. doi: 10.1017/S0305004103006662.  Google Scholar

[9]

K. J. Palmer, On bounded solutions of almost periodic linear differential systems, J. Math. Anal. Appl., 103 (1984), 16-25. doi: 10.1016/0022-247X(84)90152-5.  Google Scholar

[10]

R. J. Sacker and G. R. Sell, Existence of dichotomies and invariant splittings for linear differential systems. I, J. Differential Equations, 15 (1974), 429-458. doi: 10.1016/0022-0396(74)90067-9.  Google Scholar

[11]

R. J. Sacker and G. R. Sell, Existence of dichotomies and invariant splittings for linear differential systems. II, J. Differential Equations, 22 (1976), 478-496. doi: 10.1016/0022-0396(76)90042-5.  Google Scholar

[12]

R. J. Sacker and G. R. Sell, Existence of dichotomies and invariant splittings for linear differential systems. III, J. Differential Equations, 22 (1976), 497-522. doi: 10.1016/0022-0396(76)90043-7.  Google Scholar

[13]

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.  Google Scholar

[14]

M. Tarallo, Fredholm's alternative for a class of almost periodic linear systems, Discrete Contin. Dyn. Syst., 32 (2012), 2301-2313. doi: 10.3934/dcds.2012.32.2301.  Google Scholar

[15]

M. Tarallo, The Favard separation condition as a purely dimensional fact, J. Dyn. Diff. Equations, 25 (2013), 291-304. doi: 10.1007/s10884-013-9309-2.  Google Scholar

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