# American Institute of Mathematical Sciences

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

## Recurrent equations with sign and Fredholm alternative

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.
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., (). [2] J. Campos and M. Tarallo, Almost automorphic linear dynamics by Favard theory,, J. Differential Equations, 256 (2014), 1350. doi: 10.1016/j.jde.2013.10.018. [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. [4] H. Dym, Linear Algebra in Action,, Graduate Studies in Mathematics, (2007). [5] J. Favard, Sur les equations différentielles linéairesà coefficients presque-périodiques,, Acta Math. , 51 (1928), 31. doi: 10.1007/BF02545660. [6] J. K. Hale, Ordinary Differential Equations,, Pure and Applied Mathematics, (1969). [7] J. C. Lillo, Approximate similarity and almost periodic matrices,, Proc. Amer. Math. Soc., 12 (1961), 400. [8] R. Ortega and M. Tarallo, Almost periodic equations and conditions of Ambrosetti-Prodi type,, Math. Proc. Camb. Phil. Soc., 135 (2003), 239. doi: 10.1017/S0305004103006662. [9] K. J. Palmer, On bounded solutions of almost periodic linear differential systems,, J. Math. Anal. Appl., 103 (1984), 16. doi: 10.1016/0022-247X(84)90152-5. [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. doi: 10.1016/0022-0396(74)90067-9. [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. doi: 10.1016/0022-0396(76)90042-5. [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. doi: 10.1016/0022-0396(76)90043-7. [13] R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems,, J. Differential Equations, 27 (1978), 320. doi: 10.1016/0022-0396(78)90057-8. [14] M. Tarallo, Fredholm's alternative for a class of almost periodic linear systems,, Discrete Contin. Dyn. Syst., 32 (2012), 2301. doi: 10.3934/dcds.2012.32.2301. [15] M. Tarallo, The Favard separation condition as a purely dimensional fact,, J. Dyn. Diff. Equations, 25 (2013), 291. doi: 10.1007/s10884-013-9309-2.

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##### References:
 [1] J. Campos, R. Obaya and M. Tarallo, Favard theory for the adjoint equation and Fredholm Alternative,, preprint., (). [2] J. Campos and M. Tarallo, Almost automorphic linear dynamics by Favard theory,, J. Differential Equations, 256 (2014), 1350. doi: 10.1016/j.jde.2013.10.018. [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. [4] H. Dym, Linear Algebra in Action,, Graduate Studies in Mathematics, (2007). [5] J. Favard, Sur les equations différentielles linéairesà coefficients presque-périodiques,, Acta Math. , 51 (1928), 31. doi: 10.1007/BF02545660. [6] J. K. Hale, Ordinary Differential Equations,, Pure and Applied Mathematics, (1969). [7] J. C. Lillo, Approximate similarity and almost periodic matrices,, Proc. Amer. Math. Soc., 12 (1961), 400. [8] R. Ortega and M. Tarallo, Almost periodic equations and conditions of Ambrosetti-Prodi type,, Math. Proc. Camb. Phil. Soc., 135 (2003), 239. doi: 10.1017/S0305004103006662. [9] K. J. Palmer, On bounded solutions of almost periodic linear differential systems,, J. Math. Anal. Appl., 103 (1984), 16. doi: 10.1016/0022-247X(84)90152-5. [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. doi: 10.1016/0022-0396(74)90067-9. [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. doi: 10.1016/0022-0396(76)90042-5. [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. doi: 10.1016/0022-0396(76)90043-7. [13] R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems,, J. Differential Equations, 27 (1978), 320. doi: 10.1016/0022-0396(78)90057-8. [14] M. Tarallo, Fredholm's alternative for a class of almost periodic linear systems,, Discrete Contin. Dyn. Syst., 32 (2012), 2301. doi: 10.3934/dcds.2012.32.2301. [15] M. Tarallo, The Favard separation condition as a purely dimensional fact,, J. Dyn. Diff. Equations, 25 (2013), 291. doi: 10.1007/s10884-013-9309-2.
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