American Institute of Mathematical Sciences

Advanced
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

[1]

Juan Campos, Rafael Obaya, Massimo Tarallo. Favard theory and fredholm alternative for disconjugate recurrent second order equations. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1199-1232. doi: 10.3934/cpaa.2017059
[2]

Massimo Tarallo. Fredholm's alternative for a class of almost periodic linear systems. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2301-2313. doi: 10.3934/dcds.2012.32.2301
[3]

Koichiro Naito. Recurrent dimensions of quasi-periodic solutions for nonlinear evolution equations II: Gaps of dimensions and Diophantine conditions. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 449-488. doi: 10.3934/dcds.2004.11.449
[4]

Tomás Caraballo, David Cheban. Almost periodic and almost automorphic solutions of linear differential equations. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1857-1882. doi: 10.3934/dcds.2013.33.1857
[5]

Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete & Continuous Dynamical Systems, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703
[6]

Mo Chen. Recurrent solutions of the Schrödinger-KdV system with boundary forces. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 5149-5170. doi: 10.3934/dcdsb.2020337
[7]

Tomás Caraballo, David Cheban. Almost periodic and asymptotically almost periodic solutions of Liénard equations. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 703-717. doi: 10.3934/dcdsb.2011.16.703
[8]

Gaston N'Guerekata. On weak-almost periodic mild solutions of some linear abstract differential equations. Conference Publications, 2003, 2003 (Special) : 672-677. doi: 10.3934/proc.2003.2003.672
[9]

Rémi Leclercq. Spectral invariants in Lagrangian Floer theory. Journal of Modern Dynamics, 2008, 2 (2) : 249-286. doi: 10.3934/jmd.2008.2.249
[10]

Barry Simon. Equilibrium measures and capacities in spectral theory. Inverse Problems & Imaging, 2007, 1 (4) : 713-772. doi: 10.3934/ipi.2007.1.713
[11]

Denis Pennequin. Existence of almost periodic solutions of discrete time equations. Discrete & Continuous Dynamical Systems, 2001, 7 (1) : 51-60. doi: 10.3934/dcds.2001.7.51
[12]

Xianhua Huang. Almost periodic and periodic solutions of certain dissipative delay differential equations. Conference Publications, 1998, 1998 (Special) : 301-313. doi: 10.3934/proc.1998.1998.301
[13]

Nguyen Minh Man, Nguyen Van Minh. On the existence of quasi periodic and almost periodic solutions of neutral functional differential equations. Communications on Pure & Applied Analysis, 2004, 3 (2) : 291-300. doi: 10.3934/cpaa.2004.3.291
[14]

Yong Li, Zhenxin Liu, Wenhe Wang. Almost periodic solutions and stable solutions for stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5927-5944. doi: 10.3934/dcdsb.2019113
[15]

Robert Carlson. Spectral theory for nonconservative transmission line networks. Networks & Heterogeneous Media, 2011, 6 (2) : 257-277. doi: 10.3934/nhm.2011.6.257
[16]

Xiongping Dai, Yu Huang, Mingqing Xiao. Realization of joint spectral radius via Ergodic theory. Electronic Research Announcements, 2011, 18: 22-30. doi: 10.3934/era.2011.18.22
[17]

Álvaro Pelayo, San Vű Ngọc. First steps in symplectic and spectral theory of integrable systems. Discrete & Continuous Dynamical Systems, 2012, 32 (10) : 3325-3377. doi: 10.3934/dcds.2012.32.3325
[18]

Kung-Ching Chang, Xuefeng Wang, Xie Wu. On the spectral theory of positive operators and PDE applications. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3171-3200. doi: 10.3934/dcds.2020054
[19]

Leonid Golinskii, Mikhail Kudryavtsev. An inverse spectral theory for finite CMV matrices. Inverse Problems & Imaging, 2010, 4 (1) : 93-110. doi: 10.3934/ipi.2010.4.93
[20]

Arunima Bhattacharya, Micah Warren. $ C^{2, \alpha} $ estimates for solutions to almost Linear elliptic equations. Communications on Pure & Applied Analysis, 2021, 20 (4) : 1363-1383. doi: 10.3934/cpaa.2021024

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (73)
  • HTML views (1)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences