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

Recurrent equations with sign and Fredholm alternative

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.
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.

show all references

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.

[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 - A, 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 - A, 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 - A, 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 - A, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703

[6]

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

[7]

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

[8]

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

[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]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

Paolo Perfetti. Hamiltonian equations on $\mathbb{T}^\infty$ and almost-periodic solutions. Conference Publications, 2001, 2001 (Special) : 303-309. doi: 10.3934/proc.2001.2001.303

[18]

Yoshihiro Hamaya. Stability properties and existence of almost periodic solutions of volterra difference equations. Conference Publications, 2009, 2009 (Special) : 315-321. doi: 10.3934/proc.2009.2009.315

[19]

Bin Chen, Xiongping Dai. On uniformly recurrent motions of topological semigroup actions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 2931-2944. doi: 10.3934/dcds.2016.36.2931

[20]

Yuanhong Chen, Chao Ma, Jun Wu. Moving recurrent properties for the doubling map on the unit interval. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 2969-2979. doi: 10.3934/dcds.2016.36.2969

2017 Impact Factor: 0.561

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]