American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Well-posedness and stabilization of an Euler-Bernoulli equation with a localized nonlinear dissipation involving the $p$-Laplacian
  • DCDS Home
  • This Issue
  • Next Article
    Heat Kernel estimates for some elliptic operators with unbounded diffusion coefficients
June  2012, 32(6): 2301-2313. doi: 10.3934/dcds.2012.32.2301

Fredholm's alternative for a class of almost periodic linear systems

Massimo Tarallo 1,

1. 

Università degli Studi di Milano, Via C. Saldini 50, Milano, I–20133, Italy

Received  March 2011 Revised  May 2011 Published  February 2012

A Fredholm alternative is proposed for linear almost periodic equations which satisfy the Favard separation condition. The alternative is then tested in the special case, where all the solutions of the homogeneous part of the equation are bounded.
Keywords: Favard condition., Almost periodic differential equations, bounded solutions, Fredholm's alternative.
Mathematics Subject Classification: Primary: 34A30; Secondary: 34C2.
Citation: 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
References:
[1]

W. A. Coppel, "Dichotomies in Stability Theory," Lecture Notes in Mathematics, Vol. 629, Springer-Verlag, Berlin-New York, 1978.  Google Scholar

[2]

J. Favard, Sur les équations différentielles linéaires à coefficients presque-périodiques, (French) [On the linear differential equations with almost peridoic coefficients], Acta Math., 51 (1928), 31-81. doi: 10.1007/BF02545660.  Google Scholar

[3]

J. Favard, Sur certains systèmes différentiels scalaires linéaires et homogénes à coefficients presque-périodiques, (French) [On some scalar linear homogeneous differential systems with almost periodic coefficients], Ann. Mat. Pura Appl. (4), 61 (1963), 297-316.  Google Scholar

[4]

A. M. Fink, "Almost Periodic Differential Equations," Lecture Notes in Mathematics, Vol. 377, Springer-Verlag, Berlin-New York, 1974.  Google Scholar

[5]

J. K. Hale, "Ordinary Differential Equations," Pure and Applied Mathematics, Vol. XXI, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1969.  Google Scholar

[6]

R. A. Johnson, A linear, almost periodic equation with an almost automorphic solution, Proc. Amer. Math. Soc., 82 (1981), 199-205. doi: 10.1090/S0002-9939-1981-0609651-0.  Google Scholar

[7]

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

[8]

R. Ortega and M. Tarallo, Almost periodic linear differential equations with non-separated solutions, J. Funct. Analysis, 237 (2006), 402-426. doi: 10.1016/j.jfa.2006.03.027.  Google Scholar

[9]

K. J. Palmer, Exponential dichotomies and transversal homoclinic points, J. Differential Equations, 55 (1984), 225-256. doi: 10.1016/0022-0396(84)90082-2.  Google Scholar

[10]

K. J. Palmer, Exponential dichotomies and Fredholm operators, Proc. Amer. Mat. Soc., 104 (1988), 149-156. doi: 10.1090/S0002-9939-1988-0958058-1.  Google Scholar

[11]

H. M. Rodrigues and M. Silveira, On the relationship between exponential dichotomies and Fredholm alternative, J. Differential Equations, 73 (1988), 78-81. doi: 10.1016/0022-0396(88)90118-0.  Google Scholar

[12]

M. Tarallo, Module containment property for linear equations, J. Differential Equations, 224 (2008), 52-60. doi: 10.1016/j.jde.2007.10.006.  Google Scholar

[13]

V. V. Žhikov and B. M. Levitan, Favard theory, Uspehi Mat. Nauk, 32 (1977), 123-171, 263.  Google Scholar

show all references

References:
[1]

W. A. Coppel, "Dichotomies in Stability Theory," Lecture Notes in Mathematics, Vol. 629, Springer-Verlag, Berlin-New York, 1978.  Google Scholar

[2]

J. Favard, Sur les équations différentielles linéaires à coefficients presque-périodiques, (French) [On the linear differential equations with almost peridoic coefficients], Acta Math., 51 (1928), 31-81. doi: 10.1007/BF02545660.  Google Scholar

[3]

J. Favard, Sur certains systèmes différentiels scalaires linéaires et homogénes à coefficients presque-périodiques, (French) [On some scalar linear homogeneous differential systems with almost periodic coefficients], Ann. Mat. Pura Appl. (4), 61 (1963), 297-316.  Google Scholar

[4]

A. M. Fink, "Almost Periodic Differential Equations," Lecture Notes in Mathematics, Vol. 377, Springer-Verlag, Berlin-New York, 1974.  Google Scholar

[5]

J. K. Hale, "Ordinary Differential Equations," Pure and Applied Mathematics, Vol. XXI, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1969.  Google Scholar

[6]

R. A. Johnson, A linear, almost periodic equation with an almost automorphic solution, Proc. Amer. Math. Soc., 82 (1981), 199-205. doi: 10.1090/S0002-9939-1981-0609651-0.  Google Scholar

[7]

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

[8]

R. Ortega and M. Tarallo, Almost periodic linear differential equations with non-separated solutions, J. Funct. Analysis, 237 (2006), 402-426. doi: 10.1016/j.jfa.2006.03.027.  Google Scholar

[9]

K. J. Palmer, Exponential dichotomies and transversal homoclinic points, J. Differential Equations, 55 (1984), 225-256. doi: 10.1016/0022-0396(84)90082-2.  Google Scholar

[10]

K. J. Palmer, Exponential dichotomies and Fredholm operators, Proc. Amer. Mat. Soc., 104 (1988), 149-156. doi: 10.1090/S0002-9939-1988-0958058-1.  Google Scholar

[11]

H. M. Rodrigues and M. Silveira, On the relationship between exponential dichotomies and Fredholm alternative, J. Differential Equations, 73 (1988), 78-81. doi: 10.1016/0022-0396(88)90118-0.  Google Scholar

[12]

M. Tarallo, Module containment property for linear equations, J. Differential Equations, 224 (2008), 52-60. doi: 10.1016/j.jde.2007.10.006.  Google Scholar

[13]

V. V. Žhikov and B. M. Levitan, Favard theory, Uspehi Mat. Nauk, 32 (1977), 123-171, 263.  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]

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
[3]

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
[4]

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
[5]

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
[6]

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
[7]

Jean Mawhin, James R. Ward Jr. Guiding-like functions for periodic or bounded solutions of ordinary differential equations. Discrete & Continuous Dynamical Systems, 2002, 8 (1) : 39-54. doi: 10.3934/dcds.2002.8.39
[8]

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
[9]

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
[10]

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
[11]

Jehad O. Alzabut. A necessary and sufficient condition for the existence of periodic solutions of linear impulsive differential equations with distributed delay. Conference Publications, 2007, 2007 (Special) : 35-43. doi: 10.3934/proc.2007.2007.35
[12]

Rui Zhang, Yong-Kui Chang, G. M. N'Guérékata. Weighted pseudo almost automorphic mild solutions to semilinear integral equations with $S^{p}$-weighted pseudo almost automorphic coefficients. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5525-5537. doi: 10.3934/dcds.2013.33.5525
[13]

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
[14]

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
[15]

M. Eller. On boundary regularity of solutions to Maxwell's equations with a homogeneous conservative boundary condition. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 473-481. doi: 10.3934/dcdss.2009.2.473
[16]

Hailong Zhu, Jifeng Chu, Weinian Zhang. Mean-square almost automorphic solutions for stochastic differential equations with hyperbolicity. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 1935-1953. doi: 10.3934/dcds.2018078
[17]

Yuan Guo, Xiaofei Gao, Desheng Li. Structure of the set of bounded solutions for a class of nonautonomous second order differential equations. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1607-1616. doi: 10.3934/cpaa.2010.9.1607
[18]

Alessandro Fonda, Fabio Zanolin. Bounded solutions of nonlinear second order ordinary differential equations. Discrete & Continuous Dynamical Systems, 1998, 4 (1) : 91-98. doi: 10.3934/dcds.1998.4.91
[19]

Peter Giesl, Martin Rasmussen. A note on almost periodic variational equations. Communications on Pure & Applied Analysis, 2011, 10 (3) : 983-994. doi: 10.3934/cpaa.2011.10.983
[20]

Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (58)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences