# American Institute of Mathematical Sciences

• 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

 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.
Citation: Massimo Tarallo. Fredholm's alternative for a class of almost periodic linear systems. Discrete and 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. [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. [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. [4] A. M. Fink, "Almost Periodic Differential Equations," Lecture Notes in Mathematics, Vol. 377, Springer-Verlag, Berlin-New York, 1974. [5] J. K. Hale, "Ordinary Differential Equations," Pure and Applied Mathematics, Vol. XXI, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1969. [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. [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. [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. [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. [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. [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. [12] M. Tarallo, Module containment property for linear equations, J. Differential Equations, 224 (2008), 52-60. doi: 10.1016/j.jde.2007.10.006. [13] V. V. Žhikov and B. M. Levitan, Favard theory, Uspehi Mat. Nauk, 32 (1977), 123-171, 263.

show all references

##### References:
 [1] W. A. Coppel, "Dichotomies in Stability Theory," Lecture Notes in Mathematics, Vol. 629, Springer-Verlag, Berlin-New York, 1978. [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. [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. [4] A. M. Fink, "Almost Periodic Differential Equations," Lecture Notes in Mathematics, Vol. 377, Springer-Verlag, Berlin-New York, 1974. [5] J. K. Hale, "Ordinary Differential Equations," Pure and Applied Mathematics, Vol. XXI, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1969. [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. [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. [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. [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. [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. [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. [12] M. Tarallo, Module containment property for linear equations, J. Differential Equations, 224 (2008), 52-60. doi: 10.1016/j.jde.2007.10.006. [13] V. V. Žhikov and B. M. Levitan, Favard theory, Uspehi Mat. Nauk, 32 (1977), 123-171, 263.
 [1] Juan Campos, Rafael Obaya, Massimo Tarallo. Favard theory and fredholm alternative for disconjugate recurrent second order equations. Communications on Pure and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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] Hailong Zhu, Jifeng Chu, Weinian Zhang. Mean-square almost automorphic solutions for stochastic differential equations with hyperbolicity. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 1935-1953. doi: 10.3934/dcds.2018078 [16] M. Eller. On boundary regularity of solutions to Maxwell's equations with a homogeneous conservative boundary condition. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 473-481. doi: 10.3934/dcdss.2009.2.473 [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 and 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 and 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 and 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 and Continuous Dynamical Systems, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277

2021 Impact Factor: 1.588