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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 - A, 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, (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], 51 (1928), 31.  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], 61 (1963), 297.   Google Scholar

[4]

A. M. Fink, "Almost Periodic Differential Equations,", Lecture Notes in Mathematics, (1974).   Google Scholar

[5]

J. K. Hale, "Ordinary Differential Equations,", Pure and Applied Mathematics, (1969).   Google Scholar

[6]

R. A. Johnson, A linear, almost periodic equation with an almost automorphic solution,, Proc. Amer. Math. Soc., 82 (1981), 199.  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.  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.  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.  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.  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.  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.  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.   Google Scholar

show all references

References:
[1]

W. A. Coppel, "Dichotomies in Stability Theory,", Lecture Notes in Mathematics, (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], 51 (1928), 31.  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], 61 (1963), 297.   Google Scholar

[4]

A. M. Fink, "Almost Periodic Differential Equations,", Lecture Notes in Mathematics, (1974).   Google Scholar

[5]

J. K. Hale, "Ordinary Differential Equations,", Pure and Applied Mathematics, (1969).   Google Scholar

[6]

R. A. Johnson, A linear, almost periodic equation with an almost automorphic solution,, Proc. Amer. Math. Soc., 82 (1981), 199.  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.  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.  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.  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.  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.  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.  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.   Google Scholar

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