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

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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

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

Received March 2011 Revised May 2011 Published February 2012

Abstract
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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).
[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.
[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.
[4]	A. M. Fink, "Almost Periodic Differential Equations,", Lecture Notes in Mathematics, (1974).
[5]	J. K. Hale, "Ordinary Differential Equations,", Pure and Applied Mathematics, (1969).
[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.
[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.
[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.
[9]	K. J. Palmer, Exponential dichotomies and transversal homoclinic points,, J. Differential Equations, 55 (1984), 225. doi: 10.1016/0022-0396(84)90082-2.
[10]	K. J. Palmer, Exponential dichotomies and Fredholm operators,, Proc. Amer. Mat. Soc., 104 (1988), 149. 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. doi: 10.1016/0022-0396(88)90118-0.
[12]	M. Tarallo, Module containment property for linear equations,, J. Differential Equations, 224 (2008), 52. doi: 10.1016/j.jde.2007.10.006.
[13]	V. V. Žhikov and B. M. Levitan, Favard theory,, Uspehi Mat. Nauk, 32 (1977), 123.

show all references

References:

[1]	W. A. Coppel, "Dichotomies in Stability Theory,", Lecture Notes in Mathematics, (1978).
[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.
[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.
[4]	A. M. Fink, "Almost Periodic Differential Equations,", Lecture Notes in Mathematics, (1974).
[5]	J. K. Hale, "Ordinary Differential Equations,", Pure and Applied Mathematics, (1969).
[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.
[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.
[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.
[9]	K. J. Palmer, Exponential dichotomies and transversal homoclinic points,, J. Differential Equations, 55 (1984), 225. doi: 10.1016/0022-0396(84)90082-2.
[10]	K. J. Palmer, Exponential dichotomies and Fredholm operators,, Proc. Amer. Mat. Soc., 104 (1988), 149. 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. doi: 10.1016/0022-0396(88)90118-0.
[12]	M. Tarallo, Module containment property for linear equations,, J. Differential Equations, 224 (2008), 52. doi: 10.1016/j.jde.2007.10.006.
[13]	V. V. Žhikov and B. M. Levitan, Favard theory,, Uspehi Mat. Nauk, 32 (1977), 123.

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