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July  2017, 16(4): 1199-1232. doi: 10.3934/cpaa.2017059

Favard theory and fredholm alternative for disconjugate recurrent second order equations

Juan Campos 1, , Rafael Obaya 2, and  Massimo Tarallo 3,

1. 

Universidad de Granada, Campus Fuentenueva, 18071 Granada, Spain
2. 

Universidad de Valladolid, Paseo del Cauce s/n, 47011 Valladolid, Spain
3. 

Università di Milano, Via Saldini 50,20133 Milano, Italy

Received  June 2016 Revised  March 2017 Published  April 2017

Fund Project: This work was partially supported by MEC and FEDER MTM2014-53406-R, MTM2015-66330-P), Junta de Andalucía (FQM-954), MIUR (PRIN2012-201274FYK7) and EC (H2020-MSCA-ITN-2014).

We discuss the existence of a Fredholm–type Alternative for a recurrent second order linear equation, which is disconjugate in a strong sense. The basic result is about bounded solutions of equations with bounded coefficients: it depends on kinematic similarities that allow to reduce the problem to a pair of very simple normal forms. Then the result is specialized to recurrent equations, by means of Favard theory.

Keywords: Recurrent and almost periodic linear equations, bounded and recurrent solutions, spectral theory, Favard theory, Fredholm Alternative.
Mathematics Subject Classification: Primary: 34C27; Secondary: 34B05, 34A30.
Citation: 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
References:
[1]

J. Campos, R. Obaya and M. Tarallo, Favard theory for the adjoint equation and Fredholm Alternative, J. Differential Equations, 262 (2017), 749-802.  doi: 10.1016/j.jde.2016.09.041.  Google Scholar

[2]

J. Campos, R. Obaya and M. Tarallo, Recurrent equations with sign and Fredholm Alternative, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 959-977.  doi: 10.3934/dcdss.2016036.  Google Scholar

[3]

J. Campos and M. Tarallo, Almost automorphic linear dynamics by Favard theory, J. Differential Equations, 256 (2014), 1350-1367.  doi: 10.1016/j.jde.2013.10.018.  Google Scholar

[4]

P. Cieutat and A. Haraux, Exponential decay and existence of almost periodic solutions for some linear forced differential equations, Port. Math., 59 (2002), 141-159.   Google Scholar

[5]

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

[6]

I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.  doi: 10.1016/0022-247X(74)90025-0.  Google Scholar

[7]

J. Favard, Sur les equations différentielles linéairesà coefficients presque-périodiques, (French) [On the linear differential equations with almost periodic coefficients], Acta Math., 51 (1927), 31-81.  doi: 10.1007/BF02545660.  Google Scholar

[8]

J. K. Hale, Ordinary Differential Equations, in Pure and Applied Mathematics, Vol. ⅩⅪ, Wiley-Interscience, New York, 1969.  Google Scholar

[9]

R. A. Johnson, Minimal functions with unbounded integral, Israel J. Math., 31 (1978), 133-141.  doi: 10.1007/BF02760544.  Google Scholar

[10]

V. V. Kozlov, On a problem by Poincaré, J. Appl. Math. Mech., 40 (1976), 326-329.  doi: 10.1016/0021-8928(76)90070-8.  Google Scholar

[11]

J. C. Lillo, Approximate similarity and almost periodic matrices, Proc. Amer. Math. Soc., 12 (1961), 400-407.  doi: 10.2307/2034205.  Google Scholar

[12]

J. K. Massera and J. J. Schäffer, Linear Differential Equations and Function Spaces, in Pure and Applied Mathematics, Vol. ⅩⅪ, Academic Press, New York, 1966.  Google Scholar

[13]

N. G. Moshchevitin, Recurrence of an integral of a smooth conditionally periodic function, Math. Notes, 63 (1998), 648-657.  doi: 10.1007/BF02312847.  Google Scholar

[14]

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

[15]

K. J. Palmer, On bounded solutions of almost periodic linear differential systems, J. Math. Anal. Appl., 103 (1984), 16-25.  doi: 10.1016/0022-247X(84)90152-5.  Google Scholar

[16]

H. Poincaré, Sur le séries trigonométriques, C.R. Acad. Sci., 101 (1885), 1131-1134.   Google Scholar

[17]

R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems, J. Differential Equations, 27 (1978), 320-358.  doi: 10.1016/0022-0396(78)90057-8.  Google Scholar

[18]

R. J. Sacker and G. R. Sell, Existence of dichotomies and invariant splittings for linear differential systems. Ⅰ., J. Differential Equations, 15 (1974), 429-458.  doi: 10.1016/0022-0396(74)90067-9.  Google Scholar

[19]

R. J. Sacker and G. R. Sell, Existence of dichotomies and invariant splittings for linear differential systems. Ⅲ., J. Differential Equations, 22 (1976), 497-522.  doi: 10.1016/0022-0396(76)90043-7.  Google Scholar

[20]

B. A. Ščerbakov, The comparability of the motions of dynamical systems with regard to the nature of their recurrence, Diff. Uravnenija, 11 (1975), 1246-1255.   Google Scholar

[21]

G. R. Sell, Topological dynamics and ordinary differential equations, in Van Nostrand Reinhold, No. 33,1971.  Google Scholar

[22]

S. Siegmund, Dichotomy spectrum for nonautonomous differential equations, J. Dynam. Differential Equations, 14 (2002), 243-258.  doi: 10.1023/A:1012919512399.  Google Scholar

[23]

M. Tarallo, Fredholm's alternative for a class of almost periodic linear systems, Discrete Contin. Dyn. Syst., 32 (2012), 2301-2313.  doi: 10.3934/dcds.2012.32.2301.  Google Scholar

[24]

M. Tarallo, The Favard separation condition for almost periodic linear systems, J. Dyn. Diff. Equations, 25 (2013), 291-304.  doi: 10.1007/s10884-013-9309-2.  Google Scholar

[25]

V. V. Zhikov and B. M. Levitan, Favard theory, Russian Math. Surveys, 32 (1977), 129-180.  doi: 10.1070/RM1977v032n02ABEH001621.  Google Scholar

show all references

References:
[1]

J. Campos, R. Obaya and M. Tarallo, Favard theory for the adjoint equation and Fredholm Alternative, J. Differential Equations, 262 (2017), 749-802.  doi: 10.1016/j.jde.2016.09.041.  Google Scholar

[2]

J. Campos, R. Obaya and M. Tarallo, Recurrent equations with sign and Fredholm Alternative, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 959-977.  doi: 10.3934/dcdss.2016036.  Google Scholar

[3]

J. Campos and M. Tarallo, Almost automorphic linear dynamics by Favard theory, J. Differential Equations, 256 (2014), 1350-1367.  doi: 10.1016/j.jde.2013.10.018.  Google Scholar

[4]

P. Cieutat and A. Haraux, Exponential decay and existence of almost periodic solutions for some linear forced differential equations, Port. Math., 59 (2002), 141-159.   Google Scholar

[5]

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

[6]

I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.  doi: 10.1016/0022-247X(74)90025-0.  Google Scholar

[7]

J. Favard, Sur les equations différentielles linéairesà coefficients presque-périodiques, (French) [On the linear differential equations with almost periodic coefficients], Acta Math., 51 (1927), 31-81.  doi: 10.1007/BF02545660.  Google Scholar

[8]

J. K. Hale, Ordinary Differential Equations, in Pure and Applied Mathematics, Vol. ⅩⅪ, Wiley-Interscience, New York, 1969.  Google Scholar

[9]

R. A. Johnson, Minimal functions with unbounded integral, Israel J. Math., 31 (1978), 133-141.  doi: 10.1007/BF02760544.  Google Scholar

[10]

V. V. Kozlov, On a problem by Poincaré, J. Appl. Math. Mech., 40 (1976), 326-329.  doi: 10.1016/0021-8928(76)90070-8.  Google Scholar

[11]

J. C. Lillo, Approximate similarity and almost periodic matrices, Proc. Amer. Math. Soc., 12 (1961), 400-407.  doi: 10.2307/2034205.  Google Scholar

[12]

J. K. Massera and J. J. Schäffer, Linear Differential Equations and Function Spaces, in Pure and Applied Mathematics, Vol. ⅩⅪ, Academic Press, New York, 1966.  Google Scholar

[13]

N. G. Moshchevitin, Recurrence of an integral of a smooth conditionally periodic function, Math. Notes, 63 (1998), 648-657.  doi: 10.1007/BF02312847.  Google Scholar

[14]

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

[15]

K. J. Palmer, On bounded solutions of almost periodic linear differential systems, J. Math. Anal. Appl., 103 (1984), 16-25.  doi: 10.1016/0022-247X(84)90152-5.  Google Scholar

[16]

H. Poincaré, Sur le séries trigonométriques, C.R. Acad. Sci., 101 (1885), 1131-1134.   Google Scholar

[17]

R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems, J. Differential Equations, 27 (1978), 320-358.  doi: 10.1016/0022-0396(78)90057-8.  Google Scholar

[18]

R. J. Sacker and G. R. Sell, Existence of dichotomies and invariant splittings for linear differential systems. Ⅰ., J. Differential Equations, 15 (1974), 429-458.  doi: 10.1016/0022-0396(74)90067-9.  Google Scholar

[19]

R. J. Sacker and G. R. Sell, Existence of dichotomies and invariant splittings for linear differential systems. Ⅲ., J. Differential Equations, 22 (1976), 497-522.  doi: 10.1016/0022-0396(76)90043-7.  Google Scholar

[20]

B. A. Ščerbakov, The comparability of the motions of dynamical systems with regard to the nature of their recurrence, Diff. Uravnenija, 11 (1975), 1246-1255.   Google Scholar

[21]

G. R. Sell, Topological dynamics and ordinary differential equations, in Van Nostrand Reinhold, No. 33,1971.  Google Scholar

[22]

S. Siegmund, Dichotomy spectrum for nonautonomous differential equations, J. Dynam. Differential Equations, 14 (2002), 243-258.  doi: 10.1023/A:1012919512399.  Google Scholar

[23]

M. Tarallo, Fredholm's alternative for a class of almost periodic linear systems, Discrete Contin. Dyn. Syst., 32 (2012), 2301-2313.  doi: 10.3934/dcds.2012.32.2301.  Google Scholar

[24]

M. Tarallo, The Favard separation condition for almost periodic linear systems, J. Dyn. Diff. Equations, 25 (2013), 291-304.  doi: 10.1007/s10884-013-9309-2.  Google Scholar

[25]

V. V. Zhikov and B. M. Levitan, Favard theory, Russian Math. Surveys, 32 (1977), 129-180.  doi: 10.1070/RM1977v032n02ABEH001621.  Google Scholar

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