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A critical exponent of Joseph-Lundgren type for an Hénon equation on the hyperbolic space
Favard theory and fredholm alternative for disconjugate recurrent second order equations
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 |
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.
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. |
[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. |
[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. |
[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.
|
[5] |
W. A. Coppel, Dichotomies in Stability Theory, in Lecture Notes in Mathematics, Vol. 629, Springer-Verlag, New York, 1978. |
[6] |
I. Ekeland,
On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.
doi: 10.1016/0022-247X(74)90025-0. |
[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. |
[8] |
J. K. Hale, Ordinary Differential Equations, in Pure and Applied Mathematics, Vol. ⅩⅪ, Wiley-Interscience, New York, 1969. |
[9] |
R. A. Johnson,
Minimal functions with unbounded integral, Israel J. Math., 31 (1978), 133-141.
doi: 10.1007/BF02760544. |
[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. |
[11] |
J. C. Lillo,
Approximate similarity and almost periodic matrices, Proc. Amer. Math. Soc., 12 (1961), 400-407.
doi: 10.2307/2034205. |
[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. |
[13] |
N. G. Moshchevitin,
Recurrence of an integral of a smooth conditionally periodic function, Math. Notes, 63 (1998), 648-657.
doi: 10.1007/BF02312847. |
[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. |
[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. |
[16] |
H. Poincaré,
Sur le séries trigonométriques, C.R. Acad. Sci., 101 (1885), 1131-1134.
|
[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. |
[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. |
[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. |
[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.
|
[21] |
G. R. Sell, Topological dynamics and ordinary differential equations, in Van Nostrand Reinhold, No. 33,1971. |
[22] |
S. Siegmund,
Dichotomy spectrum for nonautonomous differential equations, J. Dynam. Differential Equations, 14 (2002), 243-258.
doi: 10.1023/A:1012919512399. |
[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. |
[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. |
[25] |
V. V. Zhikov and B. M. Levitan,
Favard theory, Russian Math. Surveys, 32 (1977), 129-180.
doi: 10.1070/RM1977v032n02ABEH001621. |
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. |
[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. |
[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. |
[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.
|
[5] |
W. A. Coppel, Dichotomies in Stability Theory, in Lecture Notes in Mathematics, Vol. 629, Springer-Verlag, New York, 1978. |
[6] |
I. Ekeland,
On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.
doi: 10.1016/0022-247X(74)90025-0. |
[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. |
[8] |
J. K. Hale, Ordinary Differential Equations, in Pure and Applied Mathematics, Vol. ⅩⅪ, Wiley-Interscience, New York, 1969. |
[9] |
R. A. Johnson,
Minimal functions with unbounded integral, Israel J. Math., 31 (1978), 133-141.
doi: 10.1007/BF02760544. |
[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. |
[11] |
J. C. Lillo,
Approximate similarity and almost periodic matrices, Proc. Amer. Math. Soc., 12 (1961), 400-407.
doi: 10.2307/2034205. |
[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. |
[13] |
N. G. Moshchevitin,
Recurrence of an integral of a smooth conditionally periodic function, Math. Notes, 63 (1998), 648-657.
doi: 10.1007/BF02312847. |
[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. |
[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. |
[16] |
H. Poincaré,
Sur le séries trigonométriques, C.R. Acad. Sci., 101 (1885), 1131-1134.
|
[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. |
[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. |
[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. |
[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.
|
[21] |
G. R. Sell, Topological dynamics and ordinary differential equations, in Van Nostrand Reinhold, No. 33,1971. |
[22] |
S. Siegmund,
Dichotomy spectrum for nonautonomous differential equations, J. Dynam. Differential Equations, 14 (2002), 243-258.
doi: 10.1023/A:1012919512399. |
[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. |
[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. |
[25] |
V. V. Zhikov and B. M. Levitan,
Favard theory, Russian Math. Surveys, 32 (1977), 129-180.
doi: 10.1070/RM1977v032n02ABEH001621. |
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