American Institute of Mathematical Sciences

Advanced
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

[1]

Juan Campos, Rafael Obaya, Massimo Tarallo. Recurrent equations with sign and Fredholm alternative. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 959-977. doi: 10.3934/dcdss.2016036
[2]

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
[3]

Koichiro Naito. Recurrent dimensions of quasi-periodic solutions for nonlinear evolution equations II: Gaps of dimensions and Diophantine conditions. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 449-488. doi: 10.3934/dcds.2004.11.449
[4]

Tomás Caraballo, David Cheban. Almost periodic and almost automorphic solutions of linear differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1857-1882. doi: 10.3934/dcds.2013.33.1857
[5]

Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703
[6]

Tomás Caraballo, David Cheban. Almost periodic and asymptotically almost periodic solutions of Liénard equations. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 703-717. doi: 10.3934/dcdsb.2011.16.703
[7]

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
[8]

Denis Pennequin. Existence of almost periodic solutions of discrete time equations. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 51-60. doi: 10.3934/dcds.2001.7.51
[9]

Rémi Leclercq. Spectral invariants in Lagrangian Floer theory. Journal of Modern Dynamics, 2008, 2 (2) : 249-286. doi: 10.3934/jmd.2008.2.249
[10]

Barry Simon. Equilibrium measures and capacities in spectral theory. Inverse Problems & Imaging, 2007, 1 (4) : 713-772. doi: 10.3934/ipi.2007.1.713
[11]

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
[12]

Nguyen Minh Man, Nguyen Van Minh. On the existence of quasi periodic and almost periodic solutions of neutral functional differential equations. Communications on Pure & Applied Analysis, 2004, 3 (2) : 291-300. doi: 10.3934/cpaa.2004.3.291
[13]

Yong Li, Zhenxin Liu, Wenhe Wang. Almost periodic solutions and stable solutions for stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5927-5944. doi: 10.3934/dcdsb.2019113
[14]

Robert Carlson. Spectral theory for nonconservative transmission line networks. Networks & Heterogeneous Media, 2011, 6 (2) : 257-277. doi: 10.3934/nhm.2011.6.257
[15]

Xiongping Dai, Yu Huang, Mingqing Xiao. Realization of joint spectral radius via Ergodic theory. Electronic Research Announcements, 2011, 18: 22-30. doi: 10.3934/era.2011.18.22
[16]

Álvaro Pelayo, San Vű Ngọc. First steps in symplectic and spectral theory of integrable systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3325-3377. doi: 10.3934/dcds.2012.32.3325
[17]

Leonid Golinskii, Mikhail Kudryavtsev. An inverse spectral theory for finite CMV matrices. Inverse Problems & Imaging, 2010, 4 (1) : 93-110. doi: 10.3934/ipi.2010.4.93
[18]

Bin Chen, Xiongping Dai. On uniformly recurrent motions of topological semigroup actions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 2931-2944. doi: 10.3934/dcds.2016.36.2931
[19]

Yuanhong Chen, Chao Ma, Jun Wu. Moving recurrent properties for the doubling map on the unit interval. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 2969-2979. doi: 10.3934/dcds.2016.36.2969
[20]

Ronald A. Knight. Compact minimal sets in continuous recurrent flows. Conference Publications, 1998, 1998 (Special) : 397-407. doi: 10.3934/proc.1998.1998.397

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (7)
  • HTML views (6)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences