# American Institute of Mathematical Sciences

October  2017, 22(8): 3063-3077. doi: 10.3934/dcdsb.2017163

## Integral conditions for nonuniform $μ$-dichotomy on the half-line

 1 Departamento de Matemática, Universidade da Beira Interior, 6201-001 Covilh˜ã, Portugal 2 Department of Mathematics, "Politehnica" University of Timişoara, Victoriei Square 2,300006 Timişoara, Romania 3 Academy of Romanian Scientists, Independenţei 54,050094 Bucharest, Romania

Received  June 2016 Revised  January 2017 Published  June 2017

We give necessary integral conditions and sufficient ones for the existence of a general concept of $μ$-dichotomy for evolution operators defined on the half-line which includes as particular cases the well-known concepts of nonuniform exponential dichotomy and nonuniform polynomial dichotomy, and also contains new situations. Additionally, we consider an adapted notion of Lyapunov function and use our results to obtain necessary and sufficient conditions for the existence of nonuniform $μ$-dichotomies using these Lyapunov functions.

Citation: António J.G. Bento, Nicolae Lupa, Mihail Megan, César M. Silva. Integral conditions for nonuniform $μ$-dichotomy on the half-line. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3063-3077. doi: 10.3934/dcdsb.2017163
##### References:
 [1] M. G. Babuţia, M. Megan and I. -L. Popa, On $(h, k)$ -dichotomies for nonautonomous linear difference equations in Banach spaces, Int. J. Differ. Equ. , (2013), Art. ID 761680, 7 pages, URL http://dx.doi.org/10.1155/2013/761680 Google Scholar [2] L. Barreira, J. Chu and C. Valls, Lyapunov functions for general nonuniform dichotomies, Milan J. Math., 81 (2013), 153-169. doi: 10.1007/s00032-013-0198-y. Google Scholar [3] L. Barreira and C. Valls, Growth rates and nonuniform hyperbolicity, Discrete Contin. Dyn. Syst. , 22 (2008), 509–528, URL http://dx.doi.org/10.3934/dcds.2008.22.509 Google Scholar [4] L. Barreira and C. Valls, Stability of Nonautonomous Differential Equations, vol. 1926 of Lecture Notes in Mathematics, Springer, Berlin, 2008, URL http://dx.doi.org/10.1007/978-3-540-74775-8 doi: 10.1007/978-3-540-74775-8. Google Scholar [5] L. Barreira and C. Valls, Polynomial growth rates, Nonlinear Anal. , 71 (2009), 5208–5219, URL http://dx.doi.org/10.1016/j.na.2009.04.005 Google Scholar [6] L. Barreira and C. Valls, Quadratic Lyapunov functions and nonuniform exponential dichotomies, J. Differential Equations, 246 (2009), 1235–1263, URL http://dx.doi.org/10.1016/j.jde.2008.06.008Google Scholar [7] A. J. G. Bento and C. Silva, Nonautonomous equations, generalized dichotomies and stable manifolds, ArXiv e-prints. URL http://arxiv.org/abs/0905.4935Google Scholar [8] A. J. G. Bento and C. Silva, Stable manifolds for nonuniform polynomial dichotomies, J. Funct. Anal. , 257 (2009), 122–148, URL http://dx.doi.org/10.1016/j.jfa.2009.01.032Google Scholar [9] A. J. G. Bento and C. M. Silva, Stable manifolds for non-autonomous equations with nonuniform polynomial dichotomies, Q. J. Math. , 63 (2012), 275–308, URL http://dx.doi.org/10.1093/qmath/haq047Google Scholar [10] A. J. G. Bento and C. M. Silva, Generalized nonuniform dichotomies and local stable manifolds, J. Dynam. Differential Equations, 25 (2013), 1139–1158, URL http://dx.doi.org/10.1007/s10884-013-9331-4 Google Scholar [11] T. Burton and L. Hatvani, Stability theorems for nonautonomous functional-differential equations by Liapunov functionals, Tohoku Math. J. , (2) 41 (1989), 65–104, URL http://dx.doi.org/10.2748/tmj/1178227868 Google Scholar [12] T. A. Burton and L. Hatvani, On nonuniform asymptotic stability for nonautonomous functional-differential equations, Differential Integral Equations, 3 (1990), 285–293, URL http://projecteuclid.org/euclid.die/1371586144 Google Scholar [13] X. Chang, J. Zhang and J. Qin, Robustness of nonuniform $(μ, ν)$ -dichotomies in Banach spaces, J. Math. Anal. Appl. , 387 (2012), 582–594, URL http://dx.doi.org/10.1016/j.jmaa.2011.09.026Google Scholar [14] C. Chicone and Y. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations, vol. 70 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1999. URL http://dx.doi.org/10.1090/surv/070 doi: 10.1090/surv/070. Google Scholar [15] R. Datko, Uniform asymptotic stability of evolutionary processes in a Banach space, SIAM J. Math. Anal. , 3 (1972), 428–445, URL http://dx.doi.org/10.1137/0503042Google Scholar [16] L. Hatvani, On the asymptotic stability for nonautonomous functional differential equations by Lyapunov functionals, Trans. Amer. Math. Soc. , 354 (2002), 3555–3571, URL http://dx.doi.org/10.1090/S0002-9947-02-03029-5Google Scholar [17] T. Li, Die Stabilitätsfrage bei Differenzengleichungen, Acta Math. , 63 (1934), 99–141, URL http://dx.doi.org/10.1007/BF02547352Google Scholar [18] N. Lupa and M. Megan, Exponential dichotomies of evolution operators in Banach spaces, Monatsh. Math. , 174 (2014), 265–284, URL http://dx.doi.org/10.1007/s00605-013-0517-y Google Scholar [19] A. M. Lyapunov, The general problem of the stability of motion, Internat. J. Control, 55 (1992), 521–790, translated by A. T. Fuller from Édouard Davaux's French translation (1907) ´ of the 1892 Russian original, With an editorial (historical introduction) by Fuller, a biography of Lyapunov by V. I. Smirnov, and the bibliography of Lyapunov's works collected by J. F. Barrett, Lyapunov centenary issue. URL http://dx.doi.org/10.1080/00207179208934253Google Scholar [20] A. D. Maǐzel0, On stability of solutions of systems of differential equations, Ural. Politehn. Inst. Trudy, 51 (1954), 20-50. Google Scholar [21] M. Megan, On $(h,k)$ -dichotomy of evolution operators in Banach spaces, Dynam. Systems Appl., 5 (1996), 189-195. Google Scholar [22] M. Megan and C. Buşe, Dichotomies and Lyapunov functions in Banach spaces, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 37 (1993), 103-114. Google Scholar [23] Y. A. Mitropolsky, A. M. Samoilenko and V. L. Kulik, Dichotomies and Stability in Nonautonomous Linear Systems, vol. 14 of Stability and Control: Theory, Methods and Applications, Taylor & Francis, London, 2003. Google Scholar [24] R. Naulin and M. Pinto, Roughness of $(h, k)$ -dichotomies, J. Differential Equations, 118 (1995), 20–35, URL http://dx.doi.org/10.1006/jdeq.1995.1065Google Scholar [25] O. Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Z. , 32 (1930), 703–728, URL http://dx.doi.org/10.1007/BF01194662Google Scholar [26] Y. Pesin, Families of invariant manifolds that corresponding to nonzero characteristic exponents, Izv. Akad. Nauk SSSR Ser. Mat. , 40 (1976), 1332–1379, (Russian) English transl. Math. USSR-Izv. , 10 (1976), 1261–1305, URL http://dx.doi.org/10.1070/IM1976v010n06ABEH001835Google Scholar [27] Y. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Uspehi Mat. Nauk, 32 (1977), 55–112, (Russian) English transl. Russ. Math. Surv. , 32 (1977), 55–114, URL http://dx.doi.org/10.1070/RM1977v032n04ABEH001639Google Scholar [28] Y. Pesin, Geodesic flows in closed Riemannian manifolds without focal points, Izv. Akad. Nauk SSSR Ser. Mat. , 41 (1977), 1252–1288, (Russian) English transl. Math. USSR-Izv. , 41 (1977), 1195–1228, URL http://dx.doi.org/doi:10.1070/IM1977v011n06ABEH001766 Google Scholar [29] M. Pinto, Discrete dichotomies, Comput. Math. Appl. , 28 (1994), 259–270, URL http://dx.doi.org/10.1016/0898-1221(94)00114-6Google Scholar [30] C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems Vol. 2002 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010, URL http://dx.doi.org/10.1007/978-3-642-14258-1 doi: 10.1007/978-3-642-14258-1. Google Scholar [31] C. Preda, P. Preda and A. Craciunescu, A version of a theorem of R. Datko for nonuniform exponential contractions, J. Math. Anal. Appl. , 385 (2012), 572–581, URL http://dx.doi.org/10.1016/j.jmaa.2011.06.082Google Scholar [32] P. Preda and M. Megan, Exponential dichotomy of evolutionary processes in Banach spaces, Czechoslovak Math. J. , 35 (1985), 312–323, URL http://dml.cz/handle/10338.dmlcz/102019Google Scholar [33] A. L. Sasu, M. G. Babut¸ia and B. Sasu, Admissibility and nonuniform exponential dichotomy on the half-line, Bull. Sci. Math. , 137 (2013), 466–484, URL http://dx.doi.org/10.1016/j.bulsci.2012.11.002Google Scholar [34] B. Sasu, Integral conditions for exponential dichotomy: A nonlinear approach, Bull. Sci. Math. , 134 (2010), 235–246, URL http://dx.doi.org/10.1016/j.bulsci.2009.06.006Google Scholar

show all references

##### References:
 [1] M. G. Babuţia, M. Megan and I. -L. Popa, On $(h, k)$ -dichotomies for nonautonomous linear difference equations in Banach spaces, Int. J. Differ. Equ. , (2013), Art. ID 761680, 7 pages, URL http://dx.doi.org/10.1155/2013/761680 Google Scholar [2] L. Barreira, J. Chu and C. Valls, Lyapunov functions for general nonuniform dichotomies, Milan J. Math., 81 (2013), 153-169. doi: 10.1007/s00032-013-0198-y. Google Scholar [3] L. Barreira and C. Valls, Growth rates and nonuniform hyperbolicity, Discrete Contin. Dyn. Syst. , 22 (2008), 509–528, URL http://dx.doi.org/10.3934/dcds.2008.22.509 Google Scholar [4] L. Barreira and C. Valls, Stability of Nonautonomous Differential Equations, vol. 1926 of Lecture Notes in Mathematics, Springer, Berlin, 2008, URL http://dx.doi.org/10.1007/978-3-540-74775-8 doi: 10.1007/978-3-540-74775-8. Google Scholar [5] L. Barreira and C. Valls, Polynomial growth rates, Nonlinear Anal. , 71 (2009), 5208–5219, URL http://dx.doi.org/10.1016/j.na.2009.04.005 Google Scholar [6] L. Barreira and C. Valls, Quadratic Lyapunov functions and nonuniform exponential dichotomies, J. Differential Equations, 246 (2009), 1235–1263, URL http://dx.doi.org/10.1016/j.jde.2008.06.008Google Scholar [7] A. J. G. Bento and C. Silva, Nonautonomous equations, generalized dichotomies and stable manifolds, ArXiv e-prints. URL http://arxiv.org/abs/0905.4935Google Scholar [8] A. J. G. Bento and C. Silva, Stable manifolds for nonuniform polynomial dichotomies, J. Funct. Anal. , 257 (2009), 122–148, URL http://dx.doi.org/10.1016/j.jfa.2009.01.032Google Scholar [9] A. J. G. Bento and C. M. Silva, Stable manifolds for non-autonomous equations with nonuniform polynomial dichotomies, Q. J. Math. , 63 (2012), 275–308, URL http://dx.doi.org/10.1093/qmath/haq047Google Scholar [10] A. J. G. Bento and C. M. Silva, Generalized nonuniform dichotomies and local stable manifolds, J. Dynam. Differential Equations, 25 (2013), 1139–1158, URL http://dx.doi.org/10.1007/s10884-013-9331-4 Google Scholar [11] T. Burton and L. Hatvani, Stability theorems for nonautonomous functional-differential equations by Liapunov functionals, Tohoku Math. J. , (2) 41 (1989), 65–104, URL http://dx.doi.org/10.2748/tmj/1178227868 Google Scholar [12] T. A. Burton and L. Hatvani, On nonuniform asymptotic stability for nonautonomous functional-differential equations, Differential Integral Equations, 3 (1990), 285–293, URL http://projecteuclid.org/euclid.die/1371586144 Google Scholar [13] X. Chang, J. Zhang and J. Qin, Robustness of nonuniform $(μ, ν)$ -dichotomies in Banach spaces, J. Math. Anal. Appl. , 387 (2012), 582–594, URL http://dx.doi.org/10.1016/j.jmaa.2011.09.026Google Scholar [14] C. Chicone and Y. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations, vol. 70 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1999. URL http://dx.doi.org/10.1090/surv/070 doi: 10.1090/surv/070. Google Scholar [15] R. Datko, Uniform asymptotic stability of evolutionary processes in a Banach space, SIAM J. Math. Anal. , 3 (1972), 428–445, URL http://dx.doi.org/10.1137/0503042Google Scholar [16] L. Hatvani, On the asymptotic stability for nonautonomous functional differential equations by Lyapunov functionals, Trans. Amer. Math. Soc. , 354 (2002), 3555–3571, URL http://dx.doi.org/10.1090/S0002-9947-02-03029-5Google Scholar [17] T. Li, Die Stabilitätsfrage bei Differenzengleichungen, Acta Math. , 63 (1934), 99–141, URL http://dx.doi.org/10.1007/BF02547352Google Scholar [18] N. Lupa and M. Megan, Exponential dichotomies of evolution operators in Banach spaces, Monatsh. Math. , 174 (2014), 265–284, URL http://dx.doi.org/10.1007/s00605-013-0517-y Google Scholar [19] A. M. Lyapunov, The general problem of the stability of motion, Internat. J. Control, 55 (1992), 521–790, translated by A. T. Fuller from Édouard Davaux's French translation (1907) ´ of the 1892 Russian original, With an editorial (historical introduction) by Fuller, a biography of Lyapunov by V. I. Smirnov, and the bibliography of Lyapunov's works collected by J. F. Barrett, Lyapunov centenary issue. URL http://dx.doi.org/10.1080/00207179208934253Google Scholar [20] A. D. Maǐzel0, On stability of solutions of systems of differential equations, Ural. Politehn. Inst. Trudy, 51 (1954), 20-50. Google Scholar [21] M. Megan, On $(h,k)$ -dichotomy of evolution operators in Banach spaces, Dynam. Systems Appl., 5 (1996), 189-195. Google Scholar [22] M. Megan and C. Buşe, Dichotomies and Lyapunov functions in Banach spaces, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 37 (1993), 103-114. Google Scholar [23] Y. A. Mitropolsky, A. M. Samoilenko and V. L. Kulik, Dichotomies and Stability in Nonautonomous Linear Systems, vol. 14 of Stability and Control: Theory, Methods and Applications, Taylor & Francis, London, 2003. Google Scholar [24] R. Naulin and M. Pinto, Roughness of $(h, k)$ -dichotomies, J. Differential Equations, 118 (1995), 20–35, URL http://dx.doi.org/10.1006/jdeq.1995.1065Google Scholar [25] O. Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Z. , 32 (1930), 703–728, URL http://dx.doi.org/10.1007/BF01194662Google Scholar [26] Y. Pesin, Families of invariant manifolds that corresponding to nonzero characteristic exponents, Izv. Akad. Nauk SSSR Ser. Mat. , 40 (1976), 1332–1379, (Russian) English transl. Math. USSR-Izv. , 10 (1976), 1261–1305, URL http://dx.doi.org/10.1070/IM1976v010n06ABEH001835Google Scholar [27] Y. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Uspehi Mat. Nauk, 32 (1977), 55–112, (Russian) English transl. Russ. Math. Surv. , 32 (1977), 55–114, URL http://dx.doi.org/10.1070/RM1977v032n04ABEH001639Google Scholar [28] Y. Pesin, Geodesic flows in closed Riemannian manifolds without focal points, Izv. Akad. Nauk SSSR Ser. Mat. , 41 (1977), 1252–1288, (Russian) English transl. Math. USSR-Izv. , 41 (1977), 1195–1228, URL http://dx.doi.org/doi:10.1070/IM1977v011n06ABEH001766 Google Scholar [29] M. Pinto, Discrete dichotomies, Comput. Math. Appl. , 28 (1994), 259–270, URL http://dx.doi.org/10.1016/0898-1221(94)00114-6Google Scholar [30] C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems Vol. 2002 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010, URL http://dx.doi.org/10.1007/978-3-642-14258-1 doi: 10.1007/978-3-642-14258-1. Google Scholar [31] C. Preda, P. Preda and A. Craciunescu, A version of a theorem of R. Datko for nonuniform exponential contractions, J. Math. Anal. Appl. , 385 (2012), 572–581, URL http://dx.doi.org/10.1016/j.jmaa.2011.06.082Google Scholar [32] P. Preda and M. Megan, Exponential dichotomy of evolutionary processes in Banach spaces, Czechoslovak Math. J. , 35 (1985), 312–323, URL http://dml.cz/handle/10338.dmlcz/102019Google Scholar [33] A. L. Sasu, M. G. Babut¸ia and B. Sasu, Admissibility and nonuniform exponential dichotomy on the half-line, Bull. Sci. Math. , 137 (2013), 466–484, URL http://dx.doi.org/10.1016/j.bulsci.2012.11.002Google Scholar [34] B. Sasu, Integral conditions for exponential dichotomy: A nonlinear approach, Bull. Sci. Math. , 134 (2010), 235–246, URL http://dx.doi.org/10.1016/j.bulsci.2009.06.006Google Scholar
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