Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 10 (2015), 61 -- 93

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.


Olivia Saierli

Abstract The long time behavior for solutions of evolution periodic equations are reviewed.

2010 Mathematics Subject Classification: 42A16; 45A05; 47A10; 47A35; 47D06; 47G10; 93D20.
Keywords: periodic evolution families; uniform exponential stability; boundedness; evolution semigroup; almost periodic functions.

Full text


  1. W. Arendt, C.J.K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc. 306(1988), 837-852. MR0933321(89g:47053). Zbl 0652.47022.

  2. W. Arendt, Ch. J. K. Batty, M. Hieber, F. Neubrabder, "Vector-Valued Laplace transforms and Cauchy problems", 2nd ed., Monographs in Mathematics, 96. Birkhäuser/Springer Basel AG, Basel, 2011. MR2798103(2012b:47109).

  3. W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. Lotz, U. Moustakas, R. Nagel, F. Neubrander, U. Schlotterbeck, One-parameter semigroups of positive operators, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1184, (1986). MR0839450(88i:47022).

  4. S. Arshad, C. Buşe, O. Saierli, Connections between exponential stability and boundedness of solutions of a couple of differential time depending and periodic systems, Electron. J. Qual. Theory Differ. Equ. 90(2011), 1-16. MR2854029.

  5. S. Arshad, C. Buşe, A. Nosheen and A. Zada, Connections between the stability of a Poincare map and boundedness of certain associate sequences, Electron. J. Qual. Theory Differ. Equ. 16(2011), pag. 1-12. MR2774099(2012c:39013). Zbl 1281.39009.

  6. S. Balint, On the Perron-Bellman theorem for systems with constant coefficients, An. Univ. Timişoara, Ştiinţe Mat. 21, (1983), 3-8. MR0742495(85m:34089). Zbl 0536.34038.

  7. E. A. Barbashin, Introduction to the theory of stability, Izd. Nauka, Moskow, Rusia, 1967. MR0264141(41 #8737).

  8. C. Buşe and D. Barbu, Some remarks about the Perron condition for C0 semigroups, An. Univ. Timişora, Ser. Mat.-Inform. 35 (1997), no. 1, 3--8. MR1875631. Zbl 1003.93043.

  9. D. Barbu and C. Buşe, Asymptotic stability and uniform boundedness with respect to parameters for discrete non-autonomous periodic systems, J. Difference Equ. Appl. 18 (2012), no. 9, 1435--1441. MR2974129. Zbl 1258.39008.

  10. D. Barbu, J. Blot, C. Buşe and O. Saierli, Stability for trajectories of periodic evolution families in Hilbert spaces Electron. J. Differ. Equ. (2014), no. 1, 13 pp. MR3159410. Zbl 1308.47049.

  11. C. Buşe, On the Perron-Bellman theorem for evolutionary processes with exponential growth in Banach spaces, New-Zealand J. of Math. 27 (1998), no. 2, 183--190. MR1706975(2001b:47063). Zbl 0972.47027.

  12. C. Buşe, S. S. Dragomir and V. Lupulescu, Characterizations of stability for strongly continuous semigroups by boundedness of its convolutions with almost periodic functions, Int. J. Differ. Equ. Appl. 2 (2001), no. 1, 103--109. MR2006440(2004f:47053).

  13. C. Buşe and C. P. Niculescu, A condition of uniform exponential stability for semigroups, Mathematical Inequalities and Applications, Vol. 11 (2008), No. 3, 529-536. MR2431216(2009j:47078). Zbl 1180.47027.

  14. C. Buşe, C. P. Niculescu and J. Pecaric, Asymptotic stability and integral inequalities for solutions of linear systems on Radon-Nikod\acuteym spaces, Mathematical Inequalities and Applications, Vol. 8, No. 2, April 2005. MR2148571(2006c:34111). Zbl 1077.35024.

  15. C. Buşe, Exponential Stability for Periodic Evolution Families of Bounded Linear Operators, Rend. Sem. Mat. Univ. Pol. Torino, 59(2001), no. 1, 17-22. MR1967496(2004a:47043). Zbl 1179.47038.

  16. C. Buşe, A spectral mapping theorem for evolution semigroups on asymptotically almost periodic functions defined on the half line, Electronic J. of Diff. Eq., USA, 2002 (2002), no. 70, 1-11. MR1921143(2003h:47078). Zbl 1010.47027.

  17. C. Buşe, Semigrupuri liniare pe spaţii Banach şi semigrupuri de evoluţie pe spaţii de funcţii aproape periodice, Editura Eubeea, Timişoara, 2003.

  18. C. Buşe, A. D. R. Choudary, S. S. Dragomir and M. S. Prajea, On Uniform Exponential Stability of Exponentially Bounded Evolution Families, Integral Equation Operator Theory, 61(2008), no. 3, 325--340. MR2417501(2009i:47088). Zbl 1165.47027.

  19. C. Buşe, P. Cerone, S. S. Dragomir and A. Sofo, Uniform stability of periodic discrete system in Banach spaces, J. Difference Equ. Appl. 11(2005), no. 12, 1081-1088. MR2179505(2006e:39008).

  20. C. Buşe, A. Khan, G. Rahmat and O. Saierli, Weak real integral characterizations for exponential stability of semigroups in reflexive spaces, Semigroup Forum 88(2014), no. 1, 195--204. MR3164159 Zbl 06288007.

  21. C. Buşe, D. Lassoued, Lan Thanh Nguyen and O. Saierli, Exponential stability and uniform boundedness of solutions for nonautonomous periodic abstract Cauchy problems. An evolution semigroup approach, Integral Equations Operator Theory, 74(2012), no. 3, 345--362. MR2992028. Zbl 1279.47063.

  22. C. Buşe and V. Lupulescu, Exponential stability of linear and almost periodic systems on Banach spaces, Electron. J. Differ. Equ. 2003(2003), no. 125, 1--7. MR2022073(2004i:47076). Zbl 1043.35022.

  23. C. Buşe, M. Megan, M. Prajea and P. Preda, The strong variant of a Barbashin theorem on stability of solutions for non-autonomous differential equations in Banach spaces, Integr. Equ. Oper. Theory, 59(2007), no. 4, 491--500. MR2370045(2009c:47063). Zbl 1160.47036.

  24. C. Buşe and A. Pogan , Individual exponential stability for evolution families of linear and bounded operators, New Zealand J. Math. 30(2001), no.1, 15--24. MR1839519(2002c:93147). Zbl 0990.35020.

  25. R. Datko, Extending a theorem of A.M. Liapunov to Hilbert space, J. Math. Anal. Appl. 32 (1970), 610--616. MR0268717(42 #3614). Zbl 0211.16802.

  26. E. B. Davies, One parameter semigroups, London Math. Soc. Mono. 15, Academic Press, London-New York, 1980. MR0591851(82i:47060).

  27. C. Chicone and Y. Latushkin, Evolution semigroups in dynamical systems and differential equations, Mathematical Surveys and Monographs, 70, American Mathematical Society, Providence R. I., (1999). MR1707332(2001e:47068). Zbl 0970.47027.

  28. S. Clark, Y. Latushkin, S. Montgomery-Smith and T. Randolph, Stability radius and internal versus external stability in Banach spaces: an evolution semigroup approach, SIAM J. Control Optim. 38 (2000), no. 6, 1757--1793. MR1776655(2001k:93085). Zbl 0978.47030.

  29. C. Corduneanu, Almost Periodic oscillations and wawes, Springer, New York, (2009). MR2460203(2009i:34002).

  30. Ju. L. Daletckii and M. G. Krein, Stability of solutions of differential equations in Banach space, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, 43, American Mathematical Society, Providence, R.I., (1974). MR0352639(50 #5126).

  31. Daniel Daners and Pablo Koch Medina, Abstract evolution equations, periodic problems and applications, Pitman Research Notes in Mathematics Series, 279, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, (1992). MR1204883(94b:34002). Zbl 0789.35001.

  32. K. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics 194, Springer, 2000. MR1721989(2000i:47075). Zbl 0952.47036.

  33. C. Foiaş, Sur une question de M. Reghiş, Ann. Univ. Timişoara Ser. Şti. Mat., 11, Vol. 9, Fasc. 2, pp. 111-114, (1973). MR0370262(51 #6489). Zbl 0334.47002.

  34. L. Gearhart, Spectral theory for contraction semigroups on Hilbert spaces, Trans. Amer. Math. Soc. 236(1978), 385--394. MR0461206(57 #1191).

  35. G. Greiner, J. Voight and M. P. Wolff, On the spectral bound of the generator of semigroups of positive operators, J. Operator Theory, 5 (1981), no. 2, 245-256. MR0617977(82h:47039). Zbl 0469.47032.

  36. E. Hille and R.S. Philips, "Functional Analysis and Semi-Groups", Third printing of the revised edition of 1957. American Mathematical Society Colloquium Publications, Vol. XXXI. American Mathematical Society, Providence, R. I., 1974. MR0423094(54 #11077).

  37. J. S. Howland, Stationary scattering theory for time-dependent Hamiltonians, Math. Ann. 207 (1974) , 315-335 MR0346559(49 #11284). Zbl 0261.35067.

  38. F. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), no.1, 43-56. MR0834231(87e:34106). Zbl 0593.34048.

  39. F. Huang,  Exponential stability of linear systems in Banach spaces, Chinese Ann. Math. Ser. B 10(1989), no. 3, 332-340. MR1027672(90k:47083). Zbl 0694.47027.

  40. F. Huang and L. Kang Sheng, A problem of exponential stability for linear dynamical systems in Hilbert spaces, Kexue Tongbao (English Ed.) 33 (1988), no. 6, 460-462. MR0961534(90g:34062)). Zbl 0679.34053.

  41. U. Krengel, Ergodic Theorems, With a supplement by Antoine Brunel, De Gruyter Studies in Mathematics, 6. Walter de Gruyter & Co., Berlin, (1985). MR0797411(87i:28001). Zbl 03918736.

  42. D. Lassoued, Exponential dichotomy of nonautonomous periodic systems in terms of the boundedness of certain periodic Cauchy problems, Electronic Journal of Difference Equations, Vol. 2013 (2013), No. 89, pp. 1-7. MR3065042. Zbl 1294.34059.

  43. B. M. Levitan and V. V. Zhikov, "Almost Periodic Functions and Differential Equations", Moscow Univ. Publ. House, 1978. English translation by Cambridge Univ. Press, Camridge, U. K., 1982. MR0690064(84g:34004). Zbl 0414.43008.

  44. Yu. I. Lyubich and Qu\hatoc Phong V\tildeu, Asymptotic stability of linear differential equations in Banach spaces, Studia Math. 88, (1988), No. 1, 37-42. MR0932004(89e:47062). Zbl 0639.34050.

  45. M.Megan, Propri\acuteet\acutees qualitatives des syst\graveems lin\graveeares contr\hatol\acutees dans les espaces de dimension infinie, Monografii Matematice (Timioara) [Mathematical Monographs (TimiÅ Ÿoara)], 32, (1988). MR0998565(90h:93053).

  46. Nguyen Van Minh, F. R\ddotabiger, R. Schnaubelt, Exponential stability, exponential expansiveness and exponential dichotomy of evolution equations on the half-line, Integral Equations Operator Theory, 32 (1998), no. 3, 332-353. MR1652689(99j:34083). Zbl 0977.34056.

  47. T. Naito and Nguyen Van Minh, Evolution semigroups and spectral criteria for almost periodic solutions of periodic evolution equations, J. Differ. Equations 152 (1999), no. 2, 358-376. MR1674561(99m:34131). Zbl 0924.34038.

  48. F. Neubrander, Laplace transform and asymptotic behavior of strongly continuous semigroups, Houston J. Math., 12 (1986), no. 4, 549-561. MR0873650(88b:47057). Zbl 0624.47031.

  49. L. T. Nguyen, On nonautonomous functional differential equations, J. Math. Anal. Appl. 239, (1999), no. 1, 158-174. MR1719064(2000h:34126). Zbl 0949.34063.

  50. L. T. Nguyen On the wellposedness of nonautonomous second order Cauchy problems, East-West J. Math. 1 (1999), no. 2, 131-146. MR1727388(2001h:34086). Zbl 0949.34045.

  51. A. Pazy, On the applicability of Lyapunov's theorem in Hilbert space, SIAM J. Math. Anal. 3 (1972), 291--294. MR0317105(47 #5653). Zbl 0242.47028.

  52. A. Pazy, Semigroups of linear operators and applications to differential equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York (1983). MR0710486(85g:47061).

  53. O. Perron, Uber eine Matrixtransformation, Math. Z. 32 (1930), no. 1, 465-473. MR1545178. Zbl 56.0105.01.

  54. Vu Quoc Phong, On stability of C0-semigroups, Proceedings of the American Mathematical Society, 129 (2001), no. 10, 2871-2879. MR1707013(2001m:47096).

  55. J. Prüss, On the spectrum of C0-semigroups, Trans. Amer. Math. Soc. 284 (1984), no. 2, 847--857. MR0743749(85f:47044)

  56. M. Reghiş, C. Buşe, On the Perron-Bellman theorem for C0-semigroups and periodic evolutionary processes in Banach spaces, Ital. J. Pure Appl. Math. (1998), no. 4, 155-166. MR1695486(2001b:47067). Zbl 0981.47030.

  57. S. Rolewicz, On uniform N-equistability, J. Math. Anal. Appl. 115 (1986), no. 2, 434-441. MR0836237(87h:47100). Zbl 0597.34064.

  58. O. Saierli, Spectral mapping theorem for an evolution semigroup on a space of vector valued almost periodic functions, Electron. J. Differential Equations, (2012), no. 175, 1-9. MR2991409. Zbl 1273.47069.

  59. R. Schnaubelt, Well-posedness and asymptotic behaviour of non-autonomous linear evolution equations, Evolution equations, semigroups and functional analysis (Milano, 2000), 311--338, Progr. Nonlinear Differential Equations Appl., 50, Birkhäuser, Basel, 2002. MR1944170(2003i:34140). Zbl 01944211.

  60. E.M. Stein and R. Shakarchi, Fourier Analysis. An Introduction, Princeton Lectures in Analysis, 1. Princeton University Press, Princeton, NJ, (2003). MR1970295(2004a:42001). Zbl 1026.42001.

  61. K. V. Storozhuk, Obstructions to the uniform stability of a C0-semigroup, Siberian Mathematical Journal, Vol. 51 (2010), No.2, pp. 330-337. MR2668108(2011g:47095). Zbl 1230.47075.

  62. K. Tanabe, On the Equations of Evolution in a Banach Space, Osaka Math. J. 12 (1960), 363-376. MR0125455(23 #A2756b). Zbl 0098.31301.

  63. A. E. Taylor, General Theory of Functions and Integration, Reprint of the 1966 second edition. Dover Publications, Inc., New York, (1985). MR0824243(87b:26001). Zbl 0135.11301.

  64. J. van Neerven, Individual stability of C0-semigroups with uniformly bounded local resolvent, Semigroup Forum, 53, (1996), no. 2, 155-161. MR1400641(97h:47036). Zbl 0892.47040.

  65. J. van Neerven, The asymptotic behaviour of semigroups of linear operators, Operator Theory: Advances and Applications, 88, Birkhäuser, Basel (1996). MR1409370(98d:47001).

  66. L. Weis and V. Wrobel, Asymptotic behavior of C0-semigroups in Banach spaces, Proc. Amer. Math. Soc. 124(1996), no. 12, 3663-3671. MR1327051(97b:47041). Zbl 0863.47027.

  67. G. Weiss, Weak Lp-stability of a linear semigroup on a Hilbert space implies exponential stability, J. Diff. Equations, 76, (1988), Issue 2, 269-285. Zbl 0675.47031.

  68. V. Wrobel,  Asymptotic behavior of C0-semigroups in B-convex spaces, Indiana Univ. Math. J. 38(1989), no. 1, 101-114. MR0982572(90b:47076). Zbl 0653.47018.

  69. K. Yosida, Functional Analysis, Grundlehen Math. Wiss., Vol 123, Springer-Verlag, 1980. MR0617913(82i:46002)

  70. J. Zabczyk,  A note on C0-semigroups , Bull. Acad. Polon. Sci. S\acuteer. Sci. Math. 23(1975), No. 8, 895-898. MR0383144(52 #4025).

  71. A. Zada, A characterization of dichotomy in terms of boundedness of solutions for some Cauchy problems, Electron. J. Differ. Equ. 2008(2008), no. 94, 1-5. MR2421170(2009e:34134). Zbl 1168.47034.

  72. A. Zada, S. Arshad , G. Rahmat and A. Khan, On the Dichotomy of Non-Autonomous Systems Over Finite Dimensional Spaces, Appl. Math. Inf. Sci. 9, No. 4, 1941-1946 (2015). MR3339917.

  73. S. D. Zaidman, Almost-periodic functions in abstract spaces, Research Notes in Mathematics, 126. Pitman (Advanced Publishing Program), Boston, MA, 1985. MR0790316(86j:42018). Zbl 0648.42006.

Olivia Saierli
"Tibiscus" University of Timişoara, Department of Computer Sciences,
Str. Lascăr Catargiu, No. 6, 300559 Timişoara, Rom ania.