Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 6 (2011), 23 -- 41


Lan Nguyen

Abstract. For the complete higher order differential equation

u(n)(t)=∑k=0n-1Aku(k)(t)+f(t), 0 ≤ t ≤ T,
on a Banach space E, we give necessary and sufficient conditions for the periodicity of mild solutions. The results, which are proved in a simple manner, generalize some well-known ones.

2010 Mathematics Subject Classification: Primary 34G10; 34K06, Secondary 47D06.
Keywords: Abstract higher order differential equations; Fourier series; Periodic mild solutions; Operator semigroups; Cosine families.

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  1. W. Arendt, C.J.K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems. Birkhäuser Verlag, Basel-Boston-Berlin 2001. MR1886588(2003g:47072). Zbl 0978.34001.

  2. W. Arendt and C. J. K. Batty, Almost periodic solutions of first- and second-order Cauchy problems. J. Differential Equations 137 (1997), no. 2, 363--383. MR1456602(98g:34099). Zbl 0879.34046.

  3. I. Cioranescu and C. Lizama, Spectral properties of cosine operator functions. Aequationes Math. 36 (1988), no. 1, 80--98. MR0959795(89i:47071). Zbl 0675.47029.

  4. J. Daleckii and M. G. Krein, Stability of solutions of differential equations on Banach spaces. Amer. Math. Soc., Providence, RI, 1974. MR0352639(50 \sharp5126).

  5. Y. S. Eidelman and I. V. Tikhonov, On periodic solutions of abstract differential equations Abstr. Appl. Anal. 6 (2001), no. 8, 489--499. MR1880990(2002j:34091). Zbl 1010.34053.

  6. L. Gearhart, Spectral theory for contraction semigroups on Hilbert space. Trans. Amer. Math. Soc. 236 (1978), 385--394. MR0461206(57 \sharp1191). Zbl 0326.47038.

  7. J. K. Hale, Ordinary differential equations. Wiley-Interscience, New York, 1969. MR0419901(54 \sharp7918). Zbl 0186.40901.

  8. L. Hatvani and T. Krisztin, On the existence of periodic solutions for linear inhomogeneous and quasi-linear functional differential equations. J. Differential Equations 97 (1992), 1--15. MR1161308(93c:34139) . Zbl 0758.34054.

  9. A. Haraux, Nonlinear evolution equations Lecture Notes in Math., vol. 841 Springer Verlag, Heidelberg 1981. MR0610796(83d:47066). Zbl 0461.35002.

  10. Y. Latushkin and S. Montgomery-Smith, Evolution semigroups and Liapunov Theorems in Banach spaces. J. Func. Anal. 127 (1995), 173--197. MR1308621(96k:47072). Zbl 0878.47024.

  11. R. Nagel and E. Sinestrari, Inhomogeneous Volterra integrodifferential equations for Hille--Yosida operators. In: K.D. Bierstedt, A. Pietsch, W. M. Ruess, D. Vogt (eds.): Functional Analysis. Proc. Essen Conference, Marcel Dekker 1993, 51-70. MR1241671(94i:34121). Zbl 0790.45011.

  12. S. Murakami, T. Naito and Nguyen Van Minh: Evolution Semigroups and Sums of Commuting Operators: A New Approach to the Admissibility Theory of Function Spaces. J. Differential Equations, 164, (2000), pp. 240-285. MR1765556(2001d:47063). Zbl 0966.34049.

  13. L. Nguyen, Periodicity of Mild Solutions to Higher Order Differential Equations in Banach Spaces Electronic J. Diff. Equ. 79 (2004), 1-12. MR2075418(2005d:34127). Zbl 1060.34026.

  14. J. Pruss, On the spectrum of C0-semigroup. Trans. Amer. Math. Soc. 284, 1984, 847--857 . MR0743749(85f:47044). Zbl 0572.47030.

  15. E. Schüler and Q. P. Vu, The operator equation AX-XB=C, admissibility and asymptotic behavior of differential equations. J. Differential Equations, 145 (1998), 394-419. MR1621042(99h:34081). Zbl 0918.34059.

  16. E. Schüler and Q. P. Vu, The operator equation AX-XD2=-δ0 and second order differential equations in Banach spaces. Semigroups of operators: theory and applications (Newport Beach, CA, 1998), 352-363, Progr. Nonlinear Differential Equations Appl., 42, Birkhauser, Basel, 2000. MR1790559(2001j:47020). Zbl 0998.47009.

  17. E. Schüler, On the spectrum of cosine functions. J. Math. Anal. Appl. 229 (1999), 376--398. MR1666408(2000c:47086). Zbl 0921.34073.

\noindentLan Nguyen
Department of Mathematics, Western Kentucky University,
Bowling Green KY 42101, USA.