Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 10 (2015), 23 -- 40

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


Nezam Iraniparast and Lan Nguyen

Abstract. For the complete higher order differential equation u(n)(t)=Σk=0n-1Aku(k)(t)+f(t), t∈ R (*) on a Banach space E, we give a new definition of mild solutions of (*). We then characterize the regular admissibility of a translation invariant subspace al M of BUC(R, E) with respect to (*) in terms of solvability of the operator equation Σj=0n-1AjXal Dj-Xal Dn = C. As application, almost periodicity of mild solutions of (*) is proved.

2010 Mathematics Subject Classification: Primary 34G10; 34K06; Secondary 47D06.
Keywords: Abstract complete higher differential equations; mild solutions; operator equations; almost periodic functions.

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  1. Arendt W., Rabiger F., Sourour A., Spectral properties of the operator equation AX-XB=Y, Quart. J. Math. Oxford 45:2 (1994), 133--149.MR1280689(95g:47060). Zbl 0826.47013.

  2. Arendt, W., Batty, C. J. K.: 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. Daleckii J., Krein M. G.: Stability of solutions of differential equations on Banach spaces, Amer. Math. Soc., Providence, RI, 1974. MR0352639(50 \sharp5126). Zbl 0960.43003.

  4. Erdelyi I., Wang S. W. A local spectral theory for closed operators, Cambridge Univ. Press, London (1985). MR1880990(2002j:34091). Zbl 0577.47035.

  5. Katznelson Y.: An Introduction to harmonic analysis, Dover Pub., New York 1976. MR2039503(2005d:43001). Zbl 1055.43001.

  6. Levitan B.M., Zhikov V.V.: Almost periodic functions and differential equations. Cambridge Univ. Press, London 1982. MR0690064(84g:34004). Zbl 0499.43005.

  7. Lizama C.: Mild almost periodic solutions of abstract differential equations, J. Math. Anal. Appl. 143 (1989), 560--571. MR1022555(91c:34064). Zbl 0698.47035.

  8. Cioranescu I., Lizama C.: Spectral properties of cosine operator functions, Aequationes Mathematicae 36 (1988), 80--98. MR0959795(89i:47071). Zbl 0675.47029.

  9. Nguyen, L.: On the Mild Solutions of Higher Order Differential Equations in Banach spaces, Abstract and Applied Analysis. 15 (2003) 865--880. MR2010941(2004h:34111). Zbl 1076.34065.

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

  11. Pruss J.: Evolutionary integral equations and applications, Birkhäuser, Berlin 2012. MR1238939(94h:45010). Zbl 1258.45008.

  12. Rosenblum M.: On the operator equation BX-XA=Q, Duke Math. J. 23 (1956), 263--269. MR0079235(18,54d). Zbl 0073.33003.

  13. Ruess, W.M., Vu Quoc Phong: Asymptotically almost periodic solutions of evolution equations in Banach spaces, J. Differential Equations 122 (1995), 282--301. MR1355893(96i:34143). Zbl 0837.34067.

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

  15. Schüler E., Vu Q. P.: 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. Schüler E., Vu Q. P.: 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. Schweiker S.: Mild solution of second-order differenttial equations on the line, Math. Proc. Cambridge Phil. Soc. 129 (2000), 129--151. MR1757784(2001d:34092). Zbl 0958.34043.

  18. Vu Quoc Phong: The operator equation AX-XB=C with unbounded operators A and B and related abstract Cauchy problems, Math. Z. 208 (1991), 567--588. MR1136476(93b:47035). Zbl 0726.47029.

  19. Vu Quoc Phong: On the exponential stability and dichotomy of C0-semigroups, Studia Mathematica 132, No. 2 (1999), 141--149. MR1669694(2000j:47076). Zbl 0926.47026.

Nezam Iraniparast Lan Nguyen
Department of Mathematics, Department of Mathematics,
Western Kentucky University, Western Kentucky University,
Bowling Green KY 42101, USA. Bowling Green KY 42101, USA
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