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Solutions Almost Periodic at Infinity to Differential Equations With Unbounded Operator Coefficients

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Abstract

The new class of functions almost periodic at infinity is defined using the subspace of functions with integrals decreasing at infinity. We obtain spectral criteria for almost periodicity at infinity of bounded solutions to differential equations with unbounded operator coefficients. For the new class of asymptotically finite operator semigroups we prove the almost periodicity at infinity of their orbits.

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References

  1. Bochner S., “Uber genisse Differential and allgemeiner Gleichungen deren Losungen fastperiodisch sind,” Math. Ann., Bd 103, 588–597 (1930).

    Article  MATH  Google Scholar 

  2. Bochner S. and Neumann J. V., “Almost periodic functions in groups. II,” Trans. Amer. Math. Soc., vol. 37, no. 1, 21–50 (1935).

    MathSciNet  MATH  Google Scholar 

  3. Levitan B. M., “On an integral equation with almost periodic solutions,” Bull. Amer. Math. Soc., vol. 43, 677–679 (1935).

    Article  MathSciNet  Google Scholar 

  4. Loomis L. H., “The spectral characterization of a class of almost periodic functions,” Ann. Math., vol. 72, no. 2, 362–368 (1960).

    Article  MathSciNet  MATH  Google Scholar 

  5. Foias C. and Zaidman S., “Almost-periodic solutions of parabolic systems,” Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), vol. 15, no. 3, 247–262 (1961).

    MathSciNet  MATH  Google Scholar 

  6. Doss R., “On the almost periodic solutions of a class of integro-differential-difference equations,” Ann. Math., vol. 81, no., 117–123 (1965).

    Article  MathSciNet  MATH  Google Scholar 

  7. Staffans O. J., “On asymptotically almost periodic solutions of a convolution equation,” Trans. Amer. Math. Soc., vol. 226, no. 2, 603–616 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  8. Baskakov A. G., “Some criteria for almost periodicity of bounded functions,” Tr. NII Mat. Voronezh Gos. Univ., vol. 11, 10–16 (1973).

    Google Scholar 

  9. Baskakov A. G., Some Problems of the Theory of Almost Periodic Vector Functions [Russian], Diss. Kand. Fiz.-Mat. Nauk, Voronezh (1973).

    Google Scholar 

  10. Baskakov A. G., “Spectral criteria for almost periodicity of solutions of functional equations,” Math. Notes, vol. 24, no. 2, 606–612 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  11. Levitan B. M. and Zhikov V. V., Almost-Periodic Functions and Differential Equations [Russian], Izdat. MGU, Moscow (1978).

    MATH  Google Scholar 

  12. Baskakov A. G., “Harmonic analysis of cosine and exponential operator-valued functions,” Math. USSR-Sb., vol. 52, no. 1, 63–90 (1985).

    Article  MATH  Google Scholar 

  13. Lyubich Yu. and Vu Q. Ph., “Asymptotic stability of linear differential equations in Banach spaces,” Studia Math., vol. 88, no. 1, 37–42 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  14. Arendt W. and Batty C. J. K., “Asymptotically almost periodic solutions of inhomogeneous Cauchy problems on the half-line,” Bull. London Math. Soc., vol. 31, 291–304 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  15. Basit B., “Harmonic analysis and asymptotic behavior of solutions to the abstract Cauchy problem,” Semigroup Forum, vol. 54, no. 1, 58–74 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  16. Basit B., “Some problems concerning different types of vector valued almost periodic functions,” Dissertation Math., vol. 338. 26 p. (1995).

    MathSciNet  MATH  Google Scholar 

  17. Ruess W. M. and Vu Q. Ph., “Asymptotically almost periodic solutions of evolution equations in Banach spaces,” J. Differential Equations, vol. 122, no. 2, 282–301 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  18. Baskakov A. G., “Representation theory for Banach algebras, Abelian groups, and semigroups in the spectral analysis of linear operators,” J. Math. Sci., vol. 137, no. 4, 4885–5036 (2006).

    Article  MathSciNet  Google Scholar 

  19. Baskakov A. G., Harmonic Analysis in Banach Modules and the Spectral Theory of Linear Operators [Russian], VSU Publ. House, Voronezh (2016).

    Google Scholar 

  20. Sobolev S. L., “On almost periodicity for solutions of a wave equation. I–III,” Dokl. Akad. Nauk SSSR, I: vol. 48, no. 8, 570–573 (1945); II: vol. 48, no. 9, 646–648 (1945); III: vol. 49, no. 1, 12–15 (1945).

    Google Scholar 

  21. Levitan B. M., “Integration of almost periodic functions with values in a Banach space,” Izv. Akad. Nauk SSSR Ser. Mat., vol. 30, no. 5, 1101–1110 (1966).

    MathSciNet  MATH  Google Scholar 

  22. Kadets M. I., “On the integration of almost periodic functions with values in a Banach space,” Funct. Anal. Appl., vol. 3, no. 3, 228–230 (1969).

    Article  MathSciNet  MATH  Google Scholar 

  23. Baskakov A. G., “Analysis of linear differential equations by methods of the spectral theory of difference operators and linear relations,” Russian Math. Surveys, vol. 68, no. 1, 69–116 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  24. Baskakov A. G., “Harmonic and spectral analysis of power bounded operators and bounded semigroups of operators on Banach spaces,” Math. Notes, vol. 97, no. 2, 164–178 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  25. Baskakov A. G. and Kaluzhina N. S., “Beurling’s theorem for functions with essential spectrum from homogeneous spaces and stabilization of solutions of parabolic equations,” Math. Notes, vol. 92, no. 5, 587–605 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  26. Baskakov A. G., Kaluzhina N. S., and Polyakov D. M., “Operator semigroups slowly varying at infinity,” Russian Math. (Iz. VUZ), vol. 58, no. 7, 1–10 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  27. Hewitt E. and Ross K., Abstract Harmonic Analysis. Vol. 2, Springer-Verlag, New York (1994).

    Book  MATH  Google Scholar 

  28. Baskakov A. G. and Krishtal I. A., “Spectral analysis of abstract parabolic operators in homogeneous function spaces,” Math. Notes, vol. 92, no. 5, 587–605 (2012).

    Article  MathSciNet  Google Scholar 

  29. Arendt W., Batty C. J. K., Hieber M., and Neubrander F., Vector-Valued Laplace Transforms and Cauchy Problems, Birkhäuser-Verlag, Basel (2011) (Monogr. Math.; vol. 96).

    Book  MATH  Google Scholar 

  30. Domar Y., “Harmonic analysis based on certain commutative Banach algebras,” Acta Math., vol. 96, 1–66 (1956).

    Article  MathSciNet  MATH  Google Scholar 

  31. Reiter H. and Stegeman I. D., Classical Harmonic Analysis and Locally Compact Groups, Clarendon Press, Oxford (1968).

    MATH  Google Scholar 

  32. Emel'yanov È. Yu., “Some conditions for a C 0-semigroup to be asymptotically finite-dimensional,” Sib. Math. J., vol. 44, no. 5, 793–796 (2003).

    Article  Google Scholar 

  33. Emel'yanov E. Yu., Non-Spectral Asymptotic Analysis of One-Parameter Operator Semigroups, Birkhäuser-Verlag, Basel (2007).

    MATH  Google Scholar 

  34. Storozhuk K. V., “A condition for asymptotic finite-dimensionality of an operator semigroup,” Sib. Math. J., vol. 52, no. 6, 1104–1107 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  35. Storozhuk K. V., “Slowly changing vectors and the asymptotic finite-dimensionality of an operator semigroup,” Sib. Math. J., vol. 50, no. 4, 737–740 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  36. Baskakov A. and Strukova I., “Harmonic analysis of functions periodic at infinity,” Eurasian Math. J., vol. 7, no. 4, 9–29 (2016).

    MathSciNet  Google Scholar 

  37. Strukova I. I., “On Wiener’s theorem for functions periodic at infinity,” Sib. Math. J., vol. 57, no. 1, 145–154 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  38. Ryzhkova A. A. and Trishina I. A., “Solutions almost periodic at infinity of difference equations,” Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., vol. 15, no. 1, 45–49 (2015).

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Voronezh State University, Voronezh, Russia

    A. G. Baskakov, I. I. Strukova & I. A. Trishina

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  1. A. G. Baskakov
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  2. I. I. Strukova
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  3. I. A. Trishina
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Correspondence to A. G. Baskakov.

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Original Russian Text Copyright © 2018 Baskakov A.G., Strukova I.I., and Trishina I.A.

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Baskakov, A.G., Strukova, I.I. & Trishina, I.A. Solutions Almost Periodic at Infinity to Differential Equations With Unbounded Operator Coefficients. Sib Math J 59, 231–242 (2018). https://doi.org/10.1134/S0037446618020052

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  • DOI: https://doi.org/10.1134/S0037446618020052

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