Integral Equations and Operator Theory

, Volume 74, Issue 3, pp 345–362

Exponential Stability and Uniform Boundedness of Solutions for Nonautonomous Periodic Abstract Cauchy Problems. An Evolution Semigroup Approach

  • Constantin Buşe
  • Dhaou Lassoued
  • Thanh Lan Nguyen
  • Olivia Saierli
Article

Abstract

Let uμ, x, s(., 0) be the solution of the following well-posed inhomogeneous Cauchy Problem on a complex Banach space X
$$\left \{\begin{array}{ll}\dot{u}(t) = A(t)u(t) + e^{i\mu t}x, \quad t > s \\ u(s) = 0. \end{array} \right.$$
Here, x is a vector in Xμ is a real number, q is a positive real number and A(·) is a q-periodic linear operator valued function. Under some natural assumptions on the evolution family \({\mathcal{U} = \{U(t, s): t \geq s\}}\) generated by the family {A(t)}, we prove that if for each μ, each s ≥ 0 and every x the solution uμ, x, s(·, 0) is bounded on R+ by a positive constant, depending only on x, then the family \({\mathcal{U}}\) is uniformly exponentially stable. The approach is based on the theory of evolution semigroups.

Mathematics Subject Classification (2010)

47A05 47A30 47D06 47A10 35B15 35B10 

Keywords

Periodic evolution families uniform exponential stability boundedness evolution semigroup almost periodic functions 

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Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  • Constantin Buşe
    • 1
  • Dhaou Lassoued
    • 2
  • Thanh Lan Nguyen
    • 3
  • Olivia Saierli
    • 1
    • 4
  1. 1.Department of MathematicsWest University of TimisoaraTimisoaraRomânia
  2. 2.Laboratoire SAMM EA543Universite Paris 1 Pantheon-SorbonneParis cedex 13France
  3. 3.Western Kentucky UniversityBowling GreenUS
  4. 4.Department of Computer SciencesTibiscus University of TimisoaraTimisoaraRomânia

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