Set-Valued Analysis

, Volume 11, Issue 4, pp 345–357 | Cite as

An Averaging Method for Singularly Perturbed Systems of Semilinear Differential Inclusions with C0-Semigroups

  • Mikhail Kamenskii
  • Paolo Nistri


We consider a system of two semilinear differential inclusions with infinitesimal generators of C0-semigroups. The nonlinear terms are of high frequency with respect to time and periodic with a specified period. Moreover, they are condensing in the state variables (x,y) with respect to a suitable measure of noncompactness. The goal of the paper is to give sufficient conditions to guarantee, for ∈>0 sufficiently small, the existence of periodic solutions and to study their behaviour as ∈→0. The main tool to achieve this is the topological degree theory for uppersemicontinuous, condensing vector fields.

periodic solutions averaging method differential inclusions singularly perturbed system C0-semigroups 


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  1. 1.
    Andreini, A., Kamenskii, M. I. and Nistri, P.: A result on the singular perturbation theory for differential inclusions in Banach spaces, Topol. Methods Nonlinear Anal. 15 (2000), 1-15.Google Scholar
  2. 2.
    Borisovich, Yu., Gel'man, B. D., Myshkis, A. D. and Obukhovskii, V. V.: Topological methods in the fixed point theory of multivalued maps, Uspekhi Mat. Nauk 35 (1980), 59-126 (Russian).Google Scholar
  3. 3.
    Couchouron, J. F. and Kamenskii, M. I.: A unified point of view for integro-differential inclusions, Lecture Notes in Nonlinear Analysis 2 (1998), 123-137.Google Scholar
  4. 4.
    Kamenskii, M. I., Obukhovskii, V. V. and Zecca, P.: Condensing multivalued maps and semilinear differential inclusions in Banach spaces, de Gruyter Ser. Nonlinear Anal. Appl. 7, Walter De Gruyter, Berlin, 2001.Google Scholar
  5. 5.
    Kamenskii, M. I. and Nistri, P.: An averaging method for singularly perturbed systems of semilinear differential inclusions with analytic semigroups, Nonlinear Anal. 53 (2003), 467-480.Google Scholar
  6. 6.
    Kamenskii, M. I. and Nistri, P.: Periodic solutions of a singularly perturbed system of differential inclusions in Banach spaces, In: Set Valued Mappings with Applications in Nonlinear Analysis, Gordon and Breach Science Publishers, London, 2001, pp. 213-226.Google Scholar
  7. 7.
    Kamenskii, M. I., Nistri, P. and Zecca, P.: On the periodic solution problem for parabolic inclusions with a large parameter, Topol. Methods Nonlinear Anal. 8 (1996), 57-77.Google Scholar
  8. 8.
    Kamenskii, M. I. and Obukhovskii, V. V.: Condensing multioperators and periodic functional-differential inclusions in Banach spaces, Nonlinear Anal. 20 (1993), 781-792.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Mikhail Kamenskii
    • 1
  • Paolo Nistri
    • 2
  1. 1.Dept. of MathematicsVoronezh State UniversityVoronezhRussia
  2. 2.Dip. di Ingegneria dell'InformazioneUniversità di SienaSienaItaly

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