Ukrainian Mathematical Journal

, Volume 64, Issue 7, pp 991–1018 | Cite as

Impulsive differential inclusions involving evolution operators in separable Banach spaces

  • M. Benchohra
  • J. J. Nieto
  • A. Ouahab
Article
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We present some results on the existence of mild solutions and study the topological structures of the sets of solutions for the following first-order impulsive semilinear differential inclusions with initial and boundary conditions:
$$ \begin{array}{*{20}{c}} {y^{\prime}(t)-A(t)y(t)\in F\left( {t,y(t)} \right)\quad \mathrm{for}\;\mathrm{a}.\mathrm{e}\quad t\in J\backslash \left\{ {{t_1},\ldots,{t_m},\ldots } \right\},} \\ {y\left( {t_k^{+}} \right)-y\left( {t_k^{-}} \right)={I_k}\left( {y\left( {t_k^{-}} \right)} \right),\quad k=1,\ldots,} \\ {y(0)=a} \\ \end{array} $$
and
$$ \begin{array}{*{20}{c}} {y^{\prime}(t)-A(t)y(t)\in F\left( {t,y(t)} \right)\quad \mathrm{for}\;\mathrm{a}.\mathrm{e}\quad t\in J\backslash \left\{ {{t_1},\ldots,{t_m},\ldots } \right\},} \\ {y\left( {t_k^{+}} \right)-y\left( {t_k^{-}} \right)={I_k}\left( {y\left( {t_k^{-}} \right)} \right),\quad k=1,\ldots,} \\ {Ly=a,} \\ \end{array} $$
where \( J={{\mathbb{R}}_{+}} \), 0 = t0 < t1 < … < tm <…, \( m\in \mathbb{N} \), limk→∞tk = ∞, A(t) is the infinitesimal generator of a family of evolution operators U(t, s) in a separable Banach space E and F is a set-valued mapping. The functions Ik characterize the jumps of solutions at the impulse points tk, k = 1, ….The mapping L: PCb→E is a bounded linear operator. We also investigate the compactness of the set of solutions, some regularity properties of the operator solutions, and the absolute retract.

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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • M. Benchohra
    • 1
  • J. J. Nieto
    • 2
  • A. Ouahab
    • 1
  1. 1.University of Sidi Bel-AbbèsSidi Bel-AbbèsAlgérie
  2. 2.University of Santiago de CompostelaSantiago de CompostelaSpain

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