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Impulsive differential inclusions involving evolution operators in separable Banach spaces

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Ukrainian Mathematical Journal Aims and scope

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 = t 0 < t 1 < … < t m <…, \( m\in \mathbb{N} \), lim k→∞ t k = ∞, 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 I k characterize the jumps of solutions at the impulse points t k , k = 1, ….The mapping L: PC b 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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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 7, pp. 867–891, July, 2012.

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Benchohra, M., Nieto, J.J. & Ouahab, A. Impulsive differential inclusions involving evolution operators in separable Banach spaces. Ukr Math J 64, 991–1018 (2012). https://doi.org/10.1007/s11253-012-0695-0

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