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Impulsive problems for semilinear differential equations with nonlocal conditions. (English) Zbl 1188.34073
This paper deals with the impulsive differential equation with nonlocal conditions $$\align u'(t) &= Au(t) + f(t, u(t)), 0\le t\le b, t\ne t_i,\\ u(0) &= u_0-g(u),\\ \Delta u(t_i) &= I_i(u(t_i)), i = 1, 2, \dots, p, 0 < t_1 < t_2 < \dots < t_p < b, \endalign$$ where $A : D(A) \subseteq X \to X$ is the infinitesimal generator of a strongly continuous semigroup $T (t), t \ge 0,$ $X$ is a real Banach space, $\Delta u(t_i) =u(t_i^+)-u(t_i^-),$ $ u(t_i^+), u(t_i^-)$ denotes the right and the left limit of $u$ at $t_i,$ respectively and $f, g, I_i $ are appropriate continuous functions. By using the Hausdorff measure of noncompactness and fixed point techniques, the author proves the existence of a mild solution without the Lipschitz continuity of the mapping $f,$ in the cases when (i) $g$ and $I_i$ are Lipschitz and the semigroup $T (t), t > 0$, generated by the linear operator $A$ is compact, and (ii) $g$ is not Lipschitz and not compact.

MSC:
34G10Linear ODE in abstract spaces
47D06One-parameter semigroups and linear evolution equations
47N20Applications of operator theory to differential and integral equations
34A37Differential equations with impulses
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References:
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