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Karakostas Fixed Point Theorem and the Existence of Solutions for Impulsive Semilinear Evolution Equations with Delays and Nonlocal Conditions

Commun. Math. Anal.
Volume 21, Number 2 (2018), 68 - 91

Karakostas Fixed Point Theorem and the Existence of Solutions for Impulsive Semilinear Evolution Equations with Delays and Nonlocal Conditions

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Abstract

We prove the existence and uniqueness of the solutions for the following impulsive semilinear evolution equations with delays and nonlocal conditions:
$$
\left\{
\begin{array}{lr}
\acute{z} =-Az +F(t,z_{t}), & z\in Z, \quad t \in (0, \tau], t \neq t_k, \\
z(s)+(g(z_{\tau_1},z_{\tau_2},\dots, z_{\tau_q}))(s) = \phi(s), & s \in [-r,0],\\
z(t_{k}^{+}) = z(t_{k}^{-})+J_{k}(z(t_{k})), & k=1,2,3, \dots, p.
\end{array}
\right.
$$where $0 < t_1 < t_2