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

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 $0 \lt t_1 \lt t_2 \lt t_3 \lt ··· \lt t_p \lt \tau, 0 \lt \tau_{1} \lt \tau_{2} \lt ··· \lt \tau_{q} \lt r \lt \tau, Z$ is a Banach space $Z$, $z_t$ defined as a function from $[−r, 0]$ to $Z^ \alpha$ by $z_{t}(s) = z(t + s),−r \le s \le 0, g : C([−r, 0];Z^{\alpha}_{q}) \rightarrow C([−r, 0];Z^{\alpha})$ and $J_{k} : Z^{\alpha} \rightarrow Z^{\alpha}, F : [0,\tau] ×C(−r,0;Z^{\alpha}) \rightarrow Z$. In the above problem, $A : D(A)\subset Z \rightarrow Z$ is a sectorial operator in $Z$ with $−A$ being the generator of a strongly continuous compact semigroup $\{T(t)\}_{t \ge 0}$, and $Z^{\alpha}= D(A^{\alpha})$. The novelty of this work lies in the fact that the evolution equation studied here can contain non-linear terms that involve spatial derivatives and the system is subjected to the influence of impulses, delays and nonlocal conditions, which generalizes many works on the existence of solutions for semilinear evolution equations in Banch spaces. Our framework includes several important partial differential equations such as the Burgers equation and the Benjamin-Bona-Mohany equation with impulses, delays and nonlocal conditions.