## Communications in Mathematical Analysis

- 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

#### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 12 January 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.cma/1547262053

**Mathematical Reviews number (MathSciNet)**

MR3896871

**Zentralblatt MATH identifier**

07002176

**Subjects**

Primary: 34K30: Equations in abstract spaces [See also 34Gxx, 35R09, 35R10, 47Jxx] 34k35 35R10: Partial functional-differential equations

Secondary: 93B05: Controllability 93C10: Nonlinear systems

**Keywords**

sectorial operator fractional power spaces semigroups semilinear evolution equations impulses delays nonlocal conditions Karakostas fixed point theorem

#### Citation

Leiva, Hugo. Karakostas Fixed Point Theorem and the Existence of Solutions for Impulsive Semilinear Evolution Equations with Delays and Nonlocal Conditions. Commun. Math. Anal. 21 (2018), no. 2, 68--91. https://projecteuclid.org/euclid.cma/1547262053