## African Diaspora Journal of Mathematics

- Afr. Diaspora J. Math. (N.S.)
- Volume 16, Number 2 (2014), 72-91.

### Topological Structure of the Solutions Set of Impulsive Semilinear Differential Inclusions with Nonconvex Right-Hand Side

M. Benchohra, J. J. Nieto, and A. Ouahab

#### Abstract

In this paper, we study the topological structure of solution sets for the following first-order impulsive evolution inclusion with initial conditions: $$ \left\{ \begin{array}{rlll} y'(t)-Ay(t) &\in& F(t,y(t)), &\hbox{ a.e. } \, t\in J\backslash \{t_{1},\ldots,t_{m}\},\\ y(t^+_{k})-y(t^-_k)&=&I_{k}(y(t_{k}^{-})), &k=1,\ldots,m\\ y(0)&=&a\in E, \end{array} \right. $$where $J:=[0,b]$ and $0 = t_0 < t_1 < \,... \,< t_m < b$, $A$ is the infinitesimal generator of a $C_0-$semigroup of linear operator $T(t)$ on a separable Banach space $E$ and $F$ is a set-valued map. The functions $I_k$ characterize the jump of the solutions at impulse points $t_k$ ($k=1, \ldots ,m$). The continuous selection of the solution set is also investigated.

#### Article information

**Source**

Afr. Diaspora J. Math. (N.S.), Volume 16, Number 2 (2014), 72-91.

**Dates**

First available in Project Euclid: 20 October 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.adjm/1413809904

**Mathematical Reviews number (MathSciNet)**

MR3270008

**Zentralblatt MATH identifier**

1332.34103

**Subjects**

Primary: 34A37: Differential equations with impulses 34A60: Differential inclusions [See also 49J21, 49K21] 34G20: Nonlinear equations [See also 47Hxx, 47Jxx]

**Keywords**

Impulsive differential inclusions semigroup solution set compactness absolute retract

#### Citation

Benchohra, M.; Nieto, J. J.; Ouahab, A. Topological Structure of the Solutions Set of Impulsive Semilinear Differential Inclusions with Nonconvex Right-Hand Side. Afr. Diaspora J. Math. (N.S.) 16 (2014), no. 2, 72--91. https://projecteuclid.org/euclid.adjm/1413809904