### Filippov-Ważewski theorems and structure of solution sets for first order impulsive semilinear functional differential inclusions

DOI: http://dx.doi.org/10.12775/TMNA.2008.049

#### Abstract

In this paper, we first present an impulsive version of Filippov's

Theorem for first-order semilinear functional differential

inclusions of the form:

$$

\cases

(y'-Ay) \in F(t,y_t) &\text{a.e. } t\in J\setminus \{t_{1},\ldots,t_{m}\},

\\

y(t^+_{k})-y(t^-_k)=I_{k}(y(t_{k}^{-})) &\text{for } k=1,\ldots,m,

\\

y(t)=\phi(t) &\text{for } t\in[-r,0],

\endcases

$$

where $J=[0,b]$, $A$ is the infinitesimal generator of a

$C_0$-semigroup 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 $t_k$ ($k=1,\ldots,m$). Then the

convexified problem is considered and a Filippov-Wa{\plr ż}ewski result

is proved. Further to several existence results, the topological

structure of solution sets -- closeness and compactness -- is also

investigated. Some results from topological fixed point theory

together with notions of measure on noncompactness are used.

Finally, some geometric properties of solution sets, AR,

$R_\delta$-contractibility and acyclicity, corresponding to

Aronszajn-Browder-Gupta type results, are obtained.

Theorem for first-order semilinear functional differential

inclusions of the form:

$$

\cases

(y'-Ay) \in F(t,y_t) &\text{a.e. } t\in J\setminus \{t_{1},\ldots,t_{m}\},

\\

y(t^+_{k})-y(t^-_k)=I_{k}(y(t_{k}^{-})) &\text{for } k=1,\ldots,m,

\\

y(t)=\phi(t) &\text{for } t\in[-r,0],

\endcases

$$

where $J=[0,b]$, $A$ is the infinitesimal generator of a

$C_0$-semigroup 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 $t_k$ ($k=1,\ldots,m$). Then the

convexified problem is considered and a Filippov-Wa{\plr ż}ewski result

is proved. Further to several existence results, the topological

structure of solution sets -- closeness and compactness -- is also

investigated. Some results from topological fixed point theory

together with notions of measure on noncompactness are used.

Finally, some geometric properties of solution sets, AR,

$R_\delta$-contractibility and acyclicity, corresponding to

Aronszajn-Browder-Gupta type results, are obtained.

#### Keywords

Impulsive functional differential inclusions; mild solution; Filippov's theorem; relaxation; solution set; compactness; AR; $R_\delta$; contractibility; acyclicity

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