Filippov-Ważewski theorems and structure of solution sets for first order impulsive semilinear functional differential inclusions
Keywords
Impulsive functional differential inclusions, mild solution, Filippov's theorem, relaxation, solution set, compactness, AR, $R_\delta$, contractibility, acyclicityAbstract
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.Downloads
Published
2008-12-01
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1.
DJEBALI, Smaïl, GÓRNIEWICZ, Lech and OUAHAB, Abdelghani. Filippov-Ważewski theorems and structure of solution sets for first order impulsive semilinear functional differential inclusions. Topological Methods in Nonlinear Analysis. Online. 1 December 2008. Vol. 32, no. 2, pp. 261 - 312. [Accessed 20 April 2024].
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