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Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions. (English) Zbl 1171.34052
The authors present the existence of integral solutions and extremal integral solutions for the following problem
\[ \begin{aligned} y'(t)\in Ay(t)+F(t,y_t), &\quad t\in [0,T],\\ \Delta y|_{t=t_k}\in I_k(y(t_k)), &\quad k=1,\ldots,m, \\ y(t)=\varphi(t),&\quad t\in[-r,0], \end{aligned} \]
where \(F: [0,T]\times {\mathcal D}\to {\mathcal P}(E)\) are a multivalued maps, \({\mathcal P}(E)\) is the family of all nonempty subsets of \(E\), \(A: D(A)\subset E\to E\) is a nondensely defined closed linear operator on \(E\), \(0<r<\infty\), \(0=t_{0}<t_1<\cdots<t_m<t_{m+1}=T\),
\[ \begin{split} {\mathcal D}= \{\psi:[-r,0]\to E: \psi\text{ is continuous everywhere except for a finite number}\\ \text{of points } \bar{t}\text{ at which }\psi(\bar{t}^-)\text{ and }\psi(\bar{t}^+)\text{ exist and satisfy }\psi(\bar{t}^-)=\psi(\bar{t})\}, \end{split} \]
\(\varphi\in {\mathcal D},\) \(I_k\in E\to {\mathcal P}(E)\) \((k=1,\ldots,m)\), \(\Delta y|_{t=t_k}= y(t_k^+)- y(t_k^-)\), \(y(t_k^+)= \lim_{h\to 0^+}y(t_k+h)\) and \(y(t_k^-)= \lim_{h\to 0^+} y(t_k-h)\) stand for the right and the left limits of \(y(t)\) at \(t=t_k\), respectively.
For any function \(y\) defined on \([-r,b]\) and any \(t\in J\), \(y_t\) refers to the element of \({\mathcal D}\) such that
\[ y_t(\theta)=y(t+\theta),\quad \theta\in[-r,0]; \]
thus the function \(y_t\) represents the history of the state from time \(t-r\) up to the present time \(t\). Also the controllability of above problem are investigated. An examples is presented.

MSC:
34K45 Functional-differential equations with impulses
34K30 Functional-differential equations in abstract spaces
34K35 Control problems for functional-differential equations
93B05 Controllability
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