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.