Existence of a mild solution of a functional integrodifferential equation with nonlocal condition (Q2712551)

The authors consider the functional integrodifferential equation with nonlocal conditions of the form NEWLINE\[NEWLINE{du\over dt}(t)+ Au(t)= f\left(t,u_t, \int^t_0k(t, \tau,u_\tau) d\tau\right), \quad t\in[0,a],\tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(s)+ \bigl(g(u_{t_1}, \dots,u_{t_p}) \bigr)(s)= \Phi(s),\quad s\in[-r,0] \tag{2}NEWLINE\]NEWLINE where \(A\) is the infinitesimal operator of \(C_0\) semigroup of operators \(T(t)\), \(t\geq 0\), on a Banach space \(E\), \(\Phi\in C([-r,0],E)\), \(f,k,g\) are given functions satisfying some assumptions, \(0<t_1< \cdots< t_p\leq a\), \(p\in N\), and \(u_t(s)= u(t+s)\), \(t\in[0,a]\), \(s\in [-r,0]\), \(r>0\).NEWLINENEWLINENEWLINEAfter five assumptions on the functions \(f,k,g\) the authors define the notion of a mild solution of the nonlocal Cauchy problem (1), (2) and prove the existence and uniqueness of a mild solution for (1), (2) using the Banach contraction theorem. The paper contains also the study of continuous dependence of a mild solution of (1), (2) and an application on the controllability of a system with control parameter.