The non-local Cauchy problem for semilinear integrodifferential equations with deviating argument (Q2716977)

Let \(A\) be the infinitesimal generator of a strongly continuous semigroup in a Banach space \(X\), \(Z(t)\) a bounded operator for \(t\in [0, T]\), \(T>0\), \(f:[0,T] \times X\to X\), \(g:X^p\to X\), \(p\in N\), be given functions. The authors consider the following integrodifferential equation with a deviating argument and nonlocal condition of the form NEWLINE\[NEWLINEu'(t)=A\bigl( u(t)+\int^t_0 Z(t-s) u(s)ds \bigr)+ f\biggl(t,u\bigl( \sigma(t)\bigr) \biggr), \quad 0\leq t\leq T,\tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(0)+g \bigl(u(t_1), \dots,u(t_p) \bigr)=u_0 \tag{2}NEWLINE\]NEWLINE where \(0\leq t_0<t_1< \cdots< t_p\leq T\), \(u_0\in X\), \(\sigma\in C([0,T], [0,T])\), \(\sigma(t)\leq t\), \(t \in [0,T]\). The solution for (1) with \(u(t_0)=u_0\) can be written as NEWLINE\[NEWLINEu(t)= R(t) \Bigl(u_0- g\bigl(u(t_1), \dots,u(t_p) \bigr)+ \int^t_0R(t-s) f\biggl( s,u \bigl( \sigma(s) \bigr)\biggr) \Bigr)dsNEWLINE\]NEWLINE where \(R\) is the resolvent operator. NEWLINENEWLINENEWLINEThe authors define the resolvent operator \(R\), the notion of a mild solution to (1), (2) and the notion of classical solution to (1), (2). Then the authors prove the existence and uniqueness of mild and classical solutions to the nonlocal Cauchy problem (1), (2). The results are established by using the method of semigroups and the contraction mapping principle.