Existence, uniqueness and convergence of approximate solutions of nonlocal functional differential equations (Q2891144)

The authors consider a class of abstract functional differential equations with nonlocal conditions in a separable Hilbert space \(H\): NEWLINE\[NEWLINE\begin{aligned} & u'(t)+au(t)=f(t,u(t),u(b(t))),\qquad t\in (0,T_0]\\ & u(0)+\sum^p_{k=1} c_k u(t_k)=u_0, \end{aligned}NEWLINE\]NEWLINE where \(0<t_1<t_2<...<t_p\leq T_0\), \(-A\) is the infinitesimal generator of a \(C_0\) semigroup on \(H\) and \(f\) is nonlinear. Under the assumption that \(A\) has pure point spectrum with a corresponding complete orthonormal system of eigenfunctions, existence, uniqueness and convergence of approximate solutions are proved. Analytic semigroups and fractional powers of the operator \(A\) are used. An example with a partial differential equation is given.