Solvability of some partial integral equations in Hilbert space (Q933146)

The existence and uniqueness of a continuous solution of the integral equation \[ \int_0^t K(t,s)u(s)\,ds+Au(t)-\lambda(t)u(t)=f(t),\qquad 0\leq t\leq T, \] where \(A\) is a selfadjoint operator on a Hilbert space and \(K(t,s)\) is a real-valued kernel, is proved under the condition that \(\lambda\) is \(C^1\), \(\{\lambda(t):0\leq t<T\}\) is contained in the resolvent set of \(A\), \(\lambda(T)\) is an isolated point of the spectrum of \(A\), and \(f\) is \(C^1\) and \(\{f(T),f^\prime(T)\}\subset[\text{ker}(A-\lambda(T))]^\perp\).