A modified quasi-boundary value method for ill-posed problems (Q705276)

Consider the final value problem (FVP) \[ u'(t)+Au(t) =0, \text{ } 0\leq t\leq T, \text{ } u(T)=f, \] in a Hilbert space \(H\), where \(A\) is a selfadjoint operator on \(H\) with eigenvalues \(\lambda_i \to \infty\). It is well known that (FVP) is an ill-posed problem. In this interesting paper, the authors perturb the final condition and reduce (FVP) to the nonlocal problem \[ u'(t)+Au(t) =0, \text{ } 0\leq t\leq T, \text{ } u(T)-\alpha u'(0)=f.\eqno(1) \] It is shown that (1) is well-posed for each \(\alpha >0\) and that their solutions \(u_{\alpha}\) converge in \(C^1([0,T], H)\) as \(\alpha \to 0\) if and only if (FVP) has a classical solution. The rates of convergence are also given.