An approximate solution for a class of ill-posed nonhomogeneous Cauchy problems (Q2666450)

Summary: In this paper, we consider a nonhomogeneous differential operator equation of first order \(u^\prime(t)+Au(t)=f(t)\). The coefficient operator \(A\) is linear unbounded and self-adjoint in a Hilbert space. We assume that the operator does not have a fixed sign. We associate to this equation the initial or final conditions \(u(0)=\Phi\) or \(u(T)=\Phi\). We note that the Cauchy problem is severely ill-posed in the sense that the solution if it exists does not depend continuously on the given data. Using a quasi-boundary value method, we obtain an approximate nonlocal problem depending on a small parameter. We show that regularized problem is well-posed and has a strongly solution. Finally, some convergence results are provided.