An Approximate Solution for a Class of Ill-Posed Nonhomogeneous Cauchy Problems

In this paper, we consider a nonhomogeneous differential operator equation of first order u′t+Aut=ft. 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 u0=Φ or uT=Φ. 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.