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

Abstract

In this paper, we consider a nonhomogeneous differential operator equation of first order $u^{\prime }\left(t\right)+Au\left(t\right)=f\left(t\right)$ . 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 $u\left(0\right)=\mathrm{\Phi }\text{\hspace{0.17em}or\hspace{0.17em}}u\left(T\right)=\mathrm{\Phi }$ . 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.