Regularization by a modified quasi-boundary value method of the ill-posed problems for differential-operator equations of the first order

In this paper, we consider the differential-operator equation d u ( t ) d t + A u ( t ) = 0 , with A a self-adjoint unbounded operator coefficient, which does not have a fixed sign. The Cauchy problem for the equation above with conditions of the form u ( 0 ) = f or u ( T ) = f , is known to be an ill-posed problem. In this work, we will use a modified quasi-boundary value method; we obtain an approximate non-local problem depending on a small parameter α ∈ ] 0 , 1 [ . We show that the approximate problems are well-posed and that their solutions converge if the original problem has a classical solution. We also obtain a convergence result for these solutions.