Sufficient conditions for convergence of the modified quasi-inverse method for generalized solution

In Almomani and Almomani [6], the authors established a priori estimation for the solution of the quasi-inverse problem. More recently, Almeflah et al. [3], the authors get some estimations of the classical solution of the same problem from the series representation of the solution [8]. Also, they establish an a priori estimation of a higher order than that in [6]. In this paper, we get the sufficient conditions for the convergence of the modified quasi-inverse method for the generalized solution. Our results play an important role in optimal control theory. Many works have been devoted to this problem, including the control problem of heat conduction with the inverse direction of time and integral boundary conditions [1, 4, 5, 7–14, 17, 18].