Uniqueness and Stability of Solutions for Certain Linear Equations of the First Kind with Two Variables

The article is devoted to the study of uniqueness and stability of solutions of linear integral equations of the first kind with two independent variables. The relevance of the problem is due to the needs in development of new approaches for the regularization and uniqueness of the solution of linear integral equations of the first kind with two independent variables. Integral and operator equations of the first kind with two independent variables arise in theoretical and applied problems.Works of A.N. Tikhonov, M.M. Lavrentyev and B.K. Ivanov, in which a new concept of correctness of setting such targets is given, different from the classical, show tools for research of ill-posed problems, which stimulated the interest to the integral equations that are of great practical importance. At the present time the theory and applications of ill-posed problems have been rapidly developing. One of the classes of such ill-posed problems are integral equations of the first kind with two independent variables. As of approximate solutions of such problems, stable to small variations of the initial data, we use the solutions derived by the method of regularization. In this article we prove the theorem of uniqueness and obtain estimates of stability for such equations in families of sets of correctnesses. For the tasks solution the methods of functional analysis and method of nonnegative quadratic forms are used. The results of the work are new.

linear, inteqral equations, first kind, two variables, solution, uniqueness and stability