On the regularizability conditions of integral equations

Solving of integral equations of the first kind is an ill-posed problem. It is known that all problems can be divided into three disjoint classes: correct problems, ill-posed regularizable problems and ill-posed not regularizable problems. Problems of the first class are so good that no regularization method for them is needed. Problems of the third class are so bad that no one regularization method is applicable to them. A natural application field of the regularization method is the problems from the second class. But how to know that a particular integral equation belongs to the second class rather than to the third class? For this purpose a large number of sufficient regularizability conditions were constructed. In this article one infinite series of sufficient conditions for regularizability of integral equations constructed with the help of duality theory of Banach spaces is investigated. This method of constructing of sufficient conditions proved to be effective in solving of ill-posed problems. It is proved that these conditions are not pairwise equivalent even if we are restricted by the equations with the smooth symmetric kernels.