On the order of singularity of the generalized solution of the Volterra integral equation of convolutional type in Banach spaces

In the study of Volterra integral equations of convolution type defined on the semi-axis with the Fredholm operator in the main part and the operator-valued kernel $\mathcal{K}=\mathcal{K}(t)$ in Banach spaces we ordinarily solved the problem of constructing a generalized $\mathcal{K}(t)$-Jordan sets. Investigation of such equations was carried out for first time by N. A. Sidorov in the assumption of completeness of Jordan structure. The problem of solvability in the class of continuous functions was solved in his papers. The series of papers of M. V. Falaleev are devoted to the problems of existence and uniqueness of generalized solutions (i. e. solutions in the class of distributions with left-bounded support). There first has been proposed approach associated with the construction of the fundamental operator-function, which is generation of the classical notion of the fundamental solution. However, using all of these methods becomes very difficult when the kernel of integral equation has a null of any order at $t=0$. In this case, it is unclear how the generalized Jordan structure is built. A similar problem is posed in the study of degenerate linear integro-differential equations in Banach spaces with the differential part of a high order, which does not have at least one term of the highest order of lower derivatives. Thus, the problem of the solvability of degenerate convolution type integral equations with such special kernel remains unresolved. It should be noted that the boundary value problems of plasma physics can be reduced to this abstract integral equations. Therefore, interest in these mathematical objects due to their applied significance. In this paper we investigate described phenomenon in the particular case of special kind integral equation. It is shown that the presence of null of the integral equation kernel at point $t=0$ leads to an increase of the order of generalized solution singularity. We have ascertained an interdependence between these two characteristics. A theorem on a form of fundamental operator-function of corresponding integral operator is proved. On this basis, we obtained sufficient conditions of the existence and uniqueness of the generalized solution. Also we considered examples illustrating obtained abstract results.