On solvability of Cauchy problem for systems of non-linear integro-differential equations in partial derivatives with parameters

It is possible to carry out the method of transforming solutions to study the problem of solvability of the Cauchy problem for non-linear integro-differential partial differential equations. The essence of this approach is the transformation of the initial Cauchy problem into an equivalent Volterra integral equation of the second kind, to which one can apply the topological method – the principle of condensed mappings. Sufficient conditions are defined for given functions for which the original problem is solvable from the conditions of contraction of the operator u.
In this paper we study the solvability of the Cauchy problem for systems of non-linear integro-differential partial differential equations of the first order with a parameter and an integral representation of the solutions obtained. Further, for a new class of systems of non-linear integro-differential partial differential equations of the third order, sufficient conditions for the existence of solutions of the Cauchy problem are found, and, in addition, an integral representation of such solutions is constructed. In view of the non-linearity of the initial problems, sufficient conditions do not guarantee the uniqueness of the solutions obtained.