We consider an integro-differential equation in convolutions of a special kind in Banach spaces with the Fredholm operator in the main part. The article concerns with the problem of unique solvability of the Cauchy-problem for this equation in the class of distributions with left-bounded support. The research is based on the theory of fundamental operator-functions of integro-differential operators in Banach spaces. The Fredholm operator from the differential part of the equation has the complete Jordan set. The kernel of the integral part of the equation is equal to zero at the starting point, which multiplicity is determined by a maximum length of Jordan chains elements of the Fredholm operator kernel and by the order of the equation’s differential operator. Under these assumptions, we prove the theorem on the structure of the fundamental operator-function (the fundamental solution) of the equation. Based on the fundamental operator-function the generalized solution is constructed. The dependence between the generalized solution and the classical (smooth) solution is considered. The abstract results are illustrated by an example of the initial-boundary value problem for the partial integrodifferential equation. The presented research continues the papers in the field, and can be generalized to other cases of a singular operator of the leading derivative (Noetherity, spectral, sectorial or radial boundedness). The results of these investigations make it possible to explore the mathematical models of the theory of oscillations in viscoelastic media and of the theory of electric chains.

MSC

34G10, 45K05, 45N05

1. Vainberg М.М., Trenogin V.А. Theory of Branching of Solutions of Nonlinear Equations. Moscow, Nauka, 1969. 528 p. (in Russian)

2. Vladimirov V.S. Generalized Functions in Mathematical Physics. Moscow, Nauka, 1979. 320 p. (in Russian)

3. Sidorov N.А., Romanova О.А. About Application some Results of Theory of Branching by Solving of Differential Equations with Degenerating. Different. Equations, 1983, vol. 19, no 9, pp. 1516-1526.

4. Falaleev М.V. Fundamental Operator-function of Singular Differential Operators in Banach Spaces. Sib. Math. Journal, 2000, vol. 41, no 5, pp. 1167-1182.

5. Falaleev М.V. Singular integro-differential equations of the special type in Banach spaces and it’s application. Izv. Irkut. gos. un-ta. Ser. Mathematics, 2013, vol. 6, no 4, pp. 128-137.

6. Sidorov N., Loginov B., Sinitsyn A., Falaleev M. Lyapunov – Schmidt Methods in Nonlinear Analysis and Applications. Dordrecht, Kluwer Academic Publ., 2002. 548 p.