Generalized Solutions of Degenerate Integro-Differential Equations in Banach Spaces

• Published in: Journal of mathematical sciences (New York, N.Y.) Vol. 279; no. 5; pp. 710 - 722
• Main Author: Falaleev, M. V.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 23.03.2024; Springer; Springer Nature B.V
• Subjects: Banach spaces; Boundary value problems; Cauchy problems; Degeneration; Differential equations; Influence functions; Mathematical analysis; Mathematics; Mathematics and Statistics; Operators (mathematics); Smoothness; generalized function; 34G10; Fredholm operator; fundamental solution; 45N05; convolution; 45K05; Cauchy–Dirichlet problem; Banach space; resolvent
• ISSN: 1072-3374; 1573-8795
• DOI: 10.1007/s10958-024-07050-y

In this paper, we present a technique for constructing generalized solutions of the Cauchy problem for abstract integro-differential equations with degeneration in Banach spaces. A generalized solution is constructed as the convolution of the fundamental operator function (fundamental solution, influence function) of the integro-differential operator of the equation with a generalized function of a special form, which involves all input data of the original problem. Based on the analysis of the representation for the generalized solution, we obtain sufficient solvability conditions for the original Cauchy problem in the class of functions of finite smoothness. Under these sufficient conditions, the generalized solution constructed turns out to be a classical solution with the required smoothness. The abstract results obtained in the paper are applied to the study of applied initial-boundary-value problems from the theory of oscillations in viscoelastic media.