Degenerate Volterra equations of convolution type in Banach spaces and their applications

S. S. Orlov

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 8 (2016) no. 3, pp. 52-63

Résumé

The article is devoted to the problem of unique solvability of linear integral and integral-differential Volterra equations in Banach spaces with irreversible operator in the main part. Operator-valued kernel has a special form, $K(t, s) = g(t - s)A$, where $g = g(t)$ is a numeric function, and $A$ is a linear operator. Abstract equations of this kind are very typical for applications. For the study of such equations it is possible to use structural stack theory of two linear operators, which has been developed by Professor G. A. Sviridyuk and his students. Another peculiarity of the studied problems is multiple zero of function $g = g(t)$ at the point $t = 0$. Fundamental operator-functions of considered integral and integral-differential operators in Banach spaces are constructed under the assumption of relative spectrally boundness of operator $A$ with respect to degenerated main part of equations. On this basis, theorems of unique existence of solutions in the class of distributions with left-bounded support are proved. The dependence between the order of singularity of generalized solutions and multiplicity of zero of integral kernel at the initial point is ascertained. Also we have obtained conditions under which generalized solutions are equal to the classical solutions. Theorems formulated for abstract equations are applied to the study of significant initial boundary value problems arising in plasma physics and mathematical theory of elasticity.

Détail

Keywords: relative spectral boundedness of linear operator, fundamental operator-function.
Mots-clés : distribution

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