Perturbation theory for Volterra integrodifferential systems
About: This article is published in Journal of Differential Equations.The article was published on 1970-11-01 and is currently open access. It has received 108 citations till now. The article focuses on the topics: Volterra integral equation & Poincaré–Lindstedt method.
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TL;DR: In this paper, conditions are given which ensure the existence of a resolvent operator for an integrodifferential equation in a Banach space, which is similar to an evolution operator for nonautonomous differential equations.
Abstract: Conditions are given which ensure the existence of a resolvent operator for an integrodifferential equation in a Banach space. The resolvent operator is similar to an evolution operator for nonautonomous differential equations in a Banach space. As in the finite dimensional case, this operator is used to obtain a variation of parameters formula which can be used to obtain results concerning the asymptotic behaviour of solutions and weak solutions.
261 citations
Cites background from "Perturbation theory for Volterra in..."
...If X is finite dimensional the work of Grossman and Miller [8] is of significance to us as they develop perturbation theory for (VE) using the resolvent operator for (VE)....
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...In the finite dimensional case this has been discussed in Grossman and Miller [8] and also in Grimmer and Seifert [7] among others....
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TL;DR: On etudie l'equation integrodifferentielle x˙(t)=A 0 x(t)+∫ 0 t B(t-s)x(s)ds+f(t), t≥ 0, x(0)=x 0 ED(A)⊂X and l'Equation integrale x(T)=∫ t a(t − s)x (s)d+f (t) dans un espace de Banach X as discussed by the authors.
135 citations
TL;DR: In this article, Liapunov stability properties of solution to a certain system of Volterra integrodifferential equations are studied, and sufficient conditions for uniform stability and uniform asymptotic stability are derived in the form of a theorem.
125 citations
TL;DR: In this paper, the authors studied the problem of initial v&e problems with initial values of T > 0 and showed that these problems almost always occur in applications with T = 0.
77 citations
60 citations
References
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31 Dec 1934
TL;DR: In this article, a generalized harmonic analysis in the complex domain of random functions has been proposed, based on Szasz's theorem and a class of singular integral equations of the exponential type.
Abstract: Introduction Quasi-analytic functions Szasz's theorem Certain integral expansions A class of singular integral equations Entire functions of the exponential type The closure of sets of complex exponential functions Non-harmonic Fourier series and a gap theorem Generalized harmonic analysis in the complex domain The harmonic analysis of random functions Bibliography Index.
1,416 citations
TL;DR: In this paper, theoremes de point fixe and d'autres moyens de l' Analyse fonctionnelle are used to compute the solutions of equations integrales non-lineaires.
Abstract: Le but du travail est l'etude du comportement global des solutions des equations integrales non-lineaires de Volterra. On utilise les theoremes de point fixe et d'autres moyens de l' Analyse fonctionnelle.
67 citations
TL;DR: Volterra integral equations linearization, discussing integral kernels, integrodifferential equations and reactor dynamics was discussed in this paper, where integral kernels and integro-linear equations were discussed.
63 citations