Resolvent Formulas for a Volterra Equation in Hilbert Space.

Abstract

The resolvent formula for a nonhomogeneous Volterra integrodifferential equation enables one to study the bahavior of solutions of the equation for large values of the time variable in terms of general properties of the forcing terms in the equation. This technique depends on having 'good' a priori estimates obtained for the resolvent kernel. When the solution takes its values in a Hilbert space, the resolvent kernel is a function whose values are operators on that space. It is important to know whether the norm of the resolvent kernel (or of its derivative) is integrable on 0, infinity. For a class of equations which includes linear models for the dynamics of viscoelastic materials, we develop sufficient conditions for the derivative of the resolvent kernel to be integrable. Results of this study and certain resolvent formulas can be used to study the asymptotic behavior of the solution y(t, x, f) as t approaches infinity. An application to a semilinear integro-partial differential equation is presented.

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Document Details

Document Type
Technical Report
Publication Date
Jan 01, 1981
Accession Number
ADA099355

Entities

People

  • Kenneth B. Hannsgen
  • Ralph W. Carr

Organizations

  • University of Wisconsin–Madison

Tags

DTIC Thesaurus Topics

  • Banach Space
  • Cauchy Problem
  • Classification
  • Contracts
  • Differential Equations
  • Equations
  • Hilbert Space
  • Identities
  • Inequalities
  • Integrals
  • Intervals
  • Mathematics
  • Partial Differential Equations
  • Two Dimensional
  • United States
  • Universities
  • Volterra Equations

Fields of Study

  • Mathematics

Readers

  • Calculus or Mathematical Analysis

Technology Areas

  • Space