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Annali di Matematica Pura ed Applicata

, Volume 118, Issue 1, pp 181–198 | Cite as

On some integral and integro-differential equations in a Hilbert space

  • Gustaf Gripenberg
Article

Summary

We consider existence of solutions of the abstract nonlinear integral equation
$$u(t) + \int\limits_0^t {k(t - s)g(s)u(s)} ds \mathrel\backepsilon f(t) , t \geqslant 0$$
(1)
as well as the nonlinear integrodifferential equation
$$\left\{ {\begin{array}{*{20}c} {u'(t) + Bu(t) + K_0 Au(t) + \int\limits_0^t {K(t - s)Au(s)ds \mathrel\backepsilon F(t), t \geqslant 0} } \\ {u(0) = u_0 } \\ \end{array} } \right.$$
(2)
in a real Hilbert space H. The operators A, B and g(s) are monotone, the kernels k and K are operator-valued and K0 is nonnegative. We give sufficient conditions for the existence of solutions and in the case of(1) also for uniqueness. The results concerning(1) are related to earlier work by Londen and Londen & Staffans whereas those concerning(2) are related to work by Barbu and Crandall, Londen & Nohel.

Keywords

Hilbert Space Integral Equation Real Hilbert Space Integrodifferential Equation Nonlinear Integral Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. [1]
    B. Barbu,Nonlinear Volterra integro-differential equations in Hilbert space, to appear in Sem. Matem. Univ. Bari.Google Scholar
  2. [2]
    H. Brezis,Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North Holland / American Elsevier, Amsterdam (1973).Google Scholar
  3. [3]
    M. G. Crandall -S-O. Londen -J. A. Nohel,An abstract nonlinear Volterra integro-differential equations, MRC Technical Summary Report 1684, The University of Wisconsin (1976).Google Scholar
  4. [4]
    G. Gripenberg,An existence result for a nonlinear Volterra integral equation in a Hilbert space, to appear in SIAM J. Math. Anal.Google Scholar
  5. [5]
    G. Gripenberg,On a nonlinear Volterra integral equation in a Banach space, Report HTKK-MAT-A95, Helsinki University of Technology (1976).Google Scholar
  6. [6]
    S-O. Londen,On an integral equation in a Hilbert space, to appear in SIAM J. Math. Anal.Google Scholar
  7. [7]
    S-O. Londen -O. Staffans,A note on Volterra equations in a Hilbert space, Report HTKK-MAT-A90, Helsinki University of Technology (1976).Google Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1978

Authors and Affiliations

  • Gustaf Gripenberg
    • 1
  1. 1.HelsinkiFinland

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