## An existence result on a Volterra equation in a Banach space

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- by Stig-Olof Londen PDF
- Trans. Amer. Math. Soc.
**235**(1978), 285-304 Request permission

## Abstract:

Let*W*be a real reflexive Banach space, dense in a Hilbert space

*H*and with dual $W’$. Let the injection $W \to H$ be continuous and compact. We consider the nonlinear integral equation \begin{equation}\tag {$1$} u’(t) + \int _0^t {a(t - \tau )Au(\tau )d\tau = f(t),\quad t \geqslant 0,} \end{equation} where

*a, f, A*are given and

*u*is the unknown. The kernel $a(t)$ maps ${R^ + }$ into

*R*and

*f*takes values in

*H*. The nonlinear function

*A*is a maximal monotone mapping $W \to W’$. Making use of the theory of maximal monotone operators we prove an existence result on (1). This result is used to obtain approximate solutions to the related nonlinear hyperbolic differential equation $u''(t) + Au(t) = f’(t),t \geqslant 0$.

## References

- Viorel Barbu,
*Nonlinear semigroups and differential equations in Banach spaces*, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976. Translated from the Romanian. MR**0390843** - Viorel Barbu,
*Nonlinear Volterra equations in a Hilbert space*, SIAM J. Math. Anal.**6**(1975), 728–741. MR**377620**, DOI 10.1137/0506064 - H. Brézis,
*Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert*, North-Holland Mathematics Studies, No. 5, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973 (French). MR**0348562**
M. G. Crandall, S.-O. Londen and J. A. Nohel, - Stig-Olof Londen,
*On an integral equation in a Hilbert space*, SIAM J. Math. Anal.**8**(1977), no. 6, 950–970. MR**511229**, DOI 10.1137/0508073
L. Tartar, MRC Technical Summary Reports 1571, 1589, Univ. Wisconsin, Madison, 1975.

*An abstract nonlinear Volterra integrodifferential equation*, MRC Technical Summary Report 1684, Univ. Wisconsin, Madison, 1976.

## Additional Information

- © Copyright 1978 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**235**(1978), 285-304 - MSC: Primary 45N05; Secondary 45D05, 47H15
- DOI: https://doi.org/10.1090/S0002-9947-1978-0473770-7
- MathSciNet review: 0473770