Stability theorems for some functional equations

Authors:
R. C. MacCamy and J. S. W. Wong

Journal:
Trans. Amer. Math. Soc. **164** (1972), 1-37

MSC:
Primary 45M05

MathSciNet review:
0293355

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Abstract | References | Similar Articles | Additional Information

Abstract: Functional-differential equations of the form

*g*an operator with domain in .

*g*may be unbounded.

*A*is called strongly positive if there exists a semigroup exp

*St*, where

*S*is symmetric and , such that

*St*is positive, that is,

*A*is strongly positive, and

*g*and

*f*are suitably restricted, then any solution which is weakly bounded and uniformly continuous must tend weakly to zero. Examples are given of both ordinary and partial differential-functional equations.

**[1]**Constantine M. Dafermos,*An abstract Volterra equation with applications to linear viscoelasticity*, J. Differential Equations**7**(1970), 554–569. MR**0259670****[2]**Avner Friedman,*Partial differential equations of parabolic type*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR**0181836****[3]**M. E. Gurtin and I. Herrera,*On dissipation inequalities and linear viscoelasticity*, Quart. Appl. Math.**23**(1965), 235–245. MR**0189346****[4]**A. Halanay,*On the asymptotic behavior of the solutions of an integro-differential equation*, J. Math. Anal. Appl**10**(1965), 319–324. MR**0176304****[5]**Kenneth B. Hannsgen,*Indirect abelian theorems and a linear Volterra equation*, Trans. Amer. Math. Soc.**142**(1969), 539–555. MR**0246058**, 10.1090/S0002-9947-1969-0246058-1**[6]**Kenneth B. Hannsgen,*On a nonlinear Volterra equation*, Michigan Math. J.**16**(1969), 365–376. MR**0249984****[7]**Einar Hille and Ralph S. Phillips,*Functional analysis and semi-groups*, American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, Providence, R. I., 1957. rev. ed. MR**0089373****[8]**J. J. Levin,*The asymptotic behavior of the solution of a Volterra equation*, Proc. Amer. Math. Soc.**14**(1963), 534–541. MR**0152852**, 10.1090/S0002-9939-1963-0152852-8**[9]**J. J. Levin and J. A. Nohel,*Note on a nonlinear Volterra equation*, Proc. Amer. Math. Soc.**14**(1963), 924–929. MR**0157201**, 10.1090/S0002-9939-1963-0157201-7**[10]**J. J. Levin and J. A. Nohel,*Perturbations of a nonlinear Volterra equation*, Michigan Math. J.**12**(1965), 431–447. MR**0182854****[11]**Michel Loève,*Probability theory*, 2nd ed. The University Series in Higher Mathematics. D. Van Nostrand Co., Inc., Princeton, N. J.-Toronto-New York-London, 1960. MR**0123342****[12]**R. C. MacCamy,*Exponential stability for a class of functional differential equations.*, Arch. Rational Mech. Anal.**40**(1970/1971), 120–138. MR**0268467****[13]**Heinz König and Josef Meixner,*Lineare Systeme und lineare Transformationen*, Math. Nachr.**19**(1958), 265–322 (German). MR**0108695****[14]**John A. Nohel,*Qualitative behaviour of solutions of nonlinear Volterra equations*, Stability problems of solutions of differential equations (Proc. NATO Advanced Study Inst., Padua, 1965) Edizioni “Oderisi”, Gubbio, 1966, pp. 177–210. MR**0405024****[15]**M. M. Vaĭnberg,*Variational methods for the study of non-linear operators*, GITTL, Moscow, 1956; English transl., Holden-Day, San Francisco, 1964. MR**19**, 567;**31**#638.

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DOI:
http://dx.doi.org/10.1090/S0002-9947-1972-0293355-X

Keywords:
Functional-differential equation,
Gårding's inequality,
Laplace transform,
partial differential-functional equation,
positivity,
strong ellipticity,
strong positivity,
symmetric,
weak boundedness,
weak convergence,
weak stability,
weak uniform continuity

Article copyright:
© Copyright 1972
American Mathematical Society