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

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

MathSciNet review:
0293355

Full-text PDF Free Access

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]**C. M. Dafermos,*An abstract Volterra equation with applications to linear viscoelasticity*, J. Differential Equations**7**(1970), 554-569. MR**41**#4305. MR**0259670 (41:4305)****[2]**A. Friedman,*Partial differential equations of parabolic type*, Prentice-Hall, Englewood Cliffs, N. J., 1964. MR**31**#6062. MR**0181836 (31:6062)****[3]**M. E. Gurtin and I. Herrera,*On dissipation inequalities and linear viscoelasticity*, Quart. Appl. Math.**23**(1965), 235-245. MR**32**#6772. MR**0189346 (32:6772)****[4]**A. Halanay,*On the asymptotic behavior of the solutions of an integro-differential equation*, J. Math. Anal. Appl.**10**(1965), 319-324. MR**31**#579. MR**0176304 (31:579)****[5]**K. B. Hannsgen,*Indirect abelian theorems and a linear Volterra equation*, Trans. Amer. Math. Soc.**142**(1969), 539-555. MR**39**#7364. MR**0246058 (39:7364)****[6]**-,*On a nonlinear Volterra equation*, Michigan Math. J.**16**(1969), 365-376. MR**40**#3225. MR**0249984 (40:3225)****[7]**E. Hille and R. S. Phillips,*Functional analysis and semi-groups*, rev. ed., Amer. Math. Soc. Colloq. Publ., vol. 31, Amer. Math. Soc., Providence, R. I., 1957. MR**19**, 664. MR**0089373 (19:664d)****[8]**J. J. Levin,*The asymptotic behavior of the solution of a Volterra equation*, Proc. Amer. Math. Soc.**14**(1963), 534-541. MR**27**#2824. MR**0152852 (27:2824)****[9]**J. J. Levin and J. A. Nohel,*Note on a nonlinear Volterra equation*, Proc. Amer. Math. Soc.**14**(1963), 924-929. MR**28**#437. MR**0157201 (28:437)****[10]**-,*Perturbations of a nonlinear Volterra equation*, Michigan Math. J.**12**(1965), 431-447. MR**32**#336. MR**0182854 (32:336)****[11]**M. Loève,*Probability theory. Foundations. Random sequences*, 2nd rev. ed., University Series in Higher Math., Van Nostrand, Princeton, N. J., 1960. MR**23**#A670. MR**0123342 (23:A670)****[12]**R. C. MacCamy,*Exponential stability for a class of functional differential equations*, Arch. Rational Mech. Anal.**40**(1971), 120-138. MR**0268467 (42:3364)****[13]**H. König and J. Meixner,*Lineare Systeme und lineare Transformationen*, Math. Nachr.**19**(1958), 256-322. MR**21**#7410. MR**0108695 (21:7410)****[14]**J. A. Nohel,*Qualitative behavior of solutions of nonlinear Volterra equations*, Stability Problems of Solutions of Differential Equations, Proceedings of a NATO Advanced Study Institute, Oderisi, Gubbis, 1966, pp. 177-210. MR**0405024 (53:8820)****[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.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
45M05

Retrieve articles in all journals with MSC: 45M05

Additional Information

DOI:
https://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