Operator semigroups for functional-differential equations with delay

Authors:
W. M. Ruess and W. H. Summers

Journal:
Trans. Amer. Math. Soc. **341** (1994), 695-719

MSC:
Primary 34K30; Secondary 47H20, 47N20

DOI:
https://doi.org/10.1090/S0002-9947-1994-1214785-X

MathSciNet review:
1214785

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Abstract | References | Similar Articles | Additional Information

Abstract: We show that a strongly continuous operator semigroup can be associated with the functional differential delay equation (FDE)

**[1]**F. V. Atkinson and J. R. Haddock,*On determining phase spaces for functional differential equations*, Funkcial. Ekvac.**31**(1988), 331-347. MR**987790 (90f:45009)****[2]**V. Barbu,*Nonlinear semigroups and differential equations in Banach spaces*, Noordhoff, Leyden, 1976. MR**0390843 (52:11666)****[3]**M. Bardi,*A nonautonomous nonlinear functional differential equation arising in the theory of population dynamics*, J. Math. Anal. Appl.**109**(1985), 492-508. MR**802909 (86m:92023)****[4]**D. W. Brewer,*A nonlinear semigroup for a functional differential equation*, Trans. Amer. Math. Soc.**236**(1978), 173-191. MR**0466838 (57:6712)****[5]**-,*A nonlinear contraction semigroup for a functional differential equation*, Volterra Equations, (S.-O. Londen and O. J. Staffans, eds.), Lecture Notes in Math., vol. 737, Springer, 1979, pp. 35-44. MR**551026 (81h:47057)****[6]**-,*The asymptotic stability of a nonlinear functional differential equation of infinite delay*, Houston J. Math.**6**(1980), 321-330. MR**597773 (81k:34081)****[7]**-,*Locally Lipschitz continuous functional differential equations and nonlinear semigroups*, Illinois J. Math.**26**(1982), 374-381. MR**658448 (83i:34091)****[8]**M. G. Crandall,*A generalized domain for semigroup generators*, Proc. Amer. Math. Soc.**37**(1973), 434-440. MR**0313873 (47:2426)****[9]**M. G. Crandall and T. M. Liggett,*Generation of semi-groups of nonlinear transformations on general Banach spaces*, Amer. J. Math.**93**(1971), 265-298. MR**0287357 (44:4563)****[10]**M. G. Crandall and A. Pazy,*Nonlinear evolution equations in Banach spaces*, Israel J. Math.**11**(1972), 57-94. MR**0300166 (45:9214)****[11]**J. Dyson and R. Villella-Bressan,*Functional differential equations and non-linear evolution operators*, Proc. Roy. Soc. Edinburgh**75A**(1975/76), 223-234. MR**0442402 (56:784)****[12]**-,*Semigroups of translations associated with functional and functional differential equations*, Proc. Roy. Soc. Edinburgh**82A**(1979), 171-188. MR**532900 (80i:34113)****[13]**H. Flaschka and M. J. Leitman,*On semigroups of nonlinear operators and the solution of the functional differential equation*, J. Math. Anal. Appl.**49**(1975), 649-658. MR**0361959 (50:14401)****[14]**J. A. Goldstein,*Semigroups of linear operators and applications*, Oxford Math. Monographs, Oxford Univ. Press, New York, 1985. MR**790497 (87c:47056)****[15]**J. R. Haddock and W. E. Hornor,*Precompactness and convergence in norm of positive orbits in a certain fading memory space*, Funkcial. Ekvac.**31**(1988), 349-361. MR**987791 (90a:34160)****[16]**J. K. Hale and J. Kato,*Phase space for retarded equations with infinite delay*, Funkcial. Ekvac.**21**(1978), 11-41. MR**0492721 (58:11793)****[17]**F. Kappel and W. Schappacher,*Some considerations to the fundamental theory of infinite delay equations*, J. Differential Equations**37**(1980), 141-183. MR**587220 (81j:34117)****[18]**N. MacDonald,*Time lag in a model of a biochemical reaction sequence with end product inhibition*, J. Theoret. Biol.**67**(1977), 549-556. MR**0490019 (58:9382)****[19]**A. Pazy,*Semigroups of linear operators and applications to partial differential equations*, Appl. Math. Sci., vol. 44, Springer, New York, 1983. MR**710486 (85g:47061)****[20]**A. T. Plant,*Nonlinear semigroups of translations in Banach space generated by functional differential equations*, J. Math. Anal. Appl.**60**(1977), 67-74. MR**0447745 (56:6055)****[21]**W. M. Ruess,*The evolution operator approach to functional differential equations with delay*, Proc. Amer. Math. Soc. (to appear). MR**1154248 (93m:34132)****[22]**C. C. Travis and G. F. Webb,*Existence and stability for partial functional differential equations*, Trans. Amer. Math. Soc.**200**(1974), 395-418. MR**0382808 (52:3690)****[23]**G. F. Webb,*Autonomous nonlinear functional differential equations and nonlinear semigroups*, J. Math. Anal. Appl.**46**(1974), 1-12. MR**0348224 (50:722)****[24]**-,*Functional differential equations and nonlinear semigroups in*-*spaces*, J. Differential Equations**20**(1976), 71-89.**[25]**-,*Asymptotic stability for abstract nonlinear functional differential equations*, Proc. Amer. Math. Soc.**54**(1976), 225-230. MR**0402237 (53:6058)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1994-1214785-X

Article copyright:
© Copyright 1994
American Mathematical Society