Operator semigroups for functional-differential equations with delay

Ruess, W. M.; Summers, W. H.

doi:10.1090/S0002-9947-1994-1214785-X

Operator semigroups for functional-differential equations with delay
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by W. M. Ruess and W. H. Summers PDF

Trans. Amer. Math. Soc. 341 (1994), 695-719 Request permission

Abstract:

We show that a strongly continuous operator semigroup can be associated with the functional differential delay equation (FDE) \[ \left \{ {\begin {array}{*{20}{c}} {x\prime (t) + ax(t) + Bx(t) \backepsilon F({x_t}),} \hfill & {t \geq 0} \hfill \\ {x{|_{{\mathbb {R}^ - }}} = \varphi \in E} \hfill & {} \hfill \\ \end {array} } \right .\] under local conditions which give wide latitude to those subsets of the state space $X$ and initial data space $E$, respectively, where (a) the (generally multivalued) operator $B \subseteq X \times X$ is defined and accretive, and (b) the historyresponsive function $F:D(F) \subseteq E \to X$ is defined and Lipschitz continuous. The associated semigroup is then used to investigate existence and uniqueness of solutions to (FDE). By allowing the domain of the solution semigroup to be restricted according to specific local properties of $B$ and $F$, moreover, our methods automatically lead to assertions on flow invariance. We illustrate our results through applications to the Goodwin oscillator and a single species population model.

References

F. V. Atkinson and J. R. Haddock, On determining phase spaces for functional-differential equations, Funkcial. Ekvac. 31 (1988), no. 3, 331–347. MR 987790
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, DOI 10.1007/978-94-010-1537-0
Martino Bardi, A nonautonomous nonlinear functional-differential equation arising in the theory of population dynamics, J. Math. Anal. Appl. 109 (1985), no. 2, 492–508. MR 802909, DOI 10.1016/0022-247X(85)90164-7
Dennis W. Brewer, A nonlinear semigroup for a functional differential equation, Trans. Amer. Math. Soc. 236 (1978), 173–191. MR 466838, DOI 10.1090/S0002-9947-1978-0466838-2
Dennis W. Brewer, A nonlinear contraction semigroup for a functional differential equation, Volterra equations (Proc. Helsinki Sympos. Integral Equations, Otaniemi, 1978) Lecture Notes in Math., vol. 737, Springer, Berlin, 1979, pp. 35–44. MR 551026
Dennis W. Brewer, The asymptotic stability of a nonlinear functional differential equation of infinite delay, Houston J. Math. 6 (1980), no. 3, 321–330. MR 597773
Dennis W. Brewer, Locally Lipschitz continuous functional-differential equations and nonlinear semigroups, Illinois J. Math. 26 (1982), no. 3, 374–381. MR 658448
Michael G. Crandall, A generalized domain for semigroup generators, Proc. Amer. Math. Soc. 37 (1973), 434–440. MR 313873, DOI 10.1090/S0002-9939-1973-0313873-1
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 287357, DOI 10.2307/2373376
M. G. Crandall and A. Pazy, Nonlinear evolution equations in Banach spaces, Israel J. Math. 11 (1972), 57–94. MR 300166, DOI 10.1007/BF02761448
Janet Dyson and Rosanna Villella Bressan, Functional differential equations and non-linear evolution operators, Proc. Roy. Soc. Edinburgh Sect. A 75 (1975/76), no. 3, 223–234. MR 442402, DOI 10.1017/S0308210500017959
Janet Dyson and Rosanna Villella Bressan, Semigroups of translations associated with functional and functional-differential equations, Proc. Roy. Soc. Edinburgh Sect. A 82 (1978/79), no. 3-4, 171–188. MR 532900, DOI 10.1017/S030821050001115X
H. Flaschka and M. J. Leitman, On semigroups of nonlinear operators and the solution of the functional differential equation $\dot x(t)=F(x_{t})$, J. Math. Anal. Appl. 49 (1975), 649–658. MR 361959, DOI 10.1016/0022-247X(75)90204-8
Jerome A. Goldstein, Semigroups of linear operators and applications, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985. MR 790497
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), no. 3, 349–361. MR 987791
Jack K. Hale and Junji Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac. 21 (1978), no. 1, 11–41. MR 492721
F. Kappel and W. Schappacher, Some considerations to the fundamental theory of infinite delay equations, J. Differential Equations 37 (1980), no. 2, 141–183. MR 587220, DOI 10.1016/0022-0396(80)90093-5
N. MacDonald, Time lag in a model of a biochemical reaction sequence with end product inhibition, J. Theoret. Biol. 67 (1977), no. 3, 549–556. MR 490019, DOI 10.1016/0022-5193(77)90056-X
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486, DOI 10.1007/978-1-4612-5561-1
Andrew T. Plant, Nonlinear semigroups of translations in Banach space generated by functional differential equations, J. Math. Anal. Appl. 60 (1977), no. 1, 67–74. MR 447745, DOI 10.1016/0022-247X(77)90048-8
Wolfgang M. Ruess, The evolution operator approach to functional-differential equations with delay, Proc. Amer. Math. Soc. 119 (1993), no. 3, 783–791. MR 1154248, DOI 10.1090/S0002-9939-1993-1154248-8
C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc. 200 (1974), 395–418. MR 382808, DOI 10.1090/S0002-9947-1974-0382808-3
G. F. Webb, Autonomous nonlinear functional differential equations and nonlinear semigroups, J. Math. Anal. Appl. 46 (1974), 1–12. MR 348224, DOI 10.1016/0022-247X(74)90277-7

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G. F. Webb, Asymptotic stability for abstract nonlinear functional differential equations, Proc. Amer. Math. Soc. 54 (1976), 225–230. MR 402237, DOI 10.1090/S0002-9939-1976-0402237-0