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Translation semigroups and their linearizations on spaces of integrable functions


Author: Annette Grabosch
Journal: Trans. Amer. Math. Soc. 311 (1989), 357-390
MSC: Primary 47H20; Secondary 34G20, 47B38, 47D05, 92A15
DOI: https://doi.org/10.1090/S0002-9947-1989-0974781-2
MathSciNet review: 974781
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Abstract: Of concern is the unbounded operator $ {A_\Phi }f = f'$ with nonlinear domain $ D({A_\Phi }) = \{ f \in {W^{1,1}}:f(0) = \Phi (f)\} $ which is considered on the Banach space $ E$ of Bochner integrable functions on an interval with values in a Banach space $ F$. Under the assumption that $ \Phi$ is a Lipschitz continuous operator from $ E$ to $ F$, it is shown that $ {A_{\Phi}}$ generates a strongly continuous translation semigroup $ {({T_\Phi }(t))_{t \geq 0}}$. For linear operators $ \Phi$ several properties such as essential-compactness, positivity, and irreducibility of the semigroup $ {({T_\Phi }(t))_{t \geq 0}}$ depending on the operator $ \Phi$ are studied. It is shown that if $ F$ is a Banach lattice with order continuous norm, then $ {({T_\Phi }(t))_{t \geq 0}}$ is the modulus semigroup of $ {({T_\Phi }(t))_{t \geq 0}}$. Finally spectral properties of $ {A_\Phi}$ are studied and the spectral bound $ s({A_\Phi })$ is determined. This leads to a result on the global asymptotic behavior in the case where $ \Phi$ is linear and to a local stability result in the case where $ \Phi$ is Fréchet differentiable.


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  • [1] C. D. Aliprantis and O. Burkinshaw, On weakly compact operators on Banach lattices, Proc. Amer. Math. Soc. 83 (1981), 573-578. MR 627695 (82j:47057)
  • [2] O. Arino and M. Kimmel, Asymptotic analysis of a cell cycle model based on unequal division, SIAM J. Appl. Math. 47 (1987), 128-145. MR 873240 (88e:92019)
  • [3] I. Becker and G. Greiner, On the modulus of one-parameter semigroups, Semigroup Forum 34 (1986), 185-201. MR 868254 (88b:47054)
  • [4] G. Di Blasio, K. Kunisch and E. Sinestrari, Stability for abstract linear functional differential equations, Israel J. Math. 50 (1985), 231-263. MR 793856 (86j:35152)
  • [5] G. Chen and R. Grimmer, Semigroups and integral equations, J. Integral Equations 2 (1980), 133-154. MR 572484 (81f:45026)
  • [6] Ph. Clément, O. Diekmann, M. Gyllenberg, H. J. A. M. Heijmans and H. R. Thieme, Perturbation theory for dual semigroups. I-V, Preprints (1986/1987).
  • [7] C. Corduneanu and V. Lakshmikantham, Equations with unbounded delay: a survey, Nonlinear Analysis 4 (1980), 831-877. MR 586852 (81i:34061)
  • [8] R. Datko, Linear autonomous neutral differential equations in a Banach space, J. Differential Equations 25 (1977), 258-274. MR 0447743 (56:6053)
  • [9] W. Desch and W. Schappacher, Linearized stability for nonlinear semigroups, Differential Equations in Banach Spaces (Bologna 1985), Lecture Notes in Math., vol. 1223, Springer-Verlag, Berlin and New York, 1986, pp. 61-73. MR 872517 (88k:47083)
  • [10] O. Diekmann, Volterra integral equations and semigroups of operators, Math. Centre Report Amsterdam TW 197/80 (1980).
  • [11] -, Perturbed dual semigroups and delay equations, Preprint (1986).
  • [12] N. Dunford and J. T. Schwartz, Linear operators, Part I: General theory, Wiley, New York, 1958. MR 1009162 (90g:47001a)
  • [13] J. Dyson and R. Villella-Bressan, Semigroups of translations associated with functional and functional differential equations, Proc. Roy. Soc. Edinburgh Sect. A 67 (1979), 171-188. MR 532900 (80i:34113)
  • [14] 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 0361959 (50:14401)
  • [15] J. A. Goldstein, Nonlinear semigroups and nonlinear partial differential equations, Atas do Décimo Colóquio Brasileiro de Matemática, Vol. I, Inst. Mat. Pura Appl., Rio de Janeiro, 1978, pp. 213-248. MR 528083 (80e:47056)
  • [16] -, Semigroups of operators and applications. II, Preprint (1985).
  • [17] A. Grabosch, A two phase transition probability model of the cell cycle, Semesterbericht Funktionalanalysis Tübingen, 1985/86, pp. 71-99.
  • [18] -, Die Làsungshalbgruppe abstrakter retardierter Gleichungen, Dissertation, Universität Tübingen, 1986.
  • [19] G. Gripenberg, Asymptotic behaviour of resolvents of abstract Volterra equations, J. Math. Anal. Appl. 122 (1987), 427-438. MR 877825 (88b:45002)
  • [20] G. Greiner, Perturbing the boundary conditions of a generator, Houston J. Math 13 (1987). MR 904952 (88i:47023)
  • [21] -, Semilinear boundary conditions for evolution equations, Semesterbericht Funktionalanalysis Tübingen, 1987.
  • [22] G. Greiner and R. Nagel, Growth of cell populations via one-parameter semigroups of positive operators, Semesterbericht Funktionalanalysis Tübingen, 1987. MR 934944 (89f:92041)
  • [23] J. K. Hale, Theory of functional differential equations, Springer-Verlag, New York and Berlin, 1977. MR 0508721 (58:22904)
  • [24] T. Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan 19 (1967), 508-520. MR 0226230 (37:1820)
  • [25] W. Kerscher, Retardierte Cauchyprobleme: Ordnunseigenschaften und Stabilität unabhängig von der Verzàgerung, Dissertation, Universität Tübingen, 1986.
  • [26] W. Kerscher and R. Nagel, Asymptotic behavior of one-parameter semigroups of positive operators, Acta Appl. Math. 2 (1984), 297-309. MR 753697 (86e:47047)
  • [27] -, Positivity and stability for Cauchy problems with delay, Semesterbericht Funktionalanalysis Tübingen, 1986, pp. 35-54.
  • [28] M. Kimmel, Z. Darzynkiewicz, O. Arino and F. Traganos, Analysis of a cell cycle model based on unequal division of metabolic constituents to daughter cells during cytokinesis, J. Theoret. Biol. 101 (1984), 637-664.
  • [29] A. Kufner, O. John and S. Fučik, Function spaces, Noordhoff, Leyden, 1977.
  • [30] F. Martello, Semigruppi di translazioni nello spazio $ {L^1}([ - r,0],X)$, Rend. Sem. Mat. Univ. Padova 72 (1984), 307-317.
  • [31] J. A. J. Metz and O. Diekmann (Eds.), The dynamics of physiologically structured populations, Lecture Notes in Biomath., vol. 68, Springer-Verlag, Berlin and New York, 1986. MR 860959 (88b:92049)
  • [32] R. Nagel (Ed.), One-parameter semigroups of positive operators, Lecture Notes in Math., vol. 1184, Springer-Verlag, Berlin and New York, 1986. MR 839450 (88i:47022)
  • [33] A. T. Plant, Nonlinear semigroups of linear operators and applications in Banach spaces, J. Math. Anal. Appl. 60 (1977), 67-74. MR 0447745 (56:6055)
  • [34] J. Prüß, Equilibrium solutions of age-specific population dynamics of several species, J. Math. Biol. 11 (1981), 65-84. MR 617881 (83a:92056)
  • [35] H. H. Schaefer, Banach lattices and positive operators, Springer-Verlag, Berlin and New York, 1974. MR 0423039 (54:11023)
  • [36] F. R. Sharpe and A. J. Lotka, A problem in age distributions, Philos. Mag. 21 (1911), 435-438.
  • [37] O. J. Staffans, The initial function and forcing function semigroups generated by a functional equation, Preprint (1984).
  • [38] -, Semigroups generated by a convolution equation, Proc. Conf. on Operator Semigroups and Applications (Graz, 1983), Lecture Notes in Math., vol. 1076, Springer-Verlag, Berlin and New York, 1984, pp. 209-226. MR 763365 (86e:45007)
  • [39] 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)
  • [40] R. Villella-Bressan, Functional equation of delay type in $ {L^1}$-spaces, Ann. Polon. Math. 45 (1985), 93-104. MR 799984 (86j:47087)
  • [41] G. F. Webb, Volterra integral equations and nonlinear semigroups, Nonlinear Analysis 1 (1977), 415-427. MR 511684 (80b:45005)
  • [42] -, Theory of nonlinear age-dependent population dynamics, Dekker, New York, 1985. MR 772205 (86e:92032)
  • [43] -, An operator theoretic formulation of asynchronous exponential growth, Preprint (1985).
  • [44] -, Random transitions, size control, and inheritance in cell population dynamics, Preprint (1986).
  • [45] G. F. Webb and A. Grabosch, Asynchronous exponential growth in transition probability models of the cell cycle, SIAM J. Math. Anal. 18 (1987). MR 892478 (88k:92081)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1989-0974781-2
Keywords: Linear and nonlinear strongly continuous semigroups, translation semigroups, first derivatives, compactness, positivity, irreducibility, domination, spectral properties, asymptotics, linearization, stability of equilibrium solutions, functional equations, Volterra equations
Article copyright: © Copyright 1989 American Mathematical Society

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