June  2020, 25(6): 2271-2292. doi: 10.3934/dcdsb.2019227

Functional differential equation with infinite delay in a space of exponentially bounded and uniformly continuous functions

1. 

School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

2. 

Univ. Bordeaux, IMB, UMR 5251, F-33400 Talence, France

3. 

CNRS, IMB, UMR 5251, F-33400 Talence, France

* Corresponding author: Pierre Magal

Revised  March 2019 Published  September 2019

Fund Project: Research was partially supported by National Natural Science Foundation of China (Grant Nos. 11871007 and 11811530272) and the Fundamental Research Funds for the Central Universities

In this article we study a class of delay differential equations with infinite delay in weighted spaces of uniformly continuous functions. We focus on the integrated semigroup formulation of the problem and so doing we provide a spectral theory. As a consequence we obtain a local stability result and a Hopf bifurcation theorem for the semiflow generated by such a problem.

Citation: Zhihua Liu, Pierre Magal. Functional differential equation with infinite delay in a space of exponentially bounded and uniformly continuous functions. Discrete & Continuous Dynamical Systems - B, 2020, 25 (6) : 2271-2292. doi: 10.3934/dcdsb.2019227
References:
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M. Adimy, Bifurcation de Hopf locale par des semi-groupes intégrés, C. R. Acad. Sci. Paris Sér. I, 311 (1990), 423-428.   Google Scholar

[2]

M. Adimy, Integrated semigroups and delay differential equations, J. Math. Anal. Appl., 177 (1993), 125-134.  doi: 10.1006/jmaa.1993.1247.  Google Scholar

[3]

M. Adimy and O. Arino, Bifurcation de Hopf globale pour des équations à retard par des semi-groupes intégrés, C. R. Acad. Sci. Paris Sér. I, 317 (1993), 767-772.   Google Scholar

[4]

W. Arendt, Vector valued Laplace transforms and Cauchy problems, Israel J. Math., 59 (1987), 327-352.  doi: 10.1007/BF02774144.  Google Scholar

[5]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, Birkhä user, Basel, 2001.  Google Scholar

[6]

O. Arino and E. Sanchez, A theory of linear delay differential equations in infinite dimensional spaces, Delay Differential Equations and Applications, 285–346, NATO Sci. Ser. Ⅱ Math. Phys. Chem., 205, Springer, Dordrecht, 2006. doi: 10.1007/1-4020-3647-7_8.  Google Scholar

[7]

P. Auger and A. Ducrot, A model of fishery with fish stock involving delay equations, Phi. Trans. Roy. Soc. A, 367 (2009), 4907-4922.  doi: 10.1098/rsta.2009.0147.  Google Scholar

[8]

E. Bocchi, On the return to equilibrium problem for axisymmetric floating structures in shallow water, Submitted, https://hal.archives-ouvertes.fr/hal-01971965. Google Scholar

[9]

F. E. Browder, On the spectral theory of elliptic differential operators, Math. Ann., 142 (1961), 22-130.  doi: 10.1007/BF01343363.  Google Scholar

[10]

O. DiekmannP. Getto and M. Gyllenberg, Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars, SIAM J. Math. Anal., 39 (2007), 1023-1069.  doi: 10.1137/060659211.  Google Scholar

[11]

O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations, Function-, Complex-, and Nonlinear Analysis, Springer-Verlag, New York, 1995.  Google Scholar

[12]

O. Diekmann and M. Gyllenberg, Equations with infinite delay: Blending the abstract and the concrete, J. Differential Equations, 252 (2012), 819-851.  doi: 10.1016/j.jde.2011.09.038.  Google Scholar

[13]

A. DucrotZ. Liu and P. Magal, Essential growth rate for bounded linear perturbation of non-densely defined Cauchy problems, J. Math. Anal. Appl., 341 (2008), 501-518.  doi: 10.1016/j.jmaa.2007.09.074.  Google Scholar

[14]

A. DucrotZ. Liu and P. Magal, Projectors on the generalized eigenspaces for neutral functional differential equations in $L^p$ spaces, Canadian Journal of Mathematics, 62 (2010), 74-93.  doi: 10.4153/CJM-2010-005-2.  Google Scholar

[15]

K.-J. Engel and R. Nagel, One Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000.  Google Scholar

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K. Ezzinbi and M. Adimy, The basic theory of abstract semilinear functional differential equations with non-dense domain, Delay Differential Equations and Applications, 347–407, NATO Sci. Ser. Ⅱ Math. Phys. Chem., 205, Springer, Dordrecht, 2006. doi: 10.1007/1-4020-3647-7_9.  Google Scholar

[17]

M. V. S. Frasson and S. M. Verduyn Lunel, Large time behaviour of linear functional differential equations, Integral Equations Operator Theory, 47 (2003), 91-121.  doi: 10.1007/s00020-003-1155-x.  Google Scholar

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S. A. GourleyG. Rost and H. Thieme, Uniform persistence in a model for bluetongue dynamics, SIAM J. Math. Anal., 46 (2014), 1160-1184.  doi: 10.1137/120878197.  Google Scholar

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J. K. Hale, Functional Differential Equations, Springer-Verlag, New York, 1971.  Google Scholar

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J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.  Google Scholar

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J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs 25, American Mathematical Society, Providence, RI, 1988.  Google Scholar

[22]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, Applied Mathematical Sciences, 99. Springer-Verlag, New York, 1993.  Google Scholar

[23]

J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41.   Google Scholar

[24] B. D. HassardN. D. Kazarinoff and Y.-H. Wan, Theory and Applications of Hopf Bifurcaton, London Mathematical Society Lecture Note Series, vol. 41. Cambridge University Press, Cambridge, 1981.   Google Scholar
[25]

Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Math., vol. 1473, Springer-Verlag, Berlin, Heidelberg, New York, 1991. doi: 10.1007/BFb0084432.  Google Scholar

[26]

Y. HinoS. MurakamiT. Naito and N. V. Minh, A variation-of-constants formula for abstract functional differential equations in phase space, J. Differential Equations, 179 (2002), 336-355.  doi: 10.1006/jdeq.2001.4020.  Google Scholar

[27]

M. A. Kaashoek and S. M. Verduyn Lunel, Characteristic matrices and spectral properties of evolutionary systems, Trans. Amer. Math. Soc., 334 (1992), 479-517.  doi: 10.1090/S0002-9947-1992-1155350-0.  Google Scholar

[28]

F. Kappel, Linear autonomous functional differential equations, Delay Differential Equations and Applications, 41–139, NATO Sci. Ser. Ⅱ Math. Phys. Chem., 205, Springer, Dordrecht, 2006. doi: 10.1007/1-4020-3647-7_3.  Google Scholar

[29]

H. Kellermann and M. Hieber, Integrated semigroups, J. Funct. Anal., 84 (1989), 160-180.  doi: 10.1016/0022-1236(89)90116-X.  Google Scholar

[30]

Z. LiuP. Magal and S. Ruan, Projectors on the generalized eigenspaces for functional differential equations using integrated semigroups, Journal of Differential Equations, 244 (2008), 1784-1809.  doi: 10.1016/j.jde.2008.01.007.  Google Scholar

[31]

Z. LiuP. Magal and S. Ruan, Hopf bifurcation for non-densely defined Cauchy problems, Z. Angew. Math. Phys., 62 (2011), 191-222.  doi: 10.1007/s00033-010-0088-x.  Google Scholar

[32]

Z. LiuP. Magal and S. Ruan, Normal forms for semilinear equations with non-dense domain with applications to age structured models, J. Differential Equations, 257 (2014), 921-1011.  doi: 10.1016/j.jde.2014.04.018.  Google Scholar

[33]

P. Magal, Compact attractors for time periodic age-structured population models, Electr. J. Differential Equations, 2001 (2001), 1-35.   Google Scholar

[34]

P. Magal and S. Ruan, On integrated semigroups and age structured models in Lp spaces, Differential and Integral Equations, 20 (2007), 197-139.   Google Scholar

[35]

P. Magal and S. Ruan, Center manifolds for semilinear equations with non-dense domain and applications to hopf bifurcation in age structured models, Memoirs of the American Mathematical Society, 202 (2009), vi+71 pp.  Google Scholar

[36]

P. Magal and S. Ruan, On semilinear cauchy problems with non-dense domain, Advances in Differential Equations, 14 (2009), 1041-1084.   Google Scholar

[37]

P. Magal and S. Ruan, Theory and Applications of Abstract Semilinear Cauchy Problems, Applied Mathematical Sciences, 201, Springer International Publishing, 2018.  Google Scholar

[38]

P. Magal and X.-Q. Zhao, Global attractors in uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275.  doi: 10.1137/S0036141003439173.  Google Scholar

[39]

R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.  doi: 10.2307/2001590.  Google Scholar

[40]

H. MatsunagaS. MurakamiY. Nagabuchi and N. Van Minh, Center manifold theorem and stability for integral equations with infinite delay, Funkcialaj Ekvacioj, 58 (2015), 87-134.  doi: 10.1619/fesi.58.87.  Google Scholar

[41]

C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 6 (2009), 603-610.  doi: 10.3934/mbe.2009.6.603.  Google Scholar

[42]

G. Rost and J. Wu, SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 5 (2008), 389-402.  doi: 10.3934/mbe.2008.5.389.  Google Scholar

[43]

S. Ruan and G. S. K. Wolkowicz, Bifurcation analysis of a chemostat model with a distributed delay, Journal of Mathematical Analysis and Applications, 204 (1996), 786-812.  doi: 10.1006/jmaa.1996.0468.  Google Scholar

[44]

W. R. Ruess, Flow invariance for nonlinear partial differential delay equations, Trans. Amer. Math. Soc., 361 (2009), 4367-4403.  doi: 10.1090/S0002-9947-09-04833-8.  Google Scholar

[45]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer, New York, 2002.  Google Scholar

[46]

H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differential Integral Equations, 3 (1990), 1035-1066.   Google Scholar

[47]

H. R. Thieme, Integrated semigroups and integrated solutions to abstract Cauchy problems, J. Math. Anal. Appl., 152 (1990), 416-447.  doi: 10.1016/0022-247X(90)90074-P.  Google Scholar

[48]

H. R. Thieme, Quasi-compact semigroups via bounded perturbation, Advances in Mathematical Population Dynamics–Molecules, Cells and Man (Houston, TX, 1995), 691–711, Ser. Math. Biol. Med., 6, World Sci. Publishing, River Edge, NJ, 1997.  Google Scholar

[49]

C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc., 200 (1974), 395-418.  doi: 10.1090/S0002-9947-1974-0382808-3.  Google Scholar

[50]

C. C. Travis and G. F. Webb, Existence, stability, and compactness in the $\alpha-$norm for partial functional differential equations, Trans. Amer. Math. Soc., 240 (1978), 129-143.  doi: 10.2307/1998809.  Google Scholar

[51]

H.-O. Walther, Differential equations with locally bounded delay, Journal of Differential Equations, 252 (2012), 3001-3039.  doi: 10.1016/j.jde.2011.11.004.  Google Scholar

[52]

G. F. Webb, Functional differential equations and nonlinear semigroups in $L^p$-spaces, J. Differential Equations, 20 (1976), 71-89.  doi: 10.1016/0022-0396(76)90097-8.  Google Scholar

[53]

G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, New York, 1985.  Google Scholar

[54]

G. F. Webb, An operator-theoretic formulation of asynchronous exponential growth, Trans. Amer. Math. Soc., 303 (1987), 751-763.  doi: 10.1090/S0002-9947-1987-0902796-7.  Google Scholar

show all references

References:
[1]

M. Adimy, Bifurcation de Hopf locale par des semi-groupes intégrés, C. R. Acad. Sci. Paris Sér. I, 311 (1990), 423-428.   Google Scholar

[2]

M. Adimy, Integrated semigroups and delay differential equations, J. Math. Anal. Appl., 177 (1993), 125-134.  doi: 10.1006/jmaa.1993.1247.  Google Scholar

[3]

M. Adimy and O. Arino, Bifurcation de Hopf globale pour des équations à retard par des semi-groupes intégrés, C. R. Acad. Sci. Paris Sér. I, 317 (1993), 767-772.   Google Scholar

[4]

W. Arendt, Vector valued Laplace transforms and Cauchy problems, Israel J. Math., 59 (1987), 327-352.  doi: 10.1007/BF02774144.  Google Scholar

[5]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, Birkhä user, Basel, 2001.  Google Scholar

[6]

O. Arino and E. Sanchez, A theory of linear delay differential equations in infinite dimensional spaces, Delay Differential Equations and Applications, 285–346, NATO Sci. Ser. Ⅱ Math. Phys. Chem., 205, Springer, Dordrecht, 2006. doi: 10.1007/1-4020-3647-7_8.  Google Scholar

[7]

P. Auger and A. Ducrot, A model of fishery with fish stock involving delay equations, Phi. Trans. Roy. Soc. A, 367 (2009), 4907-4922.  doi: 10.1098/rsta.2009.0147.  Google Scholar

[8]

E. Bocchi, On the return to equilibrium problem for axisymmetric floating structures in shallow water, Submitted, https://hal.archives-ouvertes.fr/hal-01971965. Google Scholar

[9]

F. E. Browder, On the spectral theory of elliptic differential operators, Math. Ann., 142 (1961), 22-130.  doi: 10.1007/BF01343363.  Google Scholar

[10]

O. DiekmannP. Getto and M. Gyllenberg, Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars, SIAM J. Math. Anal., 39 (2007), 1023-1069.  doi: 10.1137/060659211.  Google Scholar

[11]

O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations, Function-, Complex-, and Nonlinear Analysis, Springer-Verlag, New York, 1995.  Google Scholar

[12]

O. Diekmann and M. Gyllenberg, Equations with infinite delay: Blending the abstract and the concrete, J. Differential Equations, 252 (2012), 819-851.  doi: 10.1016/j.jde.2011.09.038.  Google Scholar

[13]

A. DucrotZ. Liu and P. Magal, Essential growth rate for bounded linear perturbation of non-densely defined Cauchy problems, J. Math. Anal. Appl., 341 (2008), 501-518.  doi: 10.1016/j.jmaa.2007.09.074.  Google Scholar

[14]

A. DucrotZ. Liu and P. Magal, Projectors on the generalized eigenspaces for neutral functional differential equations in $L^p$ spaces, Canadian Journal of Mathematics, 62 (2010), 74-93.  doi: 10.4153/CJM-2010-005-2.  Google Scholar

[15]

K.-J. Engel and R. Nagel, One Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000.  Google Scholar

[16]

K. Ezzinbi and M. Adimy, The basic theory of abstract semilinear functional differential equations with non-dense domain, Delay Differential Equations and Applications, 347–407, NATO Sci. Ser. Ⅱ Math. Phys. Chem., 205, Springer, Dordrecht, 2006. doi: 10.1007/1-4020-3647-7_9.  Google Scholar

[17]

M. V. S. Frasson and S. M. Verduyn Lunel, Large time behaviour of linear functional differential equations, Integral Equations Operator Theory, 47 (2003), 91-121.  doi: 10.1007/s00020-003-1155-x.  Google Scholar

[18]

S. A. GourleyG. Rost and H. Thieme, Uniform persistence in a model for bluetongue dynamics, SIAM J. Math. Anal., 46 (2014), 1160-1184.  doi: 10.1137/120878197.  Google Scholar

[19]

J. K. Hale, Functional Differential Equations, Springer-Verlag, New York, 1971.  Google Scholar

[20]

J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.  Google Scholar

[21]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs 25, American Mathematical Society, Providence, RI, 1988.  Google Scholar

[22]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, Applied Mathematical Sciences, 99. Springer-Verlag, New York, 1993.  Google Scholar

[23]

J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41.   Google Scholar

[24] B. D. HassardN. D. Kazarinoff and Y.-H. Wan, Theory and Applications of Hopf Bifurcaton, London Mathematical Society Lecture Note Series, vol. 41. Cambridge University Press, Cambridge, 1981.   Google Scholar
[25]

Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Math., vol. 1473, Springer-Verlag, Berlin, Heidelberg, New York, 1991. doi: 10.1007/BFb0084432.  Google Scholar

[26]

Y. HinoS. MurakamiT. Naito and N. V. Minh, A variation-of-constants formula for abstract functional differential equations in phase space, J. Differential Equations, 179 (2002), 336-355.  doi: 10.1006/jdeq.2001.4020.  Google Scholar

[27]

M. A. Kaashoek and S. M. Verduyn Lunel, Characteristic matrices and spectral properties of evolutionary systems, Trans. Amer. Math. Soc., 334 (1992), 479-517.  doi: 10.1090/S0002-9947-1992-1155350-0.  Google Scholar

[28]

F. Kappel, Linear autonomous functional differential equations, Delay Differential Equations and Applications, 41–139, NATO Sci. Ser. Ⅱ Math. Phys. Chem., 205, Springer, Dordrecht, 2006. doi: 10.1007/1-4020-3647-7_3.  Google Scholar

[29]

H. Kellermann and M. Hieber, Integrated semigroups, J. Funct. Anal., 84 (1989), 160-180.  doi: 10.1016/0022-1236(89)90116-X.  Google Scholar

[30]

Z. LiuP. Magal and S. Ruan, Projectors on the generalized eigenspaces for functional differential equations using integrated semigroups, Journal of Differential Equations, 244 (2008), 1784-1809.  doi: 10.1016/j.jde.2008.01.007.  Google Scholar

[31]

Z. LiuP. Magal and S. Ruan, Hopf bifurcation for non-densely defined Cauchy problems, Z. Angew. Math. Phys., 62 (2011), 191-222.  doi: 10.1007/s00033-010-0088-x.  Google Scholar

[32]

Z. LiuP. Magal and S. Ruan, Normal forms for semilinear equations with non-dense domain with applications to age structured models, J. Differential Equations, 257 (2014), 921-1011.  doi: 10.1016/j.jde.2014.04.018.  Google Scholar

[33]

P. Magal, Compact attractors for time periodic age-structured population models, Electr. J. Differential Equations, 2001 (2001), 1-35.   Google Scholar

[34]

P. Magal and S. Ruan, On integrated semigroups and age structured models in Lp spaces, Differential and Integral Equations, 20 (2007), 197-139.   Google Scholar

[35]

P. Magal and S. Ruan, Center manifolds for semilinear equations with non-dense domain and applications to hopf bifurcation in age structured models, Memoirs of the American Mathematical Society, 202 (2009), vi+71 pp.  Google Scholar

[36]

P. Magal and S. Ruan, On semilinear cauchy problems with non-dense domain, Advances in Differential Equations, 14 (2009), 1041-1084.   Google Scholar

[37]

P. Magal and S. Ruan, Theory and Applications of Abstract Semilinear Cauchy Problems, Applied Mathematical Sciences, 201, Springer International Publishing, 2018.  Google Scholar

[38]

P. Magal and X.-Q. Zhao, Global attractors in uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275.  doi: 10.1137/S0036141003439173.  Google Scholar

[39]

R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.  doi: 10.2307/2001590.  Google Scholar

[40]

H. MatsunagaS. MurakamiY. Nagabuchi and N. Van Minh, Center manifold theorem and stability for integral equations with infinite delay, Funkcialaj Ekvacioj, 58 (2015), 87-134.  doi: 10.1619/fesi.58.87.  Google Scholar

[41]

C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 6 (2009), 603-610.  doi: 10.3934/mbe.2009.6.603.  Google Scholar

[42]

G. Rost and J. Wu, SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 5 (2008), 389-402.  doi: 10.3934/mbe.2008.5.389.  Google Scholar

[43]

S. Ruan and G. S. K. Wolkowicz, Bifurcation analysis of a chemostat model with a distributed delay, Journal of Mathematical Analysis and Applications, 204 (1996), 786-812.  doi: 10.1006/jmaa.1996.0468.  Google Scholar

[44]

W. R. Ruess, Flow invariance for nonlinear partial differential delay equations, Trans. Amer. Math. Soc., 361 (2009), 4367-4403.  doi: 10.1090/S0002-9947-09-04833-8.  Google Scholar

[45]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer, New York, 2002.  Google Scholar

[46]

H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differential Integral Equations, 3 (1990), 1035-1066.   Google Scholar

[47]

H. R. Thieme, Integrated semigroups and integrated solutions to abstract Cauchy problems, J. Math. Anal. Appl., 152 (1990), 416-447.  doi: 10.1016/0022-247X(90)90074-P.  Google Scholar

[48]

H. R. Thieme, Quasi-compact semigroups via bounded perturbation, Advances in Mathematical Population Dynamics–Molecules, Cells and Man (Houston, TX, 1995), 691–711, Ser. Math. Biol. Med., 6, World Sci. Publishing, River Edge, NJ, 1997.  Google Scholar

[49]

C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc., 200 (1974), 395-418.  doi: 10.1090/S0002-9947-1974-0382808-3.  Google Scholar

[50]

C. C. Travis and G. F. Webb, Existence, stability, and compactness in the $\alpha-$norm for partial functional differential equations, Trans. Amer. Math. Soc., 240 (1978), 129-143.  doi: 10.2307/1998809.  Google Scholar

[51]

H.-O. Walther, Differential equations with locally bounded delay, Journal of Differential Equations, 252 (2012), 3001-3039.  doi: 10.1016/j.jde.2011.11.004.  Google Scholar

[52]

G. F. Webb, Functional differential equations and nonlinear semigroups in $L^p$-spaces, J. Differential Equations, 20 (1976), 71-89.  doi: 10.1016/0022-0396(76)90097-8.  Google Scholar

[53]

G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, New York, 1985.  Google Scholar

[54]

G. F. Webb, An operator-theoretic formulation of asynchronous exponential growth, Trans. Amer. Math. Soc., 303 (1987), 751-763.  doi: 10.1090/S0002-9947-1987-0902796-7.  Google Scholar

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