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

Zhihua Liu; Pierre Magal

doi:10.3934/dcdsb.2019227

Article Contents

2020, Volume 25, Issue 6: 2271-2292. Doi: 10.3934/dcdsb.2019227

This issue Previous Article Global dynamics of an age-structured model with relapse Next Article Optimal intervention strategies of a SI-HIV models with differential infectivity and time delays

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

Zhihua Liu^1, and
Pierre Magal^2,3, ,

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
^* Corresponding author: Pierre Magal

Revised: February 28, 2019

Early access: September 2019

Published: June 2020

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

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: 34K18, 34K20, 37L10.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	M. Adimy, Bifurcation de Hopf locale par des semi-groupes intégrés, C. R. Acad. Sci. Paris Sér. I, 311 (1990), 423-428.
[2]	M. Adimy, Integrated semigroups and delay differential equations, J. Math. Anal. Appl., 177 (1993), 125-134. doi: 10.1006/jmaa.1993.1247.
[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.
[4]	W. Arendt, Vector valued Laplace transforms and Cauchy problems, Israel J. Math., 59 (1987), 327-352. doi: 10.1007/BF02774144.
[5]	W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, Birkhä user, Basel, 2001.
[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.
[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.
[8]	E. Bocchi, On the return to equilibrium problem for axisymmetric floating structures in shallow water, Submitted, https://hal.archives-ouvertes.fr/hal-01971965.
[9]	F. E. Browder, On the spectral theory of elliptic differential operators, Math. Ann., 142 (1961), 22-130. doi: 10.1007/BF01343363.
[10]	O. Diekmann, P. 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.
[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.
[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.
[13]	A. Ducrot, Z. 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.
[14]	A. Ducrot, Z. 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.
[15]	K.-J. Engel and R. Nagel, One Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000.
[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.
[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.
[18]	S. A. Gourley, G. Rost and H. Thieme, Uniform persistence in a model for bluetongue dynamics, SIAM J. Math. Anal., 46 (2014), 1160-1184. doi: 10.1137/120878197.
[19]	J. K. Hale, Functional Differential Equations, Springer-Verlag, New York, 1971.
[20]	J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.
[21]	J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs 25, American Mathematical Society, Providence, RI, 1988.
[22]	J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, Applied Mathematical Sciences, 99. Springer-Verlag, New York, 1993.
[23]	J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41.
[24]	B. D. Hassard, N. 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.
[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.
[26]	Y. Hino, S. Murakami, T. 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.
[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.
[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.
[29]	H. Kellermann and M. Hieber, Integrated semigroups, J. Funct. Anal., 84 (1989), 160-180. doi: 10.1016/0022-1236(89)90116-X.
[30]	Z. Liu, P. 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.
[31]	Z. Liu, P. 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.
[32]	Z. Liu, P. 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.
[33]	P. Magal, Compact attractors for time periodic age-structured population models, Electr. J. Differential Equations, 2001 (2001), 1-35.
[34]	P. Magal and S. Ruan, On integrated semigroups and age structured models in L^p spaces, Differential and Integral Equations, 20 (2007), 197-139.
[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.
[36]	P. Magal and S. Ruan, On semilinear cauchy problems with non-dense domain, Advances in Differential Equations, 14 (2009), 1041-1084.
[37]	P. Magal and S. Ruan, Theory and Applications of Abstract Semilinear Cauchy Problems, Applied Mathematical Sciences, 201, Springer International Publishing, 2018.
[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.
[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.
[40]	H. Matsunaga, S. Murakami, Y. 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.
[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.
[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.
[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.
[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.
[45]	G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer, New York, 2002.
[46]	H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differential Integral Equations, 3 (1990), 1035-1066.
[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.
[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.
[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.
[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.
[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.
[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.
[53]	G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, New York, 1985.
[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.