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

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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

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

Revised March 2019 Published June 2020 Early access 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.

Keywords: Functional differential equations, infinite delay, integrated semigroup, stability, Hopf bifurcation.

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

Citation: Zhihua Liu, Pierre Magal. Functional differential equation with infinite delay in a space of exponentially bounded and uniformly continuous functions. Discrete and Continuous Dynamical Systems - B, 2020, 25 (6) : 2271-2292. doi: 10.3934/dcdsb.2019227

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.

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.
[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.

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