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  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.

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. 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.

[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. 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.

[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.

[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. 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.

[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. 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. 
[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. 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.

[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. 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.

[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.

[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.

[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 Lp 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. 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.

[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. 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.

[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. 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.

[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.

[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. 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.

[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. 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. 
[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. 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.

[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. 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.

[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.

[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.

[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 Lp 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. 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.

[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.

[1]

Abdelhai Elazzouzi, Aziz Ouhinou. Optimal regularity and stability analysis in the $\alpha-$Norm for a class of partial functional differential equations with infinite delay. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 115-135. doi: 10.3934/dcds.2011.30.115

[2]

Pham Huu Anh Ngoc. New criteria for exponential stability in mean square of stochastic functional differential equations with infinite delay. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021040

[3]

Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with state-dependent delay. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3167-3197. doi: 10.3934/dcdsb.2017169

[4]

Xiuli Sun, Rong Yuan, Yunfei Lv. Global Hopf bifurcations of neutral functional differential equations with state-dependent delay. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 667-700. doi: 10.3934/dcdsb.2018038

[5]

Jean-François Couchouron, Mikhail Kamenskii, Paolo Nistri. An infinite dimensional bifurcation problem with application to a class of functional differential equations of neutral type. Communications on Pure and Applied Analysis, 2013, 12 (5) : 1845-1859. doi: 10.3934/cpaa.2013.12.1845

[6]

R. Ouifki, M. L. Hbid, O. Arino. Attractiveness and Hopf bifurcation for retarded differential equations. Communications on Pure and Applied Analysis, 2003, 2 (2) : 147-158. doi: 10.3934/cpaa.2003.2.147

[7]

Fuke Wu, Shigeng Hu. The LaSalle-type theorem for neutral stochastic functional differential equations with infinite delay. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 1065-1094. doi: 10.3934/dcds.2012.32.1065

[8]

Ya Wang, Fuke Wu, Xuerong Mao, Enwen Zhu. Advances in the LaSalle-type theorems for stochastic functional differential equations with infinite delay. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 287-300. doi: 10.3934/dcdsb.2019182

[9]

Vitalii G. Kurbatov, Valentina I. Kuznetsova. On stability of functional differential equations with rapidly oscillating coefficients. Communications on Pure and Applied Analysis, 2018, 17 (1) : 267-283. doi: 10.3934/cpaa.2018016

[10]

Shigui Ruan, Junjie Wei, Jianhong Wu. Bifurcation from a homoclinic orbit in partial functional differential equations. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1293-1322. doi: 10.3934/dcds.2003.9.1293

[11]

Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084

[12]

Ovide Arino, Eva Sánchez. A saddle point theorem for functional state-dependent delay differential equations. Discrete and Continuous Dynamical Systems, 2005, 12 (4) : 687-722. doi: 10.3934/dcds.2005.12.687

[13]

Ismael Maroto, Carmen NÚÑez, Rafael Obaya. Dynamical properties of nonautonomous functional differential equations with state-dependent delay. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3939-3961. doi: 10.3934/dcds.2017167

[14]

Zalman Balanov, Meymanat Farzamirad, Wieslaw Krawcewicz, Haibo Ruan. Applied equivariant degree. part II: Symmetric Hopf bifurcations of functional differential equations. Discrete and Continuous Dynamical Systems, 2006, 16 (4) : 923-960. doi: 10.3934/dcds.2006.16.923

[15]

Leonid Berezansky, Elena Braverman. Stability of linear differential equations with a distributed delay. Communications on Pure and Applied Analysis, 2011, 10 (5) : 1361-1375. doi: 10.3934/cpaa.2011.10.1361

[16]

Hernán R. Henríquez, Claudio Cuevas, Alejandro Caicedo. Asymptotically periodic solutions of neutral partial differential equations with infinite delay. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2031-2068. doi: 10.3934/cpaa.2013.12.2031

[17]

Jiaohui Xu, Tomás Caraballo. Long time behavior of fractional impulsive stochastic differential equations with infinite delay. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2719-2743. doi: 10.3934/dcdsb.2018272

[18]

Ferenc Hartung, Janos Turi. Linearized stability in functional differential equations with state-dependent delays. Conference Publications, 2001, 2001 (Special) : 416-425. doi: 10.3934/proc.2001.2001.416

[19]

Jan Čermák, Jana Hrabalová. Delay-dependent stability criteria for neutral delay differential and difference equations. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4577-4588. doi: 10.3934/dcds.2014.34.4577

[20]

Hildebrando M. Rodrigues, J. Solà-Morales, G. K. Nakassima. Stability problems in nonautonomous linear differential equations in infinite dimensions. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3189-3207. doi: 10.3934/cpaa.2020138

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (277)
  • HTML views (329)
  • Cited by (1)

Other articles
by authors

[Back to Top]