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January  2014, 34(1): 51-77. doi: 10.3934/dcds.2014.34.51

On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems

1. 

Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Campus Reina Mercedes, Apdo. de Correos 1160, 41080 Sevilla

2. 

Department d'Economia Aplicada, Facultat d'Economia, Universitat de València, Campus del Tarongers s/n, 46022-València, Spain

3. 

Centro de Investigación Operativa, Universidad Miguel Hernández de Elche, Avda. de la Universidad, s/n, 03202 Elche

Received  November 2012 Revised  January 2013 Published  June 2013

In this paper we first prove a rather general theorem about existence of solutions for an abstract differential equation in a Banach space by assuming that the nonlinear term is in some sense weakly continuous.
    We then apply this result to a lattice dynamical system with delay, proving also the existence of a global compact attractor for such system.
Citation: Tomás Caraballo, Francisco Morillas, José Valero. On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems. Discrete & Continuous Dynamical Systems, 2014, 34 (1) : 51-77. doi: 10.3934/dcds.2014.34.51
References:
[1]

V. S. Afraĭmovich and V. I. Nekorkin, Chaos of traveling waves in a discrete chain of diffusively coupled maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 4 (1994), 631-637. doi: 10.1142/S0218127494000459.  Google Scholar

[2]

A. Y. Abdallah, Exponential attractors for first-order lattice dynamical systems, J. Math. Anal. Appl., 339 (2008), 217-224. doi: 10.1016/j.jmaa.2007.06.054.  Google Scholar

[3]

J. M. Amigó, Á. Giménez, F. Morillas and J. Valero, Attractors for a lattice dynamical system generated by non-Newtonian fluids modeling suspensions, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2681-2700. doi: 10.1142/S0218127410027295.  Google Scholar

[4]

V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces," Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976.  Google Scholar

[5]

P. W. Bates and A. Chmaj, A discrete convolution model for phase transitions, Arch. Ration. Mech. Anal., 150 (1999), 281-305. doi: 10.1007/s002050050189.  Google Scholar

[6]

P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stochastics & Dynamics, 6 (2006), 1-21. doi: 10.1142/S0219493706001621.  Google Scholar

[7]

P. W. Bates, K. Lu and B. Wang, Attractors for lattice dynamical systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 143-153. doi: 10.1142/S0218127401002031.  Google Scholar

[8]

J. Bell, Some threshhold results for models of myelinated nerves, Mathematical Biosciences, 54 (1981), 181-190. doi: 10.1016/0025-5564(81)90085-7.  Google Scholar

[9]

J. Bell and C. Cosner, Threshold behaviour and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons, Quarterly Appl. Math., 42 (1984), 1-14.  Google Scholar

[10]

W.-J. Beyn and S. Yu. Pilyugin, Attractors of reaction diffusion systems on infinite lattices, J. Dynam. Differential Equations, 15 (2003), 485-515. doi: 10.1023/B:JODY.0000009745.41889.30.  Google Scholar

[11]

T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise, Front. Math. China, 3 (2008), 317-335. doi: 10.1007/s11464-008-0028-7.  Google Scholar

[12]

T. Caraballo, F. Morillas and J. Valero, Random Attractors for stochastic lattice systems with non-Lipschitz nonlinearity, J. Diff. Equat. App., 17 (2011), 161-184. doi: 10.1080/10236198.2010.549010.  Google Scholar

[13]

T. Caraballo, F. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities, J. Differential Equations, 253 (2012), 667-693. doi: 10.1016/j.jde.2012.03.020.  Google Scholar

[14]

S.-N. Chow and J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems. I, IEEE Trans. Circuits Syst., 42 (1995), 746-751. doi: 10.1109/81.473583.  Google Scholar

[15]

S.-N. Chow, J. Mallet-Paret and W. Shen, Traveling waves in lattice dynamical systems, J. Differential Equations, 149 (1998), 248-291. doi: 10.1006/jdeq.1998.3478.  Google Scholar

[16]

S.-N. Chow, J. Mallet-Paret and E. S. Van Vleck, Pattern formation and spatial chaos in spatially discrete evolution equations, Random Computational Dynamics, 4 (1996) 109-178.  Google Scholar

[17]

S.-N. Chow and W. Shen, Dynamics in a discrete Nagumo equation: Spatial topological chaos, SIAM J. Appl. Math., 55 (1995), 1764-1781. doi: 10.1137/S0036139994261757.  Google Scholar

[18]

L. O. Chua and T. Roska, The CNN paradigm, IEEE Trans. Circuits Syst., 40 (1993), 147-156. Google Scholar

[19]

L. O. Chua and L. Yang, Cellular neural networks: Theory, IEEE Trans. Circuits Syst., 35 (1988), 1257-1272. doi: 10.1109/31.7600.  Google Scholar

[20]

L. O. Chua and L. Yang, Cellular neural neetworks: Applications, IEEE Trans. Circuits Syst., 35 (1988), 1273-1290. doi: 10.1109/31.7601.  Google Scholar

[21]

A. Pérez-Muñuzuri, V. Pérez-Muñuzuri, V. Pérez-Villar and L. O. Chua, Spiral waves on a 2-D array of nonlinear circuits, IEEE Trans. Circuits Syst., 40 (1993), 872-877. Google Scholar

[22]

R. Dogaru and L. O. Chua, Edge of chaos and local activity domain of Fitz-Hugh-Nagumo equation, Internat. J. Bifur. Chaos, 8 (1988), 211-257. doi: 10.1142/S0218127498000152.  Google Scholar

[23]

T. Erneux and G. Nicolis, Propagating waves in discrete bistable reaction diffusion systems, Physica D, 67 (1993), 237-244. doi: 10.1016/0167-2789(93)90208-I.  Google Scholar

[24]

M. Gobbino and M. Sardella, On the connectedness of attractors for dynamical systems, J. Differential Equations, 133 (1997), 1-14. doi: 10.1006/jdeq.1996.3166.  Google Scholar

[25]

A. M. Gomaa, On existence of solutions and solution sets of differential equations and differential inclusions with delay in Banach spaces, J. Egyptian Math. Soc., 20 (2012), 79-86. doi: 10.1016/j.joems.2012.08.007.  Google Scholar

[26]

X. Han, Random attractors for stochastic sine-Gordon lattice systems with multiplicative white noise, J. Math. Anal. Appl., 376 (2011), 481-493. doi: 10.1016/j.jmaa.2010.11.032.  Google Scholar

[27]

X. Han, W. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266. doi: 10.1016/j.jde.2010.10.018.  Google Scholar

[28]

R. Kapral, Discrete models for chemically reacting systems, J. Math. Chem., 6 (1991), 113-163. doi: 10.1007/BF01192578.  Google Scholar

[29]

S. Kato, On existence and uniqueness conditions for nonlinear ordinary differential equations in Banach spaces, Funkcialaj Ekvacioj., 19 (1976), 239-245.  Google Scholar

[30]

J. P. Keener, Propagation and its failure in coupled systems of discrete excitable cells, SIAM J. Appl. Math., 47 (1987), 556-572. doi: 10.1137/0147038.  Google Scholar

[31]

J. P. Keener, The effects of discrete gap junction coupling on propagation in myocardium, J. Theor. Biol., 148 (1991), 49-82. Google Scholar

[32]

O. A. Ladyzhenskaya, "Attractors for Semigroups and Evolution Equations," Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511569418.  Google Scholar

[33]

J. P. Laplante and T. Erneux, Propagating failure in arrays of coupled bistable chemical reactors, J. Phys. Chem., 96 (1992), 4931-4934. Google Scholar

[34]

V. Lakshmikantham, A. R. Mitchell and R. W. Mitchell, On the existence of solutions of differential equations of retarde type in a Banach space,, Annales Polonici Mathematici, 35 (): 253.   Google Scholar

[35]

Y. Lv and J. Sun, Dynamical behavior for stochastic lattice systems, Chaos, Solitons and Fractals, 27 (2006), 1080-1090. doi: 10.1016/j.chaos.2005.04.089.  Google Scholar

[36]

J. Mallet-Paret, The global structure of traveling waves in spatially discrete dynamical systems, J. Dynam. Differential Equations, 11 (1999), 49-127. doi: 10.1023/A:1021841618074.  Google Scholar

[37]

V. S. Melnik and J. Valero, On attractors of multi-valued semi-flows and differential inclusions, Set-Valued Anal., 6 (1998), 83-111. doi: 10.1023/A:1008608431399.  Google Scholar

[38]

F. Morillas and J. Valero, Peano's theorem and attractors for lattice dynamical systems, Internat. J. Bifur. Chaos, 19 (2009), 557-578. doi: 10.1142/S0218127409023196.  Google Scholar

[39]

F. Morillas and J. Valero, On the connectedness of the attainability set for lattice dynamical systems, J. Diff. Equat. App., 18 (2012), 675-692. doi: 10.1080/10236198.2011.574621.  Google Scholar

[40]

N. Rashevsky, "Mathematical Biophysics: Physico-Mathematical Foundations of Biology," Third revised edition, Dover Publications, Inc., Vol. 1, New York, 1960.  Google Scholar

[41]

A. C. Scott, Analysis of a myelinated nerve model, Bull. Math. Biophys., 26 (1964), 247-254. Google Scholar

[42]

W. Shen, Lifted lattices, hyperbolic structures, and topological disorders in coupled map lattices, SIAM J. Appl. Math., 56 (1996), 1379-1399. doi: 10.1137/S0036139995282670.  Google Scholar

[43]

A. Sikorska-Nowak, Retarded functional differential equations in Banach spaces and Henstock-Kurzweil-Pettis integrals, Discuss. Math. Differ. Incl. Control Optim., 27 (2007), 315-327. doi: 10.7151/dmdico.1087.  Google Scholar

[44]

B. Wang, Dynamics of systems of infinite lattices, J. Differential Equations, 221 (2006), 224-245. doi: 10.1016/j.jde.2005.01.003.  Google Scholar

[45]

B. Wang, Asymptotic behavior of non-autonomous lattice systems, J. Math. Anal. Appl., 331 (2007), 121-136. doi: 10.1016/j.jmaa.2006.08.070.  Google Scholar

[46]

X. Wang, Sh. Li and D. Xu, Random attractors for second-order stochastic lattice dynamical systems, Nonlinear Anal., 72 (2010), 483-494. doi: 10.1016/j.na.2009.06.094.  Google Scholar

[47]

W. Yan, Y. Li and Sh. Ji, Random attractors for first order stochastic retarded lattice dynamical systems, J. Math. Phys., 51 (2010), 17 pp. doi: 10.1063/1.3319566.  Google Scholar

[48]

C. Zhao and S. Zhou, Attractors of retarded first order lattice systems, Nonlinearity, 20 (2007), 1987-2006. doi: 10.1088/0951-7715/20/8/010.  Google Scholar

[49]

C. Zhao and Sh. Zhou, Sufficient conditions for the existence of global random attractors for stochastic lattice dynamical systems and applications, J. Math. Anal. Appl., 354 (2009), 78-95. doi: 10.1016/j.jmaa.2008.12.036.  Google Scholar

[50]

C. Zhao, S. Zhou and W. Wang, Compact kernel sections for lattice systems with delays, Nonlinear Analysis TMA, 70 (2009), 1330-1348. doi: 10.1016/j.na.2008.02.015.  Google Scholar

[51]

S. Zhou, Attractors for first order dissipative lattice dynamical systems, Physica D, 178 (2003), 51-61. doi: 10.1016/S0167-2789(02)00807-2.  Google Scholar

[52]

S. Zhou, Attractors and approximations for lattice dynamical systems, J. Differential Equations, 200 (2004), 342-368. doi: 10.1016/j.jde.2004.02.005.  Google Scholar

[53]

S. Zhou and W. Shi, Attractors and dimension of dissipative lattice systems, J. Differential Equations, 224 (2006), 172-204. doi: 10.1016/j.jde.2005.06.024.  Google Scholar

[54]

B. Zinner, Existence of traveling wavefront solutions for the discrete Nagumo equation, J. Differential Equations, 96 (1992), 1-27. doi: 10.1016/0022-0396(92)90142-A.  Google Scholar

show all references

References:
[1]

V. S. Afraĭmovich and V. I. Nekorkin, Chaos of traveling waves in a discrete chain of diffusively coupled maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 4 (1994), 631-637. doi: 10.1142/S0218127494000459.  Google Scholar

[2]

A. Y. Abdallah, Exponential attractors for first-order lattice dynamical systems, J. Math. Anal. Appl., 339 (2008), 217-224. doi: 10.1016/j.jmaa.2007.06.054.  Google Scholar

[3]

J. M. Amigó, Á. Giménez, F. Morillas and J. Valero, Attractors for a lattice dynamical system generated by non-Newtonian fluids modeling suspensions, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2681-2700. doi: 10.1142/S0218127410027295.  Google Scholar

[4]

V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces," Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976.  Google Scholar

[5]

P. W. Bates and A. Chmaj, A discrete convolution model for phase transitions, Arch. Ration. Mech. Anal., 150 (1999), 281-305. doi: 10.1007/s002050050189.  Google Scholar

[6]

P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stochastics & Dynamics, 6 (2006), 1-21. doi: 10.1142/S0219493706001621.  Google Scholar

[7]

P. W. Bates, K. Lu and B. Wang, Attractors for lattice dynamical systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 143-153. doi: 10.1142/S0218127401002031.  Google Scholar

[8]

J. Bell, Some threshhold results for models of myelinated nerves, Mathematical Biosciences, 54 (1981), 181-190. doi: 10.1016/0025-5564(81)90085-7.  Google Scholar

[9]

J. Bell and C. Cosner, Threshold behaviour and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons, Quarterly Appl. Math., 42 (1984), 1-14.  Google Scholar

[10]

W.-J. Beyn and S. Yu. Pilyugin, Attractors of reaction diffusion systems on infinite lattices, J. Dynam. Differential Equations, 15 (2003), 485-515. doi: 10.1023/B:JODY.0000009745.41889.30.  Google Scholar

[11]

T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise, Front. Math. China, 3 (2008), 317-335. doi: 10.1007/s11464-008-0028-7.  Google Scholar

[12]

T. Caraballo, F. Morillas and J. Valero, Random Attractors for stochastic lattice systems with non-Lipschitz nonlinearity, J. Diff. Equat. App., 17 (2011), 161-184. doi: 10.1080/10236198.2010.549010.  Google Scholar

[13]

T. Caraballo, F. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities, J. Differential Equations, 253 (2012), 667-693. doi: 10.1016/j.jde.2012.03.020.  Google Scholar

[14]

S.-N. Chow and J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems. I, IEEE Trans. Circuits Syst., 42 (1995), 746-751. doi: 10.1109/81.473583.  Google Scholar

[15]

S.-N. Chow, J. Mallet-Paret and W. Shen, Traveling waves in lattice dynamical systems, J. Differential Equations, 149 (1998), 248-291. doi: 10.1006/jdeq.1998.3478.  Google Scholar

[16]

S.-N. Chow, J. Mallet-Paret and E. S. Van Vleck, Pattern formation and spatial chaos in spatially discrete evolution equations, Random Computational Dynamics, 4 (1996) 109-178.  Google Scholar

[17]

S.-N. Chow and W. Shen, Dynamics in a discrete Nagumo equation: Spatial topological chaos, SIAM J. Appl. Math., 55 (1995), 1764-1781. doi: 10.1137/S0036139994261757.  Google Scholar

[18]

L. O. Chua and T. Roska, The CNN paradigm, IEEE Trans. Circuits Syst., 40 (1993), 147-156. Google Scholar

[19]

L. O. Chua and L. Yang, Cellular neural networks: Theory, IEEE Trans. Circuits Syst., 35 (1988), 1257-1272. doi: 10.1109/31.7600.  Google Scholar

[20]

L. O. Chua and L. Yang, Cellular neural neetworks: Applications, IEEE Trans. Circuits Syst., 35 (1988), 1273-1290. doi: 10.1109/31.7601.  Google Scholar

[21]

A. Pérez-Muñuzuri, V. Pérez-Muñuzuri, V. Pérez-Villar and L. O. Chua, Spiral waves on a 2-D array of nonlinear circuits, IEEE Trans. Circuits Syst., 40 (1993), 872-877. Google Scholar

[22]

R. Dogaru and L. O. Chua, Edge of chaos and local activity domain of Fitz-Hugh-Nagumo equation, Internat. J. Bifur. Chaos, 8 (1988), 211-257. doi: 10.1142/S0218127498000152.  Google Scholar

[23]

T. Erneux and G. Nicolis, Propagating waves in discrete bistable reaction diffusion systems, Physica D, 67 (1993), 237-244. doi: 10.1016/0167-2789(93)90208-I.  Google Scholar

[24]

M. Gobbino and M. Sardella, On the connectedness of attractors for dynamical systems, J. Differential Equations, 133 (1997), 1-14. doi: 10.1006/jdeq.1996.3166.  Google Scholar

[25]

A. M. Gomaa, On existence of solutions and solution sets of differential equations and differential inclusions with delay in Banach spaces, J. Egyptian Math. Soc., 20 (2012), 79-86. doi: 10.1016/j.joems.2012.08.007.  Google Scholar

[26]

X. Han, Random attractors for stochastic sine-Gordon lattice systems with multiplicative white noise, J. Math. Anal. Appl., 376 (2011), 481-493. doi: 10.1016/j.jmaa.2010.11.032.  Google Scholar

[27]

X. Han, W. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266. doi: 10.1016/j.jde.2010.10.018.  Google Scholar

[28]

R. Kapral, Discrete models for chemically reacting systems, J. Math. Chem., 6 (1991), 113-163. doi: 10.1007/BF01192578.  Google Scholar

[29]

S. Kato, On existence and uniqueness conditions for nonlinear ordinary differential equations in Banach spaces, Funkcialaj Ekvacioj., 19 (1976), 239-245.  Google Scholar

[30]

J. P. Keener, Propagation and its failure in coupled systems of discrete excitable cells, SIAM J. Appl. Math., 47 (1987), 556-572. doi: 10.1137/0147038.  Google Scholar

[31]

J. P. Keener, The effects of discrete gap junction coupling on propagation in myocardium, J. Theor. Biol., 148 (1991), 49-82. Google Scholar

[32]

O. A. Ladyzhenskaya, "Attractors for Semigroups and Evolution Equations," Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511569418.  Google Scholar

[33]

J. P. Laplante and T. Erneux, Propagating failure in arrays of coupled bistable chemical reactors, J. Phys. Chem., 96 (1992), 4931-4934. Google Scholar

[34]

V. Lakshmikantham, A. R. Mitchell and R. W. Mitchell, On the existence of solutions of differential equations of retarde type in a Banach space,, Annales Polonici Mathematici, 35 (): 253.   Google Scholar

[35]

Y. Lv and J. Sun, Dynamical behavior for stochastic lattice systems, Chaos, Solitons and Fractals, 27 (2006), 1080-1090. doi: 10.1016/j.chaos.2005.04.089.  Google Scholar

[36]

J. Mallet-Paret, The global structure of traveling waves in spatially discrete dynamical systems, J. Dynam. Differential Equations, 11 (1999), 49-127. doi: 10.1023/A:1021841618074.  Google Scholar

[37]

V. S. Melnik and J. Valero, On attractors of multi-valued semi-flows and differential inclusions, Set-Valued Anal., 6 (1998), 83-111. doi: 10.1023/A:1008608431399.  Google Scholar

[38]

F. Morillas and J. Valero, Peano's theorem and attractors for lattice dynamical systems, Internat. J. Bifur. Chaos, 19 (2009), 557-578. doi: 10.1142/S0218127409023196.  Google Scholar

[39]

F. Morillas and J. Valero, On the connectedness of the attainability set for lattice dynamical systems, J. Diff. Equat. App., 18 (2012), 675-692. doi: 10.1080/10236198.2011.574621.  Google Scholar

[40]

N. Rashevsky, "Mathematical Biophysics: Physico-Mathematical Foundations of Biology," Third revised edition, Dover Publications, Inc., Vol. 1, New York, 1960.  Google Scholar

[41]

A. C. Scott, Analysis of a myelinated nerve model, Bull. Math. Biophys., 26 (1964), 247-254. Google Scholar

[42]

W. Shen, Lifted lattices, hyperbolic structures, and topological disorders in coupled map lattices, SIAM J. Appl. Math., 56 (1996), 1379-1399. doi: 10.1137/S0036139995282670.  Google Scholar

[43]

A. Sikorska-Nowak, Retarded functional differential equations in Banach spaces and Henstock-Kurzweil-Pettis integrals, Discuss. Math. Differ. Incl. Control Optim., 27 (2007), 315-327. doi: 10.7151/dmdico.1087.  Google Scholar

[44]

B. Wang, Dynamics of systems of infinite lattices, J. Differential Equations, 221 (2006), 224-245. doi: 10.1016/j.jde.2005.01.003.  Google Scholar

[45]

B. Wang, Asymptotic behavior of non-autonomous lattice systems, J. Math. Anal. Appl., 331 (2007), 121-136. doi: 10.1016/j.jmaa.2006.08.070.  Google Scholar

[46]

X. Wang, Sh. Li and D. Xu, Random attractors for second-order stochastic lattice dynamical systems, Nonlinear Anal., 72 (2010), 483-494. doi: 10.1016/j.na.2009.06.094.  Google Scholar

[47]

W. Yan, Y. Li and Sh. Ji, Random attractors for first order stochastic retarded lattice dynamical systems, J. Math. Phys., 51 (2010), 17 pp. doi: 10.1063/1.3319566.  Google Scholar

[48]

C. Zhao and S. Zhou, Attractors of retarded first order lattice systems, Nonlinearity, 20 (2007), 1987-2006. doi: 10.1088/0951-7715/20/8/010.  Google Scholar

[49]

C. Zhao and Sh. Zhou, Sufficient conditions for the existence of global random attractors for stochastic lattice dynamical systems and applications, J. Math. Anal. Appl., 354 (2009), 78-95. doi: 10.1016/j.jmaa.2008.12.036.  Google Scholar

[50]

C. Zhao, S. Zhou and W. Wang, Compact kernel sections for lattice systems with delays, Nonlinear Analysis TMA, 70 (2009), 1330-1348. doi: 10.1016/j.na.2008.02.015.  Google Scholar

[51]

S. Zhou, Attractors for first order dissipative lattice dynamical systems, Physica D, 178 (2003), 51-61. doi: 10.1016/S0167-2789(02)00807-2.  Google Scholar

[52]

S. Zhou, Attractors and approximations for lattice dynamical systems, J. Differential Equations, 200 (2004), 342-368. doi: 10.1016/j.jde.2004.02.005.  Google Scholar

[53]

S. Zhou and W. Shi, Attractors and dimension of dissipative lattice systems, J. Differential Equations, 224 (2006), 172-204. doi: 10.1016/j.jde.2005.06.024.  Google Scholar

[54]

B. Zinner, Existence of traveling wavefront solutions for the discrete Nagumo equation, J. Differential Equations, 96 (1992), 1-27. doi: 10.1016/0022-0396(92)90142-A.  Google Scholar

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