March  2018, 38(3): 1365-1403. doi: 10.3934/dcds.2018056

Hölder-Lebesgue regularity and almost periodicity for semidiscrete equations with a fractional Laplacian

1. 

Departamento de Matemática y Ciencia de la Computación, Facultad de Ciencias, Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile

2. 

BCAM -Basque Center for Applied Mathematics, Alameda de Mazarredo 14,48009 Bilbao, Spain

* Corresponding author: Carlos Lizama.

Received  May 2017 Published  December 2017

Fund Project: The first author is partially supported by FONDECYT grant number 1140258 and CONICYTPIA-Anillo ACT1416. The second author is partially supported by grant MTM2015-65888-C04-4-P from the Government of Spain, by the Basque Government through the BERC 2014-2017 program, by Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-2013-0323 and by a 2017 Leonardo Grant for Researchers and Cultural Creators, BBVA Foundation. The Foundation accepts no responsibility for the opinions, statements and contents included in the project and/or the results thereof, which are entirely the responsibility of the authors.

We study the equations
$\begin{align}\partial_t u(t, n) = L u(t, n) + f(u(t, n), n); \partial_t u(t, n) = iL u(t, n) + f(u(t, n), n)\end{align}$
and
$\begin{align}\partial_{tt} u(t, n) =Lu(t, n) + f(u(t, n), n), \end{align}$
where
$n∈ \mathbb{Z}$
,
$t∈ (0, ∞)$
, and
$L$
is taken to be either the discrete Laplacian operator
$Δ_\mathrm{d} f(n)=f(n+1)-2f(n)+f(n-1)$
, or its fractional powers
$-(-Δ_{\mathrm{d}})^{σ}$
,
$0<σ<1$
. We combine operator theory techniques with the properties of the Bessel functions to develop a theory of analytic semigroups and cosine operators generated by
$Δ_\mathrm{d}$
and
$-(-Δ_\mathrm{d})^{σ}$
. Such theory is then applied to prove existence and uniqueness of almost periodic solutions to the above-mentioned equations. Moreover, we show a new characterization of well-posedness on periodic Hölder spaces for linear heat equations involving discrete and fractional discrete Laplacians. The results obtained are applied to Nagumo and Fisher-KPP models with a discrete Laplacian. Further extensions to the multidimensional setting
$\mathbb{Z}^N$
are also accomplished.
Citation: Carlos Lizama, Luz Roncal. Hölder-Lebesgue regularity and almost periodicity for semidiscrete equations with a fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1365-1403. doi: 10.3934/dcds.2018056
References:
[1]

L. AbadíasM. de León-Contreras and J. L. Torrea, Non-local fractional derivatives. Discrete and continuous, J. Math. Anal. Appl., 449 (2017), 734-755.  doi: 10.1016/j.jmaa.2016.12.006.  Google Scholar

[2]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables National Bureau of Standards Applied Mathematics Series, no. 55, U. S. Government Printing Office, Washington, DC, 1964.  Google Scholar

[3]

E. C. Aifantis, Continuum Nanomechanics for Nanocrystalline and Ultrafine Grain Materials Materials Science and Engineering, 63, 6th International Conference on Nanomaterials by Severe Plastic Deformation, 2014. doi: 10.1088/1757-899X/63/1/012129.  Google Scholar

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W. ArendtC. Batty and S. Bu, Fourier multipliers for Hölder continuous functions and maximal regularity, Studia Math., 160 (2004), 23-51.  doi: 10.4064/sm160-1-2.  Google Scholar

[5]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace transforms and Cauchy problems. Second edition. Monographs in Mathematics, 96. Birkhäuser/Springer Basel AG, Basel, 2011. xii+539 pp.  Google Scholar

[6]

H. Bateman, Some simple differential difference equations and the related functions, Bull. Amer. Math. Soc., 49 (1943), 494-512.  doi: 10.1090/S0002-9904-1943-07927-X.  Google Scholar

[7]

J. J. Betancor, A. J. Castro, J. C. Fariña and L. Rodríguez-Mesa, Discrete harmonic analysis associated with ultraspherical expansions, preprint, arXiv:1512.01379. Google Scholar

[8]

S. Bochner, Curvature and Betti numbers in Real and complex vector bundles, Univ. e Politec. di Torino. Rend. Sem. Mat., 15 (1955/56), 225-253.   Google Scholar

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J. Campbell, The SMM model as a boundary value problem using the discrete diffusion equation, Theor. Population Biol., 72 (2007), 539-546.  doi: 10.1016/j.tpb.2007.08.001.  Google Scholar

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S-N. ChowJ. 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

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Ó. CiaurriT. A. GillespieL. RoncalJ. L. Torrea and J. L. Varona, Harmonic analysis associated with a discrete Laplacian, J. Anal. Math., 132 (2017), 109-131.  doi: 10.1007/s11854-017-0015-6.  Google Scholar

[12]

Ó. CiaurriC. LizamaL. Roncal and J. L. Varona, On a connection between the discrete fractional Laplacian and superdiffusion, Appl. Math. Letters, 49 (2015), 119-125.  doi: 10.1016/j.aml.2015.05.007.  Google Scholar

[13]

Ó. Ciaurri, L. Roncal, P. R. Stinga, J. L. Torrea and J. L. Varona, Fractional discrete Laplacian versus discretized fractional Laplacian, preprint, arXiv:1507.04986. Google Scholar

[14]

Ó. Ciaurri, L. Roncal, P. R. Stinga, J. L. Torrea and J. L. Varona, Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications, preprint, arXiv:1608.08913. Google Scholar

[15]

C. Corduneanu, Almost Periodic Functions Interscience Tracts in Pure and Applied Mathematics, New York-London-Sydney, 1968.  Google Scholar

[16]

O. DefterliM. D'EliaQ. DuM. GunzburgerR. Lehoucq and M. M. Meerschaert, Fractional diffusion on bounded domains, Fract. Calc. Appl. Anal., 18 (2015), 342-360.   Google Scholar

[17]

R. E. Edwards, Fourier Series: A Modern Introduction, Vol. 2, Second ed. , Springer-Verlag, 1982.  Google Scholar

[18]

K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, vol. 194, Springer, New York, 2000.  Google Scholar

[19]

G. Fath, Propagation failure of traveling waves in a discrete bistable medium, Phys. D, 116 (1998), 176-190.  doi: 10.1016/S0167-2789(97)00251-0.  Google Scholar

[20]

H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces North-Holland Mathematics Studies, 108. Notas de Matemática, 99. North-Holland Publishing Co. , Amsterdam, 1985.  Google Scholar

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A. FaviniR. LabbasS. MaingotH. Tanabe and A. Yagi, Necessary and sufficient conditions for maximal regularity in the study of elliptic differential equations in Hölder spaces, Discrete Contin. Dyn. Syst., 22 (2008), 973-987.  doi: 10.3934/dcds.2008.22.973.  Google Scholar

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I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products, Seventh Edition Elsevier Academic Press, New York, 2007.  Google Scholar

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F. A. Grünbaum, "The bispectral problem: An overview", in Special functions 2000: current perspective and future directions (Tempe, AZ), 129-140, NATO Sci. Ser. Ⅱ Math. Phys. Chem. 30, Kluwer Acad. Publ., Dordrecht, 2001.  Google Scholar

[27]

F. A. Grünbaum and P. Iliev, Heat kernel expansions on the integers, Math. Phys. Anal. Geom., 5 (2002), 183-200.  doi: 10.1023/A:1016258207606.  Google Scholar

[28]

J.-S. Guo and C.-C. Wu, The existence of traveling wave solutions for a bistable three-component lattice dynamical system, J. Differential Equations, 260 (2016), 1445-1455.  doi: 10.1016/j.jde.2015.09.036.  Google Scholar

[29]

C. Hu and B. Li, Spatial dynamics for lattice differential equations with a shifting habitat, J. Differential Equations, 259 (2015), 1967-1989.  doi: 10.1016/j.jde.2015.03.025.  Google Scholar

[30]

I. Lasiecka and M. Wilke, Maximal regularity and global existence of solutions to a quasilinear thermoelastic plate system, Discrete Contin. Dyn. Syst., 33 (2013), 5189-5202.  doi: 10.3934/dcds.2013.33.5189.  Google Scholar

[31]

N. N. Lebedev, Special Functions and Its Applications Dover, New York, 1972.  Google Scholar

[32]

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

[33]

J. Mallet-Paret, Traveling waves in spatially discrete dynamical systems of diffusive type, Dynamical Systems, Lecture Notes in Math., Springer, Berlin, 1822 (2003), 231-298.   Google Scholar

[34]

C. Martínez Carracedo and M. Sanz Alix, The Theory of Fractional Powers of Operators North-Holland Mathematics Studies, vol. 187, North-Holland Publishing Co. , Amsterdam, 2001.  Google Scholar

[35]

F. W. J. Olver and L. C. Maximon, Bessel Functions, NIST handbook of mathematical functions (edited by F. W. F. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark), Chapter 10, National Institute of Standards and Technology, Washington, DC, and Cambridge University Press, Cambridge, 2010. Available online in http://dlmf.nist.gov/10. Google Scholar

[36]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983.  Google Scholar

[37]

C. E. Pearson, Asymptotic behavior of solutions to the finite-difference wave equation, Math. Comp., 23 (1969), 711-715.  doi: 10.1090/S0025-5718-1969-0264862-4.  Google Scholar

[38]

A. P. Prudnikov, A. Y. Brychkov and O. I. Marichev, Integrals and Series. Vol. 1. Elementary Functions Gordon and Breach Science Publishers, New York, 1986.  Google Scholar

[39]

A. P. Prudnikov, A. Y. Brychkov and O. I. Marichev, Integrals and Series. Vol. 2. Special Functions Gordon and Breach Science Publishers, New York, 1990.  Google Scholar

[40]

E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces Princeton Univ. Press, 1971.  Google Scholar

[41]

P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122.  doi: 10.1080/03605301003735680.  Google Scholar

[42]

V. E. Tarasov and G. M. Zaslavsky, Fractional dynamics of coupled oscillators with long-range interaction, Chaos, 16 (2006), 023110, 13 pp.  Google Scholar

[43]

V. E. Tarasov, Large lattice fractional Fokker-Planck equation, J. Stat. Mech., 2014 (2014).   Google Scholar

[44]

V. E. Tarasov, Lattice fractional calculus, Appl. Math. Comp., 257 (2015), 12-33.  doi: 10.1016/j.amc.2014.11.033.  Google Scholar

[45]

V. E. Tarasov, Fractional-order difference equations for physical lattices and some applications, J. Math. Phys. , 56 (2015), 103506, 19 pp.  Google Scholar

[46]

V. E. Tarasov, Fractional Liouville equation on lattice phase-space, Phys. A, 421 (2015), 330-342.  doi: 10.1016/j.physa.2014.11.031.  Google Scholar

[47]

E. S. Van Vleck and A. Zhang, Competing interactions and traveling wave solutions in lattice differential equations, Commun. Pure Appl. Anal., 15 (2016), 457-475.  doi: 10.3934/cpaa.2016.15.457.  Google Scholar

[48]

J. B. Walsh, Estimating the time to the most recent common ancestor for the Y chromosome or mitochondrial DNA for a pair of individuals, Genetics, 158 (2001), 897-912.   Google Scholar

[49]

G. A. Watson, A Treatise on the Theory of Bessel Functions Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944.  Google Scholar

[50]

K. Yosida, Functional Analysis reprint of the sixth (1980) edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.  Google Scholar

[51]

L. Zhang and S. Guo, Existence and multiplicity of wave trains in 2D lattices, J. Differential Equations, 257 (2014), 759-783.  doi: 10.1016/j.jde.2014.04.016.  Google Scholar

[52]

L. Zhou and W. Zhang, Admissibility and roughness of nonuniform exponential dichotomies for difference equations, J. Funct. Anal., 271 (2016), 1087-1129.  doi: 10.1016/j.jfa.2016.06.005.  Google Scholar

show all references

References:
[1]

L. AbadíasM. de León-Contreras and J. L. Torrea, Non-local fractional derivatives. Discrete and continuous, J. Math. Anal. Appl., 449 (2017), 734-755.  doi: 10.1016/j.jmaa.2016.12.006.  Google Scholar

[2]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables National Bureau of Standards Applied Mathematics Series, no. 55, U. S. Government Printing Office, Washington, DC, 1964.  Google Scholar

[3]

E. C. Aifantis, Continuum Nanomechanics for Nanocrystalline and Ultrafine Grain Materials Materials Science and Engineering, 63, 6th International Conference on Nanomaterials by Severe Plastic Deformation, 2014. doi: 10.1088/1757-899X/63/1/012129.  Google Scholar

[4]

W. ArendtC. Batty and S. Bu, Fourier multipliers for Hölder continuous functions and maximal regularity, Studia Math., 160 (2004), 23-51.  doi: 10.4064/sm160-1-2.  Google Scholar

[5]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace transforms and Cauchy problems. Second edition. Monographs in Mathematics, 96. Birkhäuser/Springer Basel AG, Basel, 2011. xii+539 pp.  Google Scholar

[6]

H. Bateman, Some simple differential difference equations and the related functions, Bull. Amer. Math. Soc., 49 (1943), 494-512.  doi: 10.1090/S0002-9904-1943-07927-X.  Google Scholar

[7]

J. J. Betancor, A. J. Castro, J. C. Fariña and L. Rodríguez-Mesa, Discrete harmonic analysis associated with ultraspherical expansions, preprint, arXiv:1512.01379. Google Scholar

[8]

S. Bochner, Curvature and Betti numbers in Real and complex vector bundles, Univ. e Politec. di Torino. Rend. Sem. Mat., 15 (1955/56), 225-253.   Google Scholar

[9]

J. Campbell, The SMM model as a boundary value problem using the discrete diffusion equation, Theor. Population Biol., 72 (2007), 539-546.  doi: 10.1016/j.tpb.2007.08.001.  Google Scholar

[10]

S-N. ChowJ. 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

[11]

Ó. CiaurriT. A. GillespieL. RoncalJ. L. Torrea and J. L. Varona, Harmonic analysis associated with a discrete Laplacian, J. Anal. Math., 132 (2017), 109-131.  doi: 10.1007/s11854-017-0015-6.  Google Scholar

[12]

Ó. CiaurriC. LizamaL. Roncal and J. L. Varona, On a connection between the discrete fractional Laplacian and superdiffusion, Appl. Math. Letters, 49 (2015), 119-125.  doi: 10.1016/j.aml.2015.05.007.  Google Scholar

[13]

Ó. Ciaurri, L. Roncal, P. R. Stinga, J. L. Torrea and J. L. Varona, Fractional discrete Laplacian versus discretized fractional Laplacian, preprint, arXiv:1507.04986. Google Scholar

[14]

Ó. Ciaurri, L. Roncal, P. R. Stinga, J. L. Torrea and J. L. Varona, Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications, preprint, arXiv:1608.08913. Google Scholar

[15]

C. Corduneanu, Almost Periodic Functions Interscience Tracts in Pure and Applied Mathematics, New York-London-Sydney, 1968.  Google Scholar

[16]

O. DefterliM. D'EliaQ. DuM. GunzburgerR. Lehoucq and M. M. Meerschaert, Fractional diffusion on bounded domains, Fract. Calc. Appl. Anal., 18 (2015), 342-360.   Google Scholar

[17]

R. E. Edwards, Fourier Series: A Modern Introduction, Vol. 2, Second ed. , Springer-Verlag, 1982.  Google Scholar

[18]

K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, vol. 194, Springer, New York, 2000.  Google Scholar

[19]

G. Fath, Propagation failure of traveling waves in a discrete bistable medium, Phys. D, 116 (1998), 176-190.  doi: 10.1016/S0167-2789(97)00251-0.  Google Scholar

[20]

H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces North-Holland Mathematics Studies, 108. Notas de Matemática, 99. North-Holland Publishing Co. , Amsterdam, 1985.  Google Scholar

[21]

A. FaviniR. LabbasS. MaingotH. Tanabe and A. Yagi, Necessary and sufficient conditions for maximal regularity in the study of elliptic differential equations in Hölder spaces, Discrete Contin. Dyn. Syst., 22 (2008), 973-987.  doi: 10.3934/dcds.2008.22.973.  Google Scholar

[22]

W. Feller, An Introduction to Probability Theory and Its Applications. Vol. 1. Third edition. John Wiley & Sons, Inc. , New York-London-Sydney, 1968.  Google Scholar

[23]

W. Feller, An Introduction to Probability Theory and Its Applications. Vol. 2. Second edition. John Wiley & Sons, Inc. , New York-London-Sydney, 1971. Google Scholar

[24]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order. Second Edition Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 224. Springer-Verlag, Berlin, 1983.  Google Scholar

[25]

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products, Seventh Edition Elsevier Academic Press, New York, 2007.  Google Scholar

[26]

F. A. Grünbaum, "The bispectral problem: An overview", in Special functions 2000: current perspective and future directions (Tempe, AZ), 129-140, NATO Sci. Ser. Ⅱ Math. Phys. Chem. 30, Kluwer Acad. Publ., Dordrecht, 2001.  Google Scholar

[27]

F. A. Grünbaum and P. Iliev, Heat kernel expansions on the integers, Math. Phys. Anal. Geom., 5 (2002), 183-200.  doi: 10.1023/A:1016258207606.  Google Scholar

[28]

J.-S. Guo and C.-C. Wu, The existence of traveling wave solutions for a bistable three-component lattice dynamical system, J. Differential Equations, 260 (2016), 1445-1455.  doi: 10.1016/j.jde.2015.09.036.  Google Scholar

[29]

C. Hu and B. Li, Spatial dynamics for lattice differential equations with a shifting habitat, J. Differential Equations, 259 (2015), 1967-1989.  doi: 10.1016/j.jde.2015.03.025.  Google Scholar

[30]

I. Lasiecka and M. Wilke, Maximal regularity and global existence of solutions to a quasilinear thermoelastic plate system, Discrete Contin. Dyn. Syst., 33 (2013), 5189-5202.  doi: 10.3934/dcds.2013.33.5189.  Google Scholar

[31]

N. N. Lebedev, Special Functions and Its Applications Dover, New York, 1972.  Google Scholar

[32]

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

[33]

J. Mallet-Paret, Traveling waves in spatially discrete dynamical systems of diffusive type, Dynamical Systems, Lecture Notes in Math., Springer, Berlin, 1822 (2003), 231-298.   Google Scholar

[34]

C. Martínez Carracedo and M. Sanz Alix, The Theory of Fractional Powers of Operators North-Holland Mathematics Studies, vol. 187, North-Holland Publishing Co. , Amsterdam, 2001.  Google Scholar

[35]

F. W. J. Olver and L. C. Maximon, Bessel Functions, NIST handbook of mathematical functions (edited by F. W. F. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark), Chapter 10, National Institute of Standards and Technology, Washington, DC, and Cambridge University Press, Cambridge, 2010. Available online in http://dlmf.nist.gov/10. Google Scholar

[36]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983.  Google Scholar

[37]

C. E. Pearson, Asymptotic behavior of solutions to the finite-difference wave equation, Math. Comp., 23 (1969), 711-715.  doi: 10.1090/S0025-5718-1969-0264862-4.  Google Scholar

[38]

A. P. Prudnikov, A. Y. Brychkov and O. I. Marichev, Integrals and Series. Vol. 1. Elementary Functions Gordon and Breach Science Publishers, New York, 1986.  Google Scholar

[39]

A. P. Prudnikov, A. Y. Brychkov and O. I. Marichev, Integrals and Series. Vol. 2. Special Functions Gordon and Breach Science Publishers, New York, 1990.  Google Scholar

[40]

E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces Princeton Univ. Press, 1971.  Google Scholar

[41]

P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122.  doi: 10.1080/03605301003735680.  Google Scholar

[42]

V. E. Tarasov and G. M. Zaslavsky, Fractional dynamics of coupled oscillators with long-range interaction, Chaos, 16 (2006), 023110, 13 pp.  Google Scholar

[43]

V. E. Tarasov, Large lattice fractional Fokker-Planck equation, J. Stat. Mech., 2014 (2014).   Google Scholar

[44]

V. E. Tarasov, Lattice fractional calculus, Appl. Math. Comp., 257 (2015), 12-33.  doi: 10.1016/j.amc.2014.11.033.  Google Scholar

[45]

V. E. Tarasov, Fractional-order difference equations for physical lattices and some applications, J. Math. Phys. , 56 (2015), 103506, 19 pp.  Google Scholar

[46]

V. E. Tarasov, Fractional Liouville equation on lattice phase-space, Phys. A, 421 (2015), 330-342.  doi: 10.1016/j.physa.2014.11.031.  Google Scholar

[47]

E. S. Van Vleck and A. Zhang, Competing interactions and traveling wave solutions in lattice differential equations, Commun. Pure Appl. Anal., 15 (2016), 457-475.  doi: 10.3934/cpaa.2016.15.457.  Google Scholar

[48]

J. B. Walsh, Estimating the time to the most recent common ancestor for the Y chromosome or mitochondrial DNA for a pair of individuals, Genetics, 158 (2001), 897-912.   Google Scholar

[49]

G. A. Watson, A Treatise on the Theory of Bessel Functions Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944.  Google Scholar

[50]

K. Yosida, Functional Analysis reprint of the sixth (1980) edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.  Google Scholar

[51]

L. Zhang and S. Guo, Existence and multiplicity of wave trains in 2D lattices, J. Differential Equations, 257 (2014), 759-783.  doi: 10.1016/j.jde.2014.04.016.  Google Scholar

[52]

L. Zhou and W. Zhang, Admissibility and roughness of nonuniform exponential dichotomies for difference equations, J. Funct. Anal., 271 (2016), 1087-1129.  doi: 10.1016/j.jfa.2016.06.005.  Google Scholar

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