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January  2011, 29(1): 241-260. doi: 10.3934/dcds.2011.29.241

$C^{\alpha}$-Hölder classical solutions for non-autonomous neutral differential equations

Eduardo Hernández 1, and  Donal O'Regan 2,

1. 

Departamento de Física e Matemática, Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto. Universidade de São Paulo, Ribeirão Preto, SP, Cp 14040-901, Brazil
2. 

Department of Mathematics, National University of Ireland, Galway

Received  December 2009 Revised  June 2010 Published  September 2010

In this paper we discuss the existence of α-Hölder classical solutions for non-autonomous abstract partial neutral functional differential equations. An application is considered.
Keywords: Neutral equations, classical solutions, analytic semigroup..
Mathematics Subject Classification: Primary: 35R10, 34K40; Secondary: 34K3.
Citation: Eduardo Hernández, Donal O'Regan. $C^{\alpha}$-Hölder classical solutions for non-autonomous neutral differential equations. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 241-260. doi: 10.3934/dcds.2011.29.241
References:
[1]

M. Adimy and K. Ezzinbi, A class of linear partial neutral functional-differential equations with nondense domain, J. Diff. Eqns., 147 (1998), 285-332. doi: doi:10.1006/jdeq.1998.3446.  Google Scholar

[2]

K. Balachandran, G. Shija and J. Kim, Existence of solutions of nonlinear abstract neutral integrodifferential equations, Comput. Math. Appl., 48 (2004), 1403-1414. doi: doi:10.1016/j.camwa.2004.08.002.  Google Scholar

[3]

K. Balachandran and R. Sakthivel, Existence of solutions of neutral functional integrodifferential equation in Banach spaces, Proc. Indian Acad. Sci. Math. Sci., 109 (1999), 325-332. doi: doi:10.1007/BF02843536.  Google Scholar

[4]

M. Benchohra and S. Ntouyas, Nonlocal Cauchy problems for neutral functional differential and integrodifferential inclusions in Banach spaces, J. Math. Anal. Appl., 258 (2001), 573-590. doi: doi:10.1006/jmaa.2000.7394.  Google Scholar

[5]

M. Benchohra, J. Henderson and S. Ntouyas, Existence results for impulsive multivalued semilinear neutral functional differential inclusions in Banach spaces, J. Math. Anal. Appl., 263 (2001), 763-780. doi: doi:10.1006/jmaa.2001.7663.  Google Scholar

[6]

J. P. Dauer and K. Balachandran, Existence of solutions of nonlinear neutral integrodifferential equations in Banach spaces, J. Math. Anal. Appl., 251 (2000), 93-105. doi: doi:10.1006/jmaa.2000.7022.  Google Scholar

[7]

P. Cannarsa and D. Sforza, Global solutions of abstract semilinear parabolic equations with memory terms, NoDEA Nonlinear Differential Equations. Appl., 10 (2003), 399-430.  Google Scholar

[8]

Ph. Clément and J. A. Nohel, Asymptotic behavior of solutions of nonlinear Volterra equations with completely positive kernels, SIAM J. Math. Anal., 12 (1981), 514-535. doi: doi:10.1137/0512045.  Google Scholar

[9]

Ph. Clément and J. Prüss, Global existence for a semilinear parabolic Volterra equation, Math. Z., 209 (1992), 17-26. doi: doi:10.1007/BF02570816.  Google Scholar

[10]

R. Datko, Linear autonomous neutral differential equations in a Banach space, J. Diff. Equations, 25 (1977), 258-274. doi: doi:10.1016/0022-0396(77)90204-2.  Google Scholar

[11]

H. Fang and J. Li, On the existence of periodic solutions of a neutral delay model of single-species population growth, J. Math. Anal. Appl., 259 (2001), 8-17. doi: doi:10.1006/jmaa.2000.7340.  Google Scholar

[12]

X. Fu and X. Liu, Existence of periodic solutions for abstract neutral non-autonomous equations with infinite delay, J. Math. Anal. Appl., 325 (2007), 249-267. doi: doi:10.1016/j.jmaa.2006.01.048.  Google Scholar

[13]

H. I. Freedman and Y. Kuang, Some global qualitative analyses of a single species neutral delay differential population model, "Second Geoffrey J. Butler Memorial Conference in Differential Equations and Mathematical Biology" (Edmonton, AB, 1992), Rocky Mountain J. Math., 25 (1995), 201-215. doi: doi:10.1216/rmjm/1181072278.  Google Scholar

[14]

M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speed, Arch. Rat. Mech. Anal., 31 (1968), 113-126. doi: doi:10.1007/BF00281373.  Google Scholar

[15]

J. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations," Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993.  Google Scholar

[16]

J. K. Hale, Partial neutral functional-differential equations, Rev. Roumaine Math Pures Appl, 39 (1994), 339-344.  Google Scholar

[17]

H. Henriquez, Periodic solutions of abstract neutral functional differential equations with infinite delay, Acta Math. Hungar., 121 (2008), 203-227. doi: doi:10.1007/s10474-008-7009-x.  Google Scholar

[18]

E. Hernández and D. O'Regan, Existence results for abstract partial neutral differential equations, Proc. Amer. Math. Soc., 137 (2009), 3309-3318. doi: doi:10.1090/S0002-9939-09-09934-1.  Google Scholar

[19]

E. Hernández and H. Henríquez, Existence results for partial neutral functional differential equation with unbounded delay, J. Math. Anal. Appl., 221 (1998), 452-475. doi: doi:10.1006/jmaa.1997.5875.  Google Scholar

[20]

E. Hernández, Existence results for partial neutral integrodifferential equations with unbounded delay, J. Math. Anal. Appl., 292 (2004), 194-210. doi: doi:10.1016/j.jmaa.2003.11.052.  Google Scholar

[21]

E. Hernández and H. Henríquez, Existence of periodic solution of partial neutral functional differential equation with unbounded delay, J. Math. Anal. Appl., 221 (1998), 499-522. doi: doi:10.1006/jmaa.1997.5899.  Google Scholar

[22]

Y. Kuang, Qualitative analysis of one- or two-species neutral delay population models, SIAM J. Math. Anal., 23 (1992), 181-200. doi: doi:10.1137/0523009.  Google Scholar

[23]

Q. Li, J. Cao and S. Wan, Positive periodic solution for a neutral delay model in population, J. Biomath., 13 (1998), 435-438.  Google Scholar

[24]

A. Lunardi, On the linear heat equation with fading memory, SIAM J. Math. Anal., 21 (1990), 1213-1224. doi: doi:10.1137/0521066.  Google Scholar

[25]

A. Lunardi., "Analytic Semigroups and Optimal Regularity in Parabolic Problems," PNLDE, 16, Birkhäuser Verlag, Basel, 1995.  Google Scholar

[26]

J. W. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math., 29 (1971), 187-204.  Google Scholar

[27]

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

[28]

J. Wu and Xia Huaxing, Self-sustained oscillations in a ring array of coupled lossless transmission lines, J. Differential Equations, 124 (1996), 247-278. doi: doi:10.1006/jdeq.1996.0009.  Google Scholar

show all references

References:
[1]

M. Adimy and K. Ezzinbi, A class of linear partial neutral functional-differential equations with nondense domain, J. Diff. Eqns., 147 (1998), 285-332. doi: doi:10.1006/jdeq.1998.3446.  Google Scholar

[2]

K. Balachandran, G. Shija and J. Kim, Existence of solutions of nonlinear abstract neutral integrodifferential equations, Comput. Math. Appl., 48 (2004), 1403-1414. doi: doi:10.1016/j.camwa.2004.08.002.  Google Scholar

[3]

K. Balachandran and R. Sakthivel, Existence of solutions of neutral functional integrodifferential equation in Banach spaces, Proc. Indian Acad. Sci. Math. Sci., 109 (1999), 325-332. doi: doi:10.1007/BF02843536.  Google Scholar

[4]

M. Benchohra and S. Ntouyas, Nonlocal Cauchy problems for neutral functional differential and integrodifferential inclusions in Banach spaces, J. Math. Anal. Appl., 258 (2001), 573-590. doi: doi:10.1006/jmaa.2000.7394.  Google Scholar

[5]

M. Benchohra, J. Henderson and S. Ntouyas, Existence results for impulsive multivalued semilinear neutral functional differential inclusions in Banach spaces, J. Math. Anal. Appl., 263 (2001), 763-780. doi: doi:10.1006/jmaa.2001.7663.  Google Scholar

[6]

J. P. Dauer and K. Balachandran, Existence of solutions of nonlinear neutral integrodifferential equations in Banach spaces, J. Math. Anal. Appl., 251 (2000), 93-105. doi: doi:10.1006/jmaa.2000.7022.  Google Scholar

[7]

P. Cannarsa and D. Sforza, Global solutions of abstract semilinear parabolic equations with memory terms, NoDEA Nonlinear Differential Equations. Appl., 10 (2003), 399-430.  Google Scholar

[8]

Ph. Clément and J. A. Nohel, Asymptotic behavior of solutions of nonlinear Volterra equations with completely positive kernels, SIAM J. Math. Anal., 12 (1981), 514-535. doi: doi:10.1137/0512045.  Google Scholar

[9]

Ph. Clément and J. Prüss, Global existence for a semilinear parabolic Volterra equation, Math. Z., 209 (1992), 17-26. doi: doi:10.1007/BF02570816.  Google Scholar

[10]

R. Datko, Linear autonomous neutral differential equations in a Banach space, J. Diff. Equations, 25 (1977), 258-274. doi: doi:10.1016/0022-0396(77)90204-2.  Google Scholar

[11]

H. Fang and J. Li, On the existence of periodic solutions of a neutral delay model of single-species population growth, J. Math. Anal. Appl., 259 (2001), 8-17. doi: doi:10.1006/jmaa.2000.7340.  Google Scholar

[12]

X. Fu and X. Liu, Existence of periodic solutions for abstract neutral non-autonomous equations with infinite delay, J. Math. Anal. Appl., 325 (2007), 249-267. doi: doi:10.1016/j.jmaa.2006.01.048.  Google Scholar

[13]

H. I. Freedman and Y. Kuang, Some global qualitative analyses of a single species neutral delay differential population model, "Second Geoffrey J. Butler Memorial Conference in Differential Equations and Mathematical Biology" (Edmonton, AB, 1992), Rocky Mountain J. Math., 25 (1995), 201-215. doi: doi:10.1216/rmjm/1181072278.  Google Scholar

[14]

M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speed, Arch. Rat. Mech. Anal., 31 (1968), 113-126. doi: doi:10.1007/BF00281373.  Google Scholar

[15]

J. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations," Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993.  Google Scholar

[16]

J. K. Hale, Partial neutral functional-differential equations, Rev. Roumaine Math Pures Appl, 39 (1994), 339-344.  Google Scholar

[17]

H. Henriquez, Periodic solutions of abstract neutral functional differential equations with infinite delay, Acta Math. Hungar., 121 (2008), 203-227. doi: doi:10.1007/s10474-008-7009-x.  Google Scholar

[18]

E. Hernández and D. O'Regan, Existence results for abstract partial neutral differential equations, Proc. Amer. Math. Soc., 137 (2009), 3309-3318. doi: doi:10.1090/S0002-9939-09-09934-1.  Google Scholar

[19]

E. Hernández and H. Henríquez, Existence results for partial neutral functional differential equation with unbounded delay, J. Math. Anal. Appl., 221 (1998), 452-475. doi: doi:10.1006/jmaa.1997.5875.  Google Scholar

[20]

E. Hernández, Existence results for partial neutral integrodifferential equations with unbounded delay, J. Math. Anal. Appl., 292 (2004), 194-210. doi: doi:10.1016/j.jmaa.2003.11.052.  Google Scholar

[21]

E. Hernández and H. Henríquez, Existence of periodic solution of partial neutral functional differential equation with unbounded delay, J. Math. Anal. Appl., 221 (1998), 499-522. doi: doi:10.1006/jmaa.1997.5899.  Google Scholar

[22]

Y. Kuang, Qualitative analysis of one- or two-species neutral delay population models, SIAM J. Math. Anal., 23 (1992), 181-200. doi: doi:10.1137/0523009.  Google Scholar

[23]

Q. Li, J. Cao and S. Wan, Positive periodic solution for a neutral delay model in population, J. Biomath., 13 (1998), 435-438.  Google Scholar

[24]

A. Lunardi, On the linear heat equation with fading memory, SIAM J. Math. Anal., 21 (1990), 1213-1224. doi: doi:10.1137/0521066.  Google Scholar

[25]

A. Lunardi., "Analytic Semigroups and Optimal Regularity in Parabolic Problems," PNLDE, 16, Birkhäuser Verlag, Basel, 1995.  Google Scholar

[26]

J. W. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math., 29 (1971), 187-204.  Google Scholar

[27]

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

[28]

J. Wu and Xia Huaxing, Self-sustained oscillations in a ring array of coupled lossless transmission lines, J. Differential Equations, 124 (1996), 247-278. doi: doi:10.1006/jdeq.1996.0009.  Google Scholar

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