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

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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

Abstract
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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 - A, 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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,, Rocky Mountain J. Math., 25 (1995), 201. 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. doi: doi:10.1007/BF00281373. Google Scholar
[15]	J. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations,", Applied Mathematical Sciences, 99 (1993). Google Scholar
[16]	J. K. Hale, Partial neutral functional-differential equations,, Rev. Roumaine Math Pures Appl, 39 (1994), 339. Google Scholar
[17]	H. Henriquez, Periodic solutions of abstract neutral functional differential equations with infinite delay,, Acta Math. Hungar., 121 (2008), 203. 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. 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. 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. 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. 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. 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. Google Scholar
[24]	A. Lunardi, On the linear heat equation with fading memory,, SIAM J. Math. Anal., 21 (1990), 1213. doi: doi:10.1137/0521066. Google Scholar
[25]	A. Lunardi., "Analytic Semigroups and Optimal Regularity in Parabolic Problems,", PNLDE, 16 (1995). Google Scholar
[26]	J. W. Nunziato, On heat conduction in materials with memory,, Quart. Appl. Math., 29 (1971), 187. Google Scholar
[27]	A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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,, Rocky Mountain J. Math., 25 (1995), 201. 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. doi: doi:10.1007/BF00281373. Google Scholar
[15]	J. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations,", Applied Mathematical Sciences, 99 (1993). Google Scholar
[16]	J. K. Hale, Partial neutral functional-differential equations,, Rev. Roumaine Math Pures Appl, 39 (1994), 339. Google Scholar
[17]	H. Henriquez, Periodic solutions of abstract neutral functional differential equations with infinite delay,, Acta Math. Hungar., 121 (2008), 203. 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. 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. 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. 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. 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. 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. Google Scholar
[24]	A. Lunardi, On the linear heat equation with fading memory,, SIAM J. Math. Anal., 21 (1990), 1213. doi: doi:10.1137/0521066. Google Scholar
[25]	A. Lunardi., "Analytic Semigroups and Optimal Regularity in Parabolic Problems,", PNLDE, 16 (1995). Google Scholar
[26]	J. W. Nunziato, On heat conduction in materials with memory,, Quart. Appl. Math., 29 (1971), 187. Google Scholar
[27]	A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (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. doi: doi:10.1006/jdeq.1996.0009. Google Scholar

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