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

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

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.
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

[1]

Okihiro Sawada. Analytic rates of solutions to the Euler equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1409-1415. doi: 10.3934/dcdss.2013.6.1409

[2]

Mustafa Hasanbulli, Yuri V. Rogovchenko. Classification of nonoscillatory solutions of nonlinear neutral differential equations. Conference Publications, 2009, 2009 (Special) : 340-348. doi: 10.3934/proc.2009.2009.340

[3]

Hernán R. Henríquez, Claudio Cuevas, Juan C. Pozo, Herme Soto. Existence of solutions for a class of abstract neutral differential equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2455-2482. doi: 10.3934/dcds.2017106

[4]

Małgorzata Migda, Ewa Schmeidel, Małgorzata Zdanowicz. Periodic solutions of a $2$-dimensional system of neutral difference equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 359-367. doi: 10.3934/dcdsb.2018024

[5]

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

[6]

Miguel V. S. Frasson, Patricia H. Tacuri. Asymptotic behaviour of solutions to linear neutral delay differential equations with periodic coefficients. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1105-1117. doi: 10.3934/cpaa.2014.13.1105

[7]

Nguyen Minh Man, Nguyen Van Minh. On the existence of quasi periodic and almost periodic solutions of neutral functional differential equations. Communications on Pure & Applied Analysis, 2004, 3 (2) : 291-300. doi: 10.3934/cpaa.2004.3.291

[8]

Josef Diblík, Zdeněk Svoboda. Existence of strictly decreasing positive solutions of linear differential equations of neutral type. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 67-84. doi: 10.3934/dcdss.2020004

[9]

Bingyuan Huang, Shijin Ding, Huanyao Wen. Local classical solutions of compressible Navier-Stokes-Smoluchowski equations with vacuum. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1717-1752. doi: 10.3934/dcdss.2016072

[10]

Peixin Zhang, Jianwen Zhang, Junning Zhao. On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1085-1103. doi: 10.3934/dcds.2016.36.1085

[11]

Mikhail Kamenskii, Boris Mikhaylenko. Bifurcation of periodic solutions from a degenerated cycle in equations of neutral type with a small delay. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 437-452. doi: 10.3934/dcdsb.2013.18.437

[12]

Kai Liu. Stationary solutions of neutral stochastic partial differential equations with delays in the highest-order derivatives. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3915-3934. doi: 10.3934/dcdsb.2018117

[13]

Tomás Caraballo, Gábor Kiss. Attractivity for neutral functional differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1793-1804. doi: 10.3934/dcdsb.2013.18.1793

[14]

Eugenia N. Petropoulou, Panayiotis D. Siafarikas. A functional-analytic technique for the study of analytic solutions of PDEs. Conference Publications, 2015, 2015 (special) : 923-935. doi: 10.3934/proc.2015.0923

[15]

Yaobin Ou, Pan Shi. Global classical solutions to the free boundary problem of planar full magnetohydrodynamic equations with large initial data. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 537-567. doi: 10.3934/dcdsb.2017026

[16]

Bingkang Huang, Lan Zhang. A global existence of classical solutions to the two-dimensional Vlasov-Fokker-Planck and magnetohydrodynamics equations with large initial data. Kinetic & Related Models, 2019, 12 (2) : 357-396. doi: 10.3934/krm.2019016

[17]

Nicola Guglielmi, Christian Lubich. Numerical periodic orbits of neutral delay differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 1057-1067. doi: 10.3934/dcds.2005.13.1057

[18]

R.H. Fabiano, J. Turi. Making the numerical abscissa negative for a class of neutral equations. Conference Publications, 2003, 2003 (Special) : 256-262. doi: 10.3934/proc.2003.2003.256

[19]

T. Candan, R.S. Dahiya. Oscillation of mixed neutral differential equations with forcing term. Conference Publications, 2003, 2003 (Special) : 167-172. doi: 10.3934/proc.2003.2003.167

[20]

Juhi Jang, Ning Jiang. Acoustic limit of the Boltzmann equation: Classical solutions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 869-882. doi: 10.3934/dcds.2009.25.869

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]