Existence of Solutions for Abstract Non-Autonomous Neutral Differential Equations

Hernández, Eduardo ; O’Regan, Donal

Canadian mathematical bulletin, Tome 55 (2012) no. 4, pp. 736-751

Résumé

In this paper we discuss the existence of mild and classical solutions for a class of abstract non-autonomous neutral functional differential equations. An application to partial neutral differential equations is considered.

Détail

DOI : 10.4153/CMB-2011-111-1

Mots-clés : 35R10, 34K40, 34K30, neutral equations, mild solutions, classical solutions

Bibliographie

[1] [1] Adimy, M. and Ezzinbi, K., A class of linear partial neutral functional-differential equations with nondense domain. J. Differential Equations 147(1998), no. 2, 285–332. Google Scholar | DOI

[2] [2] Balachandran, K., Shija, G., and Kim, J. H., Existence of solutions of nonlinear abstract neutral integrodifferential equations. Comput. Math. Appl. 48(2004), no. 10-11, 1403–1414. Google Scholar | DOI

[3] [3] Balachandran, K. and Sakthivel, R., Existence of solutions of neutral functional integrodifferential equation in Banach spaces. Proc. Indian Acad. Sci. Math. Sci. 109(1999), no. 3, 325–332. Google Scholar | DOI

[4] [4] Benchohra, M. and Ntouyas, S. K., Nonlocal Cauchy problems for neutral functional differential and integrodifferential inclusions in Banach spaces. J. Math. Anal. Appl. 258(2001), no. 2, 573–590. Google Scholar | DOI

[5] [5] Benchohra, M., Henderson, J., and Ntouyas, S. K., Existence results for impulsive multivalued semilinear neutral functional differential inclusions in Banach spaces. J. Math. Anal. Appl. 263(2001), no. 2, 763–780. Google Scholar | DOI

[6] [6] Cannarsa, P. and Sforza, D., Global solutions of abstract semilinear parabolic equations with memory terms. NoDEA Nonlinear Differential Equations. Appl. 10(2003), no. 4, 399–430. Google Scholar | DOI

[7] [7] Clément, P. and Nohel, J. A., Asymptotic behavior of solutions of nonlinear Volterra equations with completely positive kernels. SIAM J. Math. Anal. 12(1981), no. 4, 514–535. Google Scholar | DOI

[8] [8] Clément, P. and Prüss, J., Global existence for a semilinear parabolic Volterra equation. Math. Z. 209(1992), no. 1, 17–26. Google Scholar | DOI

[9] [9] Datko, R., Linear autonomous neutral differential equations in a Banach space. J. Diff. Equations 25(1977), no. 2, 258–274. Google Scholar | DOI

[10] [10] Dauer, J. P. and Balachandran, K., Existence of solutions of nonlinear neutral integrodifferential equations in Banach spaces. J. Math. Anal. Appl. 251(2000), no. 1, 93–105. Google Scholar | DOI

[11] [11] Fang, H. and Li, J., On the existence of periodic solutions of a neutral delay model of single-species population growth. J. Math. Anal. Appl. 259(2001), no. 1, 8–17. Google Scholar | DOI

[12] [12] Freedman, H. I. and Kuang, Y., Some global qualitative analyses of a single species neutral delay differential population model. Rocky Mountain J. Math. 25(1995), no. 1, 201–215. Google Scholar | DOI

[13] [13] Fu, X. and Liu, X., Existence of periodic solutions for abstract neutral non-autonomous equations with infinite delay. J. Math. Anal. Appl. 325(2007), no. 1, 249–267. Google Scholar | DOI

[14] [14] Granas, A.. and Dugundji, J., Fixed Point Theory. Springer-Verlag, New York, 2003. Google Scholar

[15] [15] Gurtin, M. E. and Pipkin, A. C., A general theory of heat conduction with finite wave speed. Arch. Rational Mech. Anal. 31(1968), no. 2, 113–126. Google Scholar | DOI

[16] [16] Hale, J. K. Partial neutral functional-differential equations. Rev. Roumaine Math. Pures Appl. 39(1994), no. 4, 339–344. Google Scholar

[17] [17] Hale, J. K. and Lunel, S. M. Verduyn, Introduction to Functional-Differential Equations. Applied Mathematical Sciences 99. Springer-Verlag, New York, 1993. Google Scholar

[18] [18] Henríquez, H. R., Periodic solutions of abstract neutral functional differential equations with infinite delay. Acta Math. Hungar. 121(2008), no. 3, 203–227. Google Scholar | DOI

[19] [19] Hernández, E., Existence results for partial neutral integrodifferential equations with unbounded delay. J. Math. Anal. Appl. 292(2004), no. 1, 194–210. Google Scholar | DOI

[20] [20] Hernández, E. and O’Regan, D., Existence results for abstract partial neutral differential equations. To appear in Proc. Amer. Math. Soc. Google Scholar

[21] [21] Hernández, E. and McKibben, M., Some comments on: “Existence of solutions of abstract nonlinear second-order neutral functional integrodifferential equations.” [Comput. Math. Appl. 46 (2003), no. 8-9] Comput. Math. Appl. 50(2005), no. 5-6, 655–669. Google Scholar | DOI

[22] [22] Hernández, E. and Henríquez, H., Existence results for partial neutral functional-differential equation with unbounded delay. J. Math. Anal. Appl. 221(1998), no. 2, 452–475. Google Scholar | DOI

[23] [23] Hernández, E. and Henríquez, H., Existence of periodic solution of partial neutral functional-differential equation with unbounded delay. J. Math. Anal. Appl. 221(1998) no. 2, 499–522. Google Scholar | DOI

[24] [24] Kuang, Y., Qualitative analysis of one- or two-species neutral delay population models. SIAM J. Math. Anal. 23(1992), no. 1, 181–200. Google Scholar | DOI

[25] [25] Li, Q., Cao, J., and Wan, S., Positive periodic solution for a neutral delay model in population. J. Biomath. 13(1998), no. 4, 435–438. Google Scholar

[26] [26] Lunardi, A., On the linear heat equation with fading memory. SIAM J. Math. Anal. 21(1990), no. 5, 1213–1224. Google Scholar | DOI

[27] [27] Lunardi, A., Analytic semigroups and optimal regularity in parabolic problems. Progress in Nonlinear Differential Equations and their Applications 16. Birkhäauser Verlag, Basel, 1995. Google Scholar

[28] [28] Nunziato, J. W., On heat conduction in materials with memory. Quart. Appl. Math. 29(1971), 187–204. Google Scholar

[29] [29] Wu, J. and Xia, H., Self-sustained oscillations in a ring array of coupled lossless transmission lines. J. Differential Equations 124(1996), no. 1, 247–278. Google Scholar | DOI