Abstract
In this paper we propose to study, in the \(\alpha \)-norm, a class of neutral partial functional integrodifferential equations with infinite delay. We assume that the linear part generates an analytic and compact semigroup and the nonlinear part of the system involve spatial derivatives. At the end, an example is provided to illustrate the application of the obtained results.
Similar content being viewed by others
References
Adimy, M., Ezzinbi, K.: Existence and stability in the \(\alpha \)-norm for partial functional differential equations of neutral type. Ann. Mate. Pura Appl. 185(3), 437–460 (2006)
Chang, J., Lui, H.: Existence of solutions for a class of neutral partial differential equations with nonlocal conditions in the \(\alpha \)-norm. Nonlinear Anal. RWA 71, 3759–3768 (2009)
Chang, J., Lui, H., Kui, N.: Existence of solutions for impulsive neutral partial integrodifferential inclusions with nonlocal conditions via fractional operators. Numer. Funct. Anal. Optim. 30 (2009)
Desch, W., Grimmer, R., Schappacher, W.: Some considerations for linear integrodifferential equtions. J. Math. Anal. Appl. 104, 219–234 (1984)
Prato, D., Lannelli, G.: Existence and regularity for a class of integrodifferential equations of parabolic type. J. Math. Anal. Appl. 1, 36–55 (1984)
Prato, D., Lunardi, G.: Solvability on the real line of a class of linear Volterra integrodifferential equations of parabolic type. Ann. Mathe. Pura Appl. 150(4), 67–117 (1998)
Diao, B., Ezzinbi, K., Sy, M.: Existence, regularity and compactness properties, in the \(\alpha \)-norm, for some partial functional integrodifferential equations with finite delay. J. Differ. Equ. Dyn. Syst. 233 (2013). doi:10.1007/s12591-014-0233-7
Diao, B., Ezzinbi, K., Sy, M.: Existence and regularity in the \(\alpha \)-norm for some partial functional integrodifferential equations with infinite delay (2013, preprint)
Diao, B., Ezzinbi, K., Sy, M.: Some results in the \(\alpha \)-norm for a class of neutral partial functional integrodifferential equations with finite delay. Afr. Mathe. 0313 (2014). doi:10.1007/s13370-014-0313-4
Ezzinbi, K., Benkhalti, R.: Existence and stability in the \(\alpha \)-norm for some partial functional differential equations with infinite delay. Differ. Integr. Equ. 19(5), 545–572 (2006)
Ezzinbi, K., Ghnimi, S.: Local existence and global continuation for some partial functional integrodifferential equations. Afr. Diaspora J. Math. 12(1), 34–45 (2011)
Ezzinbi, K., Ghnimi, S.: Existence and regularity of solution for neutral partial functional integrodifferential equations. Nonlinear Anal. RWA 11, 2335–2344 (2010)
Ezzinbi, K., Ghnimi, S., Taoudi, M.: Existence and regularity of solutions for neutral partial functional integrodifferential equations with infinite delay. Nonlinear Anal. 4, 54–64 (2010)
Ezzinbi, K., Touré, H., Zabsonre, I.: Existence and regularity of solutions for some partial functional integrodifferential equations in Banach spaces. Nonlinear Anal. Theory Methods Appl. 70(7), 2761–2771 (2009)
Grimmer, R.: Resolvent operators for integral equations in a Banach space. Trans. Am. Math. Soc. 273, 333–349 (1982)
Grimmer, R., Pritchard, A.J.: Analytic resolvent operators for integral equations in a Banach space. J. Differ. Equ. 50(2), 234–259 (1983)
Grimmer, R., Pruss, R.: On linear Voltera equations in Banach spaces, hyperbolic partial differential equations II. Comput. Math. Appl. 11(1–3), 189–205 (1985)
Hale, J.K., Kato, J.: Phase space for retarded equations with infinite delay. Funkc. Ekvac 21, 11–41 (1978)
Hernández, E., Henríquez, H., Santos, J.P.C.: Existence results for abstract partial neutral integrodifferential equation with unbounded delay. Electron. J. Qual. Theory Differ. Equ. 29, 1–23 (2009)
Hino, Y., Murakami, S., Naito, T.: Functional differential equations with infinite delay. In: Lecture Notes in Mathematics, vol. 1473. Springer, Berlin (1991)
Lunardi, A.: Alessandra Laplace transform methods in integrodifferential equations, integrodifferential evolution equations and applications. J. Integr. Equ. 1–3 (1985)
Miller, R.K.: An integrodifferential equation for rigid heat conductors with memory. J. Math. Anal. Appl. 66, 313–332 (1978)
Pazy, A.: Semigroups of Linear Operators and Application to Partial Differential Equations, Applied Mathematical Sciences, vol. 44. Springer, New York (2001)
Sakthivel, R., Choi, Q.H., Anthoni, S.M.: Controllability of nonlinear neutral evolution integrodiffrential equations. J. Math. Anal. Appl. 275, 402–417 (2002)
Travis, C.C., Webb, G.F.: Existence, stability, and compactness in the \(\alpha \)-norm for partial functional differential equations. Trans. Am. Math. Soc. 240, 129–143 (1978)
Travis, C.C., Webb, G.F.: Existence and stability for partial functional differential equations. Trans. Am. Math. Soc. 200, 395–418 (1974)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Diao, B., Ezzinbi, K. & Sy, M. Existence results in the \(\alpha \)-norm for neutral partial functional integro-differential equations with infinite delay. Afr. Mat. 27, 457–468 (2016). https://doi.org/10.1007/s13370-015-0342-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13370-015-0342-7