Abstract
We establish the existence of mild solutions and periodic mild solutions for a class of abstract first-order non-autonomous neutral functional differential equations with infinite delay in a Banach space.
Similar content being viewed by others
References
M. Adimy and K. Ezzinbi, A class of linear partial neutral functional-differential equations with nondense domain, J. Diff. Eqns., 147 (1998), 285–332.
M. Adimy and K. Ezzinbi, Strict solutions of nonlinear hyperbolic neutral differential equations, Applied Maths. Letters, 12 (1999), 107–112.
M. Adimy, K. Ezzinbi and M. Laklach, Existence of solutions for a class of partial neutral differential equations, C.R. Acad. Sci. Paris Ser. I Math., 330 (2000), 957–962.
M. Adimy, H. Bouzahir and K. Ezzinbi, Existence and stability for some partial neutral functional differential equations with infinite delay, J. Math. Anal. Appl., 294 (2004), 438–461.
K. Balachandran and R. Sakthivel, Controllability of neutral functional integrodifferential systems in Banach space, Comp. Maths. Appl., 39 (2000), 117–126.
K. Balachandran and A. Leelamani, Null controllability of neutral evolution integrodifferential systems with infinite delay, Mathematical Problems in Engineering, (2006), 1–18.
Y. Chen, The existence of periodic solutions for a class of neutral differential difference equations, J. Aust. Math. Soc. Ser. B, 33 (1992), 507–516.
R. Datko, Representation of solutions and stability of linear differential-difference equations in a Banach space, J. Diff. Eqns., 29 (1978), 105–166.
A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, Inc. (New York, 1969).
X. Fu, Controllability of neutral functional differential systems in abstract space, Appl. Math. Comp., 141 (2003), 281–296.
X. Fu and K. Ezzinbi, Existence of solutions for neutral functional differential evolution equations, Nonlinear Anal., TMA, 54 (2003), 215–227.
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.
K. Gopalsamy and B. G. Zhang, On a neutral delay logistic equation, Dynamics and Stability of Systems, 2 (1988), 183–195.
K. Gopalsamy and P-X. Weng, On the stability of a neutral integro-partial differential system, Bull. Inst. Maths. Acad. Sinica, 20 (1992), 267–284.
J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, Springer-Verlag, Applied Mathematical Sciences, 99 (New York, 1993).
J. K. Hale, Partial neutral functional-differential equations, Rev. Roumaine Math. Pures Appl., 39 (1994), 339–344.
J. K. Hale, Coupled oscillators on a circle, Resenhas, 1 (1994), 441–457.
E. Hernández and H. R. Henríquez, Existence of periodic solutions of partial neutral functional-differential equations with unbounded delay, J. Math. Anal. Appl., 221 (1998), 499–522.
E. Hernández and H. R. Henríquez, Existence results for partial neutral functional-differential equations with unbounded delay, J. Math. Anal. Appl., 221 (1998), 452–475.
Y. Hino, S. Murakami and T. Naito, Functional-Differential Equations with Infinite Delay, Springer-Verlag, Lecture Notes in Mathematics, 1473 (Berlin, 1991).
V. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, Kluwer Acad. Publ. (Dordrecht, 1999).
Y. Kuang and H. L. Smith, Global stability in diffusive delay Lotka-Volterra systems, Diff. and Integral Eqns., 4 (1991), 117–128.
V. Lakshmikantham, L. Wen and B. Zhang, Theory of Differential Equations with Unbounded Delay, Kluwer Acad. Publ. (Dordrecht, 1994).
N. I. Mahmudov and S. Zorlu, Approximate controllability of semilinear neutral systems in Hilbert spaces, J. Applied Maths. & Stoch. Anal., 16 (2003), 1–10.
C-M. Marle, Mesures et Probabilités, Hermann (Paris, 1974).
R. H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Robert E. Krieger Publ. Co. (Florida, 1987).
M. C. Memory, Stable and unstable manifolds for partial functional differential equation, Nonlinear Analysis, TMA, 16 (1991), 131–142.
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag (New York, 1983).
R. Schnaubelt, Asymptotic behaviour of parabolic nonautonomous evolution equations, Springer-Verlag, Lect. Notes in Mathematics, 1855 (Berlin, 2004), pp. 401–472.
H. Tanabe, Equations of Evolution, Pitman (London, 1979).
K. Wang, On the unique existence of periodic solutions of neutral Volterra integrodifferential equations, Periodica Math. Hungar., 21 (1990), 21–29.
P-X. Weng, Oscillation in periodic neutral parabolic differential system, Bull. Inst. Maths. Acad. Sinica, 24 (1996), 33–47.
J. Wu and H. Xia, The existence of periodic solutions to integro-differential equations of neutral type via limiting equations, Math. Proc. Camb. Phil. Soc., 112 (1992), 403–418.
J. Wu, Global continua of periodic solutions to some difference-differential equations of neutral type, Tôhoku Math. J., 45 (1993), 67–88.
J. Wu and H. Xia, Self-sustained oscillations in a ring array of coupled lossless transmission lines, J. Diff. Eqns., 124 (1996), 247–278.
J. Wu and H. Xia, Rotating waves in neutral partial functional-differential equations, J. Dynam. Diff. Eqns., 11 (1999), 209–238.
J. Wu, H. Xia and B. Zhang, Topological transversality and periodic solutions of neutral functional differential equations, Proc. Roy. Soc. Edin., 129 A (1999), 199–220.
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported by FONDECYT-CONICYT, Grant 1050314.
Rights and permissions
About this article
Cite this article
Henriquez, H.R. Periodic solutions of abstract neutral functional differential equations with infinite delay. Acta Math Hung 121, 203–227 (2008). https://doi.org/10.1007/s10474-008-7009-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-008-7009-x