MR2332082 Satoru MURAKAMI and Yutaka NAGABUCHI Satoru MURAKAMI and Yutaka NAGABUCHI Invariant Manifolds for Abstract Functional Differential Equations and Related Volterra Difference Equations in a Banach space Funkcialaj Ekvacioj. Serio Internacia 50 2007 133--170 http://fe.math.kobe-u.ac.jp/FE/FullPapers/50-1/50_133.pdf http://www.ams.org/mathscinet-getitem?mr=MR2332082 For abstract functional differential equations (FDE) and Volterra difference equations (VDE) in a Banach space, the local existence and smoothness of invariant manifolds, such as stable/unstable manifolds, center-stable/center-unstable manifolds and center manifolds, are established by means of the variation of constants formula in the phase space in [18] and [12]. Also, it is shown that in a neighborhood of the zero solution the behavior of solutions of FDE (resp. VDE) is described, in some sense, by a certain ordinary differential equation (resp. first order difference equation) in a finite dimensional space. As a corollaly, the principle of linearized stability is derived. Abstract functional differential equations, Volterra difference equations, Invariant manifolds, Principle of linearized stability. Primary 39A10, 39A11; Secondary 34K20, 35B35. 50-133 2007 Invariant Manifolds for Abstract Functional Differential Equations and Related Volterra Difference Equations in a Banach space Satoru MURAKAMI and Yutaka NAGABUCHI Satoru MURAKAMI and Yutaka NAGABUCHI 1 Agarwal, R. P. Difference Equations and Inequalities, 2nd ed., Revised and Expanded: Theory, Methods and Applications Marcel Dekker, New York 2000 MR1740241
2 Aullbach, B.; Nguyen, V. M. Nonlinear semigroups and the existence and stability of solutions of semilinear nonautonomous evolution equations Abstract and Applied Analysis 1 1996 351-380 MR1481548
3 Carr, J. L. Applications of Centre Manifold Theory Springer-Verlag, New York 1981 MR0635782
4 Cuevas, C.; Pinto, M. Asymptotic properties of solutions to nonautonomous Volterra difference systems with infinite delay Advances in difference equations, III, Comput. Math. Appl. 42 2001 671-685 MR1838023
5 Cuevas, C.; Pinto, M. Convergent solutions of linear functional difference equations in phase space J. Math. Anal. Appl. 277 2003 324-341 MR1954479
6 Diekmann, O.; van Gils, S. A.; Verduyn Lunel, S. M.; Walther, H.-O. Delay Equations Springer-Verlag, Berlin, New York 1995 MR1345150 7 Elaydi, S. An Introduction to Difference Equations, 2nd ed. Springer-Verlag, New York 1999 MR1711587
8 Elaydi, S.; Murakami, S. Asymptotic stability versus exponential stability in linear Volterra difference equations of convolution type J. Difference Equations and Appl. 2 1996 401-410 MR1444269
9 Elaydi, S.; Murakami, S. Uniform asymptotic stability in linear Volterra difference equations J. Difference Equations and Appl. 3 1998 203-218 MR1616085
10 Furumochi, T.; Murakami, S.; Nagabuchi, Y. Volterra difference equations on a Banach space and abstract differential equations with piecewise continuous delays Japanese J. Math. 30 2004 387-412 MR2111555
11 Furumochi, T.; Murakami, S.; Nagabuchi, Y. A generalization of Wiener's lemma and its application to Volterra difference equations on a Banach space J. Difference Equations and Appl. 10 2004 1201-1214 MR2100722
12 Furumochi, T.; Murakami, S.; Nagabuchi, Y. Stabilities in Volterra difference equations on a Banach space Fields Institute Communications 42 2004 159-175 MR2089608
13 Hale, J. K. Ordinary Differential Equations John Wiley and Sons, New York 1969 MR0419901 14 Hale, J. K.; Kato, J. Phase space for retarded equations with infinite delay Funkcial. Ekvac. 21 1978 11-41 MR0492721 0492721 15 Hale, J. K.; Verduyn Lunel, S. M. Introduction to Functional Differential Equations Springer-Verlag, Berlin, New York 1993 MR1243878 16 Henry, D. Geometric Theory of Semilinear Parabolic Equations Lecture Notes in Math., 804, Springer-Verlag, Berlin, New York 1981 MR0610244 17 Hino, Y.; Murakami, S.; Naito, T. Functional Differential Equations with Infinite Delay Lecture Notes in Math., 1473, Springer-Verlag, Berlin, New York 1991 MR1122588
18 Hino, Y.; Murakami, S.; Naito, T.; Nguyen, V. M. A variation of constants formula for abstract functional differential equations in Banach spaces J. Differential Equations 179 2002 336-355 MR1883747
19 Hino, Y.; Murakami, S.; Nguyen, V. M. Decomposition of variation of constants formula for abstract functional differential equations Funkcial. Ekvac. 45 2002 341-372 MR1975051
20 Hirsh, M. W.; Pugh, C.; Shub, M. Invariant Manifolds Lecture Notes in Math., 583, Springer-Verlag, Berlin, New York 1977 MR0501173
21 Krisztin, T. Invariance and noninvariance of center manifolds of time-$t$ maps with respect to the semiflow SIAM J. Math. Anal. 36 2004/2005 717-739 (electronic) MR2111913
22 Krisztin, T.; Walther, H.-O.; Wu, J. Shape, Smoothness, and Invariant Stratification of an Attracting Set for Delayed Monotone Positive Feedback Field Institute Monographs, 11, Amer. Math. Soc., Providence, RI 1999 MR1719128 23 Lang, S. Fundamentals of Differential Geometry GTM 191, Springer-Verlag, New York, Berlin, Heidelberg 1999 MR1666820
24 Matsunaga, H.; Murakami, S. Some invariant manifolds for functional difference equations with infinite delay J. Difference Equations and Appl. 10 2004 661-689 MR2064815
25 Matsunaga, H.; Murakami, S. Asymptotic behavior of solutions of functional difference equations J. Math. Anal. Appl. 305 2005 391-410 MR2130709
26 Memory, M. C. Stable and unstable manifolds for partial functional differential equations Nonlinear Anal. TMA 16 1991 131-142 MR1090786
27 Nguyen, V. M.; Wu, J. Invariant manifolds of partial functional differential equations J. Differential Equations 198 2004 381-421 MR2039148
28 Murakami, S.; Naito, T.; Nguyen, V. M. Massera's theorem for almost periodic solutions of functional differential equations J. Math. Soc. Japan. 56 2004 247-268 MR2027625
29 Murakami, S.; Nguyen, V. M. Some invariant manifolds for abstract functional differential equations and linearized stabilities Vietnam J. Math. 30 2002 437-458 MR1964235
30 Nagabuchi, Y. Decomposition of phase space for linear Volterra difference equations in a Banach space Funkcial. Ekvac. 49 2006 269-290 MR2271235 2271235 31 Shub, M. Global Stability of Dynamical Systems Springer-Verlag, Berlin, New York 1987 MR0869255 32 Wu, J. Theory and Applications of Partial Functional Differential Equations Applied Math. Sci., 119, Springer-Verlag, Berlin, New York 1996 MR1415838