## Table of Contents

**Table of Contents**

Preface

Chapter 1. Preliminaries

Chapter 2. Almost Periodic Functions

Chapter 3. Almost Automorphic Functions and Sequences

Chapter 4. Spectrum of Bounded Functions

Chapter 5. Applications to Differential Equations

Appendix

References

Author’s Contact Information

Index

**Reviews**

“This monograph is devoted to the theory of spectra of bounded functions in abstractspaces. Spectral theory is very important as it gives important qualitative properties ofsolutions to evolution equations. It is particularly useful in the study of the asymptoticbehavior of mild solutions of evolution equations. The monograph is divided into fivechapters. Chapter 1 is about some basic defnitions and notations. Chapter 2 is devotedto the theory of almost periodic functions. These functions have important applicationsin celestial mechanics, control theory and other fields. In chapter 3, the author discussesthe concepts of almost automorphic functions and sequences, which are generalizationsof almost periodic functions and sequences. After these chapters, the author moves onto defining the spectrum of bounded functions, in chapter 4. The concepts of Carleman,Beurling, uniform and circular spectra are defined. The last chapter is devoted to thestudy of difference and first- and second-order differential equations.Overall, this monograph is useful for researchers working in the field of spectraltheory of bounded functions and the qualitative theory of differential equations inabstract spaces. The monograph is written for graduate students, and to read it oneneeds to first go through the basics of abstract spaces and functional analysis.” – Dr. Syed Abbas, Professor, Indian Institute of Technology Mandi, India

**References**

[1] M. Adamczak, C(n)-almost periodic functions, Comm. Math. Prace Mat. 37(1997), 1-12.[2] M. Adamczak, S. Sto´ınski, On the (NC(n)) almost periodic functions, in: R. Grza´slewicz, Cz. Ryll-Nardzewski, H. Hudzik, J. Musielak (Eds.), Proceedings of the 6th. Conference on Functions Spaces, World Scientiﬁc Publishing, New Jersey, 2003, pp. 39-4[3] L. Amerio, G. Prouse, Almost Periodic Functions and Functional Equations, Van Nostrand Reinhold, New York (1971).[4] D. Araya, R. Castro, C. Lizama, Almost automorphic solutions of difference equations, Advances in Difference Equations, Hindawi Publishing Corporation, Vol. 2009, Art. ID 591380, 15 pp.[5] W. Arendt, S. Schweiker, Discrete spectrum and almost periodicity,Taiwanese J. Math. (1999), 475-490[6] W. Arendt, F. Ra¨biger, A. Sourour, Spectral properties of the operators equations AX + XB = Y, Quart. J. Math. Oxford (2)(1994), 133-149.[7] W. Arendt, C.J.K. Batty, M. Hieber, F. Neubrander, Vector-valued Laplace transforms and Cauchy problems, Monographs in Mathematics, 96, Birkhuser Verlag, Basel, 2001.[8] J-B. Baillom, J. Blot, G.M. N’Gu´re´kata, D. Pennequin, On C(n)-almost periodic solutions to some nonautonomous differential equations in Banach spaces,Comment. Math. Prace Mat.46(2) (2006), 263-273. [9] B. Basit, Generalization of two theorems of M.I. Kadets concerning the indeﬁnite integral of abstract almost periodic functions, Mat. Zametki 9 (1971), 311321.[10] B. Basit, A.J. Pryde, Ergodicity and stability of orbits of unbounded semigroup representations, J. Aust. Math. Soc. (2004), 209-232.[11] B. Basit, Harmonic analysis and asymptotic behavior of solutions to the abstract Cauchy problem, Semigroup Forum, 54 (1997), 58-74.[12] B. Basit, Hans Gunzler, Relations between different types of spectra and spectral characterization, Semigroup Forum, (2008), 217-233.[13] A.G. Baskakov, Spectral tests for for the almost periodicity of the solutions of functional equations, Mat. Zametki, (1978), 195-206 (in Russian).[14] C.J.K. Batty, W. Hutter, F. R¨ abiger, Almost periocity of mild solutions of inhomogeneous periodic Cauchy problems, J. Diff. Eq. 156 (1999), 3090327.[15] C.J.K. BattyJ.van Neerven, F. Rabiger, Local psectra and individual stability of uniformly bounded C0-semigroups. Trans. Amer. Math. Soc. (1998), 2071-2085.[16] A. Berger, S. Siegmund, Y. Yi, On almost automorphic dynamics in symbolic lattices, Ergodic Theory Dynam. Systems 24 (2004), 677-696.[17] A.S. Besicovitch, Almost Periodic Functions, Dover Publications (1954).[18] S. Bochner, Abstrakte Fastperiodsche Funktiones, Acta Mathematica, 61(1) (1933), 149-184.[19] S. Bochner, Uniform convergence of monotone sequences of functions, Proc. Nat. Acad. Sci. U.S.A., (1961), 582-585.[20] S. Bochner, A new approach to almost periodicity, Proc. Nat. Acad. Sci. U.S.A., 48, (1962), 2039-2043.[21] S. Bochner, Continuous mappings of almost automorphic and almost automorphic functions, Proc. Nat. Acad. Sci. U.S.A. (1964), 907-910. [22] D. Bugajewski, G.M. N’Gue´re´kata, On some classes of almost periodic functions in abstract spaces, Intern. J. Math. Math. Sci., 61 (2004), 32373247.[23] D. Bugajewski, G.M. N’Gue´re´kata, Almost periocicity in Fre´chet spaces, J. Math. Anal. Appl.299(2004)534-549.[24] R. Chill, E. Fasangova, Equality of two spectra arising in harmonic analysis and semigroup theory, Proc. Amer. Math. Soc. 130 (2002), 675-681.[25] C. Corduneanu, Almost Periodic Functions, Chelsea Publishing Co., New York, 1989.[26] T. Diagana, G. N’gue´rekata, Nguyen Van Minh, Almost automorphic solutions of evolution equations.Proc. Amer. Math. Soc. (2004), 3289– 3298.[27] H-S. Ding, T-J. Xiao, J. Liang, Asymptotically almost solutions for some integrodifferential equations with nonlocal conditions, J. Math. Analysis Appl., 338 (2008), 141-151.[28] J. Favard, Lec¸ons sur les FonctionsPresque-Pe´riodiques, Gauthier-Villars, Paris, 1933.[29] A. Favini, A. Yagi, Abstract second order differential equations with applications, Funkc. Ekv. (1995), 81-99.[30] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., Springer-Verlag, Berlin-New York, 1981.[31] A.M. Fink,Almost Periodic Differential Equations, Springer-Verlag, Berlin-Heidelberg-New York, 1974.[32] J.A. Goldstein,Semigroups of Linear Operators and Applications, Oxford University Press, 1985.[33] E. Hewitt, K. Ross, Abstract Harmonic Analysis, V.1 Springer, Berlin, 1979. [34] Y. Hino, T. Naito, N.V. Minh, J.S. Shin, Almost Periodic Solutions of Differential Equations in Banach Spaces. Taylor & Francis, London – New York, 2002.[35] Y. Katznelson, An Introduction to Harmonis Analysis, Dover, New York, 1976.[36] J. Kopec´, On vector-valued almost periodic functions, Ann. Sci. Polon. Math(1952), 100-105.[37] J. Gil de Lamadrid, L.N. Argabright, Almost Periodic Meausres, Memoirs of the Amer. Math. Soc., Vol.85 (1990)[38] B.M. Levitan, V.V. Zhikov, Almost Periodic Functions and Differential Equations, Moscow Univ. Publ. House 1978. English translation by Cambridge University Press 1982.[39] L.H. Loomis, The spectral characterization of a class of almost periodic functions, Ann. Math, 72 (2) (1960), 362-368.[40] J. Liang, L. Maniar, G.M. N’Gue´re´kata, T.-J. Xiao, Existence and uniqueness of C(n)-almost periodic solutions to some ordinary differential equations, Nonlinear Analys,, 66 (2007), 1899-1910.[41] J. Liang, J. Zhang, T-J. Xiao, Composition of pseudo almost automorphic and asymptotically almost automorphic functions, J. Math. Anal. Appl.340(2), (2008), 1493-1499.[42] J. Liu, G. N’Gue´re´kata, Nguyen van Minh, A Massera type theorem for almost automorphic solutions of differential equations. J. Math. Anal. Appl.299 (2004), no. 2, 587-599.[43] J. Liu, G. N’Gue´re´kata, Nguyen van Minh, Topics on Stability and Periodicity in Abstract Differential Equations, Series on Concrete and Applicable Mathematics 6, World Scientiﬁc Publishing Co. Pte. Ltd. Hackensack, New Jersey, 2008.[44] J. Liu, G. N’Gue´re´kata, Nguyen van Minh, Almost automorphic solutions of second order evolution equations, Appl. Analysis 84 (11)(2005), 1173-1184. [45] J. Liu, G. N’Gue´re´kata, Nguyen van Minh, V.Q Phong, Bounded solutions of parabolic equations in conyinuous function spaces, Funkcialaj Ekvacioj, 49 (2006), 337-355.[46] Q, Liu, Nguyen Van Minh, G. N’Gue´re´kata, R. Yuan, Massera type theorem for abstract functional differential equations, Funkcialaj Ekvacioj, 51 (2006), 329-350.[47] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birhauser, Basel, 1995.[48] Yu. I. Lyubich, V. Q. Phong, A spectral criterion for the almost periodicity of one-parameter semigroup, Funktsional. Anal. i Prilozhen, 47 (1987), 36-41.[49] Y. Meyer, Quasicrystals, almost periodic patterns, mean-periodic functions and irregular samplings, African Diaspora J. Math. 13(1) (2012), 1-45.[50] V.M. Nguyen, A spectral theory of continouos functions and the Loomis-Arendt-Batty-Vu theory on the asymtotic behavior of saolutions of evolution equations, J. Differential Eq. 247 (4) (2009), 1249-1274.[51] V.M.Nguyen, Corrigendum to ”A spectral theory of continouos functions and the Loomis-Arendt-Batty-Vu theory on the asymtotic behavior of solutions of evolution equations, J. Differential Eq. 247 (4) (2009), 12401274., J. Differential Eq. 249 (6) (2010), 1516-1517[52] V.M. Nguyen, G.M. N’Gue´re´kata, S. Sigmund, Circular spectrum and bounded solutions of periodic evolution equations, J. Differential Eq. 246 (8), (2009), 3089-3108.[53] N.V. Minh, T. Naito, G. N’Gue´re´kata, A spectral countability condition for almost automorphy of solutions of differential equations, Proc. Amer. Math. Soc. 134 (11), (2006),3257-3266.[54] N.V. Minh, G.M. N’Gue´re´kata, R. Yuan, Lectures on the asymptotic behavior of solutions of differential equations, Nova Science Publishers Onc. New York, (2008). [55] N.V. Minh, G. Mophou, G.M. N’Gue´re´kata, On the uniform spectrum of bounded functions and applications to differential equations, J. Concr. Appl. Math.8(2)(2010), 246-260.[56] X. Mora, Semilinear parabolic problems deﬁne semiﬂows on Ck spaces. Trans. Amer. Math. Soc.278 (1983), 21–55.[57] S. Murakami, T. Naito, N.V. Minh, Evolution semigroups and sums of commuting operators: a new approach to the admissibility theory of function spaces, J. Differential Equations164 (2000), 240-285.[58] H. and J. Musielakowie, Mathematical Analysis, vol. I, WN UAM, Poznan´.[59] T. naito, Nguyen Van Minh, J.S. Shin, New spectral criteria for almost periodic solutions of evolution equations, Studia Math. 145 (2001), 97-111.[60] T. Naito, N.V. Minh, R. Mitazaki, Y. Hamaya, Boundedness and almost periodicity in dynamical systems, J. Difference Equ. Appl. 7 (2001), 507-527.[61] T. Naito, Nguyen Van Minh, J. Liu, On the bounded solutions of Volterra equations, plicable Analysis, (2004), 433-446.[62] G. M. N’Gue´re´kata, Almost Automorphic and Almost Periodic Functions in Abstract Spaces, Kluwer Academic/Plenum Publishing, New York-Boston-Dordrecht-London-Moscow, 2001.[63] G. M. N’Gue´re´kata, Topics in Almost Automorphy, Springer-Verlag, New York, 2005.[64] G. M. N’Gue´re´kata, Comments on almost automorphic and almost periodic functions in Banach spaces, Far East. J. Math. Sci. (FJMS), 17 (2005), no.3, 337-334.[65] G. M. N’Gue´re´kata, An extension of the Bohr-Neugebauer theorem, Dynamic Systems and Applicatiobs 10 (2001) 451-454. [66] G. M. N’Gue´re´kata, On almost automorphic solutions of linear operational-differential equations, Intern. J. Math. and Math. Sci., 22 (2004), 1179-1182.[67] G. M. N’Gue´re´kata, Sur les solutions presqu’automorphes d’e´quations diffe´rentielles abstraites Ann. Sci. Math. Que´bec, 5 (1) (1981), 69-79.[68] G. M. N’Gue´re´kata, Quelques remarques sur les fonctions presqu’automorphes, Ann. Sci. Math. Que´bec,7 (2) (1983), 185-191.[69] G.M. N’Gue´re´kata, Almost automorphic solutions to second-order semilinear evolution equations, Nonlinear Analysis, (2009), e432-e435.[70] N. V. Minh, G. N’Gue´re´kata, S. Siegmund, Circular spectrumand bounded solutions of periodic evolution equations, J. Diff. Equ.246 (2009), 3089-3108.[71] G. N’Gue´re´kata, A. Pankov, Integral operators in spaces of bounded, almost periodic and almost automorphic functions, Diff. Integral Equ. 21 (11-12) (2008), 1155-1176.[72] G. N’Gue´re´kata, A. Pankov, Stepanov-like almost automorphic functions and monotone evolution equations, Nonlinear Analysis, (2008), 2658-2667.[73] A. Pankov, Almost periodic functions, Bohr compactiﬁcation, and differential equations, Rend. Semin. mat. Fis Milano, 64 (1998), 149-158.[74] A. Pankov, Bounded and Almost Periodic Solutions of Nonlinear Operator Differential Equations, Kluwer, Dordrecht, 1990.[75] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Math. Sci. 44, Spriger-Verlag, Berlin-New York 1983.[76] J. Pru¨ss, Bounded solutions of Volterra equations, SIAM Math. Anal.(1987), 133-149.[77] W. Shen, Y. Yi, Almost Automorphic and Almost Periodic Dynamics in Skew-product Semiﬂows, Memoirs of the Amer. Math. Soc. 136 (1998) [78] S. Schweiker, Mild solutions of second-order differential equations on the line. Math. Proc. Cambridge Philos. Soc.129 (2000), 129–151. [79] E. Schuler, Vu Quoc Phong, The operator equation AX −XD2 = −δ0 and second order differential equations in Banach spaces, Semigroups of operators: theory and applications, Newport Beach, CA, (1998), 352– 363.<br>[80] W. Shen, Y. Yi, Almost Automorphic and Almost Periodic Dynamics in Skew-Product Flows, Memoirs of the Amer. Math. Soc., number 647 (1998).[81] M.A. Shubin, Almost periodic functions and partial differential operators, Russian Math. Surveys33(2) (1978), 1-52.[82] E. Sinestrari, On the abstract Cauchy problem of parabolic type in spaces of continuous functions, J. Math. Anal. Appl.(1985), 16-66.[83] B. Stewart, Generation of analytic semigroups by strongly elliptic operators. Trans. Amer. Math. Soc.199 (1974), 141-162.[84] Q.P. Vu, Stability and almost periodicity of trajectories of periodic processes, J. Differential Eq. 115 (1995), 402-415. [85] Q.P. Vu, E. Schu¨ler, The operator equation AX − XB = C, stability and asymptotic behaviour of differential equations, J. Differential Eq. 145 (1998), 394-419.[86] W.A. Veech, Almost automorphic functions on groups, Amer. J. Math. 87 (1965), 719-751.[87] A. Wintener, On Fourier averages, Amer. J. Math., 63 (4), (1941), 698-704.[88] M. Yamaguchi, Existence of periodic solutions of second order nonlinear evolution equations and applications. Funkc. Ekv.38 (1995), 519-538.[89] S. Zaidman, Topics in Abstract Differential Equations, I, II, Pitman Research Notes in Mathematics Series, Longman Scientiﬁc and Technical, New York, 1994. [90] S. Zaidman, Abstract Differential Equations, Pitman Publishing, San Francisco-London-Melbourne, 1979.[91] S. Zaidman, Almost automorphic solutions of some abstract evolution equations, Istituto Lombardo Accademia di Scienze e Lettere, 110 (2), (1976), 578-588.[92] S. Zaidman, Existence of asymptotically almost periodic and almost automorphic solutions for some classes of abstract differential equations, Annales des Sci. Math. du Que´bec, 13 (1) (1989), 79-88.[93] M. Zaki, Almost automorphic solutions of certain abstract differential equations, Annali di Mat. Pura ed Appl., 101 (1), (1074), 91-114.[94] Z-M. Zheng, H-S. Ding, G. M. N’Gue´re´kata, The space of continuous periodic functions is a set of ﬁrst category in AP(X), J. Funct. Spaces Appl., 2013, Art. ID 275702.[95] M. Zaki, Almost automorphic solutions of certain abstract differential equations, Ann. Mat. Pura Appl. series 4, 101 (1974), 91-114.

This book is designed for beginning researchers and graduate students in harmonic analysis, functional analysis and evolution equations