Spectral Theory for Bounded Functions and Applications to Evolution Equations

Gaston Mandata N’Guerekata
University Distinguished Professor of Mathematics, The World Academy of Sciences (TWAS) Research Professor, School of Computer, Mathematical and Natural Sciences, Morgan State University Baltimore, MD, USA

Series: Mathematics Research Developments
BISAC: MAT007000




Digitally watermarked, DRM-free.
Immediate eBook download after purchase.

Product price
Additional options total:
Order total:



One of the central questions in the qualitative theory of difference and differential equations is to find the conditions of existence and asymptotic behavior of bounded solutions. For equations with almost periodic coefficients, the problem goes back to Favard and Perron. A remarkable theory has been developed in harmonic analysis with outstanding contributions by Loomis, Arendt, Batty, Lyubic, Phong, Naito, Minh and many others, when the Carleman spectrum of the functions is countable. Uniform continuity in this case plays a key role. In the absence of this condition, the theory does not apply. This lead to the introduction over the last decade of new types of spectrum of functions which helped solve the problem, especially in the case of almost automorphic functions, using the theory of commutating operators.

This monograph presents in a unique and unified manner recent developments in the theory of spectra of bounded continuous functions including the space of (Bohr) almost periodic functions and some of their generalizations, and the spaces of (Bochner) almost automorphic functions and almost automorphic sequences. Classical concepts from harmonic analysis such as the Bohr spectrum, Beurling spectrum and Carleman spectrum are also presented with some examples. A special attention is devoted to the recently introduced concepts of uniform spectrum and circular spectrum of bounded functions derived from the study of the existence of solutions of linear differential equations whose forcing terms are not necessarily uniformly continuous. Connections between these various types of spectrum are also investigated. As applications the book provides a semigroup-free study of the existence and asymptotic behavior of mild solutions of evolution equations of the first and second order, and difference equations. Bibliographical and historical notes complete the major chapters. An appendix on basic results on the theory of commutating operators is given. The content is presented in a way that is easily accessible to readers who are working in differential equations but are not familiar with harmonic analysis and advanced functional analysis. It’s our hope that this first monograph ever on this topic will attract more researchers. (Imprint: Novinka)


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



Author's Contact Information


“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

[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. 39-48.
[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.,

3 (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),

45(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.

77 (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,

76 (2008), 217-233.
[13] A.G. Baskakov, Spectral tests for for the almost periodicity of the solutions of functional equations, Mat. Zametki,

24 (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.,

47, (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.,

52 (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.

[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.

132 (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.

38 (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.,

25(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.,


(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 Equations

164 (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, Applicable Analysis,

83 (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,

71(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,

68 (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.

19(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.
[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. Surveys

33(2) (1978), 1-52.
[82] E. Sinestrari, On the abstract Cauchy problem of parabolic type in spaces of continuous functions, J. Math. Anal. Appl.

107(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

You have not viewed any product yet.