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Almost periodic solutions of differential equations in Banach spaces. (English) Zbl 1026.34001
Stability and Control: Theory, Methods and Applications. 15. London: Taylor & Francis. viii, 250 p. (2002).
The objective of this monograph is to extend classical results on the existence of periodic solutions to the case of almost-periodic solutions to both linear and nonlinear equations in Banach spaces. The two theorems in question are formulated as follows. Theorem A. A $$T$$-periodic nonhomogeneous linear equation has a unique $$T$$-periodic solution if and only if the monodromy operator does not have 1 as an eigenvalue. Theorem B. A $$T$$-periodic nonhomogeneous linear equation has a unique $$T$$-periodic solution if and only if it has a bounded solution.
The book contains three chapters. Chapter 1, “$$C_0$$-semigroups, well posed evolution equations, spectral theory and almost periodicity of functions,” has an auxiliary character. In this chapter, strongly continuous semigroups of linear operators are introduced. Then, applications of this theory to evolution equations are presented, and spectral theory and almost-periodicity of bounded uniformly continuous functions are discussed. In the second chapter, entitled “Spectral criteria for periodic and almost-periodic solutions,” the authors are concerned with the extension of Theorems A and B to the infinite-dimensional case. First, evolution semigroups and almost-periodic solutions are discussed. Then, the framework of evolution semigroups and sums of commuting operators is introduced. It is useful for stating spectral conditions in terms of spectral properties of operator coefficients when autonomous equations are studied. A fundamental decomposition technique for the critical case is presented in section 3. The traditional approach for the study of periodic solutions based on the fixed-point theory is presented in section 4. Then, boundedness and almost-periodicity in discrete systems are studied. Finally, the methods introduced earlier in this chapter are extended to semilinear and nonlinear equations. The conditions are given in terms of dissipativeness of the equations under study. Making use of evolution semigroups, one can give simple proofs for results on the finite-dimensional case and extend them to infinite-dimensional case. Chapter 3, “Stability methods for semilinear evolution equations and nonlinear evolution equations,” deals with existence of almost-periodic solutions to almost-periodic evolution equations by using stability properties of nonautonomous dynamical systems. First, the concept of skew product flow of processes is extended to a more general concept for quasi-processes generated by abstract functional-differential equations. Several stability concepts (BC-stability and $$\rho$$-stability) are introduced and equivalent relationships between them are discussed. In section 5, existence of almost-periodic solutions to abstract almost-periodic evolution equations is studied by using results established in section 2. Finally, several applications to functional partial differential equations are discussed in section 6. Basic facts on Fredholm operators, closed range theorems, essential spectrum, measures of noncompactness, sums of commuting operators, and Lipschitz operators are presented in appendices. An extensive bibliography contains 242 entries.
The monograph has collected recent results on the existence of almost-periodic solutions to linear and nonlinear differential equations in Banach spaces and a relationship between the existence of almost-periodic solutions and their stability properties. It can be useful for graduate students and researchers in differential equations, dynamical systems, stability theory, and control theory.

##### MSC:
 34-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations 34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations 34G10 Linear differential equations in abstract spaces 47D06 One-parameter semigroups and linear evolution equations 34C29 Averaging method for ordinary differential equations 34G20 Nonlinear differential equations in abstract spaces 39A11 Stability of difference equations (MSC2000)