## Almost periodic type functions and ergodicity.(English)Zbl 1068.34001

Dordrecht: Kluwer Academic Publishers; Beijing: Science Press (ISBN 1-4020-1158-X/hbk; 7-03-010489-7/hbk). xi, 355 p. (2003).
In this monograph, the function classes considered are: Bohr almost periodic (= ap) $$\text{AP}(J,X)$$, asymptotic ap AAP, Eberlein weakly ap WAP, pseudo ap PAP (introduced by the author) and (uniformly) ergodic $$E(J,X)$$, with $$J={\mathbb R}$$ or $${\mathbb R}_+=[0,\infty)$$ and $$X$$ a Banach space. These are introduced and discussed in some detail; then the author addresses their application to the asymptotic study of solutions of differential equations, also in $$X$$, nonlinear systems and functional-differential equations.
In Chapter 1 (80 pages), the above function classes are defined, and their properties as needed later are deduced, including existence of an ergodic mean with corresponding convolution, Fourier series and summation methods, also for $$f(s,t), t\in J$$, parameter $$s\in\Omega\subset{\mathbb C}^n$$ or $$X$$. Furthermore, one has the decomposition $$\text{AAP}=\text{AP}+C_0$$ (functions vanishing at infinity), $$\text{WAP}= \text{AP}+ \text{WAP}_0$$, and by definition $$\text{PAP}= \text{AP}+ \text{PAP}_0$$, where $$\text{PAP}_0$$ contains all continuous bounded $$f$$ with $$M(|f|)= \lim_{T\to\infty} {1\over2T}\int^T_{-T}\| f(t)\| \,dt=0$$ for $$J=\mathbb R$$. Generalized pseudo ap functions are given by $$\text{GPAP}= \text{AP}+ \text{GPAP}_0$$ with $$f\in\text{GPAP}_0$$ if only $$f\in L^1_{\text{loc}}(J,X)$$ with $$M(|f|)=0$$. One has $$\text{AP}\subset \text{AAP}\subset \text{WAP}\subset \text{PAP}\subset \text{GPAP}$$ if $$X=\mathbb C$$.
Also ap, asymptotically ap and pseudo ap sequences and their relations to such functions on $$\mathbb R$$ are discussed.
In Chapter 2 (115 pages), first the system $y'=Ay+f \tag{1}$ is treated; if $$A$$ has no eigenvalue with real part 0, then $$f\in\text{APT}({\mathbb R},{\mathbb C}^n)$$ implies that (1) has a unique solution in APT, where APT can be AP, AAP, WAP or PAP. For small $$\mu$$ this is extended to $$y'=Ay+f+\mu G(t,y(t))$$ with suitable $$G$$. Such results hold also for (1) and $$X$$-valued solutions if $$A$$ is a “dissipative” bounded linear operator from $$X$$ into itself. In analogy to the Bohl-Bohr-Kadets theorem for $$f\in\text{AP}(J,X)$$, if $$f=g+\phi\in\text{PAP}$$ and $$F(t)=\int^t_0f(s)\,ds$$ with (i) $$F$$ bounded and $$X$$ contains no isomorphic copy of $$c_0$$ or (ii) $$f({\mathbb R})$$ is relatively weakly compact, the $$F$$ is in PAP if and only if there is $$a\in X$$ such that $$\int^t_0\phi \,ds-a\in\text{PAP}_0$$ (the proof assumes $$J={\mathbb R}$$).
For the Dirichlet problem for $$\Delta u=0$$ on the half-plane $${\mathbb R}\times[0,\infty)$$, if $$u(\cdot,0)\in\text{APT}({\mathbb R},{\mathbb C})$$, then the solution $$u(\cdot,t)\in\text{APT}$$, locally uniformly in $$t>0$$ – and similarly for the heat equation $$u_{xx}=u_t+f(x,t,u)$$ if $$f$$ satisfies a Lipschitz condition in $$u$$ (no mention of the many extensions as e.g. in [L. Amerio and G. Prouse, Almost-periodic functions and functional equations. New York etc.: Van Nostrand Reinhold, Company VIII (1971; Zbl 0215.15701)]. The nonlinear equation $$x''-\nabla V(x)=f$$ with suitable $$V$$ has an ap solution if $$f$$ is ap.
As a typical result in the general nonlinear case we mention: given an ap $$F(x,\cdot)$$, locally uniformly in $$x\in\Omega\subset{\mathbb R}^n$$. If all bounded solutions of the system $x'=G(x,\cdot) \tag{2}$ have positive distance, $$G\in H(F)=\{$$uniform limit of $$F(x,\cdot+t_n)$$, locally uniformly in $$x,(t_n)$$ from $${\mathbb R}\}$$, then all these solutions are ap. If $$x'=F(x,\cdot)$$ has a pseudo ap solution, then the ap component of $$x$$ is a solution of some (2).
If in (1) $$A=A(t)$$ is (Eberlein weakly) ap and $$Ly:=y'-A(t)y$$ satisfies an exponential dichotomy, then, for (Eberlein weakly) ap $$f$$, $$Ly=f$$ has a unique (Eberlein weakly) ap solution, similarly for $$f\in\text{PAP}_0$$ or (unbounded) $$f\in\text{GPAP}$$.
For the differential system $y'(t)=A(t)y(t)+B(t)y([t])+g(t,y(t),y([t]), \quad t\in\mathbb R, \tag{3}$ via ap difference equations it is shown that for ap $$A,B$$ with exponential dichotomy and ap $$g(t,u,v)$$ with a small Lipschitz constant with respect to $$u,v$$, (3) has a unique ap solution, similarly for periodic $$A,B,g$$.
Almost periodic structurally stable systems $$x'=f(t,x)$$ are introduced and their connection with the linearized system and exponential dichotomy is studied.
Chapter 3 (76 pages) treats ergodicity and abstract differential equations. Here, a continuous bounded $$f: J\to X$$ is called ergodic, if there is an $$a\in X$$ such that $${1\over T}\int^T_0f(x+t)\,dt\to a$$ uniformly in $$x\in J$$ as $$T\to\infty$$. As a first application, if $$a$$ is ergodic, then $$y'+ay=f$$ has for each bounded continuous $$f:{\mathbb R}\to\mathbb C$$ a unique bounded solution if and only if $$M(\text{Re}(a))\not=0$$; if $$f\in\text{PAP}$$, the unique bounded solution is also in PAP. This can be extended to certain systems $$y'+A(t)y=f$$, and (with (asymptotic) ap instead of pseudo ap) to $$X$$-valued solutions of $$y'=A(t,y)+f$$ with an ergodic condition $$(K_3)$$ on A (p. 218–222).
After an introduction to strongly continuous $$(C_0)$$ semigroups $$(T(t))_{t\geq0}$$ on a Banach space $$X$$ and their infinitesimal generators $$A$$, for $$x'=Ax+f$$ with $$f\in\text{APT}({\mathbb R},X)$$ with bounded continuous derivative $$f'$$, if $$(T(t))$$ is exponentially stable, i.e., $$\| T(t)\|\leq Me^{-ct}$$ for $$t>0$$ with positive $$c$$, then there is exactly one bounded solution $$x$$ on $$\mathbb R$$ which is in APT. For the autonomous functional-differential equation $$x'(t)=Lx_t$$ with $$x_0\in C$$, where $$x_t(s)=x(t+s)$$, $$-r\leq s\leq0$$, $$C:= \{f:[-r,0]\to{\mathbb R}^n$$ continuous, bounded}, $$L$$ continuous linear from $$C$$ into $${\mathbb R}^n$$, the solution operator $$T(t)$$ gives a $$C_0$$ semigroup with compact $$T(t)$$; if its infinitesimal generator $$A$$ has no purely imaginary eigenvalues, for $$f\in\text{APT}({\mathbb R},{\mathbb C})^n$$ there is a unique bounded solution of $$x'=Lx_t+f$$ which is in APT. The nonautonomous case is treated also. If $$V$$ is a translation-invariant closed linear subspace of $$C_b:=\{f:{\mathbb R}\to X$$ continuous bounded}and $$\phi\in C_b$$, then the spectrum of $$\phi$$ with respect to $$V$$ is defined by $$\text{sp}_V(\phi):=\{\mu\in{\mathbb R}:\widehat f(\mu)=0$$ for all $$f\in L^1({\mathbb R},{\mathbb C})$$ with $$\phi*f\in V\}$$; $$V=\{0\}$$ gives the Beurling spectrum of $$\phi$$. For this $$\text{sp}_V$$ some results of B. Basit are discussed, e.g.: If $$\phi\in C_b$$ is uniformly continuous, then $$\phi\in V$$ if and only if $$\text{sp}_V\phi=\theta$$; if additionally $$V$$ contains the constants, $$\phi\in V$$ is uniformly continuous and its indefinite integral $$\theta$$ is ergodic, then $$\theta\in V$$. (Here, Remark 5.8, Theorem 5.17 and Corollaries 5.18, 5.19 are not correctly formulated, one has to add “$$f$$ is uniformly continuous”.) As a corollary one gets: If $$U$$ is a mild solution of (4) $$u'=Au+f$$ on $$\mathbb R$$ where $$A$$ is the infinitesimal generator of a $$C_0$$ semigroup, $$\sigma(A)\cap i\mathbb R$$ is countable, $$f|J\in V$$ with $$V$$ being a closed linear translation-invariant subspace of $$\{f\in C_b(J,X): f$$ uniformly continuous}, $$V$$ containing the constants, $$J={\mathbb R}_+$$ or $${\mathbb R},U$$ uniformly continuous and $$U|J$$ totally ergodic, then $$U|J\in V$$. Here, $$g$$ totally ergodic means all $$e^{i\omega t}g$$ are ergodic, $$\omega\in\mathbb R$$ and for $$f=0$$, a bounded uniformly continuous solution of (4) is asymptotic ap if and only if it is totally ergodic. (In Corollary 6.10 it should read “$$T(t)$$ is asymptotic ap”.) If $$(T(t))$$ is exponentially stable, $$V$$ as before, invariant under multiplication by $$e^{i\omega t}, \omega\in{\mathbb R}, V$$ translation-invariant and $$BV\subset V$$ for $$B$$ continuous linear from $$X$$ into $$X$$, then every mild solution of (4) on $${\mathbb R}_+$$ is in $$V$$ if $$f\in V$$.
In the last chapter (56 pages), ergodicity and averaging methods in perturbation theory are studied. As a sample we mention only: If with $$C$$ as above the continuous function $$f:{\mathbb R}\times C\to{\mathbb R}^n$$ satisfies a Lipschitz condition in the second variable and $$f_0(\phi):=\lim_{T\to\infty}{1\over T}\int^{t+T}_tf(s,\phi)$$ ds exists uniformly in $$t\in{\mathbb R}, \phi\in C$$, and if $$x'=\varepsilon f(t,x_t)$$ with $$x_0=\phi$$, $$z'=\varepsilon f_0(z_t)$$ with $$z_0=\phi$$, then $$|x(t)-z(t)|\leq Q(\varepsilon)$$ with (explicit) $$Q(\varepsilon)\to0$$ as $$\varepsilon\to0$$.
There is a short index, a (too short) list of notations, and a bibliography containing 200 articles and books. The presentation in the first half is reasonably selfcontained, while later it gets a bit rough sometimes, with occasionally undefined terms or the use of concepts or results of functional analysis without reference. Also, this reviewer had difficulties with the formulation of some results or definitions (e.g., definitions 2.7.8, 4.1.4, 4.5.1, exponential dichotomy on page 249). Additionally, more explicit examples would have been helpful. Nevertheless, one should be grateful for the wealth of material presented, collecting, unifying and sometimes generalizing many recent results. Consequently, the book can serve as an introduction to this field and it would be especially good for seminars.

### 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 34K14 Almost and pseudo-almost periodic solutions to functional-differential equations 34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations 42A75 Classical almost periodic functions, mean periodic functions 35B15 Almost and pseudo-almost periodic solutions to PDEs 37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010) 47A35 Ergodic theory of linear operators 11K70 Harmonic analysis and almost periodicity in probabilistic number theory

Zbl 0215.15701